Modeling and Forecasting Implied Volatility

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Ihsan Ullah Badshah

Modeling and Forecasting Implied Volatility

Implications for Trading, Pricing, and Risk Management

Helsinki 2010 <

Modeling and Forecasting Implied Volatility: Implications for Trading, Pricing, and Risk Management

Key words: Asymmetric Volatility, Modeling Implied Volatility, Options, Principal Component Analysis, Quantile Regression, Spillovers, Value-at-Risk, Volatility Index, Volatility Smirk/Skew, Volatility Surface

© Hanken School of Economics & Ihsan Ullah Badshah

Ihsan Ullah Badshah Hanken School of Economics Department of Finance and Statistics P.O.Box 287, 65101 Vaasa, Finland

Distributor: Library Hanken School of Economics P.O.Box 479 00101 Helsinki, Finland Telephone: +358-40-3521 376, +358-40-3521 265 Fax: +358-40-3521 425 E-mail: [email protected] http://www.hanken.fi

ISBN 978-952-232-106-0 (printed) ISBN 978-952-232-107-7 (PDF) ISSN 0424-7256

Edita Prima Ltd, Helsinki 2010

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Dedicated to my parents

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ACKNOWLEDGMENTS

This PhD thesis would not have been possible without the help, guidance, and support of many individuals, all of whom deserve special thanks and appreciation. First, I must express my deepest gratitude to my thesis supervisor—Johan Knif—for providing me the opportunity to explore the area of volatility modeling and risk management. He has been kind and helpful and has always provided valuable suggestions and feedback on my thesis. He has extended his support to me on various occasions. For example, his encouragement led me to teach advanced (MSc) level courses at the department of finance, Hanken School of Economics, an experience that opened my eyes to the world of teaching and bolstered my confidence. Johan, I am grateful and indebted to you for all your help and support during my PhD. I am also very thankful to Kenneth Hogholm, who has always been obliging, provided timely help in various matters, including administrative matters, and offered useful comments on my research. I would also like to thank my thesis reviewers—Mika Vaihekoski and George Skiadopoulos—for their thorough and valuable comments on my work. These significantly improved the thesis, and I greatly appreciate and acknowledge all the effort that went into them. Eva Liljeblom deserves special mention for funding my PhD studies and trips abroad. She has also been kind enough to comment on my articles and has always made available her support. I will always be indebted to my uncle Dr. Rahman Wali, who inspired and motivated me to pursue a PhD. Thank you, uncle, for your support and guidance all these years. My friends and colleagues have always been there for me, and I am grateful to them all. Sheraz Ahmed and Kashif Saleem, both currently teaching at the LUT School of Business, Finland, have supported and encouraged me throughout this journey and even before. Muhammad Ali, Helsinki University Central Hospital, has always given me invaluable advices. Sohail Farooq, currently studying in the University of Vaasa, has been a good friend and discussion partner. I owe special thank to Alireza Tourani-Rad, who has provided valuable suggestions, and comments on my thesis articles and has always been supportive, so I will always be indebted to you for all these. I would also like to thank Bart Frijns, Hossein Asgharian, Seppo Pynnonen, Gregory Koutmos, David Simon, Jussi Nikkinen, Yakup Arisoy, and Dudley Gilder for providing constructive comments on my thesis articles at various international and local conferences. I must mention the director of the Graduate School of Finance (GSF), Mikko Leppamaki, for his efforts in organizing research workshops for the doctoral students to present their work at. I would like to thank my colleagues at the department of finance: Benny Jern, Mujahid Hussain, Anand Gulati, Hilal Butt, Kari Harju, Christian Johansson, Peter Nyberg, Nadir Virk, Saint Kuttu, Nasib Nabulsi, and David Gonzalez. Their companionship and constructive feedback at the internal presentations have been a great help. I also wish to thank Khalid Bhatti, Mats Engsbo, and Chouki Sfandla for their time and the stimulating discussions.

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Finally, a very special thanks to my family—my parents, sisters, and brothers—to whom I owe so much. You stood by me and gave me the confidence to take on such a demanding and challenging project. Most important of all, I thank my dearest wife Riffat. Your patience, support, and love never fail to amaze me. You give me the strength to carry on, and it is your encouragement that helped me through the final thesis submission. The financial support from the Evald & Hilda Nissi Foundation, Center of Financial Research (CEFIR), Hanken Foundation, Ella and Georg Ehrnrooth Foundation, NASDAQ-OMX Foundation, and Marcus Wallenberg Foundation is greatly acknowledged. September 27, 2010 Ihsan Ullah Badshah

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CONTENTS

1 INTRODUCTION……………………………………………………………………….........................1

2 OPTION MARKETS’ EMPIRICAL REGULARITIES AND MODELING…………........8

3 METHODOLOGY……………………………………………………………………..........................15

4 SUMMARIES OF THE ESSAYS…………………………………………………….....................28

5 REFERENCES……………………………………………………………………….......................….34

THE ESSAYS

1. Badshah, I., (2010). “Modeling the Dynamics of Implied Volatility Surfaces”.

Manuscript, Hanken School of Economics, Finland........................................................41

2. Badshah, I., (2010). “Quantile Regression Analysis of Asymmetric Return-Volatility

Relation”. Manuscript, Hanken School of Economics, Finland......................................73

3. Badshah, I., (2010). “The Information Content of VDAX Volatility Index and

Backtesting Daily Value-at-Risk Models”. Manuscript, Hanken School of Economics,

Finland............................................................................................................................107

4. Badshah, I., (2010). “Modeling Risk Factors Driving the EUR, USD, and GBP

Swaption Volatilities”. Manuscript, Hanken School of Economics, Finland.................135

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PART I: BACKGROUND, METHODOLOGY AND FINDINGS

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1 Introduction

Modeling and forecasting of volatility are important to both financial practitioners and

academics, especially in risk management activities, option pricing, option trading,

hedging of derivative positions, constructing volatility indices, portfolio construction

and diversification, and policy making, all of which require the accurate modeling and

forecasting of volatility. We have seen tremendous development in volatility forecasting

models since the introduction of the autoregressive conditional heteroskedasticity

(ARCH) model proposed by Engle (1982). Since then, many ARCH models have been

developed that attempt to forecast volatility using historical information. In 1986,

Bollerslev (1986) extended the ARCH model to a generalized ARCH model called

GARCH. Many extensions have also been made to the GARCH model (see Poon and

Granger, 2003, 2005); for example, Glosten, Jagannathan and Runkle (hereafter GJR)

(1993) have proposed the GJR model, which accounts for asymmetric volatility.

However, a growing body of literature supports the use of implied volatility (hereafter

IV) instead, calling IV the best forecast of future realized volatility (hereafter RV).

These studies empirically document that the information content of IV is richer than

and superior to that of historical volatility (hereafter HV) when forecasting the future

RV of the underlying asset (e.g., Day and Lewis, 1992; Christensen and Prabhala, 1998;

Fleming, 1998; Dumas, Fleming and Whaley, 1998; Blair et al., 2001; Ederington and

Guan, 2002; Poon and Granger, 2003; Mayhew and Stivers, 2003; Martens and Zein,

2004).1

Moreover, these previous studies assert that HV is backward-looking and develop

expectations about future volatility based on the past behavior of stock prices and other

relevant information. In contrast, IV is forward-looking, i.e., implied by the market

prices of options. Option prices reflect market participants’ consensus view (aggregated

beliefs updated through option trades) about the expected future volatility of an

underlying asset over the remaining life of an option.2 These volatility expectations can

1 Modeling of implied volatility is a less trite area of research in quantitative finance even today; however, it has received far less attention from academic circles in comparison to volatility modeling using historical information; for instance, there is an abundance of research articles on GARCH-type models. 2 Through option prices, investors update their beliefs about the underlying processes such as dividend growth, macroeconomic variables, etc., thereby altering IVs and their shape [i.e., inducing smirk (skew), or surface]. However, their dynamics are driven by latent risk factors (see, for economic explanations, studies such as David and Veronesi, 2000; Guidolin and Timmermann, 2003; Garcia et al., 2003; and Chalamandaris and Tsekrekos, 2009).

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be backed out by inverting option-pricing models such as Black and Scholes’ (hereafter

BS) (1973) model for equity options and Black’s model (1976) for interest rate options

(examples of these are swaptions and caps).

Since the 1987 market crash, one of the strongest empirical regularities in the U.S.

stock index option has appeared: IV recovered from stock index options appears to

differ between moneyness (the strikes dimension), called volatility smirk (skew), and

the term structure (the time-to-maturity dimension), called the volatility term

structure, if the two are examined at the same time. This implies that IVs present a rich

implied volatility surface (hereafter IVS) whose dynamics across strikes and over time

warrant further investigation. In fact, the assumptions behind the BS model do not

hold empirically, as asset prices are mostly influenced by risk factors such as stochastic

volatility or time-varying volatility, jumps, net-buying pressure of options, and

transaction costs (see, e.g., Carr et al., 2001; Heston, 1993; Bates, 1991; Leland, 1985;

David and Veronesi, 2000; Garcia et al., 2003; Bollen and Whaley, 2004).3 To account

for these deviations, practitioners employ different IVs for different strikes and

maturities. Consequently, the IV backed out using the BS model with the help of the

bisection method may reflect determinants of the option price that are not captured by

the BS model. Here, by repeating this procedure, we obtain IVs for every strike and

maturity; i.e., the volatility structure shows discrepancies between theoretical prices

and market prices. Therefore, an IVS that incorporates these features is essential in

practice, particularly for forecasting future volatility, pricing illiquid options, option

trading, pricing and hedging exotic derivatives, and risk management of options

portfolios.4

Additionally, it is of paramount importance for the practitioners to generate an IVS

from the IVs, which are implied by options representing the majority of market

sentiments (beliefs about bearish and bullish markets). Examples of this are accounting

for the put options in a volatility measure, particularly out-of-the-money (hereafter

OTM) put options that embed beliefs about market crashes, and, on the other hand,

OTM call options on market run-ups (see, e.g., Pan, 2002; Liu et al., 2005; Doran et al.,

2007; Bates, 2008; Camara and Heston, 2008). Because investors (particularly

3 Most of the BS model’s assumptions fail empirically; for instance, the BS model assumes that underlying assets follow a geometric Brownian motion and constant volatility, implying that options with different strikes and maturities on the same underlying asset should have the same IV. 4 See Skiadopoulos (2001) for the literature review on the implied volatility smiles and smile consistent models.

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institutional investors) mostly hedge their portfolios’ downside risk with put options

and in market turmoil drive a greater buying demand for OTM puts than OTM calls,

higher IVs are generated (see, e.g., Bollen and Whaley, 2004). Therefore, it is essential

that the IVS be generated using the IVs implied by both put and call options (the most

informed options), thus leading the IVS to move with changes in options prices.5

Furthermore, Liu et al. (2005) also support the use of OTM put and call options and

thus argue that the rare-event premia play an important role in generating the volatility

smirk (skew) pattern observed for options across moneyness and that these rare-events

are embedded in the OTM options.6 Camara and Heston (2008) have proposed an

option model that accounts for both OTM put and call options. They derive the extreme

negative events from OTM puts and extreme positive events from OTM calls. The IV

measure that accounts for OTM options thus contains a broader set of information and

thereby should be more robust; as a result, the IV measure should be the perfect

candidate for generating IVS.

Furthermore, IVS can be viewed as highly correlated, as IVs present a high degree of

correlation among the underlying risk factors, thereby indicating a high dependence

therein. When few important sources of information are common to the risk factors, we

observe a high degree of correlation among the risk factors. By using principal

component analysis (hereafter PCA), most important risk factors can be extracted from

the correlated IVS, which can explain most of the dynamic therein (see, e.g.,

Skiadodopulos et al., 1999; Alexander, 2001; Cont and Da Fonseca, 2002). PCA helps in

reducing dimensionality and enhancing computational efficiency. The extracted

independent risk factors can be used for a variety of purposes, such as vega-hedging,

projecting IVS, value-at-risk (hereafter VaR) forecasting, and model calibration.

Later, Britten-Jones and Neuberger (2000) and Jiang and Tian (2005) derived a

model-free implied volatility (hereafter MFIV) under the pure diffusion assumption

and asset price processes with jumps.7 This MFIV measure has now been adapted by

5 For example, a negative or positive shock to the market induces adjustments in hedging and trading strategies, consequently triggering changes in the prices of one type (i.e., put or call) of option. The IV then changes in the direction of the market demand of a particular type of option and the underlying asset (see Bollen and Whaley, 2004). 6 Similarly, Pan (2002) shows that the volatility smirk (skew) pattern is primarily due to investors’ fear of large, undesirable jumps. 7 They show that the information content of MFIV is superior to that of the Black-Scholes implied volatility (hereafter BSIV) because the MFIV measure accounts for all strikes when computing IV at a particular point in time, whereas the BSIV measure is a point-based IV and does not account for all strikes in computation; i.e., each strike has a separate IV. Moreover, BSIV is subject to both model and market efficiency, while MFIV is only subject to market efficiency (see Poon and Granger, 2003).

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the major IV indices, which used to employ at-the-money (ATM) IV measures in their

methodologies.8 The MFIV index is forward-looking and captures market sentiments

(aggregated beliefs), that are often referred to as the “investors’ fear gauge” (e.g.,

Whaley, 2000).9 Similarly, The MFIV measure contains a broader set of information

and is thereby more robust; the MFIV index is, as a result, an excellent tool for some of

the challenging volatility modeling and forecasting issues.10The first of these is the

strong daily negative asymmetric return-volatility relationship, where the volatility

index reacts differently to negative and positive returns—the important stylized fact

empirically observed in the market. The second is daily implied-VaR forecasting, i.e.

incorporating the information content of MFIV into different specifications of daily

VaR models, and then comparing and backtesting their daily forecasts with other

horse-race VaR models afterwards. Referring to the former, there are two main existing

hypotheses in the literature that characterizes the negative asymmetric return-volatility

relationship: the leverage effect and feedback effect hypotheses. However, both the

leverage and feedback hypotheses have been unable to explain the observed strong

negative asymmetric return-volatility relationship at daily frequencies (see, e.g., French

et al., 1987; Breen et al., 1989; Schwert, 1989, 1990). Similarly, a recent study by

Hibbert et al. (2008) found a very strong contemporaneous negative asymmetric

return-volatility relationship using daily data, thereby empirically rejecting both the

leverage and volatility feedback hypotheses.11 Further empirical investigations are

indeed required here to characterize an asymmetric return-volatility relationship using

a daily MFIV index measure.12 Importantly, we believe that the estimation technique

currently being used for this relationship is the standard ordinary least squares (OLS),

which may not fully characterize this relationship because it ignores the volatility 8 The motives for adopting MFIV measures are the following. First, the MFIV index measure is economically appealing and robust, as it accounts for OTM put and call options (i.e., volatility smirk/skew). Second, the previous IV index measure (now called VXO) was upward-biased, induced by trading-day conversion, which is now omitted from the new VIX measure. Finally, with the new robust MFIV index measure, it is possible to replicate volatility derivatives (e.g., variance swaps), which was not possible with the previous measure. 9 Major option exchanges, including the Chicago Board of Exchange (CBOE) and the Deutsche Börse, have launched IV indices that robustly provide information on options using MFIV measures; examples of this are the VIX index for the S&P 500 index, VXN for the NASDAQ 100 index, VDAX for the DAX 30 index and VSTOXX for the Dow Jones (DJ) EURO STOXX 50 index. 10 A recent study by Konstantinidi et al. (2008) attempted to forecast the evolution of implied volatility index. 11 Other studies by Simon (2003) and Giot (2005) have also found a very strong negative asymmetric return-volatility relationship using daily data. 12 We also believe that the asymmetric return-volatility relationship should be more pronounced with the new robust MFIV volatility index measure in contrast to the old BSIV volatility index measure. Therefore, it is important to compare the asymmetric responses of both MFIV and BSIV measures to negative and positive returns.

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responses at both tails of the IV changes’ distributions. Therefore, to obtain a complete

picture of this relation, the conditional quantile regression should be preferred over

OLS regression, especially to investigate the asymmetric responses of volatility at the

uppermost quantiles.13 Few well-known studies exist showing a significant negative and

asymmetric relations between stock index returns and changes in the IV index using

OLS (or mean) regressions (e.g., Fleming et al., 1995; Whaley, 2000; Giot, 2005;

Simon, 2003; Skiadopoulos, 2004; Low, 2004; Dennis et al., 2006).

However, referring to the latter, i.e., modeling and backtesting of implied-VaR, as we

believe that VaR modeling is all about extreme events, the MFIV measure should be

important as input for any VaR model because it is implied by OTM options, which are

informed on future events and contain information on negative and positive jumps

(see, e.g., Liu et al., 2005; Doran et al., 2007; Camara and Heston, 2008; Bates, 2008).

Therefore, it is possible to use the MFIV measure as an input to implied-VaR or

implied-VaR augmented by GJR model and then to compare the daily-VaR forecasts

with the forecasts of other horse-race VaR models such as the filtered historical

simulation (hereafter FHS) proposed by Barone-Adesi et al. (1998) and RiskMetrics.

Consequently, a gap in the literature needs to be filled by further investigation in this

direction as well.14

However, in addition to equity index option markets, there is an enormous over-the-

counter (OTC) interest rate option market (such as swaptions) that also needs

consideration. In particular, investigation of the IV dynamics of the larger swaption

markets (e.g., the EUR-, USD-, and GBP-denominated swaption markets) is indeed

indispensable. In fact, European swaption contracts are quoted to be consistent with

the Black (1976) model’s IVs. The IVs recovered from swaptions contain important

information as well; therefore, their dynamics need further modeling and exploration.

Example questions are how many risk factors are driving each of the EUR, USD, and

GBP swaption IVs, whether these risk factors are linked across markets, and whether

they respond to each other’s shocks. Furthermore, is it possible to reproduce (or

forecast) the whole swaption maturity IVS by calibrating a multifactor model to each of

these swaption markets separately?

13 Quantile regression estimates are robust to outliers, non-normal error distributions, etc. 14 To our knowledge, no study yet compared and backtested forecasts of implied-VaR’s with those of FHS-VaR.

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This PhD thesis is thus positioned within the area of modeling and forecasting implied

volatility discussed above. The first essay in this thesis is modeling IVS and identifying

risk factors that account for most of the randomness in the IVS. The approach extends

the Dumas, Fleming and Whaley (DFW) (1998) framework; for instance, using

moneyness in the implied forward price and OTM put-call options on the FTSE 100

index, a nonlinear parametric optimization technique is used to estimate different DFW

models and thereby produce rich, smooth IVSs. Here, the constant-volatility model fails

to explain the variations in the rich IVS. Next, PCA is applied to the IVS, and risk

factors are extracted that account for most of the dynamics in the shape of the IVS; it is

found that the first three PCs can explain about 69-88% of the variance in the IVS. Of

this, on average, 56% is explained by the level factor, 15% by the term-structure factor,

and the additional 7% by the jump-fear factor. The second essay proposes a quantile

regression model (QRM) for modeling the strong negative asymmetric return-volatility

relationship with newly adapted MFIV indices, especially to quantify the effects of the

positive and negative stock index returns at various quantiles of the IV changes’

distribution. This model is the generalization of standard mean regression models, such

as those of Simon (2003), Giot (2005) and Hibbert et al. (2008). The results show a

pronounced negative asymmetric return-volatility relationship that is monotonically

increasing when moving from the median quantile to the uppermost quantile (i.e.,

95%); therefore, OLS underestimates this relationship at upper quantiles. Additionally,

the asymmetric relationship is more pronounced with the MFIV volatility index

measure in comparison to the old BSIV volatility index measure. Nonetheless, the

volatility indices are ranked in terms of asymmetric volatility as follows: VIX, VSTOXX,

VDAX, and VXN. The third essay examines the information content of the new VDAX

volatility index to forecast daily VaR estimates and compares its VaR forecasts with the

forecasts of the FHS and RiskMetrics models. These daily VaR models are then

backtested from January 1992 through May 2009 using unconditional coverage,

independence, conditional coverage, and quadratic score tests. It is found that the

VDAX volatility index subsumes almost all information required for the volatility of

daily VaR forecasts for a portfolio of the DAX30 stock index; implied-VaR models

outperform VaR models of the FHS (GJR) and RiskMetrics.

Three essays discuss modeling and forecasting of IV for equity markets. However, the

fourth essay emphasizes modeling and exploring of IV dynamics for the important

swaption markets; the essay models risk factors driving the EUR, USD, and GBP

swaption IVs. Two main purposes are then achieved. First, the common underlying

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implied factors are identified that affect IV movements in the EUR, USD, and GBP

swaption markets. Then, the dynamic interactions between the implied factors are

examined using techniques such as Granger causality and the generalized impulse

response function. We find that the first three implied factors explain 94-97% of the

variation in each of the EUR, USD, and GBP swaption IVs. There are significant

linkages across factors, and bi-directional causality is at work between the factors

implied by EUR and USD swaption IVs. Furthermore, in innovation-accounting

investigations, the factors implied by EUR and USD swaption IVs respond to each

others’ shocks; however, surprisingly, GBP does not affect them. Second, the string

market model (SMM) is calibrated for each of the swaption markets using multivariable

nonlinear optimization. The calibration results show that the SMM can efficiently

reproduce (or forecast) the entire swaption volatility matrix (or surface) for each of the

EUR, USD, and GBP markets.

The rest of the introductory part to the thesis is organized as follows. Section 2

discusses option markets’ empirical regularities and modeling. Section 3 discusses the

methodology. Section 4 discusses the four essays, their contribution to the literature

and the implications of their results.

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2 Options Markets’ Empirical Regularities and Modeling

One of the strongest empirical regularities of all time encountered with U.S. stock index

options appeared after the U.S stock market crash in 1987; when IVs are plotted against

any moneyness measure (along the strike dimension), volatility smirk (or skew)

appears (see, e.g., the empirical evidence of Bates, 1991; Rubinstein, 1994; Ait-Sahalia

and Lo, 1998; Foresi and Wu, 2005, Zhang and Xiang, 2008; and Carverhill et al.,

2009).15 In fact, there are three commonly observed shapes of the plot of IVs against

moneyness: smile, skew and smirk. The volatility smile is a pattern; IV increases as the

strike price increases (or decreases). The volatility skew is a pattern; there is a high IV

for low strike prices and a low IV for high strike prices. Finally, the volatility smirk is a

pattern; IV increases more quickly at low strike prices than at high strike prices (see,

for further details, e.g., Jiang and Tian, 2005; and Foresi and Wu, 2005).16 Rubinstein

(1985) documented that the volatility smile pattern existed in U.S. stock index options

before the 1987 crash. However, it has changed its pattern, following the 1987 market

crash, to a typical volatility smirk (or skew) pattern. On the other hand, we have

another stylized fact regarding the IV: the IV shows a volatility term structure pattern

when plotted against time to maturity (time dimension). Therefore, when the IVs (of

different strike prices and times to maturity) on a particular day are plotted against

moneyness and time dimensions, the implied volatility surface is viewed. As we have

empirically observed a rich IVS, it has two features that have attracted the interest of

academics and practitioners. First, the IVs systematically change with the strike price

and maturity dimensions, which induce an instantaneously non-flat IVS. Second, the IV

changes dynamically as time passes by because with the arrival of new information in

the option market, option prices are later updated based on the new information, i.e.,

the influences beliefs or expectations of investors.17 The dynamics of the IVS are

empirically confirmed to be driven mostly by the underlying risk factors (see the

empirical evidence of Dumas et al., 1998; Skiadopoulos et al., 1999; Cont and da

Fonseca, 2002; Mixon, 2002; and Gocalves and Guidolin, 2006; and Chalamandaris

and Tsekrekos, 2009).

15 The volatility smirk (skew) can be interpreted in terms of a premium that the market is willing to pay for insurance against any future crash. 16 The volatility smirk pattern is observed when OTM put options are much more expensive than the corresponding OTM call options (see Foresi and Wu, 2005). 17 See, for further discussion, the studies of David and Veronesi, 2000; Guidolin and Timmermann, 2003; Chalamandaris and Tsekrekos, 2009.

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An important contribution of the David and Veronesi (2000), and others such as Garcia

et al. (2003), and Guidolin and Timmermann (2003) is that they use a general

equilibrium approach, attributing these empirical regularities about IVS dynamics to

investors’ beliefs and updating from the option prices about the processes of

fundamentals that increase IVs, e.g., the time-varying volatility or stochastic volatility,

jumps, net-buying pressure of options, and transaction costs. Therefore, the shape of

the IVS incorporates the aggregated underlying beliefs (usually, most option traders are

institutional investors who posses professional skills) of the investors; therefore, IVS is

very informative.18 Consequently, modeling and generating IVS is of paramount

importance for forecasting future volatility, option trading, pricing illiquid options,

hedging and pricing exotic derivatives, and the risk management of an options

portfolio.

Figure 1 shows the average weekly profile of the implied volatility surface for March

2004 implied from FTSE 100 index options, i.e., the market IVs (pink dots) versus a

constant-BS model projection (green). As can be seen, a rich market IVS exists, but the

BS model projects a flat IVS and constant IVs at every segment of the IVS. This

empirical evidence clearly rejects the constant BS model and further supports the

existence of rich IVS, which is a stylized fact regarding IVs.

Figure 1. The market IVS versus BS model projection for March 2004 implied by

FTSE100 index options.

18 Chalamandaris and Tsekrekos (2009) discuss different approaches to modeling IVS dynamics.

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Figures 2 and 3 provide the proportion of factors and their loadings, respectively, that

drive the IVS for March 2004 implied by FTSE 100 index options. The first three

factors (or PCs) can explain, on average, approximately 82% of the variation in the IVS.

1 2 3 4 5 6 7 8 90

10

20

30

40

50

60

70

80

90

100

Principal Components

Var

ianc

e E

xpla

ined

(%)

Average Implied Volatility Surface, March- 2004

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Figure 2. Principal components driving the market IVS for March 2004.

5 10 15 20 25 30

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Three Factor Loadings of the Implied Surface, March-2004

PC3PC1

PC2

Figure 3. Factor loadings of the first three PCs from the market IVS for March 2004.

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The first factor, which is common to all IVs, is systematic for the entire IVS in the same

direction. It is a volatility-level factor (or parallel shifts) that explains about 64% of the

variations in IVS. It is important for IVs implied from the short- and long-maturity

options because it alters the steepness of the volatility pattern of the near-term OTM

put-call options and the effect is greater on the OTM options than on the ATM options.

In particular, OTM puts are affected the most, featuring a mean-reverting stochastic

volatility after a shock that increases instantaneous volatility, which then persists for

some time (the correlation between stochastic volatility and jump).19 The second factor,

which can be interpreted as a term-structure factor (or tilt, which generates shifts in the

slope of the term structure of IVs), explains about 10% of the variations in IVS and

affects the short- and long-maturity IVs differently (i.e., the sign of the effects differs

between the short- and long-maturity IVs). The term-structure factor is also consistent

with the notion that the segments in the IVS that are implied from long-term options

do not completely respond to the segments of the IVS that are implied from short-term

options; therefore, this factor can be attributed to the long-term risks such as risks

related to the market and the macroeconomy. Moreover, the second factor affects IVs

across the moneyness dimension identically. The third factor can be interpreted as a

jump-fear factor (or curvature, which changes the steepness of IV smirk/skew) and

explains about 8% of the variations in IVs. The effects of the jump-fear factor on the

OTM put and call options significantly differ, thereby inducing steepness in IV smirk

(skew). For empirical evidence on the dynamics of the factors driving the IVS of index

options, see the studies of Skiadopoulos et al. (1999), Mixon (2002), Cont and da

Fonseca (2002) and Chalamandaris and Tsekrekos (2009).

Another stylized fact is the strong negative asymmetric return-volatility relationship,

which implies that the IV reacts to negative and positive returns differently. The studies

of Simon (2003), Giot (2005) and Hibbert et al. (2008) have documented a very strong

negative asymmetric return-volatility relationship using daily data. The results of these

studies and many others put in question the two main hypotheses in the existing

literature that characterize the asymmetric return-volatility relationship: the leverage

effect and feedback effect hypotheses (see also, e.g., French et al., 1987; Breen et al.,

1989; Schwert, 1989, 1990). The leverage hypothesis proposed by Black (1976) and

Christie (1982) attributes asymmetric volatility to the financial leverage of a firm; i.e.,

19 Factor 1 can be also interpreted as being consistent with the view that the change in IVs changes the skewness of the corresponding implied risk-neutral density (see, for further discussion, the study of Mixon, 2002; Zhang and Xiang, 2008).

12

when a firm’s debt increases, the firm’s value declines, triggering the value of its stock

to decline further because the equity of a firm is more exposed to the firm’s total risk,

thereby increasing the volatility of equity.20 On the other hand, the volatility feedback

hypothesis is proposed by French et al. (1987), Campbell and Hentschel (1992), Poterba

and Summers (1986) and Bekaert and Wu (2000), who attribute asymmetric volatility

to a volatility feedback effect. In contrast to the leverage effect hypothesis, the volatility

feedback hypothesis states that increases in volatility induce negative stock returns.

The economic explanation should be that time-varying risk premia induce volatility

feedback because they represent the linkage between fluctuations in volatility and

returns. Increases in volatility imply that the required expected future returns will

increase as well; as a result, current stock prices decline. Hibbert et al. (2008) and

Breen et al. (1989) empirically found that the volatility feedback hypothesis does not

always hold; i.e., we do not always find positive correlations between current volatility

and expected future returns. Nevertheless, the accounting and economic explanations

might be important for characterizing an asymmetric return-volatility relationship at

lower frequencies, such as quarterly or monthly, but not for daily or higher frequencies.

Further investigations are important at a daily frequency to characterize the strong

negative asymmetric return-volatility relationship using robust MFIV (smirk-adjusted)

volatility indices. Importantly, an estimation technique such as quantile regression

should be preferred over OLS to capture the responses of the entire distribution of IV

changes to the negative and positive stock returns. Because the OLS will characterize

this relationship just at the mean of the response variable not at the other parts of the

response variable’s distribution, however, which could be nicely characterized by using

quantile regression.

Figure 4 provides a daily time-series plot (levels) of the stock market indices (S&P 500,

NASDAQ 100, DAX 30, and DJ Euro STOXX 50) versus the corresponding volatility

indices (VIX, VXN, VDAX, and VSTOXX) from February 2, 2001 to May 29, 2009. It is

evident that volatility indices are moving in the opposite direction to the stock market

indices. This phenomenon is stronger in crisis periods, particularly the credit-crunch

and liquidity crises in the year 2008, where historically high volatility levels of the

volatility indices are witnessed; for instance, VIX level twice surpassed 80%, and the

corresponding stock market crashed afterwards. Therefore, the inverse relationship is

20 However, Schwert (1990) argues that it is too strong for leverage hypothesis to completely explain asymmetric volatility.

13

the empirical evidence of a daily strongly negative return-volatility relationship, which

needs further examination.

Figure 4. Stock indices versus MFIV indices from February 2, 2001 to May 29, 2009.

14

Nonetheless, these empirically observed stylized facts in the option markets also bring

challenges for modeling: the first is determining a method to model IVS and explore the

dynamics of the factors driving IVS. The second is to incorporate the rich information

of the MFIV measure with the quantile regression framework to model and investigate

the strong daily negative asymmetric return-volatility relationship, i.e., how the entire

IV changes’ distribution would react to the negative and positive returns, particularly to

examine the responses of the upper-most quantiles of the IV changes’ distribution.

Moreover, there is the question of whether the relationship would differ between the

BSIV and MFIV measures in a quantile regression framework. The third is to

incorporate the information content from the MFIV measure (smirk-adjusted) into our

daily VaR models for VaR forecasts and, furthermore, to compare and backtest the

daily implied-VaR models with other benchmark VaR models. The fourth is to

determine how the factors implied from swaption maturity IVS are dynamically linked

to each other and across markets and, furthermore, to calibrate a multifactor model to

the swaption market and thereby reproduce the entire swaption maturity IVS.

The following section briefly discusses our methodology for the four essays, such as

how we approach modeling and forecasting of implied volatility in the presence of

above regularities.

15

3 Methodology

3.1 Calculating Black-Scholes Implied Volatility (BSIV)

Merton (1973) extended the Black-Scholes (1973) model for European call and put

options on dividend paying, tS stock or stock index with strike price K , time-to-

maturity tT� �� , call with payoff K,0)(SMax t � , and put ,0)S(KMax t� . The BS

formulae for the values of these call and put options are

� � � � � �,d�Ked�eS,K,�K,SC 2r�

1q�

tt�� (1)

� � � � � �,1q�

t2r�

t d�eSd�Ke,K,�K,SP ��

(2)

� � � �,

��2�qr�KSln

d2

t1

��

(3)

� � � �.

��2�qr�KSln

d2

t2

��

(4)

where q is the dividend rate on the underlying asset, r is the risk-free interest rate, and

� �� is the standard-normal cumulative distribution function. Now consider a market

where the BS assumptions do not hold, as previously empirically found by many

researchers (see the empirical evidence of Bates, 1991; Rubinstein, 1994; Ait-Sahalia

and Lo, 1998); the quoted market prices of put and call options are t

~C and t

~P . Then,

there is a unique implied volatility BS

t� that equates the theoretical option price with

the market option price given all other observed parameters:

,~

tBSttBS C),K,�K,(SC �

(5)

t~

BSttBS P),K,�K,(SP � . (6)

Nonetheless, with the help of the Bisection Method, all IVs are obtained for each strike

price and time to maturity, i.e., for the OTM put and call as well as the ATM options.

The bisection method is efficient and fast, and the advantage is that IVs can be

estimated without the knowledge of Vega. In the first essay, IVs are backed out from

OTM put and call options as well as from ATM options for modeling of the dynamics of

implied volatility surfaces for the FTSE 100 stock index.

16

3.2 Model-Free Implied Volatility (MFIV) Index

Britten-Jones and Neuberger (2000) derived a model-free risk-neutral implied

volatility under the pure diffusion assumptions

� � � ��

��

�

�

��

��

�

02

0rTT

0

2

t

tQ0 dK,

KK,0SmaxT,KeC

2S

dS�

(7)

The risk-neutral integrated implied volatility between the current date and a future

date is completely specified by a group of options expiring on the future date. The price

and volatility processes are completely model-free. In fact, in the option markets, the

cross-sections of the strike price are available in discrete periods; therefore, the above

continuous equation cannot be implemented in this situation. To overcome this issue,

Jiang and Tian (2005) have derived a discrete form of the MFIV that holds under a

variety of assumptions, such as diffusion, stochastic volatility and jumps:

� � � �,��

��

�

�

� ��

��

�

�

�

��

��

� � m

mi i

i0rT

i1n

0i

2

ti

ti1tiQ0 K

,0KSmaxeT,KCu1u

SSS

E

(8)

where

,n2,.....,1,0,iforih,ti �� ,nTh � m,2,.....,1,0,ifor,uSK i

0i ��

� � .m1

k1u �� The CBOE introduced a new VIX volatility index in September 2003

based on options on the S&P 500 stock index. The VIX is based on a similar MFIV

measure. The VIX is then determined from the bid-ask prices of the options underlying

the S&P 500 index; it provides an estimate of expected future realized stock market

volatility for the 22 subsequent trading days (over 30 calendar days). However, the old

VIX index, based on options on the S&P 100 index, was introduced in 1993 and has

now moved to the new ticker symbol VXO. In contrast to the VXO, which is based on

near-the-money BSIV options on the S&P 100 index, the new VIX uses market prices of

options on the S&P 500 index.21 This new model-free VIX methodology accounts for

both OTM put and call options (i.e., volatility smirk/skew). The new methodology is

thus more appealing and robust. The CBOE’s introduction of the new VIX was

motivated by both theoretical and practical deliberations. First, the new VIX is 21 The options on the S&P 500 index, in comparison with the options on the S&P 100 index, contain a much broader set of implied information; the new VIX is thus a more informative measure than the old VIX (now VXO).

17

economically more appealing, as it is based on a portfolio of options, whereas the old

VIX was based on the ATM option prices. Second, the new VIX makes it easy to

replicate variance swap payoffs while using static positions in a range of options and

dynamic positions in futures trading. Third, the new VIX has removed the induced

upward bias of the old VIX in the trading day conversion (see, e.g., Carr and Wu,

2006). Similarly, in September 2003, the CBOE introduced the VXN using the same

MFIV methodology as that of VIX. The CBOE has calculated price histories for VIX and

VXN back to the years 1986 and 2001, respectively. 22 The model-free formula for the

new VIX volatility index calculation is

� � � � ,1KF

tT1,TKOe

K�K

tT2t,T�

2

0

tit

rT

i2i

i2��

��

��

��

��

(9)

where T is time to expiry date of all S&P 500 index options (best bid and best ask of all

options), tF is the forward price derived from the prices of the S&P 500 index options,

iK is the strike price of the ith OTM option, � �K,TOt is the mid-price (of the bid-ask

spread) of the OTM option at strike iK , 0K is the first strike price below the forward

price tF , 2

KK�K i1i

i�

� �

is the interval between the strike prices, and r is the risk-

free rate at time .t Finally, the above equation is used to calculate � �t,T� 2 at two of the

nearest maturities of the available options. Then, via linear interpolation between the

two nearest maturities, the VIX estimate is obtained, leading to the estimation of the

annualized 30-day VIX

.30

365

TT

T30212

TT

30TT211t N

NNNNN

�TNN

NN�T100VIX

12

1

12

2

��

�

�

��

�

��

��

�

�

��

��

��

�

�

��

(10)

where 1TN and

2TN denote the number of days to maturities 1T and 2T of the two S&P

500 index options, respectively, and 365N is the time for a standard year.23

22 Later, the Deutsche Börse and Goldman Sachs jointly developed a similar model-free methodology for the new VDAX and VSTOXX indices. The VDAX is based on options on the DAX 30 stock index, whereas VSTOXX is based on options on the Dow Jones (DJ) Euro STOXX 50 stock index, which consists of the Eurozone’s 50 largest blue-chip stocks. The price histories for both VDAX and VSTOXX were calculated back to the years 1992 and 1999, respectively. 23See, for further detail on the VIX construction, the CBOE VIX white paper: www.cboe.com/micro/VIX/vixwhite.pdf.

18

3.3 Modeling Implied Volatility Smirk (Skew) and Surface

To model and obtain smooth implied volatility surfaces for FTSE100 stock index

options, we use a parsimonious parametric approach similar to the approach proposed

by Dumas et al. (1998). Their model is known as the practitioner BS model (see, e.g.,

Christoffersen and Jacobs, 2004). However, our approach somewhat differs from the

original work of Dumas et al. (1998). First, we incorporate most informed and liquid

options, i.e., OTM put and call options. Because put options represent the natural

hedging instrument for the underlying stock index portfolio, institutional investors

always hedge their portfolios using OTM puts, particularly when their forecast shows

any market declines in future (see, e.g., Pan, 2002; Bollen and Whaley, 2004; Foresi

and Wu, 2005; Bates, 2008; Camara and Heston, 2008).24 Second, we obtain IVS with

a practitioner-friendly moneyness measure that takes into account forward prices as

well as time dependence because the IVS should be able to be updated continuously on

the changes in both IVs and stock prices. We use a moneyness in implied forward price

that is similar to that of Gross and Waltner (1995). Moneyness in implied forward price,

m, is defined as the logarithm of the implied forward price �rtSe over the strike price

K , normalized by square root of time to maturity � , i.e.,

,�

/K)elog(Sm

t�rt�

(11)

Four different structural forms of model specifications are estimated, leading to smooth

IVSs, each of which differs in the parametric structure that serves to characterize the

IVs:

Model 1: � � ,��m,�I 0 �� (12)

Model 2: � � ,�m�m��m,�I 2210 �� (13)

Model 3: � � ,�m��m�m��m,�I 432

210 �� (14)

Model 4: � � .��m��m�m��m,�I 2543

2210 �� (15)

24 However, if investors who forecast any market spikes in the future go long in the OTM call positions with limited downside risks which remain to the premiums of the options only, instead taking positions in the underlying stock index.

19

Model 1, the flat IV function, is a constant in the BS model that gives volatility equal to

ATMIV. This does not account for volatility skew or smirk and term structure IV effects.

Model 2 attempts to capture volatility skew and volatility smirk across moneyness;

here, the slope, 1� , represents volatility skew, and the curvature, 2� , represents

volatility smirk effects.25 Model 3 captures an additional variation attributable to time

(e.g., time to maturity slope), 3� , and a combined effect of the skew and time

dimensions, 4� . Finally, Model 4 adds a parameter that captures the time-to-maturity

curvature effect, 5� . Further clarifications for the rest of notations are: � �m,�I , which

represents the dependence of IV on the moneyness level as well as time to maturity,

and 0� , which a parameter that is constant for all models.

In the first essay, the four structural models are estimated, and smooth surfaces are

obtained using the nonlinear optimization technique. In fact, the choice of the

nonlinear optimization technique is obvious because of the following reasons: first, it is

more common in the market practices, particularly in multifactor models such as the

well-known LIBOR Market Models. Second, optimization is more flexible. Third,

parameters need to be continuously calibrated to the market information. Fourth, this

technique can capture the nonlinear effects nicely. Nevertheless, we optimize the

surface with a nonlinear least square function (lsqnonlin) in MATLAB.26 However, to

test for the goodness of fit of these models, we employ two important loss functions

proposed by Christoffersen and Jacobs (2004): relative root mean squared errors

(%RMSE) and implied volatility root mean squared errors (IVRMSE). The former

minimizes the relative difference between the market (i.e., BSIVs) and practitioner BS

model IVs; it assigns greater weight to the OTM options, as these values are small. The

latter function minimizes the error between the market IVs and model IVs, and it

assigns equal weights to IVs.

In the first essay of the thesis, aside from parametric structural specifications for IVS,

we proceed to explore the movements in the IVS without parametric specifications.

Principal component analysis (PCA) is used to reduce the whole surface dimension to

just a few independent underlying risk factors that drive the IVS; the rationale is to

extract independent factors that explain most of the dynamics of the IVS. With PCA, 25 In Model 2, when the slope is zero, it becomes a pure smile. 26 This is a built-in function. It follows an algorithm that minimizes the sum of squared errors between the actual value and the prediction for a given vector of a parameter.

20

independent risk factors can be indentified without any prior assumptions. Many

researchers have previously attempted to extract risk factors that explain most of the

variations in the IV smirk (skew) or surface: for FTSE 100 index IV, Alexander (2001)

and Cont and Da Fonseca (2002); for S&P 500 index IV, Cont and Da Fonseca (2002),

and Skiadopoulos et al. (1999); and for DAX index IV, Fengler et al. (2003). It was

confirmed that about 70-90% of the total variation in the IVS or volatility smirk (skew)

can be attributed to just three risk factors: parallel shift, tilt, and curvature. Our

approach is, therefore, similar to that of Skiadopoulos et al. (1999); however, our data

differ. We used FTSE 100 stock index IVs, whereas they used S&P 500 stock index IVs.

We thus compute the first-difference IVs across different moneyness levels and times to

maturity.27 PCA is applied to the pool data of IVs to analyze the dynamics of the IVS.

The data are then assembled into different groups on the basis of trading days left to

maturity, i.e., 8-30, 30-60, 60-90, 90-150, 150-220, and 220 days and above. After

grouping, PCA is applied to each group for a detailed investigation of the dynamics of

the IVS.

PCA is a method of matrix decomposition into eigenvectors and eigenvalue matrices. It

is applied to a pool of IVs across moneyness levels and term structures, effectively

decomposing the covariance matrix as T�� , where the diagonal elements of � are

the eigenvalues and the columns � are the associated eigenvectors. The choice of

column labeling in � allows the ordering of PCs such that 1e belongs to the largest

eigenvalue 1� , 2e belongs to second largest eigenvalue 2� , and so forth. In a highly

correlated IVS, the first eigenvalue would be much larger than the others.

Consequently, the first PC can explain much of the variation in the IVS. If the first three

PCs explain most of the variations in the IVS, then these PCs can replace the original IV

variables without loss of much information (see, e.g., Skiadopoulos et al., 1999;

London, 2004).

3.4 Modeling the Asymmetric Return-Volatility Relationship

In the second essay, we propose a quantile regression model for the negative

asymmetric relationship between returns on the stock market index and IV changes in

27 This is because the data input to PCA must be stationary, as IVs are often nonstationary; see, for more discussion on the data issues, Skiadopoulos et al. (1999).

21

the volatility index. Our model is the generalization of the standard mean-regression

models (MRM) of Simon (2003), Giot (2005) and Hibbert et al. (2008), who have

empirically confirmed the asymmetric return-volatility relationship. However, we

extend their MRM by modeling the asymmetric return-volatility relationship using the

conditional QRM to examine how negative and positive stock index returns vary across

different quantiles of IV changes, i.e., how much this asymmetric relationship changes

across different quantiles of IV changes. Their MRM is considered a standard model in

our analysis specified in the second essay. Their MRM assumes that the effects of both

types of returns are static across different IV changes (i.e., response variables);

therefore, an MRM would miss important information across quantiles of the IV

changes’ distribution that we could otherwise model using our QRM, particularly when

determining how the median or perhaps the 95th or 5th percentiles of the response

variable (IV changes) are affected by negative and positive stock return variables

(regressor variables).28

Nonetheless, first we regress the daily volatility index changes (denoted it�VI , where

i=�VIX, �VXN, �VDAX, �VSTOX) on the daily percentage continuously compounded

returns of the stock market index (denoted ,Rit where i=S&P 500, NASDAQ, DAX, DJ

Euro STOXX 50), �itR was used for positive returns, and �

itR was used for negative

returns). For the positive returns, itit RR �� if 0Rit � , and 0Rit �� otherwise. On the

other hand, for the negative returns, itit RR �� if 0Rit � , and 0Rit �� otherwise.

Koenker and Bassett (1978) were the first to introduce quantile regressions, which

could effectively model the quantiles of the distribution.29 QRM is a generalization of

the MRM and is therefore a robust regression, especially in situations where errors are

non-normally distributed, i.e., are skewed and leptokurtic. Nonetheless, the QRM is

used to examine the asymmetric return-volatility relationship; for instance, the QRM,

the specification of the qth QRM is

� � � � � � � � ,tuRR�VI��VI3

0LLit

qiL

3

0LLit

qiL

3

1LLit

qiL

qit ��

�

��

�

��

��

(16)

28 See for further discussion Meligkotsidou et al. (2009). 29 Koenker (2005) provides mathematical details on the quantile regression as well its different extensions.

22

where � �q� is the intercept, � �qiL� represents the coefficients for the lagged IV changes in

a volatility index i , 3to1L � , � �qiL represents the coefficients for positive returns and

� �qiL represents the coefficients for negative returns of a stock market index i, where

3to0L � for both type of returns and the errors tu are assumed to be independent

from an error distribution )(u� tq with the qth quantile equal to zero. The above QRM

implies that the qth conditional quantile of the dependent variable i�VI given

��

��

��

��

��

��

�� 3it2it1itit3it2it1itit3it2it1it ,R,R,R,R,R,R,R,R,��V,��V�VI and

denoting � �,��

��

�� 3itit3itit3it1itiq ,..,R,R,..,R,R,...�..�VI�VIQ

, is equal to

� � � � � � � ��

��

�

��

��

3

0LLit

qiL

3

0LLit

qiL

3

1LLit

qiL

q .RR�VI��

The main feature of this quantile

regression framework is that the effects of the variables captured by (q)iL� , � �q

iL ,and

� �qiL vary for each qth quantile within the range (0,1).q � Furthermore, the framework

allows for heteroskedasticity in error ,ut and the coefficients are different for different

quantiles. Consequently, a quantile regression provides a broader set of information

about the asymmetric return-volatility relationship here (i.e., the effects of negative and

positive returns not only on the mean of the volatility changes but it captures the effects

on all parts of the distribution of the volatility changes) than an OLS regression would,

particularly when the error distribution is not symmetric.30 Nevertheless, the QRM is

estimated using the quantile regression method proposed by Koenker and Bassett

(1978), which minimizes the asymmetric sum of absolute residuals and robustly models

the conditional quantiles of the response variable, i.e., changes in the volatility index in

our case:31

��

��

�

��

�

�

��

��

��

��

��

��

��

��

��

��

LitiLLitiLLitiLit

LitiLLitiLLitiLit

RR�VI��t:�:�LitiLLitiLLitiLit


RR�VI��VIq)(1

RR�VI��VIqmin

ˆˆˆˆ

ˆˆˆˆ

ˆˆˆˆ

ˆˆˆˆ

(17)

30 Because the differences between the mean and the median produce asymmetric distributions, see, for a more detailed explanation, Meligkotsidou et al. (2009). 31 For a discussion of quantile models and their estimation techniques, see Koenker (2005).

23

3.5 Modeling and Backtesting Daily Value-at-Risk Models

Value at risk (VaR) is defined as the maximum expected loss in the value of a portfolio.

It has a certain probability over a certain holding period (for details on VaR, see, e.g.,

Duffie and Pan, 1997; Jorion, 2000; Dowd, 2005; Christoffersen, 2008). VaR forecasts

are fundamental to financial risk management and risk regulation. However, the

importance and recognition of VaR as a risk management tool came from the Market

Risk Amendment (1996) to the Basel Capital Accord of 1998 and also because of the

popularity of RiskMetrics introduced by J.P. Morgan (see Jorion, 2000; Dowd, 2005).

Afterwards, VaR became widely accepted by banks and was also imposed by regulators.

The aim of these two groups was to supervise and manage market risk, as market-

exposure risk was arising due to unfavorable movements in equity, interest rates, and

exchange rates, among other indicators.32 Because VaR has become a standard measure

for market risk due to the immense trading activities and market positions taken by

large banks, most financial institutions and trading houses currently use VaR models to

assess their daily portfolio losses from significant trading activities. This practice leads

to backtesting of VaR models by observing when the portfolio returns exceed the VaR

forecasts, i.e., when the VaR forecasts match their expectations. As a result, accurate

VaR forecasts are crucial for market risk management. Given that accurate VaR

forecasts are heavily dependent on the accurate forecasting of the volatility of a

portfolio, this is an important parameter for any VaR model. For instance, the level of

the VaR over a one-day holding period is defined as the solution

,�)�VaRP(r 1tPt

Pt ��

(18)

where � is 1 minus the VaR confidence level (i.e., 99%) and Ptr is the return of a

portfolio over a one-day holding period. Having conditional volatility specification th

such that ,ttP

t �hr � where the residuals are distributed as � �tt 0,h~N� , then a one-

period VaR at time t is

� � ,t1

t h��VaR ��

(19)

where � denotes the standard-normal cumulative distribution function. In fact, we

need to accommodate heavy tails in the VaR estimation; therefore, VaR needs to be

32 The Basel committee for banking supervision allows banks to use VaR as a benchmark to determine how much additional capital is needed to cover market risk.

24

estimated using a Student’s t density function.33 We assume that t� are distributed as

follows:

� �� , 21

tt ~�2 h� �

(20)

where �� is the standardized Student’s t-distribution with � degrees of freedom.

To estimate the �%daily VaR using Student’s t density function, we follow the method

described by Dowd (2005):34

� � � � ,t

21

1 t,� h

2 ��VaR ��

��

(21)

where the shape � is to be estimated. In the third essay, we used four different types of

volatility forecasts [implied volatility (VDAX), implied volatility (VDAX) with GJR, FHS

(GJR), and RiskMetrics] as parameters for the daily VaR models.

For the implied-VaR model, we considered the new VDAX index (MFIV index), which

is a robust and more informed volatility measure. The rationale for considering the new

VDAX is that we believe that VaR modeling is about extreme events; implying volatility

from the extreme outcomes in the options market is of paramount importance to VaR

forecasting. These extreme events are embedded in the IV derived from OTM options

(see, for instance, Liu et al., 2005; Camara and Heston, 2008; and Bates, 2008), i.e.,

the IV smirk-adjusted volatility measure. Likewise, the importance of the new VDAX

measure increases because IV smirk accounts for the net buying pressure of the put

options as well (see Bollen and Whaley, 2004). Volatility smirk (skew) is an obvious

phenomenon previously documented by many other researchers and is important to

capture in any volatility measure (e.g., Skiadopoulos et al., 1999; Alexander, 2001; Cont

and Da Fonseca, 2002; Low, 2004; Goncalves and Guidolin, 2006; Badshah, 2008).

Furthermore, information from trading strategies and other shocks are well absorbed

into the new VDAX index, as it accounts for a cross-section of options. Finally, as the

majority of option traders is very informed and possesses professional skills, the new

VDAX represents the beliefs of the informed traders (see, e.g., Low, 2004 and

Chakravarty et al., 2004). As a result, the new VDAX is a good candidate for a volatility

parameter in the daily VaR model to quantify a daily VaR forecast for the DAX 30 stock 33 Most previous studies have concluded that distribution functions accounting for fat tails are fundamental to VaR modeling. See, for instance, Huisman et al. (1998), Alexander and Sheedy (2008) and Giot (2005) for VaR forecasts obtained through different density functions. 34 Also, because the returns on the DAX 30 are non-normally distributed, the VaR forecasts are estimated using Student’s t distribution function.

25

index portfolio.

However, for the daily VaR model, a daily-variance parameter is needed in place of the

standard deviation because VaR uses variance as an input, as the VDAX is expressed in

annualized standard-deviation units. As a result, transformation is essential; for

instance, at time t, we insert 1�tVDAX into the daily VaR model as35

,21imp,1,tt �h ��

(22)

while

� � .2

252VDAX2

1imp,1,t1t� ��

23)

However, Granger and Poon (2003, 2005) point out that BSIV is biased and is always

higher than the actual volatility. They suggest using HV for calibration as

,21imp,1,tt ��h ��

(24)

where � and need to be estimated. However, we assert that VDAX is calculated

using the MFIV measure, whose IV value should not be subject to model risk, and that

this bias should thus not be of great concern; therefore, any of the above two variance

measures can be used equally. Furthermore, we have a combined specification for

variance using GJR-GARCH(1,1) extended with the lagged IV as

.21imp,1,t1t1t

21t2

21t10t ��hd��h ��

(25)

This equation has a dummy variable used to capture asymmetry. For instance, the

dummy variable 1td � is equal to 1 when 0� 1t �� and is equal to 0 otherwise. Therefore,

estimation of an �% daily implied (VDAX) VaR forecast or the combined VaR forecast

using implied volatility (VDAX) plus GJR can be done with the following specification:

� � � � .t

21

1 t,� h

2 ��VaR ��

��

(26)

where � is the standardized Student’s t-distribution function, the shape � parameter

needs to be estimated, and � is the quantile (which in our case is 99%, 97.5% or 95%).

Another two volatility forecasts (FHS-GJR and RiskMetrics) as parameters for the

daily-VaR models are also used for the comparison with implied-VaR models. In the

35 A similar transformation scale is used by Blair et al. (2001) for the VIX index.

26

third essay, we thus backtest the daily-VaR models [implied volatility (VDAX), implied

volatility (VDAX) with GJR, FHS (GJR), and RiskMetrics] from January 1992 through

May 2009 using unconditional coverage, independence, and conditional coverage tests.

A quadratic score was also estimated for each of the models.

3.6 Modeling Risk Factors driving the Swaption Volatilities

In the fourth essay, a PCA technique is used to reduce the swaptions maturity surface to

only a few risk factors and then further model their implied dynamics across markets in

a vector-autoregressive (VAR) framework (i.e., the EUR, USD, and GBP swaption

markets). PCA can be applied to the Black (1976) consistent swaption IVs; as in the

swaption markets, swaption IVs are quoted in a manner consistent to that used by the

Black model. Numerous studies have been conducted on the dynamics of IVS implied

from equity index options.36 However, the study by Bruggemann et al. (2008) most

closely meets this objective; in this study, the stochastic properties of the factors of IVs

(i.e., those obtained from options on the DAX 30 index) were analyzed and modeled in

a VAR framework and significant interactions were found between the factor loadings.

Therefore, in a similar vein, the VAR model of Sims (1980) is used to investigate the

dynamic interactions between the first two implied factors (i.e., PC1s, PC2s extracted

from each of the EUR, USD, and GBP Swaption IVs).37 This model successfully treats

each endogenous variable in a system as a function of the lagged values of all

endogenous variables in the dynamic simultaneous equation system. Therefore, the

VAR model for the factor loading dynamics can be represented mathematically

,uF��F t

L

1iitit ��

��

(27)

where ),z,z,z,z,z(zF GBPt2

GBPt1

USDt2

USDt1

EURt2

EURt1t �� is an 1m � vector of endogenous variables

representing the factors in the EUR, USD, and GBP swaption IVs, � L,3,2,1,i,�i ��

is an mm � matrix of coefficients, and tu is an 1m � vector of innovations that can be

36 For example, Cont and da Fonseca (2002), Mixon (2002), Skiadopoulos et al. (1999), Fengler et al. (2003), Badshah (2008) and many others. 37 The first two PCs are selected because they are able to explain on average 94% of the variation in each of the market IVs. We do not consider PC3s in our VAR framework. It helps to maintain a reasonable number of parameters, increase efficiency, and achieve parsimony.

27

contemporaneously correlated and uncorrelated with its own lagged values and with

other variables. Accordingly, the VAR model is estimated by OLS estimation.

However, to investigate the direction between PCs, we apply the Granger (1969)

causality test, which establishes a causal relationship between the PCs and hence

confirms the lead-lag relationship. The Granger causality test establishes that iPC is

Granger-caused by jPC if the information in the past and present values of jPC helps

to improve the forecasts of iPC . However, Granger causality in a VAR system using the

F-test provides information only about which variables impact the future values of each

of the variables in the VAR system. In this way, F-test values do not provide the sign of

the relationship, speed or persistence. The VAR’s impulse response functions (IRF)

could provide information about this kind of dynamic relationship (see, e.g., Brooks,

2002). Therefore, a generalized version of Pesaran and Shin (1998) is preferred, as it

does not require the orthogonalization of shocks and is invariant to the reordering of

variables in the VAR system. Therefore, an IRF measures the responses of the variables

in the dynamic VAR system (in our case, the first two factors of each the EUR, USD,

and GBP swaption IVs) when a shock is given to each factor: a one-standard-error

shock is applied to the error of a factor, and the effect on the dynamical VAR system

over a specified period of time is recorded.

Furthermore, in the fourth essay, we model and reproduce the swaption maturity IVS

(or volatility matrix) on a particular day using orthogonal PCs in an SMM framework

for each of the EUR, USD, and GBP swaption markets. Santa-Clara and Sornette

(2001), and Longstaff et al. (2001) were the first to introduce the SMM model, also

known as the high- or infinite-dimensional model. Later, Kerkhof and Pelsser (2002)

showed that the SMM and the LMM are observationally equivalent. However, for the

SMM calibration, we select a separated approach proposed by Gatarek et al. (2007) that

is a calibration technique for multifactor interest rate models used widely in the

financial industry. Separated calibration could be implemented in a straightforward

way.38 For instance, the SMM model could be calibrated to the whole swaption

maturing IVS. In fact, calibration of the model to a swaption matrix is preferred over

the caps because it is essential in a situation where the prices of derivatives instruments

are dependent on covariance/correlation structures in addition to volatilities such as

38 Technical details of the separated calibration are provided in Appendix A of the fourth essay, or see Gatarek et al. (2007).

28

the values of exotic instruments including as the Bermudan type of swaption, which are

more dependent on swaption prices than on caps.39

4 Summaries of the Essays

This PhD thesis, Modeling and Forecasting Implied Volatility: Implications for

Trading, Pricing, and Risk Management, consists of four essays. All of the four essays

are single-authored. This section provides a brief summary of the essays, their

contribution to the literature and their implications.

4.1 Essay 1: Modeling the Dynamics of Implied Volatility Surfaces

This essay models implied volatility surfaces and identifying risk factors that account

for most of the randomness in the volatility surfaces for the FTSE 100 index options. In

general, the essay attempts to model and uncover important stylized facts in the stock

index options appearing after the 1987 crash. The approach is an extension that follows

the framework by Dumas, Fleming and Whaley (1998); for instance it uses moneyness

in the implied forward price and OTM put-call options on the FTSE 100 stock index,

and a nonlinear parametric optimization technique is then used to estimate different

DFW (1998) models.

We estimated four parametric models of DFW (1998); the constant-volatility Model 1

fails to capture variations in the IVSs, yielding only flat IVSs. However, Model 2, which

captures the volatility skew and volatility smirk effects, does a good job and captures

most of the volatility skew and smirk effects. When we estimated Models 3 and 4, which

account for both the volatility term structure and volatility smirk/skew effects, they

performed better than the one-dimensional smirk/skew model. However, both models

fit the market IVS well, generating the best IVSs for the observed data. In the second

part of the study, which uses PCA (a nonparametric approach), we find that the first

three factors can explain about 69-88% of the variances in the IVS. Of this fraction, an

average of 56% is explained by the level factor, 15% is explained by the term-structure

factor, and an additional 7% is explained by the jump-fear factor. Finally, when

applying PCA to the maturity groups, we find that level factors (or parallel shifts) are

evident and thus important for shorter- and longer-maturity IVs. However, the term-

39 See Gatarek et al. (2007) for a detailed discussion.

29

structure factor is important for IVs with medium maturities, and the jump-fear factor

is important for the IVs with the shortest (8-90 days) maturities.

IVS is important in practice; for instance, in the FTSE 100 index option market, 100

options on average are traded daily with a variety of strike prices and time to

maturities. If the practitioner needs to quote more options for different strikes or

maturities, IVS can be used as a reference; the practitioner could easily quote new

options by taking estimates from IVS. Moreover, the shape of the IVS is important in

option trading; one of its most important characteristics is that it can be used to

identify mispricing in the market. Second, the IVS can be used to manage exotic

derivative positions, which could be hedged with plain-vanilla options such as

European options. Third, a robust volatility index could easily be constructed from the

IVS and consequently used to price volatility derivatives. Fourth, risk can be managed

for securities that underlie the FTSE 100 stock index by using independent risk factors

from the IVS.

4.2 Essay 2: Quantile Regression Analysis of Asymmetric Return-

Volatility Relation

Essay 2 proposes a quantile regression model (QRM) for modeling of the negative

asymmetric return-volatility relationship with newly adapted IV indices, especially to

quantify the effects of the positive and negative stock index returns at various quantiles

of the IV changes’ distribution; in particular, this essay is an attempt to model and

explore another important stylized fact: the strong negative asymmetric return-

volatility relationship at high frequencies. The quantile regression model that we

propose is the generalization of the standard mean regression models (MRM) of Simon

(2003), Giot (2005) and Hibbert et al. (2008). Nonetheless, we investigated the

asymmetric return-volatility phenomenon in the newly adapted robust volatility indices

(i.e., the VIX, VXN, VDAX, and VSTOXX) using quantile regression. In particular, we

quantified the effects of positive and negative stock index returns at different quantiles

of IV changes’ distributions, asking about the degree to which the asymmetric

responses at the uppermost quantiles are comparable to the responses of median (or

mean) regressions. Additionally, as Bollen and Whaley (2004) have documented, the

net buying pressure for stock index put options from institutional investors seeking to

hedge their portfolios induces increases in IVs. Likewise, new IV indices incorporate

both OTM put and call options and are thus highly informed and robust measures.

30

Accordingly, they should present more pronounced asymmetric return-volatility

relationships in comparison with their older counterparts.

There is noticeable evidence that the volatility indices VIX, VXN, VDAX, VDAXO and

VSTOXX from February 2001 through May 2009 responded in a pronounced

asymmetric fashion to the negative and positive returns of their corresponding stock

indices; the asymmetry monotonically increases when moving from the median

quantile to the uppermost quantile (i.e., 95%). Therefore, OLS underestimates this

relationship at upper quantiles. These IV indices thus sharply rise during market

declines (fear) and fall during market rallies (exuberance). The VIX presents the

highest asymmetry, followed by the VSTOXX, VDAX and VXN volatility indices.

Second, our argument that asymmetry with a volatility smirk (skew)-adjusted volatility

index measure (MFIV) should be pronounced is confirmed by comparing the

asymmetric responses of VDAX (MFIV) and VDAXO (BSIV); the MFIV index responds

in a pronounced fashion when compared to the BSIV index. Third, we also confirmed

that a significant amount of asymmetry occurs contemporaneously rather than with a

lag, thus rejecting the leverage hypothesis, and that a similar conclusion can be drawn

for the feedback hypothesis.

Our results have a number of implications. First, as we found that newly adapted MFIV

indices are strongly negatively correlated with their corresponding stock indices and

that the MFIV indices are important instruments for hedging stock portfolios.

Derivatives exchanges provide liquid markets for the futures and options underlying

MFIV indices. Therefore, a position in futures or options on a MFIV index can

completely hedge a stock portfolio position without consideration of complicated stock

index option-trading strategies; for instance, a call option on the MFIV index can be

seen as put option on the underlying stock index. Second, when the stock index drops,

the MFIV index rises sharply. Therefore, MFIV indices are useful for assessing not only

potential risks but also speculative transactions by risk-seeking investors. Third,

because the MFIV indices are based on the robust MFIV concept and provide better

tradability, it is easier for issuers of derivatives to engineer structured products based

on the MFIV indices. Fourth, trading strategies with regard to range could generate

profits; an example of this could be a volatility-long position in decreasing volatility

markets paired with a volatility-short position in increasing volatility markets.

31

4.3 Essay 3: The Information Content of the VDAX Volatility Index and

Backtesting Daily Value-at-Risk Models

The third essay examines the information content of the new VDAX volatility index to

forecast daily VaR estimates for a DAX 30 index portfolio and compare its VaR

forecasts with the forecasts of the other horse-race models such as Filtered Historical

Simulation (FHS) and RiskMetrics. Many researchers have recently documented that

the information content of a volatility measure implied from the OTM put and call

options contains broader market-wide sentiments (beliefs that are updated upon option

trading) about future volatility; OTM options embed negative and positive volatility

jumps, stochastic volatility, and demand-supply pressure (see, e.g., Pan, 2002; Liu et

al., 2005; Bates, 2008; Camara and Heston, 2008; Doran et al., 2007, Bollen and

Whaley, 2004). We know that VaR modeling is all about the extreme events; therefore,

the new VDAX (MFIV index) index, which incorporates most of the information

required for VaR forecasting, is a perfect candidate to consider for VaR forecasting.

The information content of the new VDAX was incorporated into daily VaR forecasts

and compared with the VaR forecasts from the FHS (GJR) and the RiskMetrics models

at various confidence levels (i.e., 99%, 97.5% and 95%), using unconditional coverage,

independence, and conditional coverage tests for backtesting of each of the VaR

models. Furthermore, a quadratic score was estimated for each VaR model for the

period from January 1, 1992 through May 29, 2009. The backtesting results showed

that the new VDAX index contains significant information about actual volatility in VaR

models. The null hypotheses of independence and conditional coverage backtests were

never rejected for implied volatility, implied volatility plus GJR, or FHS (GJR) VaR

models. The number of VaR exceptions was not significantly different from the set

coverage rates. However, the null hypotheses of the RiskMetrics model were rejected

for lower confidence levels. It was also found that implied volatility and implied

volatility-GJR VaR models presented the fewest VaR exceptions and clusters of

exceptions, in contrast to the FHS (GJR) and RiskMetrics models. On the other hand,

the quadratic score for each model suggests the following ranking of VaR models:

implied volatility (VDAX), combined (implied volatility plus GJR), FHS, and

RiskMetrics. Our findings have implications for traders who hold long positions, risk

managers (internal), and regulators (external).

32

4.4 Essay 4: Modeling Risk Factors Driving the EUR, USD, and GBP

Swaption Volatilities

The fourth essay models risk factors driving EUR, USD, and GBP swaptions’ implied

volatilities. In particular, the essay attempts to answer the following: first, how many

common underlying implied-risk factors are driving the IV maturity surfaces of the

EUR, USD, and GBP swaption markets? This essay then attempts to examine the

dynamic interactions between the implied risk factors across markets by using

techniques such as Granger causality and the generalized impulse response function.

Finally, this study aims to calibrate a multifactor model such as a string market model

(SMM) to swaption IVS by using multivariable nonlinear optimization to reproduce the

swaption surfaces for the EUR, USD, and GBP swaption markets.

First, we applied PCA to each of the EUR, USD, and GBP swaption IVs to discover the

important risk factors. We found that three risk factors explain about 94 -97% of the

variance in each of the EUR, USD, GBP swaption IVs. Second, the significant implied

factors present high correlations across swaption markets; consequently, there are

strong linkages across the three markets. Bi-directional causality is at work between the

implied factors from each of the EUR and USD swaption markets. The factors from

EUR and USD swaption markets Granger-cause the factors from the GBP swaption

market, but not vice-versa. Furthermore, in innovation-accounting investigations,

shocks to both factors implied by the EUR and USD IVs are found to be influential for

the factor implied from the GBP IVs. However, a shock to the GBP factors does not

affect the factors observed in the other two markets. Finally, there are many similar

characteristics between EUR and GBP markets, in contrast to the USD swaption

market. The whole swaption matrix for the EUR, USD and GBP markets is reproduced

using the SMM model. The fewest differences are observed between the theoretical and

market volatilities for EUR, GBP, and USD, respectively. Here too, similar

characteristics are found between EUR and GBP markets.

The identification of risk factors is important in practice. These factors can be used for

hedging of the portfolio position, particularly Vega-hedging, generating smooth implied

volatility surfaces, managing risk, and calibrating models. First, as in the interest rate

market, we find securities with a variety of maturities; therefore, it is almost impossible

for a trader to Vega-hedge portfolios against each and every individual risk. Therefore,

in the easiest way to account for most of the risks, a trader uses the major risk factors

33

instead of hedging against every underlying risk. Second, we find a limited number of

OTC-traded swaptions in the swaption markets. When a trader needs to quote options

with different maturities, he is generally restricted. This could be solved by a swaption

volatility surface, thereby allowing us to use the risk factors to generate a swaption

volatility surface. A trader can easily quote new options by taking estimates from the

volatility surface. Third, by employing these factors, a trader can manage the overall

downside risk for his portfolio, i.e., the value-at-risk for a portfolio consisting of

interest rate derivatives. Finally, market models could be easily calibrated to only few

important factors, i.e., LIBOR market models.

The SMM is an important pricing and hedging tool. First, this model enriches the LMM

by calibrating to the whole swaption matrix, whereas the LMM is calibrated to only a

few risk factors. Therefore, the SMM calibration accounts for all independent risk

factors in a parsimonious fashion. Second, once a whole swaption matrix is reproduced,

the pricing and hedging of exotic derivatives such as Bermudan swaptions could be

done using these plain-vanilla products. Finally, in conjunction with SMM,

multivariable nonlinear optimization could be used for increased accuracy and

efficiency in the pricing and hedging of exotic interest rate derivatives.

34

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Studies 3, 77-102. Simon, D., 2003. The Nasdaq volatility index during and after the bubble. Journal of

Derivatives 11, 9-24. Sims, C.A., 1980. Macroeconomics and reality. Econometrica, 48, 1-48. Skiadopoulos, G., 2001. Volatility smile consistent option models: A survey.

International Journal of Theoretical and Applied Finance 4, 403-437. Skiadopoulos, G., 2004. The Greek implied volatility index: construction and

properties. Applied Financial Economics 14, 1187-1196. Skiadopoulos, G., Hodges, S., and Clewlow, L., 1999. The dynamics of the S&P500

implied volatility surface. Review of Derivatives Research 3, 263-282. Whaley, R., 2000. The investor fear gauge. Journal of Portfolio Management 26, 12-17. Zhang, J., and Xiang, Y., 2008. The implied volatility smirk. Quantitative Finance 8,

263-284.

38

39

PART II: THE ESSAYS

40

41

Modeling The Dynamics of Implied Volatility Surfaces

Ihsan Ullah Badshah

Hanken School of Economics, Department of Finance and Statistics, P.O. Box 287, FIN-65101 Vaasa, Finland. Phone: +358-6-3533 721, Fax: +358-6-3533 703,

Email: [email protected]

September 20, 2010

Abstract

The purpose of this study is to model implied volatility surfaces and identify risk factors that account for most of the randomness in these volatility surfaces. The approach is similar to that of the study by Dumas, Fleming and Whaley (DFW) (1998). We use moneyness in the implied forward price and out-of-the-money put-call options on the FTSE 100 stock index. After adjustments, a nonlinear parametric optimization technique is used to estimate different DFW models to characterize and produce smooth implied volatility surfaces. Next, principal component analysis is applied to the implied volatility surfaces to extract principal components that account for most of the dynamics in the shape of the surfaces. We then estimate and obtain smooth implied volatility surfaces with the parametric models that account for both smirk (skew) and time to maturity. We find that the constant-volatility model fails to explain the variations in the surfaces. However, the first three principal components (or factors) can explain about 69-88% of the variance in the implied volatility surfaces, in which, on average, 56% is explained by the level factor, 15% is explained by the term-structure factor, and the remaining 7% is explained by the jump-fear factor. Applications of our study include options trading, hedging of derivatives positions, risk management of options, and policymaking. Keywords: implied volatility, implied volatility surface, options, principal component analysis, smirk JEL classification: C13, C53, G12, G13 The Author would like to thank Johan Knif, Mika Vaihekoski, George Skiadopoulos, Gregory Koutmos, Kenneth Högholm, David Simon, and Yakup Arisoy for providing useful comments. The author would also like to thank to the participants of the EFMA conference (2008), Athens, Greece; the Global Finance Conference (2007), Melbourne, Australia; and the Graduate School of Finance (GSF) Research workshops for their useful comments. The author acknowledges Evald & Hilda Nissi and HANKEN foundations for providing financial support.

42

1 Introduction

Modeling and exploring the dynamics of the volatility of financial assets has been an

active area of research in quantitative finance, for both academics and practitioners.

Volatility is fundamental for risk management, option pricing, option trading, hedging

derivative positions, constructing and diversifying portfolios and policy making. Most

previous research has focused on historical volatility (HV), but implied volatility (IV)

has recently received attention due to some breakthrough studies, such as those of

Christensen and Prabhala (1998), Fleming (1998), Skiadopoulos et al. (1999), Dumas et

al. (1998), Blair et al. (2001), Ederington and Guan (2002), Poon and Granger (2003)

and Gocalves and Guidolin (2006). They have shown that implied information is

superior to historical information when forecasting volatility and thereby suggest that a

precise view of the market's judgment and expectation of volatility is vital for

forecasting volatility. HV is backward looking and incorporates expectations about

future volatility based on the past behavior of stock prices and other relevant

information. In contrast, IV is forward looking; that is, it is implied by the market

option prices. Option prices are the common consensus of market participants

regarding the expected future volatility of an underlying asset over the remaining life of

an option; usually, the majority of option markets’ participants are institutional

investors who can make professional judgments on the future direction of volatility.

The volatility expectation of market participants can be recovered by inverting the

option-pricing model. However, since the 1987 stock market crash, one of the strongest

empirical regularities in the stock index options has been that IV recovered from stock

index options has appeared to differ between moneyness (strike dimension), the

volatility smirk (skew), and term structure (time-to-maturity dimension), the volatility

term structure if the two are examined at the same time, for example, on the same day.

This indicates that implied volatilities (IVs) present a rich implied volatility surface

(IVS), whose dynamics across strikes and over time warrant further investigation.

The Black and Scholes (BS) (1973) model assumes that underlying assets follow

geometric Brownian motion and constant volatility, implying that options on the same

underlying asset with different strike prices and times to maturity should have the

same IV. In fact, the assumption of constant volatility does not hold empirically

because asset prices are mostly influenced by risk factors such as jumps, stochastic

volatility or time-varying volatility, the demand/supply pressure of options, and

transaction costs (see, e.g., Carr et al., 2001; Heston, 1993; Leland 1985; David and

43

Veronesi, 2000; Garcia et al., 2003; and Bollen and Whaley, 2004). To account for

these deviations, practitioners use different IVs for different strikes and maturities.

Therefore, the IV of an option reflects the determinants of the option value that are not

captured by the constant volatility BS model – this volatility structure illustrates

discrepancies between theoretical prices and the market. Therefore, an IVS that

includes all of these features is essential in practice, particularly for forecasting future

volatility, trading options, pricing illiquid options, pricing and hedging exotic

derivatives, and managing the risk of the options portfolio.

Furthermore, it is important for practitioners to generate IVS from the IVs implied by

options that contain the majority of market sentiments (beliefs on bearish and bullish

markets). Therefore, implying volatility from the out-of-the-money (OTM) put and call

options in addition to at-the-money (ATM) options is of paramount importance to any

volatility measure or IVS. OTM put options incorporate beliefs on the market crashes,

and OTM call options incorporate beliefs on market spikes (see, e.g., Pan, 2002; Liu et

al., 2005; Doran et al., 2007; Bates, 2008; Camara and Heston, 2008). Because we

know that investors (particularly institutional investors) always hedge their stock index

portfolios with put positions, during market declines, there is greater demand for OTM

puts (with less supply) than OTM calls (with less demand), leading to higher IVs (see,

e.g., Bollen and Whaley, 2004). Therefore, it is essential that IVS be generated (or

forecasted) using IVs from both OTM puts and calls; these IVs contain a broader set of

information that includes information on future negative and positive jumps.

However, the IVS can be viewed as highly correlated because IVs present a high degree

of correlation among risk factors, indicating high dependence. When few important

sources of information are common to the risk factors, we find a high degree of

correlation among risk factors. Principal component analysis (PCA) is a tool that

extracts the most important independent risk factors from the correlated IVS and can

explain most of the dynamics. Therefore, PCA reduces the dimensionality and enhances

computational efficiency. After extracting common risk factors from the IVs, these can

be used for a variety of purposes, such as projecting IVS, Vega hedging, Value-at-Risk

modeling, and model calibration.

There are few studies that estimate and obtain IVS and further study the dynamics of

these IVSs by applying PCA. However, many studies have examined either the volatility

smirk/skew/smile (strike dimension) or the volatility term structure. One well-known

44

study on IVS was conducted by Cont and Da Fonseca (2002), who examined the

dynamics of IVSs of S&P 500 and FTSE 100 index options. The price of an index option

on a given date is represented by a corresponding IVS, which has volatility smirk (skew)

and term structure features. They used different methods to obtain smooth IVSs.

However, the main findings were, first, that an IVS has a non-flat shape in both

dimensions (i.e., strike and time). Second, the shape of the IVS changes with time.

Third, IVs present positive autocorrelation, indicating the property of mean reversion.

Fourth, variations in IVS can be explained by two or three principal components (PCs).

Finally, the movements in the underlying assets are not correlated with movements in

the IVs. In another study on IVS, le Roux (2006) developed an econometric model to

estimate IVSs and then examined the dynamics of the estimated IVS using PCA. He

used VIX index data and options on the S&P500 index and found that 75.2% of the

variations of the IVS can be explained by the first PC and that another 15.6% can be

explained by the second PC. Moreover, he found that processes for these factors appear

to be independent of the process of the VIX index. Alentorn (2004) adapted a

parametric form of the DFW (1998) approach and estimated IVS for options on the

FTSE 100 futures. He used three DFW models. He further suggests how to implement

this methodology in real time. This study differs from the above two in that it only

estimates IVS, whereas the other two additionally study the dynamics of the IVS.

Many studies have been conducted on the dynamics of either smirk/skew/smile or

surfaces using PCA, such as that by Skiadopoulos, Hodges, and Clewlow (SHC) (1999),

who investigated the factors and shape of shocks that move IV smiles and IVS of

S&P500 index options. They formed maturity buckets within the IVS, the average IVs

of options whose maturities fall into them, and applied PCA to each bucket separately.

They identified two factors that explain about 60% of the variance. They suggest that to

effectively price hedge future options, one needs only three risk factors: one for the

underlying asset and the other two for IV. In another study, Alexander (2001)

developed a new PCA model of fixed-strike volatility deviations from at-the-money

(ATM) volatility. She focused on its application to the volatility skew and used FTSE

100 index options with a maximum of three-month maturities. She investigated

whether the IVS would move continuously if the index moved and, furthermore,

whether second and higher PCs with non-zero conditional correlation with the index

changes cause non-parallel movements in the IVS as the index moves. She found that

most of the variation in the IV of FTSE 100 index options can be explained by just three

45

risk factors: parallel shifts, tilts, and curvature changes. However, Alexander (2001)

investigated only skew, whereas SCH (1999) investigated both skew and surface.

The objective of this essay is to estimate and obtain smooth IVSs using parametric

models and to study the dynamics of these IVSs. Our study is related to that of DFW

(1998) in terms of functional forms of estimated models for different IVSs, whereas the

second part of our study is related to that of SHC (1999) in terms of applying PCA to

IVSs. Our study, however, differs in some ways from the two cited studies: first, we

estimate IVS using OTM puts for low strikes and OTM calls for high strikes in addition

to ATM options, which are not accounted for in DFW (1998). Second, we study IVSs

using moneyness in the forward price and time to maturity, whereas DFW (1998)

incorporated the strike instead. Third, we employ PCA to extract those risk factors that

explain most of the dynamics in the IVSs to determine what these factors look like, how

many factors can explain these IVSs, and their interpretation. Finally, we study the

dynamics of the IVSs of FTSE 100 index options (i.e., over the most recent four years),

whereas SHC (1999) studied futures options on the S&P500 index.

Our main findings are that the DFW constant volatility model fails to capture the

variations in the IVSs and, consequently, generates only flat IVSs. Model 2, which

captures the volatility skew and volatility smirk effect, does a good job and captures

most of the volatility skew and smirk effects. When we estimated Models 3 and 4, which

account for both the volatility term structure and volatility smirk/skew effects; these

models performed better than the one-dimensional smirk/skew model. However, both

models fit the market IVS well and generate the most appropriate IVSs for the observed

data. In the second part of the study, which uses PCA (nonparametric approach), we

find that the first three factors can explain about 69-88% of the variances in the IVS:

within this, on average, 56% is explained by the level factor, 15% is explained by the

term-structure factor, and an additional is explained 7% by the jump-fear factor.

Finally, when applying PCA to the maturity groups, we find that level factor (or parallel

shifts) is evident and thus important for shorter- and longer-maturity IVs. However,

the term-structure factor is important for the IVs with medium maturities, and the

jump-fear factor is important for the IVs with the shortest (8-90 days) maturities.

This essay is organized as follows. Section 2 describes the methodology. Section 3

describes our data. Section 4 presents results, and Section 5 concludes our study.

46

2 Methodology

2.1 Model for Implied Volatilities

Merton (1973) extended the Black-Scholes (1973) model for European call and put

options on dividend-paying tS stock or stock index with a strike price of K , a time to

maturity of tT� �� , a call with a payoff of K,0)(SMax t � , and a put with a payoff of

,0)S(KMax t� . The BS formulae for the values of these call and put options are

� � � � � �,2r�

1q�

tt d�Ked�eS,K,r,�K,SC �� (1)

� � � � � �,1q�

t2r�

t d�eSd�Ke,K,r,�K,SP ��

(2)

� � � �,

��2�qr�KSln

d2

t1

��

(3)

� � � �.

��2�qr�KSln

d2

t2

��

(4)

where q is the dividend rate on the underlying asset, r is the risk-free interest rate, and

)�( � stands for the standard normal cumulative density. Consider an option market in

which the BS assumptions do not hold empirically (see the empirical evidence of Bates,

1991; Rubinstein, 1994; Ait-Sahalia and Lo, 1998); then, the quoted market prices of

put and call options are t

~C and t

~P . Thus, there is a unique implied volatility,

BSt� ,

which equates the theoretical option price with the market option price given all other

observed parameters:

,~

tBSttBS C),K,r,�K,(SC � (5)

.t~

BSttBS P),K,r,�K,(SP �

(6)

The BSIV can be found uniquely because of the monotonicity of the BS formula in the

volatility parameter,

0�

BS�

!!

. (7)

47

For a fixed value of (K,T) , (K,T)� BSt is a stochastic process, and for a fixed value of t,

its value depends on the characteristics of the option such as maturity T and strike K.40

Therefore, the function is

.(K,T)�:(K,T)� BSt

BSt " (8)

This is the implied volatility surface at date t. However, with moneyness of the option,

the IVS is a function of moneyness and time-to-maturity, that is,

� � .�)(m(S(t),t�m,�I BStt �� (9)

Nonetheless, with the help of the bisection method, IVs are obtained for each strike

price and time to maturity, that is, for the OTM put and call as well as for ATM options.

The bisection method is efficient and fast with the advantage that IVs can be estimated

without knowing Vega.

2.2 Modeling Implied Volatility Surfaces

To model and obtain smooth implied volatility surfaces, we prefer a parametric

approach similar to the approach proposed by Dumas et al. (1998). This approach is

also known as the Practitioner Black and Scholes model (PBS) (see, e.g., Christoffersen

and Jacobs, 2004; Goncalves and Guidolin, 2006). Its parsimony and application

motivated us to study it thoroughly and extend it further, particularly to fit PBS to the

FTSE100 index options’ information. However, we approach the problem somewhat

differently than the original work of DFW (1998): first, we use the most informed and

liquid options besides ATM, which are the OTM puts and calls. Because the put option

is the natural hedging instrument for the underlying stock index portfolio, institutional

investors always hedge their portfolios’ downside risk with OTM put options; this trend

is obvious when they forecast any market uncertainty (see, e.g., Pan, 2002; Bollen and

Whaley, 2004; Foresi and Wu, 2005; Bates, 2008; Camara and Heston, 2008).41

Second, we obtain IVS with more practitioner-friendly moneyness, which takes into

account forward prices and time dependence, because, for the informed IVS, daily

calibration is essential to the changes in stock price and IV. The IV as a function of

40 See, for a detailed discussion, the study of Cont and da Fonseca (2002). 41 However, when they forecast any market spikes in the future they go long in the OTM calls with limited downside risks, which remain to the premium of the options only, instead of taking positions in the underlying stock index.

48

strike does not adequately capture the market movements, whereas IV as a function of

the moneyness parameter does. Third, we use FTSE 100 index options data, whereas

DFW (1998) used S&P 500 index options data. Fourth, we use a parametric

optimization technique to estimate an IVS that is common in practice.

Fung and Hsieh (1991) and Jackwerth and Rubenstein (1996) were the first to propose

a simple moneyness measure, that is, to take the ratio of the strike price and stock

price. However, rather than absolute moneyness, practitioners express moneyness

based on the log of the implied forward price versus the strike. For a strike, K , and an

implied forward price, �rtSe , where tr is the interest rate until the maturity of the

option, and tT� �� , is the time to maturity. However, this moneyness measure is

time independent. Time dependence is important when updating IVS. There are many

measures in the market that account for different information in the market (on the

implied forward price moneyness measures, see, e.g., Foresi and Wu, 2005; Goncalves

and Guidolin, 2006; Tompkins, 2001; Alentorn, 2004). However, we use a similar

moneyness in implied forward price as proposed by Gross and Waltner (1995).

Moneyness in implied forward price, m, is defined as the logarithm of the implied

forward price, ,Se �rt over the strike price, K, normalized by the square root of the time

to maturity, tT� �� , that is,

�/K)elog(S

mt�r

t� . (10)

Four different structural forms of model specifications are estimated, leading to smooth

IVSs, each of which differs in the parametric structure that serves to characterize the

IVs:

Model 1: � � ,��m,�I 0 �� (11)

Model 2: � � ,�m�m��m,�I 2210 �� (12)

Model 3: � � ,�m��m�m��m,�I 432

210 �� (13)

Model 4: � � .��m��m�m��m,�I 2543

2210 �� (14)

In Model 1, the flat IV function is a constant, that is, the BS model that gives volatility

equal to ATMIV does not account for volatility skew, smirk or term-structure effects.

49

Model 2 attempts to capture volatility skew and volatility smirk across moneyness;

here, the slope, ,�1 represents volatility skew, and the curvature, 2� , represents

volatility smirk effects.42 Model 3 captures additional variation attributable to time

(e.g., time to maturity slope), 3� , and a combined effect of the skew and time

dimension, .4� Finally, Model 4 adds a parameter that captures the time-to-maturity

curvature effect, 5� . The remaining notations are as follows: � �m,�I represents the

dependence of IV on the moneyness level and time to maturity, and 0� is a parameter

that is constant for all models.

The four structural models were estimated and smooth IVSs were obtained using the

nonlinear optimization technique. In fact, the choice of the nonlinear optimization

technique is obvious for the following reasons: first, we chose nonlinear optimization

because this is more common in market practice, particularly for multifactor models

such as the well-known LIBOR market models. Second, optimization is more flexible.

Third, the parameters need to be continuously calibrated to the market information.

Fourth, this can capture the nonlinear effects nicely. Therefore, we optimize the IVS

with a nonlinear least square function (lsqnonlin) in MATLAB.43

However, to test for the goodness of fit of the models and for comparison, we use two

important loss functions proposed by Christoffersen and Jacobs (2004): relative root

mean squared errors (%RMSE) and IV root mean squared errors (IVRMSE). The

%RMSE loss function is as follows:

� � � � .��

��

��

N

1i

2

i

iN1

eOptionPric�e�%RMSE

(15)

The %RMSE loss function minimizes the relative difference between the market IVs

(i.e., BSIVs) and PBM model IVs. Therefore, %RMSE assigns a greater weight to the

deep OTM options because these values are small. For instance, we can see that the

denominator of the %RMSE loss function includes the market price of the option.

Therefore, when the market price of the option is small, the difference between the IVs

42 In Model 2, when the slope is zero, it becomes a pure smile. 43 This is a built-in function. It follows an algorithm that minimizes the sum of squared errors between the actual value and the prediction for a given vector of a parameter.

50

is amplified. However, the IVRMSE loss function assigns equal weights to IVs, as

follows:

� � � �� .��IVRMSEN

1i

2iiN

1 ��

��

(16)

Therefore, the IVRMSE loss function minimizes the error between the market IVs and

PBS model IVs.44

2.3 Principal Component Analysis of Implied Volatility Surfaces

In addition to the parametric approach for modeling the IVS, we proceed to examine

the dynamics of the IVS using the principal component analysis (PCA) approach.45 PCA

is used to reduce the IVS dimensions to only a few uncorrelated underlying risk factors

that drive the IVS, the rationale is to extract independent factors that explain most of

the dynamics of IVS. With PCA, independent factors can be identified without any prior

assumptions. Many researchers have attempted to extract those risk factors that

explain most of the variation in the volatility smirk (skew) or surface; Alexander (2001)

and Cont and Da Fonseca (2002) (FTSE 100 index IV); Cont and Da Fonseca (2002)

and Skiadopoulos et al.(1999) (S&P 500 index IV); and Fengler et al. (2003) (DAX

index IV). Their findings confirmed that about 70-90% of the total variations in the IVS

or skew can be attributed to just three factors: parallel shift, tilt, and curvature, which

are changes that are captured by the first three PCs. Therefore, our approach is similar

to that of Skiadopoulos et al. (1999) but our data differ. We used FTSE 100 index IV

data, whereas they used S&P500 index IV data.

We compute the first-difference IVs across different moneyness levels and times to

maturity because the data input to PCA must be stationary, and IVs are often

nonstationary (for more discussion on the data issues, see, e.g., Skiadopoulos et al.,

1999). PCA was applied to pool data of IVs to analyze the IVS dynamics. Then, the data

were assembled into different groups based on the number of trading days left to

maturity, that is, 8-30, 30-60, 60-90, 90-150, 150-220, and 220 days and above. After

44 See Rouah and Vainberg (2007) for a nice detail discussion on the above loss functions. 45 However, recently Fengler et al. (2007) proposed a modified method to the PCA, that is a semiparametric approach of modeling the dynamics of implied volatility surface, their approach might be advantageous in modeling the dynamics, however, we believe that our approach is simple, parsimonious and widely acknowledged in practice as well previously employed by many studies such as Skiadopoulos et al. (1999) etc.

51

making these groupings, PCA was applied to each group to investigate the IVS

dynamics in greater detail.

PCA is a matrix decomposition method that decomposes a matrix into eigenvectors and

eigenvalue matrices. It was applied to a pool of IVs across moneyness levels and term

structures to decompose the covariance matrix into T�� , where the diagonal

elements of � are the eigenvalues and the columns � are the associated eigenvectors.

The choice of column labeling in � allows PC ordering such that 1e belongs to the

largest eigenvalue 1� , 2e belongs to the second-largest eigenvalue, 2� , and so forth. In

a highly correlated IVS, the first eigenvalue would be much larger than the others.

Consequently, the first PC can explain much of the variation in the IVS. If the first three

PCs explain most of the variations in the IVS, then these PCs can replace the original IV

variables without losing much information (see, e.g., Skiadopoulos et al., 1999; London,

2004). That is, the original IV input data may be written as a linear combination of the

PCs, which reduces the dimensions of the system.

The first three PCs or factors can be interpreted in this study as follows: the first factor

(or PC1), which is common to the IVs and moves the entire IVS systematically in the

same direction, can be interpreted as the volatility level factor (or parallel shifts). It is

important for IVs implied from the shortest and longest maturity options. The second

factor (or PC2) can be interpreted as the term-structure factor (or tilt, which generates

shifts in the slope of the term structure of IVs), which affects the shorter- and longer-

maturity IVs differently (i.e., the sign of the effects differ for the shorter and longer

maturities IVs). The third factor (or PC3) can be interpreted as the jump-fear factor (or

curvature, which changes the steepness of IV smirk/skew); the effects of the jump-fear

factor on the OTM puts and calls significantly differ, which induces steepness in the

volatility smirk (skew). For empirical evidence on the dynamics of the factors driving

the IVS of index options, see, e.g., the studies by Skiadopoulos et al. (1999), Mixon

(2002), and Cont and da Fonseca (2002).

52

3 Data

We used daily data of European options on the FTSE 100 stock index. The data were

extracted from the Euronext database for the period from January 01, 2004 to

December 31, 2007. The end-of-the-day prices were considered and were updated daily

in the Euronext database. The database includes the following daily information for

each call and put traded option: the type (call/put), date, delivery date, strike price,

volume, and open interest; opening, high, low, and closing option prices; and

underlying price. First, we used option data; the maturity ranged from eight days up to

one year and options varied across different strike prices. Second, data on the discount

curve, which had different maturities to proxy for a risk-free rate downloaded from

Thomson DataStream. Finally, dividend data; the dividend yield for the FTSE 100 stock

index was backed out from FTSE 100 Futures prices, using the cost-of-carry approach

(See e.g., Hull, 2002), the futures data were extracted from the same Euronext

database. We considered data for ATM options and OTM put and call options: OTM

puts were used for low strikes, and OTM calls were used for high strikes. However, the

data on in-the-money (ITM) puts and calls were not considered because they are

sensitive to the non-synchronicity problem. Moreover, options with fewer than eight

days to maturity were also excluded because they are very sensitive to small errors in

the option price (e.g., Skiadopoulos et al., 1999).

Figure 1 shows the scatter graph of the FTSE 100 index (level) against the ATMIV

(level) of short maturity (i.e., one month to maturity) – the FTSE 100 stock index level

is on the vertical axis, and the ATMIV level is on the horizontal axis. We observe a

noticeable negative relationship between the two historical series. This evidence is

consistent with previous research, which has found similar evidence that the stock

returns are negatively correlated with changes in IVs.

53

4,000

4,400

4,800

5,200

5,600

6,000

6,400

6,800

8 10 12 14 16 18 20 22 24 26 28

ATM Implied Volatility(Level)

FTSE

100

Inde

x(Le

vel)

FTSE100 Index versus ATM Implied Volatility

Figure 1. A scatter plot of FTSE100 Index (level) versus ATMIV (Level) from January

1, 2004, to December 31, 2007.

Table 1 presents the descriptive statistics for the ATMIVs and the FTSE 100 index from

January 1, 2004 to December 31, 2007. Panel I of Table 1 shows the descriptive

statistics for the level. The mean ATM volatility was 12.8% annually, with a maximum

of 27.2% and a minimum of 8%, whereas for the FTSE 100 index, the mean level was

5,501 points, with a maximum level of 6,732 points and a minimum level of 4,287

points. However, Panel II of Table 1 reports the descriptive statistics on the daily

percentage changes (�) in the one-month ATMIVs and the daily percentage

continuously compounded returns of the FTSE 100 stock index. We can observe from

the skewness that �ATMIVs are positively skewed, whereas the FTSE 100 index returns

are negatively skewed; however, the kurtosis for both �ATMIVs and FTSE 100 returns

is high (i.e., greater than the normal three). The Jarque-Bera tests for overall normality

reject that both �ATMIVs and FTSE 100 returns are normally distributed. Therefore,

both �ATMIVs and FTSE100 index returns are non-normally distributed. Some

researchers have documented that non-normality in the underlying should trigger non-

constant IVs. Panel III of Table 1 shows a correlation between the FTSE 100 index and

54

one-month �ATMIVs. The correlation between the FTSE 100 index returns and

�ATMIVs is about -0.83, which is significantly negative and therefore reconciles with

the results of Figure 1.

Table 1. The descriptive statistics for the daily FTSE 100 index levels (and returns) and daily levels (and changes) in the ATMIVs of the FTSE 100 index from January 1, 2004 to December 31, 2007.

Panel I: Daily ATM implied Volatility (level) and Daily FTSE 100 Index (level)

ATM Implied Volatility FTSE100

Mean 12.865 5501.783

Median 11.995 5595.450

Minimum 8.000 4287.040

Maximum 27.260 6732.400

Panel II: Daily percentage changes of ATM implied Volatility and percentage returns of FTSE

�ATM Implied Volatility FTSE100

Mean 0.007812 0.038537

Median -0.02000 0.049429

Minimum -3.98000 -4.098649

Maximum 4.61000 3.504223

Std. Dev. 0.795208 0.786640

Skewness 0.878056 -0.373138

Kurtosis 8.834837 5.880725

J-Bera 1612.029 384.4766

Probability 0.000000 0.00000

Panel III: Correlation Analysis between ATM implied Volatility and FTSE100 index

�ATM Implied Volatility FTSE100

�ATM Implied Volatility 1.0000 -0.833891**

---------- (-48.7239)

FTSE 100(Returns) -0.833891** 1.0000

(-48.7239) ---------

T-statistics are in the parenthesis. ** Denote rejection of the null hypothesis at the 1% significance levels.

Table 2 presents descriptive statistics of the IVs recovered from the OTM put and call

options’ weekly data from 2004-2007 for the months of March and October. The mean

IV (level) is about 18% annually, whereas the mean percentage change (�) of the IVs is -

0.46. The maximum IV (level) is 41%, with a minimum of 7.5%. However, the

55

maximum �IV is 2.265, with a minimum of -6.058. Similarly, both IV (level) and �IV

are non-normally distributed as well, which is confirmed by the Jarque-Bera statistics.

Table 2. The descriptive statistics for the crossection of weekly implied volatilities recovered from the out-of-the-money put and call options from 2004-2007 for the months of March and October. The descriptive statistics for both levels and changes in IVs are reported.

OTM-PutCall Implied Volatility(Level) �OTM-PutCall Implied Volatility

Mean 18.45401 -0.465341

Median 17.47506 -0.448484

Minimum 7.562080 -6.058008

Maximum 41.33367 2.265540

St.Dev. 6.856906 0.407839

Skewness 0.607074 -0.549265

Kurtosis 2.705163 10.64076

J-Bera 497.9209** 19006.07**

T-statistics are in the parenthesis. ** Denote rejection of the null hypothesis at the 1% significance levels.

In Figure 2 the IVs recovered from OTM puts and calls are plotted against moneyness

levels from 2004 to 2007 for the months of March and October. An almost one-to-one

relationship between the IVs and moneyness levels exists--most of the data lie at a 45-

degree angle. This evidence supports the monotonic relationship between IV and the

option price; that is as volatility increases, the option price increases.

The data presents some important patterns consistent with previous empirical findings.

First, the underlying stock index returns are negatively correlated with the changes in

IVs. Second, IVs backed out from option prices with different strike prices and

maturities on the same underlying stock index differ considerably; therefore, the

assumption of BS model of constant volatility is rejected. Third, there is a one-to-one

relationship between volatility and moneyness that implies that option prices have a

monotonic relationship with volatility. This relationship has important implications for

option traders.

56

4

8

12

16

20

24

28

32

36

40

44

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Moneyness

OTM

-Put

Cal

l im

plie

d vo

latil

ity (%

)OTM-PutCall implied volatility versus Moneyness

Figure 2. A scatter plot of IVs (level) implied by OTM Put and Call options versus

Moneyness (level).

57

4 Empirical Results

4.1 Results of the Four Estimated Models of Volatility Surfaces

Table 3 reports the results of the estimated implied volatility surfaces in the months of

March and October for the years 2004, 2005, 2006, and 2007. These surfaces were

estimated with four parametric models using a nonlinear optimization technique. Panel

I of Table 3 shows the results for Model 1, the constant volatility model. The goodness

of fit criteria IVRMSE and %RMSE range from a minimum of 0.046 and 0.046 to a

maximum of 0.078 and 0.069, respectively, indicating very high levels. Figure 3 is a

visual display of the average IVS (March, 2004) (pink dots), which corresponds to

Model 1 projection (green). As shown in Figure 3, a rich market IVS exists, and the IVS

projected by the Model 1 (or BS Model) is flat; therefore, IV is constant at every point of

the IVS. We use this constant volatility model later as a reference model for comparison

purposes.

Figure 3. Estimated implied volatility surface with Model 1 for March 2004.

58

Table 3. Results for estimated implied volatility surfaces (2004-2007)

Panel-I: Model 1 � � ��m,tI 0 ��

Month 0 IVRMSE %RMSE

Mar-2004 0.2082 0.0775 0.0640

Mar-2005 0.1468 0.0562 0.0639

Mar-2006 0.1529 0.0458 0.0501

Mar-2007 0.1849 0.0577 0.0465

Oct-2004 0.1752 0.0698 0.0666

Oct-2005 0.1633 0.0627 0.0689

Oct-2006 0.1631 0.0493 0.0461

Oct-2007 0.2262 0.0635 0.0468

Panel-II: Model 2 � � �m�m��m, �I 2210 ��

Month 0 1 2 IVRMSE %RMSE

Mar-2004 0.1674 0.2156 0.0077 0.0250 0.0174

Mar-2005 0.1098 0.1671 0.0794 0.0123 0.0110

Mar-2006 0.1221 0.1466 0.0669 0.0110 0.0080

Mar-2007 0.1515 0.2184 -0.0454 0.0180 0.0144

Oct-2004 0.1309 0.1850 0.0679 0.0164 0.0159

Oct-2005 0.1256 0.1810 0.0614 0.0141 0.0119

Oct-2006 0.1285 0.1888 0.0006 0.0105 0.0073

Oct-2007 0.1877 0.2089 -0.0053 0.0156 0.0104

Panel-III: Model 3 � � �m��m�m��m, �I 432

210 ��

Month 0 1 2 3 4 IVRMSE %RMSE

Mar-2004 0.1570 0.1756 0.0362 0.0186 0.0827 0.0221 0.0152

Mar-2005 0.1024 0.1535 0.0787 0.0163 0.0384 0.0100 0.0112

Mar-2006 0.1127 0.1187 0.0937 0.0185 0.0441 0.0068 0.0063

Mar-2007 0.1611 0.2150 -0.0513 -0.0195 0.0086 0.0170 0.0132

Oct-2004 0.1223 0.1600 0.0837 0.0182 0.0501 0.0139 0.0148

Oct-2005 0.1218 0.1491 0.0854 0.0073 0.0858 0.0126 0.0106

Oct-2006 0.1200 0.1716 0.0200 0.0198 0.0245 0.0076 0.0067

Oct-2007 0.1885 0.1938 0.0016 -0.0035 0.0355 0.0153 0.0097

Panel-IV: Model 4 � � ��m��m�m��m, �I 2543

2210 ��

Month 0 1 2 3 4 5 IVRMSE %RMSE

Mar-2004 0.1438 0.1818 0.0319 0.0938 0.0739 -0.066 0.0214 0.0155

Mar-2005 0.0997 0.1552 0.0772 0.0313 0.0345 -0.0130 0.0100 0.0115

Mar-2006 0.1137 0.1185 0.0941 0.0133 0.0442 0.0046 0.0068 0.0063

Mar-2007 0.1745 0.2100 -0.0476 -0.0948 0.0218 0.0652 0.0161 0.0122

Oct-2004 0.1271 0.1588 0.0838 -0.0092 0.0525 0.0258 0.0138 0.0144

Oct-2005 0.1192 0.1503 0.0853 0.0230 0.0823 -0.0152 0.0126 0.0106

Oct-2006 0.1194 0.1717 0.0198 0.0230 0.0243 -0.0031 0.0076 0.0068

Oct-2007 0.1845 0.1947 0.0013 0.0218 0.0342 -0.024 0.0152 0.0099

59

Panel II of Table 3 displays the results for Model 2, which attempts to capture volatility

skew and volatility smirk across moneyness, the quadratic skew model, which includes

additional parameters for moneyness slope (captures volatility skew) and curvature

(captures volatility smirk).46 As is evident from the goodness-of-fit criteria, IVRMSE

and %RMSE decreased for all six months, ranging from a minimum of 0.010 and 0.007

to a maximum of 0.025 and 0.017, respectively, and improved by 0.036 to 0.053 and

0.039 to 0.050 points, respectively, compared with Model 1. Figure 4 shows the

corresponding 3D graph for the March 2004 IVS. As shown, the pink dots present the

observed term structure of the smirk/skew (also called strings), that is, skewed in the

moneyness level, and show the decline in the term structure. Figure 4 demonstrates

two important findings. First, in addition to the volatility level effect (ATMIV captured

by the Model 1), there are strong volatility skew and smirk effects (captured by the

slope and curvature parameters); therefore, model 2 fits well into the observed rich

IVS. Second, the smirk (skew) IVS of Model 2 is contrary to the BS model’s assumption

of constant volatility and therefore suggests the existence of stylized facts regarding IV.

Third, we observe high volatilities for options with shorter maturities than options with

longer maturities. Fourth, the level and volatility of moneyness have a monotonic

relationship, which is consistent with the findings of previous research.


46 The volatility smirk can be also interpreted a fear factor in the market. This volatility smirk pattern arises when OTM put options are more expensive than OTM call options (see Foresi and Wu, 2000; Zhang and Xiang, 2008; Carverhill et al., 2009).

60

Panel III of Table 3 reports results for IVSs estimated with Model 3, which is Model 2

with additional parameters for the volatility term-structure effect (time dimension) and

combined effect of the volatility skew and volatility term structure. As can be seen from

the goodness of fit criteria, IVRMSE and %RMSE decreased for all six months with

ranges from a minimum of 0.007 and 0.006 to a maximum of 0.022 and 0.015,

respectively, and improved by 0.003 to 0.0032 and 0.001 to 0.002 points, respectively,

compared with Model 2 (smirk/skew model). Figure 5 displays a 3D graph of the IVs

for March 2004. The IVS generated by Model 3 is more closely fitted to the observed

IVS, with the last corner of the IVS stretched upward in comparison with the IVS in

Figure 4. Figure 5, therefore, suggests that there is volatility term structure effect in the

IVs, which is further confirmed by the results in Panel III of Table 3.


Panel IV of Table 3 presents the results for Model 4, which is Model 3 with an

additional parameter to capture the curvature effect of the volatility term-structure

effect. As can be seen from the goodness-of-fit criterion levels, IVRMSE and %RMSE

decreased for all six months with values ranging from a minimum of 0.007 and 0.006

to a maximum of 0.021 and 0.015, respectively, and improved from 0.00 to 0.001 and

0.00 to 0.001 points, respectively, compared with Model 3. Overall, compared with the

goodness of fit of the Model 3, very small improvements can be seen. Figure 6 presents

61

the corresponding 3D IVS generated by Model 4. We observe a slight change in shape,

and the last corner of the IVS is now stretched somewhat further.

We conclude from the results presented in Table 3 that Model 1 cannot explain

variations in the IVS because it generates constant volatilities across both dimensions.

However, Model 4 explains most of the variations in the IVS. Therefore, Model 4 fits

the best among the models to the rich market IVS. However, the difference in

comparison with the fit with the Model 3 is marginal. Therefore, we conclude that the

forecasts of Model 3 and Model 4 for the IVSs are the best. Furthermore, we suggest

using the %RMSE goodness-of-fit criterion, particularly when IVs are implied by OTM

options such as in our study, because if we compare their levels, we find that %RMSE

throughout presents minimum levels in contrast to IVRMSE. Therefore, %RMSE is a

more suitable criterion because it assigns greater weight to OTM options.


62

4.2 Results for Principal Component Analysis of Implied Volatility

Surfaces

Table 4 presents the results of PCA on the average IVSs for March and October for the

years 2004, 2005, 2006, and 2007. Columns 2, 3 and 4 show the results for the three

factors (or PCs) extracted from the IVS. Figure 7 presents the proportions of the factors

that explain the dynamics in the IVS for the March 2004, and the corresponding three

factor loadings for the same IVS are plotted in Figure 8.47 The first factor, which is

common to the entire IVS and moves the entire IVS systematically in the same

direction, can be interpreted as the volatility level factor (or parallel shift), which, on

average, explains about 56% of the variations in the implied volatility surfaces for

March and October in 2004, 2005, 2006, and 2007. It is important for IVs implied

from the shortest and longest maturity options because it alters the steepness of the

volatility structure of the near-term OTM put and call options, The effect is stronger on

the OTM options than ATM options; OTM puts are affected the most and feature mean-

reverting stochastic volatility after a shock that trigger instantaneous volatility. The

effect then persists for some time (the correlation between stochastic volatility and

jump)48. The second factor can be interpreted as a term-structure factor (or tilt, which

generates shifts in the slope of the term structure of IVs), which, on average, explains

about 15% of the variations in IVSs for all eight months and affects the shorter-and

longer-maturity IVs differently: the sign of the effects differs between the shorter- and

longer-maturity IVs. The term-structure effect factor is also consistent with the notion

that those segments in the IVS that are implied from long-term options do not

correspond entirely with those segments of the IVS that are implied from short-term

options; this factor can thus be attributed to long-term macroeconomic risks.

Moreover, factor 2 affects IVs across the moneyness dimension identically. The third

factor can be interpreted as the jump-fear factor (or curvature, which changes the

steepness of IV smirk/skew), which, on average, explains about 7% of the variation in

IVS for all eight months. The effects of jump-fear factor on the OTM put and call

options significantly differ and induce steepness in the IV smirk (skew). For empirical

evidence on the dynamics of the factors driving the IVS of index options, see the studies

by Skiadopoulos et al. (1999), Mixon (2002), and Cont and Da Fonseca (2002).

47 Appendix includes Figures for the PCA of the other IVSs. 48 The level factor can also be interpreted with the view that the change in IVs changes the skewness of the corresponding implied risk-neutral density (for further discussion, see the studies by Mixon (2002) and Zhang and Xiang (2008))

63

Table 4. Principal components in the implied volatility surfaces

Month PC1 PC2 PC3 Total explained variance(%)

bMar-2004 64.0% 10.4% 7.7% 82.1%

Mar-2005 53.6% 12.0% 10.0% 75.6%

Mar-2006 43.6% 31.3% 4.8% 79.7%

Mar-2007 61.8% 21.9% 4.7% 88.4%

Oct-2004 55.5% 13.0% 8.5% 77.0%

Oct-2005 59.3% 10.9% 7.0% 77.2%

Oct-2006 46.6% 12.5% 10.6% 69.7%

Oct-2007 67.3% 8.8% 6.5% 82.6%

Mean 56.4% 15.1% 7.4% 79.0%

1 2 3 4 5 6 7 8 90

10

20

30

40

50

60

70

80

90

100


Var

ianc

e E

xpla

ined

(%)

Average Implied Volatility Surface, March- 2004

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Figure 7. Principal components in the IVS for March 2004

64

5 10 15 20 25 30

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Three Factor Loadings of the Implied Surface, March-2004

PC3PC1

PC2

Figure 8. Three factor loadings of the IVS for March 2004.

For further investigation of the dynamics of IVs, we go beyond IVS and study each

group of maturities in the IVS, that is, 8-30, 30-60, 60-90, 90-150, 150-220 and 220

days and above. Table 5 documents the results for the different maturity groups. It is

evident that, as before, the first three factors can capture variations in each group.

Therefore, we find two distinct patterns in all six groups.

The level factor or parallel-shifts factor can explain, on average, 61% of the variance in

the first three maturity groups (from 8-90 days). However, for the remaining three

groups, the results are in complete contrast, where most of the dynamics can be

attributed to the level factor in the IVs; therefore, about 91% of the variations can be

explained by the level factor. In addition, the term structure factor, or tilt factor, can

explain about 19% of the variations in the IVs for the first three maturity groups,

whereas, for the last three groups, this factor, on average, explains 6% of the variance.

Finally, the jump-fear factor, or the curvature factor, is important for the first three

maturity groups (shorter maturities) that capture the jump-fear in the market (the

pattern implied by deep OTM options), which explain 10% of the variation in the IVs

but only 2% of the variation in the last three maturity groups.

65

Table 5. Principal components in the different maturity groups Range Month PC1 PC2 PC3 Total Month PC1 PC2 PC3 Total

8-30 Mar-04 68.03% 14.35% 7.84% 90.2% Oct-04 60.02% 25.98% 8.79% 94.7%

Mar-05 68.87% 14.97% 8.53% 92.3% Oct-05 66.63% 15.33% 9.97% 91.9%

Mar-06 72.79% 14.62% 6.00% 93.4% Oct-06 64.71% 19.01% 8.13% 91.8%

Mar-07 50.70% 35.73% 9.60% 96.0% Oct-07 65.81% 13.97% 9.59% 89.3%

Mean 64.7% 19.2% 8.5% 92.4%

30-60 Mar-04 69.72% 16.86% 8.31% 94.8% Oct-04 55.28% 22.17% 15.24 92.6%

Mar-05 63.28% 14.16v 10.99 74.2% Oct-05 52.15% 22.91% 11.00 86.0%

Mar-06 54.26% 15.23% 10.72 80.2% Oct-06 50.16% 20.39% 15.28 85.8%

Mar-07 77.50% 15.44% 4.00% 96.9% Oct-07 63.60% 15.07% 6.38% 85.0%

Mean 60.4% 17.7% 10.2% 86.9%

60-90 Mar-04 48.65% 21.24% 17.95 87.8% Oct-04 69.06% 11.99% 7.50% 88.5%

Mar-05 57.23% 20.09% 14.01 91.3% Oct-05 69.79% 14.25% 7.02% 91.0%

Mar-06 55.10% 22.22% 9.76% 87.0% Oct-06 52.63% 22.47% 14.33 89.4%

Mar-07 47.36% 32.08% 7.72% 87.1% Oct-07 77.81% 12.93% 5.31% 96.0%

Mean 59.7% 19.6% 10.4% 89.7%

90-150 Mar-04 94.74% 2.72% 1.44% 98.9% Oct-04 77.65% 15.71% 5.44% 98.8%

Mar-05 92.55% 5.62% 1.13% 99.3% Oct-05 94.87% 3.44% 1.10% 99.4%

Mar-06 90.98% 7.40% 1.03% 99.4% Oct-06 63.30% 18.64% 14.16% 96.1%

Mar-07 97.28% 1.67% 1.00% 99.5% Oct-07 88.37% 8.78% 1.23% 98.3%

Mean 87.4% 8.0% 3.3% 98.7%

150-220 Mar-04 95.90% 1.43% 1.27% 98.6% Oct-04 94.13% 2.73% 2.41% 99.2%

Mar-05 67.29% 29.78% 1.58% 98.6% Oct-05 97.91% 0.96% 0.75% 99.6%

Mar-06 90.58% 6.01% 1.93% 98.5% Oct-06 92.04% 4.08% 2.90% 99.0%

Mar-07 88.01% 10.43% 1.17% 99.6% Oct-07 96.52% 1.50% 0.92% 98.9%

Mean 90.3% 7.1% 1.6% 99.0%

220- Mar-04 93.17% 4.23% 1.65% 99.0% Oct-04 95.73% 3.15% 0.90% 99.7%

Mar-05 97.82% 1.32% 0.60% 99.7% Oct-05 96.58% 1.73% 1.10% 99.4%

Mar-06 92.52% 6.69% 0.63% 99.8% Oct-06 93.10% 3.40% 1.91% 98.4%

Mar-07 98.09% 1.28% 0.55% 99.9% Oct-07 96.51% 2.11% 1.00% 99.6%

Mean 95.4% 3.0% 1.0% 99.4%

66

To summarize Table 5, we conclude that shocks that are common to the entire IVS that

can be captured by the first factor (or PC ) are very important for the shortest

maturities (8-30 days) and for the longest (90 days-above) maturities; therefore, this

factor is of sole importance for these IVs. However, the term structure factor is

important for the IVs with medium (particularly between 60-90 days) maturities, and

the jump-fear factor is important for IVs with shorter (8-90 days) maturities.

Figure 9 is a plot of the forecasted values (blue circles) and actual values (red line) for

the March 2004 IVS; the IVS is forecasted (predicted) using a different number of

retained factors (PCs) from the market (or actual) IVS. There are four subplots in

Figure 9: the first subplot shows predicted IVS values using only the first latent factor,

the second subplot forecasts IVS values with the first two factors, the third subplot

forecasts IVS values with the first three factors, and the last uses the first 20 factors to

forecast the IVS values. It is evident that the first three factors, the level factor, the term

structure factor, and the jump-fear factor, are more than adequate for forecasting IVS,

and these factors can explain most of the dynamics of the market IVS. The results in

Figure 9 match our previous analysis.

0 20 40 600

10

20

30

40

Term structure

Impl

ied

Vol

atili

ty(%

)

One PC and Imp Surface Mar-04

0 20 40 6010

20

30

40

Term structure

Impl

ied

Vol

atili

ty(%

)

Two PCs and Imp Surface Mar-04

0 20 40 6010

20

30

40

Term structure

Impl

ied

Vol

atili

ty(%

)

Three PCs and Imp Surface Mar-04

0 20 40 6010

20

30

40

Term structure

Impl

ied

Vol

atili

ty(%

)

20 PCs and Imp Surface Mar-04

Figure 9. Forecasted IVS (blue circles) versus Market observed IVS (red line) using

PCs.

67

5 Conclusion

We estimated and obtained implied volatility surfaces for FTSE 100 index options with

parametric structural forms of models proposed by Dumas et al. (1998). We modified

these models by considering moneyness in implied forward prices and OTM put and

call options. Further, we investigated the dynamics of the IVSs using principal

component analysis.

We estimated four parametric models of DFW (1998): the constant-volatility Model 1

fails to capture variations in the IVSs and yields only flat IVSs. However, Model 2,

which captures the volatility skew and volatility smirk effects, does a good job and

captures most of the volatility skew and smirk effects. Models 3 and 4, which account

for both the volatility term structure and the volatility smirk/skew effects, performed

better than the one-dimensional smirk/skew model. However, both models fit the

market IVS well and generated the best IVSs for the observed data. In the second part

of the study, which uses PCA (nonparametric approach), we find that the first three

factors can explain about 69-88% of the variance in the IVS: on average, 56% is

explained by the level factor, 15% is explained by the term structure factor, and an

additional 7% is explained by the jump-fear factor. Finally, when applying PCA to the

maturity groups, we find that the level factor (or parallel shifts) is evident and thus

important for shorter- and longer-maturity IVs. However, the term-structure factor is

important for IVs with medium maturities, and the jump-fear factor is important for

IVs with the shortest maturities (8-90 days).

IVS is important in practice; for instance, on average, in the FTSE 100 index option

market, 100 options are traded daily with a variety of strike prices and times to

maturity. If practitioners need to quote more options for different strikes or maturities,

then IVS can be used as a reference. The practitioner could easily quote new options by

taking estimates from the IVS. Moreover, the shape of the IVS is important in option

trading: one of its most important characteristics is that it can be used to identify

mispricing in the market. Second, the IVS can be used to manage exotic derivative

positions, which could be hedged with plain-vanilla options such as European options.

Third, a robust volatility index could easily be constructed from the IVS and

consequently used to price volatility derivatives. Fourth, risk can be managed for those

securities that underlie the FTSE 100 index by using independent risk factors from the

IVS.

68

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Rubinstein, M., 1994. Implied binomial trees. Journal of Finance 49, 771-818. Skiadopoulos, G., Hodges, S., and Clewlow, L., 1999, The dynamics of the S&P500

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263- 284.

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Appendix. Estimated implied volatility surfaces with four models for Oct-2005.

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Principal component analysis of the IVS for Oct-2005

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Quantile Regression Analysis of Asymmetric Return-Volatility Relation

Ihsan Ullah Badshah



September 20, 2010

Abstract This essay uses quantile regression to investigate the asymmetric return-volatility phenomenon with the newly adapted and robust implied volatility indices VIX, VXN, VDAX and VSTOXX. A particular goal is to quantify the effects of positive and negative stock index returns at various quantiles of the IV changes’ distribution. As the level of the new volatility index increases during market declines, we believe that the negative asymmetric return-volatility relationship should be significantly more pronounced at upper quantiles of the IV changes’ distribution than is indicated by ordinary least squares (OLS) regression. We find pronounced negative and asymmetric return-volatility relationships between each volatility index and its corresponding stock market index. The asymmetry increases monotonically when moving from the median quantile to the uppermost quantile (i.e., 95%); OLS thereby underestimates this relationship at upper quantiles. Additionally, the asymmetry is pronounced with a volatility smirk (skew)-adjusted new volatility index measure in comparison to the old at-the-money volatility index measure. The VIX volatility index presents the highest asymmetric return-volatility relationship, followed by the VSTOXX, VDAX and VXN volatility indices. Our findings have implications for trading strategies, hedging portfolios, pricing and hedging volatility derivatives, and risk management. Keywords: Asymmetric return-volatility relationship, implied volatility, index options, quantile regression, volatility index. JEL Classifications: C21, G12, G13. The Author would like to thank Johan Knif, Mika Vaihekoski, George Skiadopoulos, Hossein Asgharian, and Kenneth Högholm for providing useful comments. The author would also like to thank the participants of the NFN workshop (2009), Copenhagen, Denmark; FMA conference (2009), Turin, Italy; MFS conference (2009), Crete, Greece; and EFMA conference (2010), Aarhus, Denmark. The author acknowledges CEFIR (centre for financial research) and NASDAQ OMX Nordic foundation for providing financial support.

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1 Introduction

It is widely documented that implied volatility (IV) is superior to historical volatility

(HV) when forecasting the future realized volatility (RV) of the underlying asset (e.g.,

Day and Lewis, 1992; Christensen and Prabhala, 1998; Fleming,�1998; Dumas et al.,

1998; Blair� et al., 2001; Ederington and Guan, 2002; Poon and Granger, 2003;

Mayhew and Stivers, 2003; and Martens and Zein, 2004). IV can be recovered by

inverting the Black-Scholes (1973) formula. However, Britten-Jones and Neuberger

(2000) and Jiang and Tian (2005) have derived a model-free implied volatility (MFIV)

under the pure diffusion assumption and asset price processes with jumps.49 This MFIV

measure has now been adapted by the major IV indices, which used to employ at-the-

money (ATM) BSIV measures in their methodologies.50

As MFIV index is forward looking, that is, it is implied by the market prices of options,

and as options represent the consensus (aggregated beliefs) of market participants

regarding expected future volatility, MFIV index is the market expectation about the

future RV of the underlying stock index for the 22 subsequent trading days (over 30

calendar days).51 Thus, IV indices are often referred to as the “investors’ fear gauge”

(e.g., Whaley, 2000), as the level of the IV index indicates the consensus view about the

expected future realized stock index volatility. When the level of the IV index increases,

fear increases in the market as a result; alternatively, when the level of the IV index

decreases, run-ups are triggered in the daily stock index prices.52

49 They show that the information content of MFIV is superior to that of the Black-Scholes implied volatility (BSIV) because the MFIV measure accounts for all strikes when computing IV at a particular point in time, whereas the BSIV measure is a point-based IV and does not account for all strikes in computation; i.e., each strike has a separate IV. Moreover, BSIV is subject to both model and market efficiency, while MFIV is only subject to the market efficiency (see Poon and Granger, 2003). 50 The motives for adopting MFIV measures are the following. First, the MFIV index measure is economically appealing and robust, as it accounts for out-of-the-money (OTM) options (i.e., volatility smirk/skew). Second, the previous IV index measure (now called VXO) was upward biased, the bias being induced by trading-day conversion, which is now omitted from the new VIX measure. Finally, with the new robust MFIV index measure, it is possible to replicate volatility derivatives (e.g., variance swaps), which was not possible�with the previous measure. 51 Major option exchanges, including the Chicago Board of Exchange (CBOE) and the Deutsche Börse, have launched IV indices, robustly providing information on options using MFIV measures; examples of this are the VIX index for the S&P 500 index, VXN for the NASDAQ 100 index, VDAX for the DAX 30 index and VSTOXX for the Dow Jones�(DJ) EURO STOXX 50 index. 52 Additionally, the IV index level indicates the degree of willingness of market participants to pay in terms of volatility in order to hedge the downside risk of their portfolios with put options or long positions in call options with downside risks remain to the premium of options instead of positions in the underlying�asset (see Simon, 2003, for a detail on trading strategies).

75

Likewise, the MFIV index measure incorporates both puts and calls and therefore

moves with changes in their prices; for example, a negative or positive shock to the

market induces adjustments in hedging and trading strategies, consequently triggering

changes in the prices of one type (i.e., put or call) of option. The MFIV index measure

then moves in the direction of the market demand of a particular type of option and the

underlying�asset (see Bollen and Whaley, 2004). Also, Liu et al. (2005) argue that the

rare-event premia play an important role in generating the volatility smirk (skew)

pattern observed for options across moneyness and that these rare events are

embedded in the OTM options.53 Camara and Heston (2008) derive an option model

that accounts for both OTM put and call options. They derive the extreme negative

events from OTM puts and extreme positive events from OTM calls. Thus, the MFIV

index that accounts for OTM options thus contains a broader set of information—it

contain information on the future negative and positive jumps, demand/supply of

options etc. Thereby MFIV measure is robust; as a result, an excellent tool for

examining the relationship between the market perception of volatility and returns.

Furthermore, this relationship is asymmetric, implying that the MFIV index reacts

differently to negative and positive returns. Two main hypotheses exist in the literature

that characterizes the negative asymmetric return-volatility relationship: the leverage

effect and feedback effect hypotheses. However, both the leverage and feedback

hypotheses have been unable to explain the observed strong negative asymmetric

return-volatility relationship at daily frequencies (see, e.g., French et al., 1987; Breen et

al., 1989; Schwert��1989, 1990). Similarly, a recent study by Hibbert et al. (2008) has

found a very strong contemporaneous negative asymmetric return-volatility

relationship using daily data, thereby empirically rejecting both the leverage and

volatility feedback hypotheses.54 Further empirical investigations are important to

characterize an asymmetric return-volatility relationship using a volatility smirk

(skew)-adjusted robust MFIV index measure with daily frequency.55 Importantly, we

53 Similarly, Pan (2002) showed that volatility skew is primarily due to investors’ fear of large adverse jumps. 54 Other studies by Simon (2003) and Giot (2005) have also found a very strong negative asymmetric return-volatility relationship using data of daily frequency. Nevertheless, the negative asymmetric return-volatility relationship is too strong at the daily level; these hypotheses might be interesting to characterize an asymmetric relation at lower frequencies, for instance, monthly or quarterly frequencies, but not at high frequencies. 55 We also believe that the asymmetric volatility-return relationship should be more pronounced with the new robust MFIV index in contrast to the old BSIV index measure. A possible explanation for pronounced asymmetric volatility is that a put option is a downside-hedging instrument and traders are always

76

believe that the estimation technique currently being used for characterizing this

relationship is ordinary least squares (OLS), which may not completely capture this

relationship because it only describes that how this relationship should be at the mean

of the IV changes’ (or response variable) distribution but not at the other parts of the IV

changes’ distribution. Therefore, to fully characterize the asymmetric return-volatility

relationship, the conditional quantile regression should be preferred over OLS

regression, especially to obtain a complete picture of the relationship at all parts of the

IV changes’ distribution (particularly at the uppermost quantiles) with the negative and

positive stock returns.56 Because we believe that the relationship should be more

pronounced and asymmetric at the uppermost quantiles (e.g., 95th quantile) of the IV

changes’ distribution, which OLS regression (or mean regression) would

underestimate. Few well-known studies exist showing a significant negative and

asymmetric relationship between stock index returns and BSIV index changes using

OLS regressions (or mean regressions) (e.g., Fleming et al., 1995; Whaley, 2000; Giot,

2005; Simon, 2003; Skiadopoulos, 2004; Low, 2004; Dennis et al., 2006); as OLS

ignores the responses at the tails of the IV changes’ distribution, therefore, the

asymmetric return-volatility relationship is underestimated, warrant further

investigation using conditional quantile regression analysis.

Nonetheless, the first study on the relationship between the old VIX (now VXO)

changes and S&P 100 index returns was conducted by Fleming et al. (1995). They

investigated the time-series properties of the VXO, finding a significant negative

contemporaneous asymmetric relationship between VXO changes and stock index

returns. Another well-known study is that conducted by Whaley (2000), who examined

the relationship between the weekly VXO changes and S&P 100 returns. He

documented that when the VXO falls by 100 basis points, the S&P 100 index increases

by 0.469%, whereas when the VXO increases by 100 basis points, the S&P 100 index

falls by -0.707%. He thus finds a large negative asymmetric association between VXO

changes and S&P 100 returns, calling the VXO the “investors’ fear gauge.” Simon�

(2003) studied the NASDAQ 100 volatility index (VXN) from January 1995 to May

2002, showing that the VXN is inversely related to both positive and negative index

returns. Furthermore, he found stable results across the bubble and post-bubble

concerned about the downward moments in the market, so traders are always hedging their positions with OTM puts. Consequently, we find a higher volatility for OTM puts than for calls (see, e.g., Bollen and Whaley, 2004). 56 Quantile regression estimates are robust to outliers, non-normal error distribution, etc.

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periods. A more recent study by Hibbert et al. (2008) used a different approach to

investigate the negative asymmetric return-volatility relationship using the newly

developed VIX index. They found a significant negative and asymmetric association

between VIX changes and stock index returns when incorporating both daily and

intraday data, thereby confirming that the MFIV VIX measure can better explain the

asymmetric relationship than the BSIV VIX or the RV measures.

The purpose of this essay is to investigate the negative asymmetric return-volatility

relationship between the stock market returns and the volatility index changes: (1) to

quantify the degree to which a volatility index is responding to the negative and positive

returns at different quantiles of an IV changes’ distribution; (2) to compare the

asymmetric responses of the two volatility index measures, i.e., the MFIV and BSIV

index measures; and (3) to rank volatility indices according to their asymmetries.

Related studies in terms of the volatility-return relationship include Simon (2003), Giot

(2005) and Hibbert et al. (2008). Simon (2003) studied the relationship between the

NASDAQ 100 index returns and the VXN index changes using the BSIV index measure,

while Giot (2005) and Hibbert et al. (2008) studied the relationship between the S&P

100 and the VIX and between the NASDAQ 100 and the VXN. Giot (2005) used the

BSIV index measure, whereas Hibbert et al. (2008) used the new MFIV index measure.

Our study differs from these three previous studies and therefore contributes to the

literature in a number of ways: first, this study extends their methodologies; for

instance, they used mean-regression models, whereas we use a robust conditional

quantile regression model to investigate the uppermost IV changes’ quantiles’

responses to the negative and positive returns. Second, this study uses a broader set of

data drawn from across the Atlantic, for example, the VIX, VXN, VDAX and VSTOXX

volatility indices (using new robust MFIV measures, thereby incorporating a broader

range of information) and their corresponding stock indices.57 Finally, this study

compares the asymmetries of the MFIV and BSIV volatility index measures.58

Our main findings are that from February 2001 through May 2009, the MFIV indices

VIX,VXN, VDAX and VSTOXX responded in a highly asymmetric fashion; i.e., negative

returns had a much greater impact on volatility than positive returns, particularly in the

57 Previously it was found that each equity option market presented somewhat different IV dynamics; therefore, this study is the first to investigate and compare the volatility asymmetries across the Atlantic. 58 We compare the new VDAX and old VDAX (denoted here VDAXO) volatility index measures; the former is based on the MFIV measure and the latter on the BSIV measure.

78

uppermost regression quantiles (e.g., q=0.95). Our quantile regression model (QRM) of

the asymmetric return-volatility relation thus reveals important information that is

underestimated by the mean-regression model (MRM). The VIX index presents the

highest asymmetry, followed by the VSTOXX, VDAX and VXN indices, respectively.

These volatility indices rise sharply in times of market turmoil and decline in market

rallies. Second, our view that the asymmetry with the MFIV index should have

pronounced responses is confirmed by comparing the asymmetric responses of VDAX

(MFIV) and VDAXO (BSIV); we find that the MFIV index responds in a pronounced

fashion, in contrast to the BSIV index. Third, there is a strong contemporaneous

asymmetry in comparison to the lags, thus rejecting the leverage hypothesis, and

similar conclusions can be drawn for the feedback hypothesis.

This essay is organized as follows. Section 2 discusses the asymmetric return-volatility

relationship. Section 3 discusses the data set, the volatility indices and their

construction. Section 4 describes the methodology. Section 5 presents the empirical

results. Section 6 summarizes and concludes.

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2 Asymmetric Return-Volatility Relationship

There are two existing hypotheses that characterize asymmetric volatility: the leverage

effect and the feedback effect hypotheses. The leverage hypothesis proposed by Black

(1976) and Christie (1982) attributes asymmetric volatility to the financial leverage of a

firm; i.e., when a firm’s debt increases, the firm’s value declines, triggering the value of

its equity declines further. Because the equity of a firm is more exposed to the firm’s

total risk, therefore, the volatility of the equity should increase as a result. On the other

hand, the volatility feedback hypothesis proposed by French et al. (1987), Campbell and

Hentschel (1992) and Bakaert and Wu (2000), who attribute asymmetric volatility to a

volatility feedback effect.59 Contrary to the leverage-based justification, the volatility

feedback hypothesis states that increases in volatility trigger negative stock returns. For

instance, increases in volatility imply that the required expected future returns will

increase as well; as a result, current stock prices decline. However, both hypotheses

empirically fail under the daily frequency data, being unable to fully characterize the

asymmetric return-volatility relationship; in that respect, Schwert (1990) argued that it

is too strong for the leverage hypothesis to fully characterize asymmetric volatility.

Furthermore, it is also empirically found that the feedback hypothesis is not always

consistent, and this has become a controversial subject; some studies have found that

there are not always positive correlations between current volatility and expected future

returns (e.g., Breen et al., 1989), but others support the hypothesis (e.g., French et al.,

1987; Campbell and Hentschel, 1992; Ghysels et al., 2005). Nonetheless, the economic

and accounting explanations might be important for characterizing the asymmetric

return-volatility relationship at lower frequencies, for instance, monthly or quarterly

data, but not for daily or higher frequencies. Many prior studies have documented very

strong negative asymmetric return-volatility relationships at higher frequencies,

contrary to the explanations of the two hypotheses (see, e.g., Fleming et al., 1995;

Whaley, 2000; Giot, 2005; Simon, 2003; Skiadopoulos, 2004; Low, 2004; Dennis et

al., 2006; Hibbert et al., 2008).

However, this essay considers new MFIV indices because we believe that the

asymmetric return-volatility relation should be more pronounced using the MFIV

indices. Likewise, the importance of the MFIV index measure increases because it

59 Poterba and Summers (1986) characterized the volatility feedback effect through economic explanation that time-varying risk premia induces volatility feedback because it represent the linkage between the fluctuations in volatility and stock returns.

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accounts for volatility smirk (skew), which may be induced by the net buying pressure

of the OTM put options (see Bollen and Whaley, 2004). Volatility smirk/skew is an

obvious phenomenon, previously documented by many other researchers and

important to capture in any volatility measure (e.g., Alexander, 2001; Low, 2004;

Goncalves and Guidolin, 2006; Badshah, 2008). Bollen and Whaley (2004)

documented that the net-buying pressure of the put options induces volatility smirk

(skew) patterns, therefore, suggested using a volatility smirk (skew) adjusted volatility

measures: who investigated the relationship between net buying pressure and the

shape of the IV function (IVF) for index options. They showed that the buying pressure

of put options considerably affects the changes in the IV. They asserted that when the

buying pressure of index put options (particularly from institutional investors who seek

to hedge their portfolios) increases and thus limits the ability of arbitrageurs to bring

the price back into alignment, this pressure permanently drives the sloping shape of the

IVF downward. Also, information from trading strategies and other shocks are well

absorbed into the MFIV index, as it accounts for both put and call types of options.60

The MFIV index is a robust and informed measure of stock index volatility and is

therefore a perfect choice for examining the asymmetric volatility-return relationship.

3 Data

First, the VIX, VXN, VDAX, and VSTOXX volatility indices are introduced, and their

construction is discussed. Second, the complete data set is presented, and the

descriptive statistics are thoroughly discussed.

3.1 VIX and VXN

The CBOE introduced a new VIX index in September 2003 based on options on the

S&P 500 index. The VIX index is determined from the bid and ask prices of the options

underlying the S&P 500 index. The new VIX is independent of any option pricing

model using�the MFIV measure. The VIX thus provides an estimate of expected future

realized stock market volatility for the 22 subsequent trading days (over 30 calendar

days). However, the old VIX index, based on options on the S&P 100 index and

introduced in 1993, has now moved to the new ticker symbol VXO. In contrast to the 60 The MFIV index is informed by both fear and exuberance embedded in option prices, and the majority of option markets’ traders are very informed and possess high skill levels (see Low, 2004; Chakravarty et al., 2004).

81

old VIX (now VXO), which is based on near-the-money BSIV options on the S&P 100

index, the new VIX uses market prices of options on the S&P 500 index.61 This new MF

VIX methodology accounts for both OTM put and call options (i.e., volatility

smirk/skew).62 The new methodology is thus more appealing and robust. The CBOE’s

introduction of the new VIX was motivated by both theoretical and practical

deliberations. First, the new VIX is economically more appealing as it is based on a

portfolio of options, whereas the old VIX was based on the ATM option prices. Second,

the new VIX makes it easy to replicate variance swap payoffs while using static

positions in a range of options and dynamic positions in futures trading. Third, the new

VIX has removed the induced upward bias of the old VIX in the trading day conversion

(see, e.g., Carr and Wu, 2006). Similarly, in September 2003, the CBOE introduced the

VXN using the same MFIV methodology as that of VIX. The CBOE has calculated price

histories for VIX and VXN back to the years 1986 and 2001, respectively.

3.2 VDAX and VSTOXX

The Deutsche Börse and Goldman Sachs jointly developed the methodology for the new

VDAX and VSTOXX� indices. The VDAX is based on options on the DAX 30 index,

whereas VSTOXX is based on options on the Dow Jones (DJ) Euro STOXX 50 index,

which consists of the eurozone’s�50 largest blue-chip stocks. Options on the DAX and

the DJ Euro STOXX 50 are traded on the EUREX derivatives exchange. The VDAX

measure accounts for IVs across all options of a given time to expiry (accounting for

volatility smirk/skew). The methodology of the VDAX, like that of the VIX, is based on

the MFIV measure.63 However, the main VDAX index is further based on eight

constituent volatility indices, which expire in 1, 2, 3, 6, 9, 12, 18, and 24 months,

respectively. The main VDAX is designed as a rolling index at a fixed 30 days to expiry

via a linear interpolation of the two nearest of the eight available sub-indices. The

VDAX and its eight sub-indices are updated every minute� and therefore offer great

advantages in terms of trading, hedging and introducing new derivatives on this index.

61 The options on the S&P 500 index, in comparison with the options on the S&P 100 index, contain a much broader set of implied information; the new VIX is thus a more informative measure than the old VIX (now VXO). 62See, for further detail on the VIX construction, the CBOE VIX white paper: www.cboe.com/micro/VIX/vixwhite.pdf. 63 See, for further detail on the VSTOXX/ VDAX construction, the STOXX website: www.stoxx.com; the Deutsche Börse and Goldman Sachs methodology for VSTOXX/ VDAX volatility indices.

82

The price histories for both VDAX and VSTOXX were calculated back to the years 1992

and 1999, respectively.

3.3 Data

This essay employs data from four sources. We obtained the daily time-series price data

for the S&P 500 stock index, the NASDAQ 100 index, the DAX 30 index, and the DJ

Euro STOXX 50 index from Thomson Financial DataStream. The data on the new VIX

and VXN were obtained from the CBOE, and the data on the new VDAX and VSTOXX

were obtained from the Deutsche Börse and STOXX, respectively. The daily frequency

data for both stock and volatility indices cover a period of 8 years and 4 months, from

February 2, 2001 to May 29, 2009, for a total of 2172 trading days.

Figure 1 shows the daily closing levels (%) of the volatility indices, i.e., the VIX, VXN,

VDAX and VSTOXX, and the corresponding stock market indices (levels) from

February 2, 2001 to May 29, 2009. Among the four volatility indices, the VXN index

presents the highest volatility level throughout our sample period, whereas the VIX

shows the lowest volatility level. Similarly, the volatility indices and stock indices move

inversely to one other. From the beginning of 2001 until the beginning of 2003, there

were considerably high volatility levels. However, from 2004 to late 2007, we find

upward movement in the stock market indices, whereas the corresponding volatility

indices moved in the opposite direction to the stock markets, showing the lowest

volatility levels. However, in the latter part of 2007, we again find somewhat increasing

volatility levels, with the stock markets again moving downward due to the beginning of

the credit crunch and liquidity crunch crises, which have caused markets to be

extremely volatile and the volatility indices to reach historically high levels

(particularly, in October and November of 2008, the VIX level twice surpassed 80%),

and the corresponding stock markets crashed afterward, therefore moving in

completely opposite directions. This phenomenon is evident until the end of the sample

period in May 2009.

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Figure 1. Stock indices versus MFIV indices from February 2, 2001, to May 29, 2009.

Table 1 reports the summary statistics for the daily percentage continuously

compounded returns of four stock indices and the daily percentage changes of five

84

volatility indices, as well as tests for normality, autocorrelations and unit roots. The

mean values for all nine stock index returns and volatility index changes series are not

statistically different from zero. The tests for skewness and kurtosis confirm that the

stock indices returns are positively skewed except for the S&P 500 returns, whereas all

five volatility indices’ changes are positively skewed, as they should be. Furthermore, all

nine series are highly leptokurtic with respect to the normal distribution. Likewise, the

Jarque-Bera statistics reject normality for each of the stock index and volatility index

changes series. The autocorrelation coefficients for the three lags show that the VIX,

VDAX and VXN changes series present strong autocorrelations, whereas the rest of the

volatility indices changes’ present significant autocorrelation coefficients at lags 2 and

3. Autocorrelations in the S&P 500, NASDAQ 100, and DJ Euro STOXX 50 returns

series are also evident at all three lags, consequently confirming the property of mean

reversion. We also investigated stationarity in all nine series (i.e., stock and volatility

indices) by applying the augmented Dickey-Fuller (ADF) unit-root test. The results in

Table 1 show the rejection of unit roots in each series at the 1% significance level.

Therefore, all nine series are stationary.

Table 1. Descriptive statistics of the daily data.

S&P500 NASDAQ DAX30 STOXX50 �VIX �VXN �VDAX

�VDAXO �VSTOXX

Mean -0.0184 -0.0274 -0.0140 -0.0300 0.0033 -0.0117 0.0058 0.00511 0.0051

Median 0.0016 0.0060 0.0318 0.0000 -0.0350 -0.0200 -0.0300 0.0000 -0.0600

Maximum 10.95 11.84 10.79 10.43 16.54 12.71 21.92 16.94 22.64

Minimum -9.46 -11.11 -8.87 -8.20 -17.36 -12.96 -15.05 -10.24 -13.98

Std. Dev. 1.392 1.968 1.696 1.605 1.747 1.674 1.718 1.490 1.918

Skewness -0.1095 0.0819 0.0568 0.0115 0.3423 0.2127 1.4111 1.3166 1.9082

Kurtosis 11.916 6.859 7.786 8.001 23.712 14.266 25.513 23.133 30.589

J-Bera 7199 1350 2074 2263 38867 11504 46592 37312 70202

Prob 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

1 -0.10*** -0.07*** -0.04* -0.04* -0.12*** -0.04* 0.04** -0.00 -0.03

2 -0.07*** -0.06*** -0.02 -0.04** -0.12*** -0.11*** -0.06*** -0.11*** -0.10***

3 0.05*** 0.02*** -0.03 -0.07*** 0.03*** 0.03*** -0.10*** -0.02*** -0.09***

ADF -37.51*** -36.50*** -48.55*** -22.71*** -28.14*** -37.60*** -24.08*** -24.40*** -23.85***

No. Obs 2172 2172 2172 2172 2172 2172 2172 2172 2172 This table reports the descriptive statistics for the daily percentage continuously compounded returns on S&P 500, NASDAQ 100, DAX 30, Dow Jones Euro STOXX 50 stock indices and and for the daily percentage changes in the VIX, VXN, VDAX, VDAXO and VSTOXX volatility indices. The autocorrelation coefficients �, the Jarque-Bera and the Augmented Dickey-Fuller (ADF) (an intercept is included in the test equation) test values are reported. ***, ** and * denote rejection of the null hypothesis at the 1% , 5% and 10% significance levels, respectively.

85

4 Methodology

4.1 Quantile Regression

In the ordinary least squares (OLS) regression where the conditional mean function,

the function that explains how the mean of dependent (or response) variable changes

with the vector of covariates (or independent variables)—simply this would be the

relationship we would know between dependent variable and covariates if we use least

squares regression i.e., the relationship at the mean of the dependent variable’s

distribution. What is more important to notice that it assumes that the error has the

same distribution whatever values may be taken by the components of the covariates

vector. Hence, this type of model is known as pure location-shift model because it

assumes that the covariates affect only the location (or mean) of the conditional

distribution of the response variable, not its scale (variance), or even shape (e.g.,

skewness or kurtosis) of the distribution. It implies that the mean regression analysis

would give incomplete picture of the asymmetric return-volatility relationship between

response variable (volatility changes) and covariates (negative and positive stock

returns) variables because it is focusing on estimating rates of change in the mean of

the response variable distribution as function of set of covariate variables.

Consequently, this mean regression method fails for a regression model with

heterogeneous variances implies that there are different rates of changes not just a

single rate of change in the mean which characterizes the probability distributions i.e.,

in our case volatility changes. Therefore, the mean regression methods which accounts

only changes in the means would underestimate or overestimate a response variable

with heterogeneous distributions ( see e.g., Cade et al., 1999; Koenker and Hallock,

2001; Cade and Noon, 2003; Koenker, 2005).

To account for the mentioned issues in regression method, Koenker and Bassett (1978)

proposed the quantile regression method as an extension of the mean regression

method for estimating rates of change in all parts of the distribution of a response

variable. The quantile regression model is advantageous because it does not assume

that the random part of the model, that is, the error of the model follow any specific

distribution. Hence, it overcomes most of the estimation issues related to the mean

regression model for instance it produces robust estimates in presence of non-normal

error distribution, outliers, and also account for omitted variables bias. In fact, the

86

quantile regression model can be viewed as the generalization of mean regression

model to a collection of models to different conditional quantile functions.

The mathematical illustration is, as we know that the sample mean as the solution to

the problem of minimizing a sum of squared residuals in ordinary least squares

regression. For example least squares regression model if we have random sample

� nyyy ,,, 21 �� , we solve

� � ,min2

1��

#��

n

iiy $

$

(1)

The sample mean is obtained which is an estimate of the unconditional population

mean EY . However, the estimate of conditional expectation function � �xYE is

obtained by replacing the scalar $ by a function � �%$ ,x in the equation 1 and solving

� �� 2

1,min�

�#�

�n

iii xy %$

%

(2)

Similarly, in a quantile regression, the median (quantile) as the solution to the problem

of minimizing a sum of absolute residuals, where the weights are symmetric. However,

the other conditional quantile functions are estimated by minimizing asymmetrically

weighted sum of absolute errors, where the weights are functions of the corresponding

quantile (see Koenker and Hallock, 2001; Barnes and Hughes, 2002; Koenker, 2005).

Thus, to obtain the conditional quantile regression estimator as the solution to the

minimization problem

� ��

#��

n

iiiq xy

1,min %&'

%

(3)

Or equivalently

� � � � � ��

� �

��

� (#� & &%

%&%&i iyi yi

iiii xyqxyq: :

,1,min

(4)

The minimization can be done very efficiently by linear programming methods. The

beauty of quantile regression is that all observations are used to estimate each quantile.

Finally, the standard errors can be computed by employing bootstrap method.

87

4.2 Quantile Regression Model for Asymmetric Return-Volatility

Relationship

We present a quantile regression model for assessing the negative asymmetric

relationship between the returns on the stock index and changes in the volatility index.

This model is the generalization of the standard mean-regression models of Simon

(2003), Giot (2005) and Hibbert et al. (2008), who have empirically confirmed the

asymmetric return-volatility relationship.64 However, this essay extends these standard

mean-regression models (MRM) by modeling the asymmetric return-volatility

relationship using the conditional quantile regression model (QRM) to examine how

negative and positive stock index returns vary across different quantiles of IV changes,

i.e., how much this asymmetric relationship tends to change across different quantiles

of IV changes. Before specifying our quantile-regression model for the asymmetric

return-volatility relationship, we first specified a MRM model similar to that of Simon

(2003), Giot (2005) and Hibbert et al. (2008), which is considered a standard model in

our analysis.65

For instance, we regressed the daily volatility index changes (denoted it�VI , where

i=�VIX, �VXN, �VDAX, �VSTOX) on the daily percentage continuously compounded

returns of the stock market index (denoted ,Rit where i=S&P 500, NASDAQ, DAX, DJ

Euro STOXX�50), where �itR was used for positive returns and �

itR for negative returns.

For the positive returns, itit RR �� if 0Rit � , and 0Rit �� otherwise. On the other

hand, for the negative returns, itit RR �� if 0Rit � , and 0Rit �� otherwise. The

standard MRM for assessing the negative asymmetric return-volatility relation thus has

the form

,t

3

0LLitiL

3

0LLitiL

3

1LLitiLit uRR�VI��VI ��

�

��

�

��

��

(5)

64 They showed that the relationship behaves differently for negative and positive stock index returns. 65 Hibbert et al. (2008) segmented negative and positive stock returns into quantiles and then used least squares for each quantile, which could not yield the robust results that we can find using quantile regression, i.e., the effects of negative and positive returns on the upper and lower quantiles of the dependent variable would be much different and robust using quantile regression instead of least-squares regression (for a detailed discussion, see Heckman, 1979; Koenker and Hallock, 2001; Bassett and Chen, 2001).

88

where � is the intercept, iL� represents the coefficients for the lagged IV changes in a

volatility index i , where 3to1L � , iL represents the coefficients for positive stock

returns and iL the coefficients for the negative returns of a stock market index i ,

where 3to0L � for both types of returns; and the errors tu are independently

identically distributed (iid) with zero means.66 Consequently, the standard MRM

assumes that the effects of both types of returns are static across different IV changes

(i.e., response variables); therefore, an MRM would miss important information across

quantiles of IV changes that we could otherwise detect using a QRM, particularly to

determine how the median or perhaps the 5th or 95th percentiles of the response

variable IV changes are affected by negative and positive stock return variables

(regressor variables).67 Koenker and Bassett (1978) were the first to introduce quantile

regression that could effectively model the uppermost quantiles.68 QRM is a

generalization of the MRM and is thereby a robust regression, especially in situations

where errors are non-normally distributed, i.e., are skewed and leptokurtic.

Nonetheless, the QRM is used for examining the asymmetric return-volatility

relationship; for instance, the qth QRM, which is a generalization of equation (5), has

the form

� � � � � � � �tuRR�VI��VI

3

0LLit

qiL

3

0LLit

qiL

3

1LLit

qiL

qit ��

�

��

�

��

�� (6)

Where � �q� is the intercept; (q)iL� represents the coefficients for the lagged IV changes in

a volatility index i , where 3to1L � ; � �qiL� represents the coefficients for positive

returns and � �qiL the coefficients for negative returns of a stock market index i , where

3to0L � for both type of returns; and the errors tu are assumed to be independent

from an error distribution )(u� tq with the qth quantile equal to zero. Equation (6)

implies that the qth conditional quantile of the dependent variable i�VI given

��

��

��

��

��

��

�� 3it2it1itit3it2it1itit3it2it1it ,R,R,R,R,R,R,R,R,��V,��V�VI and denoted

66 A similar specification as of Hibbert et al. (2008) and Fleming et al. (1995) is used, which is consistent with the dynamics of stochastic volatility. Moreover, the lags of negative and positive returns are inclduded in order to caputre the leverage effect. 67 See a good discussion on this issue in Meligkotsidou et al. (2009). 68 Koenker (2005) provides mathematical details on the different versions of the quantile regression models.

89

� �,��

��

�� 3itit3itit3it1itiq ,..,R,R,..,R,R,...�..�VI�VIQ

is equal to

� � � � � � � � .��

��

�

��

��

3

0LLit

qiL

3

0LLit

qiL

3

1LLit

qiL

q RR�VI�� The main feature of this quantile

regression framework is that the effects of the variables captured by (q)iL� , � �q

iL ,and

� �qiL vary for each qth quantile within the range � �0,1q � . Furthermore, the framework

allows for heteroskedasticity in error tu , and the coefficients are different for different

quantiles. Consequently, a quantile regression provides a broader set of information

about volatility changes here (i.e., the effects on all parts of the distribution of volatility

changes) than OLS regression would, particularly when the error distribution is not

symmetric.69 QRM is thus estimated for the sample period from February 2, 2001

through May 29, 2009 using the quantile regression method proposed by Koenker and

Bassett (1978), which minimizes the asymmetric sum of absolute residuals and robustly

models the conditional quantiles of the response variable, i.e., in our case, changes in

the volatility index:70

��

��

�

��

�

�

��

��

��

��

��

��

��

��

��

��

LitiLLitiLLitiLit

LitiLLitiLLitiLit



RR�VI��VIq)(1

RR�VI��VIqmin

ˆˆˆˆ

ˆˆˆˆ

ˆˆˆˆ

ˆˆˆˆ

(7)

69 Because the differences between the mean and the median produce asymmetric distributions, see, for a more detailed explanation, Meligkotsidou et al. (2009). 70 For a discussion of quantile models and their estimation techniques, see Koenker (2005).

90

5 Empirical Results

Figure2 provides quantile regression results for S&P 500 returns with the VIX index

changes, where we have 11 covariates and an intercept. For each of the 12 coefficients,

19 quantile regression estimates were plotted for q ranging along

0.95).....,0.9,(0.05,0.1,q � as the solid curve (blue) with circles. In each plot on the x-

axis, we have a quantile (or q) scale, and the y-axis indicates the covariate effect as a

percentage. For each covariate, these estimates could be interpreted as the effect of a

percentage-point change of the covariate on the volatility, holding other covariates

unchanged. The two red-dotted lines show the conventional 95% confidence level for

the quantile regression estimates.

However, more detailed results for the important upper and lower quantiles of

estimates from Figure 2 are provided in Table 2, including corresponding t-statistics (in

parentheses) for each of the estimates therein. The standard errors were obtained using

the bootstrap method; therefore, robust t-statistics were obtained for each of the

quantile estimates. On the other hand, for the OLS estimates, the standard errors were

made heteroskedasticity-consistent using Newey-West (1987) correction. As the aim

was to quantify the asymmetric return-volatility relationship, we limit our discussion to

the positive and negative returns covariates, especially to capturing the

contemporaneous effects. When we look at the estimated coefficients of covariates

�tSP500R

and �

tSP500R

in Columns 6 and 10, respectively, which represent the

contemporaneous return-volatility relationship; it is apparent from the absolute

difference that there are asymmetric effects for all quantile regression estimates,

including OLS estimates (here, OLS estimates are merely provided for comparative

purposes). The absolute values of �tSP500R

are higher than the absolute values of

.�tSP500R Moreover, the Wald test for coefficients was applied in order to find the

statistical difference between the coefficients � �qt and � �q

t in equation 6. The null

hypothesis (i.e., the coefficients for negative and positive returns are equal) for the

Wald test was significantly rejected for each of the quantile regression estimates.71

These results imply an asymmetric return-volatility relationship, indicating that the

negative returns for the stock index are linked to much higher volatilities for the VIX

71 Wald tests results are not reported here to save space.

91

index than those linked to positive returns. More specifically, looking at each row of

Table 2 (i.e., each quantile of estimates), the results indicate that the impacts of the

negative and positive S&P 500 index returns on the VIX are highly asymmetric, with

both contemporaneous coefficients being statistically significant at the 1% significance

level. The mean or OLS regression estimates are quite similar to the q = 0.5 (median)-

quantile regression estimates; however, the changing nature of the estimates at the

other quantiles provides an interesting picture of how the distribution of IV changes

depends on the positive and negative returns variables and lagged IV changes variables.

The absolute value of �tSP500R monotonically increases when moving from a median

quantile to an upper quantile; i.e., the marginal effect of the negative returns is larger in

upper quantiles (i.e., q=0.95%), and vice versa for positive returns.72 As a result, OLS

underestimates the magnitude of these effects for the highest quantiles and

overestimates for the lowest quantiles.

In detail, the coefficient estimates with q = 0.5 or median (and OLS) for the

�tSP500R variable imply that a 1% decline in S&P 500 returns is linked to a 1.040%

(1.185%) increase in the VIX level, whereas the coefficient estimates for the �tSP500R

variable imply that a 1% increase in S&P 500 returns is linked to a 0.795% (0.864%)

decrease in the VIX level.73 However, in the coefficient estimates for quantile q=0.95,

the �tSP500R

variable implies that a 1% decline in S&P 500 returns is linked to a

1.646% increase in the VIX, whereas the coefficient estimates for the �tSP500R variable

imply that a 1% increase in the S&P 500 returns is linked to a 0.401% decrease in the

VIX level. Obviously, it is apparent from the quantile regression results that the

asymmetry is much smaller in the lower and median quantiles of the distribution and

noticeably higher in the upper quantiles of the distribution. Thus, the OLS estimate,

which simply captures the mean effect, does a poor job of accounting for this

asymmetry in the upper quantiles.74

72 The equality of the coefficients across quantiles was formally tested using the Wald test. The test results significantly rejected the null hypothesis of equality of the coefficients (particularly the contemporaneous negative and positive returns) across quantiles; the Wald test is reported in Table 2. 73 Mean-regression model (or OLS) estimates are provided in parentheses for comparison. 74 The standard mean-regression models of Simon (2003), Giot (2005) and Hibbert et al. (2008) for the asymmetric return-volatility relationship ignore the higher effects of negative and positive returns on the upper quantiles of the IV changes’ distribution.

92

-.4

-.2

.0

.2

.4

.6

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

Intercept

-.20

-.15

-.10

-.05

.00

.05

.10

.15

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

VIX(-1)

-.2

-.1

.0

.1

.2

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

VIX(-2)

-.2

-.1

.0

.1

.2

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

VIX(-3)

-1.6

-1.2

-0.8

-0.4

0.0

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

SP500R+

-.6

-.4

-.2

.0

.2

.4

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

SP500R(-1)+

-.6

-.4

-.2

.0

.2

.4

.6

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

SP500R(-2)+

-.6

-.4

-.2

.0

.2

.4

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

SP500R(-3)+

-2.0

-1.6

-1.2

-0.8

-0.4

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

SP500R-

-.4

-.2

.0

.2

.4

.6

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

SP500R(-1)-

-.2

.0

.2

.4

.6

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

SP500R(-2)-

-.4

-.2

.0

.2

.4

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

SP500R(-3)-

Dependent Variable VIX-----Quanti le Process Estimates

Figure 2. Quantile Regression Estimates: Dependent variable VIX changes.

Ta

ble

2.

Qu

anti

le R

egre

ssio

n R

esu

lts:

Res

pon

se v

aria

ble

VIX

ch

ange

s Q

uant

ile

Inte

rcep

t 1

t�V

IX�

2

t�V

IX�

3

t�V

IX�

� t

R50

0SP

� � 1t

R50

0SP

� �

2t

R50

0SP

� � 3t

R50

0SP

� t

R50

0SP

� � 1t

R50

0SP

� �

2t

R50

0SP

� � 3t

R50

0SP

0.0

5 -0

.231

***

-0.0

68

-0

.025

-0

.021

-1

.30

1***

-0.2

79**

*-0

.24

2**

-0.2

81*

**

-0.6

36**

* 0

.313

***

0.3

44

***

0.1

51

(-3.

21)

(-1.

39)

(-0

.37)

(-0

.45)

(-11

.65)

(-

3.17

) (-

2.51

) (-

3.19

) (-

6.7

2)

(3.3

3)

(2.8

6)

(1.6

3)

0

.10

-0

.219

***

-0.0

30

-0.0

16

-0.0

05

-1.0

45**

* -0

.19

0**

*-0

.145

* -0

.224

**-0

.80

0**

* 0

.271

***

0.3

14**

* 0

.16

6**

(-

4.1

9)

(-0

.68

) (-

0.2

9)

(-0

.11)

(-

13.5

6)

(-2.

94

) (-

1.9

5)

(-2.

82)

(-

12.9

8)

(3.4

8)

(3.0

7)

(2.3

1)

0

.15

-0.1

57**

* -0

.04

6

-0.0

24

0.0

44

-1

.06

6**

*-0

.126

**

-0.1

82*

**

-0.0

89

-0.8

02*

**0

.18

6**

* 0

.26

0**

* 0

.153

**

(-3.

32)

(-1.

12)

(-0

.52)

(1

.16

) (-

17.6

4)

(-2.

04

)(-

2.8

4)

(-1.

38)

(-15

.42)

(2.9

8)

(3.1

3)

(2.3

1)

0

.20

-0

.134

***

-0.0

62*

-0

.035

0

.027

-0

.970

***

-0.1

25**

-0

.14

1**

-0.0

78

-0.8

60

***

0.1

19**

0.2

33**

*0

.14

7***

(-

3.0

3)

(-1.

81)

(-

0.8

8)

(0.8

2)

(-14

.11)

(-

2.56

)(-

2.4

9)

(-1.

42)

(-

17.5

1)(2

.28

) (3

.75)

(2.6

1)

0

.25

-0.1

15**

* -0

.078

**

-0.0

06

0.0

32-0

.94

3***

-0

.137

***

-0.0

68

-0

.06

1-0

.88

8**

* 0

.09

7**

0.1

82*

**

0.1

67*

**

(-3.

04

) (-

2.56

) (-

0.1

7)

(1.1

6)

(-16

.18

) (-

3.4

1)(-

1.4

2)

(-1.

36)

(-22

.78

)(2

.00

) (3

.70

) (3

.53)

Med

ian

-0

.010

-0.0

66

**

0.0

12

0.0

12

-0.7

95*

**

-0.0

87*

* 0

.029

0

.031

-1

.04

0**

* 0

.04

7 0

.09

0*

0.0

61

(-0

.35)

(-2.

22)

(0.3

8)

(0.4

8)

(-23

.86

)(-

2.14

)(0

.87)

(1

.02)

(-

29.4

5)(1

.03)

(1

.87)

(1

.44

)

0.7

5 0

.06

5*-0

.054

* -0

.023

0

.024

-0.5

79**

* 0

.06

4*0

.076

0

.06

3 -1

.28

5***

0

.012

0

.00

8

0.0

58

(1.7

0)

(-1.

69

) (-

0.6

1)

(0.8

3)

(-12

.71)

(1

.72)

(1

.44

) (1

.25)

(-

32.6

0)

(0.2

2)(0

.16

)(1

.42)

0.8

0

0.0

72*

-0.0

74**

-0

.00

7 0

.00

9

-0.5

41**

* 0

.047

0

.116

**

0.1

11**

-1

.335

***

-0.0

43

-0.0

04

0.0

34

(1.9

2)

(-2.

23)

(-0

.17)

(0

.29

)(-

9.9

9)

(1.4

2)(2

.01)

(2.1

4)

(-37

.98

)(-

0.7

6)

(-0

.07)

(0.7

2)

0

.85

0.1

49**

* -0

.09

0**

0

.020

0

.013

-0

.472

***

0.0

230

.145

**

0.1

05*

*-1

.371

***

-0.0

73

-0.0

02

0

.051

(3

.25)

(-

2.39

) (0

.42)

(0

.36

)(-

8.8

0)

(0.5

8)

(2.3

4)

(2.1

7)(-

24.2

3)(-

1.0

6)

(-0

.03)

(0.7

6)

0

.90

0

.26

7***

-0

.079

* 0

.018

0

.021

-0

.451

***

0.0

110

.18

5**

0.0

35

-1.5

33**

* -0

.18

5**

-0.0

260

.055

(4

.90

) (-

1.6

9)

(0.3

6)

(0.5

2)(-

8.7

6)

(0.2

1)

(2.5

4)

(0.6

4)

(-23

.02)

(-2.

11)

(-0

.38

)(0

.70

)

0.9

5 0

.373

***

-0.0

25

0.0

48

0

.06

6

-0.4

01*

**

0.1

33

0.2

43*

**

0.1

05

-1.6

46

***

-0.1

62

-0.0

05

0.0

13

(5.0

2)

(-0

.39

) (0

.86

) (1

.16

) (-

5.6

7)(1

.57)

(2.9

8)

(1.4

3)

(-14

.93)

(-

1.4

5)

(-0

.06

)(0

.12)

OL

S 0

.00

0

-0.1

01*

* -0

.033

-0

.034

-0

.86

4***

-0.0

89

* -0

.039

-0

.079

-1

.18

5***

-0.0

170

.06

60

.10

0

(0.0

1)

(-2.

27)

(-0

.48

) (-

0.6

9)

(-10

.75)

(-1.

71)

(-0

.70

) (-

1.15

) (-

24.1

9)

(-0

.18

) (0

.59

) (1

.61)

Qu

anti

le S

lop

e E

qual

ity

Tes

t R

esu

lts:

On

ly s

ign

ific

ant

resu

lts

of a

sym

met

ry a

re r

epor

ted

wit

h t

he

corr

esp

ond

ing

quan

tile

s �

� q1

q�

�.

0

.2-0

.4**

*

0.2

-0.4

**

0.2

-0.4

***

0.2

-0.4

***

0

.2-0

.4*

0.4

-0.5

**

0.4

-0.5

*

0

.5-0

.6*

0

.5-0

.6**

0.5

-0.6

***

0

.5-0

.6**

0

.6-0

.8**

* 0

.6-0

.8**

*0

.6-0

.8**

*0

.6-0

.8**

* 0

.6-0

.8**

Th

e ta

ble

rep

orts

res

ult

s fr

om t

he

Qu

anti

le R

egre

ssio

n a

nd

OL

S R

egre

ssio

n o

f th

e V

IX c

han

ges

on a

set

of

vari

able

s; s

pec

ific

atio

ns

2 an

d 1

are

est

imat

ed. T

-sta

tist

ics

are

pro

vid

ed in

par

enth

eses

. ***

, **

, an

d *

den

ote

reje

ctio

n o

f th

e n

ull

hyp

oth

esis

at

the

1% ,

5% a

nd

10

% s

ign

ific

ance

leve

ls, r

esp

ecti

vely

.

93

94

Figure 3 presents quantile regression results for NASDAQ 100 returns with the VXN

index changes, and the important full-sample daily upper and lower quantiles’ results

are presented in Table 3.75 The results are qualitatively similar to those found for the

S&P 500 and VIX asymmetric relationship.76 The major difference lies in the lower

asymmetric responses of the covariates (i.e., negative and positive returns) across

different quantiles of the VXN distribution in comparison with the VIX results.77

Furthermore, the significance of covariates is lower for the VXN than for the VIX. The

finding is consistent with both Giot (2005) and Hibbert et al. (2008) in that during

volatile periods, option traders react less aggressively to negative returns. As the

NASDAQ is a tech index, it inherently presents a higher volatility than the S&P 500;

therefore, the conclusion drawn by Giot (2005) and Hibbert et al. (2008) can be

applied to the NASDAQ results.78

On the other hand, Figures 4 and 5 provide quantile regression results for DAX 30

returns with the VDAX changes and the VDAXO changes, respectively; the important

daily upper and lower quantile results are reported in Table 4 and Table 5.79 The

quantile results for the DAX 30 returns with both the VDAX changes and VDAXO

changes are discussed simultaneously in order to compare the asymmetric responses of

both volatility indices to the same negative and positive returns of the DAX 30 index.80

The coefficients of the covariates �tDAXR

and �

tDAXR are shown in Columns 6 and

10, respectively, in both tables; the coefficients represent contemporaneous return-

volatility relationships. Based on the absolute difference in the coefficients’ values, it is

clear that there are asymmetric effects for all quantile regression estimates, including

the OLS estimates. The absolute values of �tDAXR are higher than the absolute values

of .�tDAXR Similarly, the null hypothesis of the Wald test that the coefficients � �qt and

75 Figure 3 and Table 3 are provided in Appendix A. 76 A detailed discussion on these results is avoided merely for space considerations. 77The Wald test for equality of the coefficients across quantiles is formally tested and reported in Table 3. Here, too, the test results significantly reject the null hypothesis of equality of the coefficients (particularly the contemporaneous negative and positive returns) across quantiles. 78 For more discussion on this point, see Hibbert et al. (2008). 79 Figures 4 and 5 and Tables 4 and 5 are provided in Appendix A. 80 Remember that VDAX is the MFIV index that incorporates volatility smirk (skew), whereas VDAXO is the BSIV index that does not account for volatility smirk (skew). Unfortunately, for the comparison of the two measures we are restricted to only the DAX 30 stock index. For the other stock indices, S&P 500, NASDAQ 100 and DJ Euro STOXX, we have no active BSIV volatility index; although the VXO, an active BSIV volatility index, is available, it cannot be compared because it is implied from the options on the S&P 100 index.

95

� �qt in equation 6 are equal is significantly rejected for each of the quantile regression

estimates. Furthermore, looking at each row of Tables 4 and 5 (i.e., each quantile

result), the results indicate that the impacts of the negative and positive DAX 30 index

returns on VDAX and VDAXO are asymmetric, with both contemporaneous coefficients

being statistically significant at the 1% significance level. The coefficient estimates for

q=0.5 or the median (and OLS) for the �tDAXR covariate imply that a 1% decline in

DAX 30 returns is linked to a 0.837% (1.022%) increase in the VDAX level and that a

similar decline in DAX 30 returns is linked to a 0.712% (0.778%) increase in VDAXO

level. On the other hand, the coefficient estimates for the �tDAXR covariate imply that

a 1% increase in DAX 30 returns is linked to a 0.515% (0.400%) decrease in the VDAX

level, and a similar increase in DAX 30 returns is linked to a 0.543% (0.370%) decrease

in the VDAXO level. However, the coefficient estimates for quantile q=0.95 of the

�tDAXR variable imply that a 1% decline in DAX 30 returns is linked to a 1.389%

(1.115%) increase in the VDAX (VDAXO), whereas the coefficient estimates for the

�tDAXR covariate imply that a 1% increase in DAX 30 returns is linked to a 0.130%

(0.156%) decrease in the VDAX (VDAXO) level. For comparison, the coefficients of

covariates �tDAXR

and �

tDAXR are listed in Tables 4 and 5. It is very clear that the

effects of the negative and positive returns are considerably different. The VDAX (MFIV

index) responds in a very asymmetric fashion in comparison to its older counterpart,

the VDAXO (BSIV index), to similar negative and positive returns, implying

pronounced asymmetric return-volatility relationship with the MFIV index.

Furthermore, the asymmetric responses are most apparent in upper-quantile

estimates, where the asymmetry is very pronounced; i.e., the asymmetries in the

absolute differences are smaller in the lower and median quantiles of the distribution

and noticeably larger in the upper quantiles of the distributions.

Figure 6 presents’ quantile regression estimates for the DJ Euro STOXX 50 returns

with the VSTOXX changes, and the important daily upper and lower quantile results

are presented in Table 6.81 Similarly, the results here are qualitatively similar to those

found for the DAX 30 and VDAX asymmetric relationships. The major difference is the

slightly more asymmetric responses of the covariates (i.e., negative and positive

returns) across different quantiles of the VSTOXX in comparison to the VDAX results. 81 Figure 6 and Table 6 are provided in Appendix A.

96

The main conclusion drawn from Tables 2 to 6 is that negative and positive stock index

returns trigger the volatility index to move in completely opposite directions and in an

asymmetric fashion; i.e., negative returns have a much greater impact on volatility than

do positive returns, particularly at the uppermost regression quantiles (e.g., q=0.95).

Our quantile regression model for the asymmetric return-volatility relation thus reveals

important missing information underestimated by the mean-regression model (OLS).

Second, our argument that the asymmetry with smirk (skew)-adjusted volatility (MFIV)

should present pronounced responses is confirmed by comparing the asymmetric

responses of VDAX (MFIV) and VDAXO (BSIV), where we found that the MFIV index

responded in a pronounced fashion in comparison with the BSIV index.82 Third, if we

look at the coefficients of the lag covariates of negative and positive returns, they are

mostly insignificant; we thus assert that at the daily level, the leverage hypothesis is

unable to quantify this strong asymmetric return-volatility relationship and that similar

conclusions could be drawn for the feedback hypothesis. Finally, the VIX volatility

index presents the strongest asymmetric return-volatility relationship, followed by the

VSTOXX, VDAX and VXN volatility indices, respectively.

82 As the MFIV volatility indices account for OTM puts, the asymmetry should be pronounced with each MFIV volatility index. Because investors hedge their downside risk by taking positions in the OTM put options, in periods of market turmoil there is greater buying demand for put options than for call options, which leads to higher volatilities than those found during market rallies. Consequently, negative stock index returns induce an increase in the levels of the volatility indices. Our results are also consistent with the net-buying-pressure hypothesis of Bollen and Whaley (2004).

97

6 Conclusion

We investigated the asymmetric return-volatility phenomenon in the newly adapted

robust volatility indices (i.e., the VIX, VXN, VDAX, VDAXO, and VSTOXX) using

quantile regression. In particular, we quantified the effects of positive and negative

stock index returns at different quantiles of IV changes’ distributions, asking about the

degree to which the asymmetric responses at the uppermost quantiles are comparable

with the responses of median (or mean) regressions. Additionally, as Bollen and

Whaley (2004) have documented, the net buying pressure for stock index put options

from institutional investors seeking to hedge their portfolios induces increases in IVs.

Likewise, new IV indices incorporate both OTM put and call options and are thus

highly informed and robust measures. Accordingly, they should present more

pronounced asymmetric return-volatility relationships in comparison to their older

counterparts.

There is noticeable evidence that the volatility indices VIX, VXN, VDAX, VDAXO and

VSTOXX from February 2001 through May 2009 responded in a pronounced

asymmetric fashion to the negative and positive returns of their corresponding stock

indices: the asymmetry monotonically increases when moving from the median

quantile to the uppermost quantile (i.e., 95%); therefore, OLS underestimates this

relationship at upper quantiles. These IV indices thus sharply rise during market

declines (fear) and fall during market rallies (exuberance). The VIX presents the

highest asymmetry, followed by the VSTOXX, VDAX and VXN volatility indices,

respectively. Second, our argument that asymmetry with the volatility smirk(skew)-

adjusted volatility index measure (MFIV) should be pronounced is confirmed by

comparing the asymmetric responses of VDAX (MFIV) and VDAXO (BSIV); the MFIV

index responds in a pronounced fashion in comparison with the BSIV index. Third, we

also confirmed that a significant amount of asymmetry occurs contemporaneously

rather than with a lag, thus rejecting the leverage hypothesis, and that a similar

conclusion can be drawn for the feedback hypothesis.

Our results have a number of implications. First, as we found that newly adapted

volatility indices are strongly negatively correlated with their corresponding stock

indices, the new volatility indices are important instruments for hedging stock

portfolios. Derivatives exchanges provide liquid markets for the futures and options

98

underlying these volatility indices. Therefore, a position in futures or options on a

volatility index can more accurately hedge a stock portfolio position without

considering complicated stock index option trading strategies. Second, when the stock

index drops, the volatility index rises sharply. Therefore, new volatility indices are

useful not only for assessing potential risks, but also for speculative transactions by

risk-seeking investors. Third, since the new volatility indices are based on the robust

MFIV concept and provide better tradability, it is easier for issuers of derivatives to

engineer structured products based on the volatility indices. Fourth, trading strategies

with regard to range could generate profits; an example of this could be a volatility-long

position in decreasing volatility markets paired with a volatility-short position in

increasing volatility�markets.

99

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101

Appendix A: Quantile Regression Estimates

-.8

-.4

.0

.4

.8

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

Intercept

-.2

-.1

.0

.1

.2

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

VXN(-1)

-.15

-.10

-.05

.00

.05

.10

.15

.20

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

VXN(-2)

-.1

.0

.1

.2

.3

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

VXN(-3)

-1.6

-1.2

-0.8

-0.4

0.0

0.4

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

NASDAQR+

-.4

-.2

.0

.2

.4

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

NASDAQR(-1)+

-.4

-.2

.0

.2

.4

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

NASDAQR(-2)+

-.3

-.2

-.1

.0

.1

.2

.3

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

NASDAQR(-3)+

-1.6

-1.2

-0.8

-0.4

0.0

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

NASDAQR-

-.4

-.2

.0

.2

.4

.6

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

NASDAQR(-1)-

-.2

.0

.2

.4

.6

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

NASDAQR(-2)-

-.2

.0

.2

.4

.6

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

NASDAQR(-3)-

Dependent Variable VXN-----Quanti le Process Estimates

Figure 3. Quantile Regression Estimates: Dependent Variable VXN changes.

-.8

-.4

.0

.4

.8

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

Intercept

-.2

-.1

.0

.1

.2

.3

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

VDAX(-1)

-.2

-.1

.0

.1

.2

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

VDAX(-2)

-.3

-.2

-.1

.0

.1

.2

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

VDAX(-3)

-1.2

-0.8

-0.4

0.0

0.4

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

DAXR+

-.6

-.4

-.2

.0

.2

.4

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

DAXR(-1)+

-.6

-.4

-.2

.0

.2

.4

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

DAXR(-2)+

-.6

-.4

-.2

.0

.2

.4

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

DAXR(-3)+

-1.6

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

DAXR-

-.4

-.2

.0

.2

.4

.6

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

DAXR(-1)-

-.2

.0

.2

.4

.6

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

DAXR(-2)-

-.4

-.2

.0

.2

.4

.6

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

DAXR(-3)-

Dependent Variable VDAX-----Quanti le Process Estimates

Figure 4. Quantile Regression Estimates: Dependent Variable VDAX changes.

102

-.4

-.2

.0

.2

.4

.6

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

Intercept

-.4

-.3

-.2

-.1

.0

.1

.2

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

VDAXO(-1)

-.2

-.1

.0

.1

.2

.3

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

VDAXO(-2)

-.2

-.1

.0

.1

.2

.3

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

VDAXO(-3)

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

DAXR+

-.3

-.2

-.1

.0

.1

.2

.3

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

DAXR(-1)+

-.3

-.2

-.1

.0

.1

.2

.3

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

DAXR(-2)+

-.4

-.2

.0

.2

.4

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

DAXR(-3)+

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

DAXR-

-.8

-.6

-.4

-.2

.0

.2

.4

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

DAXR(-1)-

-.2

.0

.2

.4

.6

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

DAXR(-2)-

-.1

.0

.1

.2

.3

.4

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

DAXR(-3)-

Dependent Variable VDAXO-----Quanti le Process Estimates

Figure 5. Quantile Regression Estimates: Dependent Variable VDAXO changes.

-.6

-.4

-.2

.0

.2

.4

.6

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

Intercept

-.2

-.1

.0

.1

.2

.3

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

VSTOXX(-1)

-.3

-.2

-.1

.0

.1

.2

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

VSTOXX(-2)

-.2

-.1

.0

.1

.2

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

VSTOXX(-3)

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

STOXXR+

-.6

-.4

-.2

.0

.2

.4

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

STOXXR(-1)+

-.8

-.6

-.4

-.2

.0

.2

.4

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

STOXXR(-2)+

-.6

-.4

-.2

.0

.2

.4

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

STOXXR(-3)+

-1.8

-1.6

-1.4

-1.2

-1.0

-0.8

-0.6

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

STOXXR-

-.4

-.2

.0

.2

.4

.6

.8

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

STOXXR(-1)

-.4

-.2

.0

.2

.4

.6

.8

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

STOXXR(-2)-

-.2

.0

.2

.4

.6

0.0 0.2 0.4 0.6 0.8 1.0

Quantile

STOXXR(-3)-

Dependent Variable VSTOXX-----Quanti le Process Estimates

Figure 6. Quantile Regression Estimates: Dependent Variable VSTOXX changes.

Ta

ble

3.

Qu

anti

le R

egre

ssio

n R

esu

lts:

Dep

end

ent

Var

iabl

e V

XN

ch

ange

s Q

uant

ile

Inte

rcep

t 1

t�V

XN

�

2t

�VX

N�

3

t�V

XN

�

� tN

ASD

AQ

R

� �1tN

ASD

AQ

R� �t

NA

SDA

QR

� �3tN

ASD

AQ

R� t

NA

SDA

QR

� �1t

NA

SDA

QR

� �2tN

ASD

AQ

R� �3t

NA

SDA

QR

0.0

5 -0

.320

**

-0.0

04

0

.00

1 0

.036

-1

.04

9**

*-0

.20

4**

-0.1

90

**

-0.1

45*

-0

.24

0**

* 0

.258

**

0.3

33**

*0

.258

**

(-2.

50)

(-0

.06

) (0

.02)

(0

.58

) (-

10.9

7)

(-2.

17)

(-1.

97)

(-

1.8

4)

(-3.

01)

(2

.51)

(4.0

7)(2

.36

)

0.1

0

-0.2

65*

**

0.0

27

-0.0

02

0.0

37-0

.854

***

-0.1

47**

-0.1

14*

-0.1

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107

The Information Content of the VDAX Volatility Index and Backtesting Daily Value-at-Risk Models

Ihsan Ullah Badshah



September 20, 2010

Abstract

This essay examines the information content of the new VDAX volatility index to forecast daily value-at-risk (VaR) estimates and compares its VaR forecasts with the VaR forecasts of the filtered historical simulation (FHS) and RiskMetrics models. The daily VaR models were backtested from January 1992 through May 2009 using unconditional coverage, independence, and conditional coverage tests. A quadratic score was also estimated for each of the models. We found that the information content of implied volatility was superior to that of the historical volatility for the daily VaR forecasts of a portfolio of the DAX 30 stock index: implied volatility (VDAX) and combined (implied volatility plus GJR) VaR models outperformed the VaR models of the FHS (GJR) and RiskMetrics. Finally, the quadratic score also supports the use of implied volatility VaR models. Our findings have implications for traders, risk managers, and regulators. Keywords: backtesting, filtered historical simulation, implied volatility, value at risk, VDAX JEL Classifications: G13, C52, C53

108

1 Introduction

Volatility forecasting is important to both financial practitioners and academics. This is

very much true in option pricing, option trading, hedging derivative positions, risk

management activities, stock selection and portfolio diversification, all of which require

accurate volatility modeling and forecasting. Fortunately, we have seen tremendous

development in volatility forecasting models since the introduction of the

autoregressive conditional heteroskedasticity (ARCH) model by Engle (1982). Since

then, many ARCH models have been developed that attempt to forecast volatility using

historical information. In 1986, Bollerslev (1986) extended the ARCH model proposed

by Engel (1982), to a generalized ARCH model called GARCH. Many extensions have

been made to the GARCH model (see Poon and Granger, 2003; Poon and Granger,

2005); for example, Glosten, Jagannathan and Runkle (hereafter GJR)(1993) proposed

the GJR model, an extension of the GARCH model that is used to account for

asymmetry. However, the growing literature supports using implied volatility (IV)

instead, calling IV the best forecast of future realized volatility (RV). These studies

empirically document that the information content of IV is richer and superior to that

of historical volatility (HV) when forecasting the future RV of the underlying asset (e.g.,

Day and Lewis, 1992; Christensen and Prabhala, 1998; Fleming, 1998; Dumas, Fleming

and Whaley, 1998; Blair et al., 2001; Ederington and Guan, 2002; Poon and Granger,

2003; Mayhew and Stivers, 2003; and Martens and Zein, 2004).

IV can be recovered by inverting the Black-Scholes (1973) formula. However, Britten-

Jones and Neuberger (2000) and Jiang and Tian (2005) derived model-free implied

volatility (MFIV) under the assumptions of pure diffusion and asset price processes

with jumps. They showed that the information content of MFIV is richer and superior

to that of Black-Scholes implied volatility (BSIV): the MFIV measure accounts for all

available strike prices, whereas the BSIV measure is a point-based IV, where each strike

price has a separate IV. Additionally, BSIV is subject to both model and market

efficiency, while MFIV is only subject to market efficiency (see Poon and Granger,

2003).

Deutsche Börse has launched a new VDAX volatility index for the German DAX 30

stock index that incorporates more robust information on options by using MFIV

109

measures.83 In addition, the new VDAX measure aligns with the consensus view of

option traders (who are usually professionals) about the future direction of the

volatility of the stock market over the next 30 days. The rationale for adopting MFIV

measures was to account for both out-of-the-money (OTM) put and call options (i.e.,

volatility smirk/skew). The new VDAX measure incorporates both types of options;

therefore, it moves with changes in option prices. For example, a negative or positive

shock to the market induces adjustments in hedging and trading strategies; this

consequently triggers changes in the prices of one type of option (i.e., put or call). It

then moves in the direction of the market demand for particular types of options and

underlying stocks (see, e.g., Bollen and Whaley, 2004).84 In addition, Liu et al. (2005)

found that the rare-event premiums play an important role in generating the volatility

skew patterns observed for options across moneyness and that these rare events are

embedded in the OTM options.85 Camara and Heston (2008) derived an option pricing

model that accounted for both OTM put and call options. They derived the extreme

negative events from OTM puts and the extreme positive events from OTM calls. Thus,

The new VDAX index should be an excellent instrument for forecasting volatility (or

VaR) for the underlying DAX 30 stock index portfolio as it is implied by OTM options,

which are informed on future events and contain information on negative and positive

jumps ( see, e.g., Pan, 2002; Liu et al., 2005; Camara and Heston, 2008; Bates, 2008).

There are some famous studies related to the topic of our essay that use the information

content of IV in their volatility forecasting models and find that IV subsumes almost all

information for forecasting future RV. For example, Blair et al. (2001) studied the

information content of the VIX (now VXO) and intraday returns for the S&P 100 stock

index and then compared them for the sample period from 1987 to 1999. Their in-

sample forecasting showed that all relevant information is provided by the IV (i.e.,

implied from the VIX) and that there is not much incremental information in intraday

index returns. However, for out-of-sample forecasts, the VIX provides superior and

accurate forecasts for all forecast performance measures and horizons (i.e., from 1 to 20

days); moreover, the incremental forecasting information in the intraday returns is

insignificant. Ederington and Guan (2002) studied the importance of IV forecasts

while using S&P 500 future options, finding that IV has superior forecasting power and

83 Chicago Board Options Exchange uses a similar MFIV measure for the VIX and VXN, which are derived from S&P 500 and NASDAQ options, respectively. 84 Whaley (2000) referred to the volatility index as an “investors’ fear gauge”. 85 Similarly, Pan (2002) showed that volatility skew is primarily due to investors’ fear of large adverse jumps.

110

subsumes the information in HV. Similarly, Martens and Zein (2004) confirmed that

the GARCH model extended with IV provides better forecasts than the GARCH model

extended with RV (obtained from high-frequency intraday returns). Finally, the study

by Giot (2005), which is slightly different from the above two studies, assesses the

information content of IV indices such as the VIX (now the VXO) and the VXN in a

daily VaR framework while studying time periods that include both bull and bear

markets. The performances of the VaR models were evaluated using unconditional

coverage, independence and conditional coverage tests. Backtesting results showed that

IV indices provide superior VaR forecasts and thus fewer VaR violations over time

relative to the time-series-based VaR forecasts. Moreover, the performance was stable

during market turmoil.

This essay contributes to the literature on market risk management, volatility

modeling, and forecasting. The field of volatility modeling and forecasting, especially

when predicting future RV by using IV, HV or a combination of the two, represents an

interesting and less trite area of research in finance. In this essay, we incorporated the

information content of the new VDAX volatility index into daily VaR models, and the

resulting VaR forecasts were compared with the VaR forecasts from the FHS model

(GJR) and the RiskMetrics model at confidence levels of 99%, 97.5% and 95%. The

daily VaR models were then backtested using unconditional coverage, independence,

and conditional coverage tests; furthermore, a quadratic score was estimated for each

of the VaR models for the period from January 1, 1992 through May 29, 2009.

However, this essay differs from the above-cited studies in several ways. First, they

focus on using the BSIV index to forecast future RV for U.S. stock markets, whereas we

focused on using a robust MFIV volatility index to forecast daily VaR for the German

stock market. Second, we compared the implied daily VaR forecasts with the forecasts

from the famous approach by Barone-Adesi et al. (1998), the filtered historical

simulation; to our knowledge, no other study has ever compared it with implied VaR

forecasts. Finally, we used much more detailed and robust backtesting techniques on

the very long data series to evaluate different VaR models for daily market risk

management.

Our main finding was that the MFIV (the new VDAX) index subsumed almost all

information required for actual volatility in VaR models. The backtesting results show

that the new VDAX index contained significant information regarding volatility in VaR

models and that the number of implied VaR violations was not significantly different

111

from the set coverage rates, thereby yielding fewer VaR exceptions and clusters of

exceptions. Consequently, the null hypotheses regarding independence and conditional

coverage tests were never rejected for implied volatility, implied volatility plus GJR,

and FHS VaR models. However, the null hypotheses for the RiskMetrics model were

rejected at lower confidence levels. Finally, the quadratic scores also favored implied

VaR models.

This essay is organized as follows. In Section 2, we discuss the new VDAX index and its

construction. In Section 3, we discuss the data. In Section 4, we specify daily VaR

models. In Section 5, we discuss backtesting techniques. The empirical results are

presented in Section 6. In Section 7, we summarize and conclude.

2 The New VDAX Volatility Index

The Deutsche Börse and Goldman Sachs jointly developed the methodology for the new

VDAX index. It is based on options on the DAX 30 index; options are traded on the

derivatives exchange EUREX. Like VIX the VDAX measure is also based on MFIV

measure. Hence, it accounts for IVs across all options of a given time to expiry (it

accounts for volatility smirk/skew). In fact, the main VDAX index is further based on

eight constituents volatility indices that expire in 1, 2, 3, 6, 9, 12, 18, and 24 months,

respectively. The main VDAX is designed as a rolling index at a fixed 30 days to expiry

via a linear interpolation of the two nearest of the eight available sub-indices. The price

history for the VDAX is calculated back to the years 1992. The VDAX and its eight sub-

indices are updated every minute. The sub-indices are calculated according to the

formula below:

,2ii �100VDAX ��

(1)

� � .1,2,...8i,1KF

T1KMR

K�K

T2�

2

i,0

i

ji,ji

j2i,j

i,j

i

2i ��

�

��

��

(2)

Where iT is time to expiry of the ith DAX 30 option (best bid and best ask of all DAX

30 options) , iF is forward price derived from the prices of the ith DAX 30 options, for

which the absolute difference between call and put prices is the smallest as

� �PCRKF iPCmini �� , i,jK the exercise price of the OTM option of the ith DAX

112

30 option expiry month , 2

KK�K 1i,j1i,j

i,j��

� is the interval between the strike

prices, i,0K the highest exercise price below forward price, iR is the refining factor

equal to ii .Tri eR � , r is the risk free rate, � �i,jKM is the price of the option i,jK ,

whereby i,oi,j KK ) , and � �i,0KM is the average of the put and call prices at exercise

price i,0K .

Finally, the main VDAX is designed as a rolling index at a fixed 30 days to expiry via a

linear interpolation of the two nearest of the eight available sub-indices as follows.

,T

365

TT

TT21i1i

TT

TT2ii N

NNN

NNVDAXT

NNNN

VDAXTVDAXi1i

i

i1i

1i ��

�

�

��

�

��

��

�

�

��

��

��

�

�

��

��

�

��

(3)

where iTN time to expiry of the ith DAX 30 index option,

1iTN�

time to expiry of the i+

1th option, TN time for next days, and 365N time for standard year.86

3 Data

This essay employs data from two sources. We obtained the daily time-series price data

for the DAX 30 stock index from Financial DataStream. The data on the new VDAX

were obtained from the Deutsche-Börse. The daily data for both the DAX 30 and the

VDAX indices cover a period of 17 years and 5 months, from January 1, 1992, to May

29, 2009, for a total of 4,543 trading days.

Figure 1 shows the daily continuously compounded returns (%) of the DAX 30 stock

index and the daily closing level (%) of the VDAX volatility index from January 1992

through May 2009. As can be seen, from January 1992 through May 1997, the VDAX

showed the lowest volatility levels and an upward moving stock index (a period in a

low-volatility bull market). From June 1997 through April 2000, there were high VDAX

levels (a bubble period where we find a high-volatility bull market). A spike in the 86 See, for further detail on the VDAX construction, the Deutsche Börse and Goldman Sachs methodology for VSTOXX on the STOXX website: www.stoxx.com, the same methodology is also used for the construction of the VDAX volatility index.

113

VDAX can be seen in 1998. This was followed by a stock market crash due to the Long-

Term Capital Management (LTCM) crisis; during this crash, the VDAX level reached as

high as 57%. A crash can also be observed from April 2000 through January 2003

when the bubble burst (a period in a high-volatility bear market).

-10

-5

0

5

10

15

92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 07 08 09

DAX 30 stock index returns

0

20

40

60

80

100

92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 07 08 09

VDAX volatility index level

Figure 1. DAX 30 stock index returns (%) and the VDAX volatility index level (%) from

January 1992 through May 2009.

However, from 2004 to August 2007 (a period in a low-volatility bull market), we

found positive returns, and the level of the VDAX decreased, showing low volatility

114

levels. From September 2007 through May 2009 (a period of extremely high volatility

and an extreme bear market), we found the highest VDAX levels, decreasing stock

index returns and some jumps due to the credit crunch and liquidity crises. In this

particular period, a spike in the VDAX level was observed in October 2008 when the

market crashed and the VDAX level reached a historical peak of 74%.

Table 1 provides the summary statistics for the daily continuously compounded returns

(in %) of the DAX 30 index as well as tests for normality and unit roots.

Table 1. Descriptive Statistics of the DAX 30 returns series. DAX 30

Mean 0.000251

Median 0.000501

Maximum 0.107975

Minimum -0.088747

Std.Dev. 0.014682

Skewness -0.111999

Kurtosis 8.228179

Jarque-Bera 5183.57**

ADF -68.75**

No. Obs 4543 Table1 report the descriptive statistics of the returns series of DAX30 index. The Jarque-Bera and the Augmented Dicky Fuller (ADF) (an intercept is included in the test equation) test values are reported. ** and * denote rejection of the null hypothesis at the 1% and 5% significance levels respectively.

The mean return on the DAX 30 index is not statistically different from zero. The tests

for skewness and kurtosis confirm that returns on the DAX 30 index are negatively

skewed and are highly leptokurtic with respect to a normal distribution. Likewise, the

Jarque-Bera statistics reject normality for the return series. The stationarity in the

return series was investigated by applying the augmented Dickey-Fuller (ADF) test for

a unit root. The ADF results reject the hypothesis of a unit root in the return series at

the 1% significance level.

Figure 2 shows a Q-Q plot of the theoretical quantiles of the normal distribution

(vertical axis) against empirical quantiles of the returns on the DAX 30 (horizontal

axis). As can be seen from this plot, the data are not normally distributed, indicated by

the fact that the empirical quantiles do not lie on a straight line

115

-.06

-.04

-.02

.00

.02

.04

.06

-.10 -.05 .00 .05 .10 .15

Quantiles of returns on DAX 30

Qua

ntile

s of

Nor

mal

Figure 2. Normal Q-Q plots returns on DAX 30.

As can be seen from the plot, there is a significant deviation from the straight line in the

tails, particularly in the lower tail, indicating that the distribution of returns on the

DAX 30 index is more heavily tailed than the normal distribution. The Q-Q plot thus

reinforces the findings of our earlier statistical tests for normality, leading to the use of

a heavy-tail distribution rather than a normal distribution in the rest of our analysis.

116

4 Daily Value-at-Risk Models

4.1 Value-at-Risk

Value at Risk (VaR) is defined as the maximum expected loss in the value of a portfolio.

It has a certain probability over a certain holding period (for details on VaR, see, e.g.,

Duffie and Pan, 1997; Jorion, 2000; Dowd, 2005; Christoffersen, 2009). VaR forecasts

are fundamental to financial risk management and risk regulation. However, the

importance and recognition of VaR as a risk management tool spring from the Market

Risk Amendment (1996) to the Basel Capital Accord of 1998 and also because of the

popularity of the RiskMetrics introduced by J.P. Morgan (see Jorion, 2000 and Dowd,

2005). After these events, VaR became widely accepted by banks and was also imposed

by regulators. The aim of these two groups was to supervise and manage market risk—

the market-exposure risk arising due to unfavorable movements in equity prices,

interest rates, exchange rates, commodity prices etc.87 Because VaR has become a

standard measure for market risk due to the trading activities and market positions

taken by large banks, most financial institutions and trading houses currently use VaR

models to assess their daily portfolio losses from significant trading activities. They also

backtest VaR models by observing when the portfolio returns exceed the VaR forecasts;

that is, the VaR forecasts match their expectations. As a result, accurate VaR forecasts

are crucial for market risk management. Given that accurate VaR forecasts rely heavily

on the accurate forecasting of the volatility of a portfolio, this is an important

parameter for any VaR model. For instance, the level of the VaR over a one-day holding

period is defined as the solution

�,)�VaRP(r 1tPt

Pt ��

(4)

where � is 1 minus the VaR confidence level (e.g. 99%), and Ptr is the return of a

portfolio over a one-day holding period. Having conditional volatility specification th

such that ,ttP

t �hr � where the residuals are distributed as � �tt 0,h~N� , then a one-

period VaR at time t is

� � ,t1

t h��VaR ��

(5)

87 The Basel committee for banking supervision allows banks to use VaR as a benchmark in order to determine how much additional capital is needed to cover daily market risk beyond that required for credit risk.

117

where � denotes the standardized normal cumulative distribution function. In fact, we

need to accommodate heavy tails in VaR estimation; therefore, VaR needs to be

estimated using a Student’s t density function.88 We assume that t� are distributed as

follows:

� �� , 21

tt ~�2 h� �

(6)

where � is the standardized Student’s t-distribution with degrees of freedom.

To estimate the �%daily VaR using Student’s t density function, we followed the

method described by Dowd (2005):89

� � � � ,t

21

1 t,� h

2 ��VaR ��

��

(7)

where the shape is to be estimated. Here, we used four different types of volatility

forecasts (implied volatility (VDAX), implied volatility (VDAX) with GJR, FHS (GJR),

and RiskMetrics) as parameters for the daily VaR models; these are specified and

discussed in the following subsections.

4.2 Implied Volatility

We considered the new VDAX index (MFIV index), which is a robust and more

informed measure. The rationale for considering the new VDAX is that we believe VaR

modeling is about extreme events; that is, implying volatility from the extreme

outcomes in the options market is of paramount importance to VaR forecasting. These

extreme events are embedded in the IV derived from OTM options (see, for instance,

Liu et al., 2005; Camara and Heston, 2008; Bates, 2008). Likewise, the importance of

the new VDAX measure increases because it accounts for volatility smirk/skew, which

may be induced by the net buying pressure of the OTM put options (see Bollen and

Whaley, 2004). Volatility skew is an obvious phenomenon previously documented by

many other researchers and is important to capture in any volatility measure (e.g.,

Alexander, 2001; Low, 2004; Goncalves and Guidolin, 2006; Badshah, 2008). In

addition, information from trading strategies and other shocks is well absorbed in the

88 Most previous studies have concluded that distribution functions accounting for fat tails are fundamental for VaR modeling. See, for instance, Huisman et al. (1998), and Alexander and Sheedy (2008) for VaR forecasts obtained through different density functions. 89 Because the returns on the DAX 30 are not normally distributed, the VaR forecasts are estimated using Student’s t density function.

118

new VDAX index because it accounts for both OTM put and call types of options.

Nevertheless, the new VDAX is implied from both the fear and exuberance embedded

in option prices, even though the majority of option traders are very informed and

possess strong skills (see, e.g., Low, 2004 and Chakravarty et al., 2004). As a result, the

new VDAX is a perfect choice to be used as a volatility parameter in the daily VaR

model to quantify a daily VaR forecast for the DAX 30 stock index portfolio. However,

for the daily VaR model, a daily-variance parameter is needed in place of the standard

deviation because VaR uses variance as an input, as the VDAX is expressed in

annualized standard deviation units. As a result, transformation is essential; for

instance, at time t, we insert 1tVDAX � into the daily VaR model as 90

,21imp,1,tt �h ��

(8)

while

� � .2

252VDAX2

1imp,1,t1t� ��

(9)

However, Granger and Poon (2003, 2005) point out that BSIV is biased and is always

higher than the actual volatility. They suggest using HV for calibration as

,21imp,1,tt ��h ��

(10)

where and � need to be estimated. However, we assert that VDAX is calculated

using the MFIV measure, whose IV value should not be subject to model risk, and that

this bias should thus not be of great concern; therefore, any of the above two variance

measures can be used equally. Furthermore, we have a combined specification for

variance using GJR-GARCH (1,1) extended with the lagged IV as

.21imp,1,t1t1t

21t2

21t10t ��hd��h ��

(11)

This equation has a dummy variable to capture asymmetry. For instance, the dummy

variable 1td � is equal to 1 when 0� 1t �� and is equal to 0 otherwise. Therefore,

estimating an �%daily implied (VDAX) VaR forecast or the combined VaR forecast

using implied volatility (VDAX) plus GJR can be done with the following specification:

� � � � ,t

21

1 t,� h

2 ��VaR ��

��

(12)

90 A similar transformation scale is used by Blair, Poon, and Taylor (2001) for the VIX index.

119

where � is the standardized Student’s t-distribution function, the shape parameter

needs to be estimated and � is the quantile (which in our case is 99%, 97.5% or 95%).

4.3 Filtered Historical Simulation

As we know, historical simulation (bootstrapping) is a model-free approach that uses

the past historical returns of each asset in a portfolio in order to generate future

scenarios; therefore, a historical simulation (HS) VaR forecast is based on the empirical

distribution of asset returns, which is more realistic and economically appealing for

measuring a portfolio’s risk. Pritsker (2006) and Boudoukh et al. (1998) thoroughly

discussed the assumptions and limitations of HS, pointing out that HS has many

disadvantages. HS-VaR forecasting may be misleading; for instance, the asset returns

are not independently, identically distributed (iid), leading to the present fat-tailed

distribution with time-varying conditional moments and volatility clustering. Barone-

Adesi et al. (1998) extend the HS and propose Filtered Historical Simulation (FHS),

which overcomes limitations of the HS and is consistent with the empirically observed

features in the financial market data.91 FHS has a great advantage in that we do not

need to make any assumptions about the distribution in advance; instead, the past data

directly tell us about the distribution. Here, we fit a GARCH-type model to the data. We

based the conditional volatility model on the assumption that the model must produce

iid-standardized returns. Finally, we came up with the following GJR-GARCH (1,1)

model specification:

� �,ttt1tt 0,h~N�,��r �� (13)

.1t1t2

1t22

1t10t �hd��h �� (14)

This model has a dummy variable to capture asymmetry. For instance, the dummy

variable 1td � is equal to 1 when 0� 1t �� and equals 0 otherwise.92 However, for

forecasting using FHS-VaR, the i.i.d.-standardized residual returns are essential. The

residuals are divided by the corresponding daily conditional volatility obtained from

the GJR model:

91 FHS combines the characteristics of parametric and non-parametric methods, which account for leptokurtosis, time-varying volatility, asymmetry, and serial correlations. 92 The MA term is inserted to remove serial correlation in the returns series.

120

.t

t

h�

tz �

(15)

The daily FHS-VaR forecast is estimated by using the VaR specification:

� �* +,�z*PerchVaR n1iittt �� , (16)

where n represents the standardized residuals and � is the quantile (which in our case

is 99%, 97.5% or 95%).

4.4 RiskMetrics

RiskMetricsTM is a set of methodologies for measuring market risk (e.g., VaR)

developed by J.P. Morgan. The RiskMetrics model assumes that the daily return of a

portfolio follows a conditional normal distribution. It assigns greater weight to recent

observations and less weight to more distant observations. This declining weighting

scheme, through the assertion that volatility tends to change over time in a stable way,

might be more reasonable than assuming that it is constant (see e.g., Christoffersen,

2003). The specification of the RiskMetrics model is

,1t2

1tt �)h(1�rh �� (17)

where � is weight, which is usually fixed at 0.94 by the RiskMetrics group. The

RiskMetrics model’s forecast for one-day-ahead volatility at time t is therefore a

weighted average of volatility at time 1t � times the squared return at 1t � . Moreover,

this model has several advantages, some of which are worth mentioning here. First, it

accounts for time variations in the variance in a manner consistent with observed

returns. Second, recent observations are given more importance than the older

observations. Third, fewer observations are needed in order to forecast a one-day-

ahead variance. Finally, there is an agreement on the parameter value 94.0�, across

assets for one-day-ahead variance forecasting (for more details, see, e.g.,

Christoffersen, 2003). Nevertheless, we improve the RiskMetrics VaR forecast by

accommodating a heavy-tailed Student’s t-distribution function instead of a normal

distribution function. Hence, the daily %� RiskMetrics VaR forecast is estimated as

� � � � ,t

21

1 t,� h

2 ��VaR ��

��

(18)

121

where � is the standardized Student’s t-distribution function, the shape parameter

needs to be estimated, and � is the quantile (which in our case is 99%, 97.5% or 95%).

5 Backtesting Daily VaR Models

The current regulatory framework requires financial institutions with massive trading

activities to have enough capital, called a market capital requirement (MRC), in order

to cover excessive portfolio losses. This MRC requirement is determined in terms of the

portfolio VaR measure. The regulatory frameworks also necessitate that financial

institutions should use their own internal VaR models to compute and provide their

99% VaR. As a result, the MRC requirements are directly linked to both the portfolio

risk level and the internal VaR model’s performance on backtests. However, based on

VaR model forecasts, the regulatory body has increased (decreased) MRC requirements

(for more details on MRC, see Campbell, 2006). Consequently, the accuracy of internal

VaR models is of paramount importance to both regulators and financial institutions.93

When examining the accuracy of a VaR model, the significance of the backtesting

technique used increases. In this respect, Kupiec (1995) was the first to propose a test

of unconditional coverage; later, Christoffersen (1998) extended the Kupiec test to the

test of conditional coverage.

Let us assume we want to backtest a VaR model; to do that, a hit sequence needs to be

defined. For instance, if ex-ante VaR forecasts and ex-post returns are observed in a

time-series manner, then a hit sequence of VaR violations (or exceptions) are

represented with the following indicative function:

� � � ��

�-.

-/0

��

��

�VaRrif0,�VaRrif1,

�IPt

Pt

Pt

Pt

t

(19)

If a portfolio’s loss on day t is greater than the forecasted VaR for day t, then the hit

sequence reports 1, implying a VaR violation; otherwise, the hit sequence reports 0,

implying no VaR violation. In this way, the hit sequence � T1ttI � across T days is

constructed, indicating the violation rate of the VaR model. In fact, Christoffersen

93 For instance, if a VaR model indicates less risk, then the regulatory body could reduce MRC requirements, and the financial institution could then use that capital for other financial activities. This is why accurate VaR forecasting is very important for a financial institution.

122

(1998) suggests that the VaR model is adequate when its ‘hit sequence’ satisfies both

unconditional coverage and independence properties.94

The unconditional coverage property: The probability that an ex-post loss exceeds VaR

forecasts must be exactly equal to the coverage rate.

The independence coverage property: The VaR violations observed at two different

periods must be independently distributed over time. The observed VaR violations do

not carry information to forecast current and future VaR violations, and this property

also holds for any other variable in the information set, for instance, past returns, past

VaR levels, etc. Therefore, the null hypothesis of the unconditional coverage property

that * + PIE t � can formally be tested using the log-likelihood ratio test is

� �� ,~�PP12lnTX

TX12lnLR 2

1XXT

XXT

UC�

�

��

��

��

��

��

��

(20)

which is asymptotically 2� -distributed with one degree of freedom. P is the target

coverage rate (i.e., 1% for a 99% VaR), T is the total number of observations, and X is

the number of violations. Unfortunately, the unconditional coverage test does not

account for the clustering of violations.95 To overcome a clustering problem, the

independence coverage property must be satisfied, thereby leading to the correct

conditional coverage test proposed by Christoffersen (1998): joint tests for both

independence and unconditional coverage tests.96 The likelihood ratio tests for these

tests are

� � � �� 21

TTTTT1

T1

T0

T0IND ~�

TX

TX12ln��1��12lnLR

11011000

11010100

��

��

��

��

��

��

��

, (21)

,INDUCCC LRLRLR �� (22)

94 Christoffersen (2009) thoroughly reviews backtesting techniques. 95 This is because the unconditional coverage test has a major problem in that it does not account for the clustering of ones or zeros in a time-dependent fashion. For instance, if a value of 1% gave exactly 1% violations, but all of these violations occurred during one-month period, then the probability of bankruptcy of a financial institution would be much higher than if the VaR violations are scattered over a longer backtesting period, such as a one or two-year period. 96 For some technical details, see, for instance, Christoffersen (1998) and Campbell (2005).

123

where ijT is the number of observations, with value i followed by j , for 0,1i,j � ;

�� ijjijij TT� are the corresponding probabilities; 1ij � indicates that a violation

has occurred; and 0ij � indicates the opposite. The CCLR statistics are asymptotically

2� distributed with two degrees of freedom. The null hypothesis in the independence

test INDLR states that the probability of a violation on a given day does not depend on

the previous day’s violation.

An alternative method for evaluating VaR forecasts could be based on the use of loss

functions that are more aligned to the regulatory requirements. Lopez (1999) was the

first to propose a loss function for evaluating VaR forecasts, here called Lopez-I.

However, this loss function has a major drawback in that it ignores the magnitude of

tail losses. To remedy this drawback, he himself proposed a second loss function called

Lopez-II. In practice, Lopez-II has a size-adjusted loss function that makes it hard to

interpret in monetary terms because it employs squared monetary returns.97 Later,

Dowd (2005) proposed a size-based loss function that avoids terms in the denominator

and the squared term in Lopez-II; therefore, we prefer Dowd’s size-based function over

the others. It takes the following form:

./0

(�

�,tt

tttt VaRLif0

VaRLifLC

(23)

where tL is the loss incurred over period t and tVaR is the forecasted VaR estimate for

that period. We then calculated the expected tail loss ET by using tC , and ET was

used as a benchmark. Finally, the quadratic score function takes the following form:

� � .2n

1ttt ETC

n2QS �

�

��

(24)

This function has advantages over others because it punishes deviations of the tail

losses from the expected value and because of its quadratic form; it gives very high tail

losses much greater weight than normal tail losses.

97 See Dowd (2005) for detailed discussions on different loss functions.

124

6 Results

6.1 Conditional Variances

Figure 3 provides a comparison of the conditional variances for implied volatility

(VDAX), GJR plus implied volatility (VDAX), GJR-GARCH (1, 1), and RiskMetrics

specifications and the squared returns that they forecast. The comparison period for

the five time series is from January 1, 1992 through May 29, 2009.

.0000

.0004

.0008

.0012

.0016

.0020

.0024

.0028

92 94 96 98 00 02 04 06 08

Implied(VDAX)

.0000

.0004

.0008

.0012

.0016

.0020

.0024

.0028

92 94 96 98 00 02 04 06 08

GJR-Implied(VDAX)

.0000

.0004

.0008

.0012

.0016

.0020

.0024

.0028

92 94 96 98 00 02 04 06 08

GJR

.0000

.0004

.0008

.0012

.0016

.0020

.0024

.0028

92 94 96 98 00 02 04 06 08

RiskMetrics

.000

.002

.004

.006

.008

.010

.012

92 94 96 98 00 02 04 06 08

Squared Returns

Figure 3. Implied, conditional variances and squared returns from January 01, 1992

to May 29, 2009.

125

It can be observed that the squared returns are very noisy and that the four variances

move closely together.

6.2 Backtesting Results

Table 2 provides backtesting results of VaR models for the DAX 30 stock index

portfolio from January 1, 1992 to May 29, 2009.98 Implied volatility (VDAX), implied

volatility–GJR, FHS (GJR) and RiskMetrics VaR models were statistically tested using

tests of unconditional coverage, independence, and conditional coverage and the

quadratic score test (using the expected tail loss) at different confidence levels, for

instance, 99%, 97.5% and 95%.99 There are three panels in Table 2. In rows 1, 2, and 3

of each panel, the results of the unconditional coverage, independence and conditional

coverage tests are provided, respectively. In rows 4, 5, 6, 7, and 8 of each panel, the

number of exceptions, the %VaR exceptions, the number of successive exceptions, the

% expected tail loss, and the quadratic score are presented, respectively.

First, we will discuss the backtesting results for our four specified VaR models at the

99% confidence level, which are provided in panel A of Table 2.100 Because the null

hypothesis of the unconditional coverage test states that the observed frequency of

exceptions should equal the frequency of expected exceptions, testing this is the first

step to take when considering any VaR model. As can be seen, the chi-square values for

the null hypothesis of the unconditional coverage tests at 10% significance cannot be

rejected for any of the four VaR models.101 Unfortunately, the unconditional test only

considers the frequency of exceptions, and it ignores the time-series independence of

those exceptions, i.e., the clustering of exceptions. As a result, the independence test is

of paramount importance for any VaR model. Conducting the independence test is the

second step to take when considering any VaR model.

98 All VaR models except FHS are estimated in conjunction with the Student’s t density function. 99 We do recognize that the daily VaR forecasts of the FHS model were based on using in-sample GJR-GARCH forecasts. However, a more appropriate way would be to estimate the daily FHS-VaR forecasts using out-of-sample GJR-GARCH model forecasts. However, this issue should not be of concent in the rest of the daily VaR models’ forecasts such as using Riskmetrics or Implied Voalatility forecasts. 100 The daily 99% VaR forecasts for each of the four VaR models are compared with the corresponding daily returns in a time-series fashion in Figure 4. 101 A chi-square critical value of 2.70 (for one degree of freedom) for a 10% significance level is used to test the null hypothesis that the frequency of observed exceptions is consistent with the frequency of expected exceptions.

126

Table 2. Backtesting results of VaR models for the DAX 30 from January 1, 1992 to May 29, 2009.

Panel A: 99% VaR-t forecasts

Model Implied (VDAX) Implied (VDAX)- FHS (GJR) RiskMetrics

LRU 1.299692 0.663502 0.919916 2.317618

LRI - -6.763572 2.115715 1.698050

LRCC - -6.100069 3.035631 4.015668

No. of Exceptions 38 51 52 56

%VaR Exceptions 0.843507 1.122606 1.144618 1.232666

No. Succ. Exceptions 0 2 2 2

Expected Tail Loss (%) -4.323369 -3.5817592 -3.71932664 -3.48212508

Quadratic Score 0.000371 0.000594 0.000635 0.000740

Panel B: 97.5% VaR-t forecasts

Model Implied(VDAX) Implied(VDAX)-GJR FHS(GJR) RiskMetrics

LRU 0.285262 0.018263 0.781421 3.912869

LRI -6.094480 -6.926530 -7.03222 -6.738680

LRCC -5.809210 -6.908270 -6.25080 -2.825810


%VaR Exceptions 2.435175 2.531367 2.707462 2.971605



Quadratic Score 0.000897 0.001082 0.001288 0.001403

Panel C: 95% VaR-t forecasts

Model Implied(VDAX) Implied(VDAX)-GJR FHS(GJR) RiskMetrics

LRU 0.123797 1.605154 1.786875 8.434941

LRI -4.187958 -1.675680 1.828643 0.047557

LRCC -4.064160 -0.070530 3.615519 8.482498


%VaR Exceptions 5.137699 5.414924 5.436936 5.965221



Quadratic Score 0.001844 0.002288 0.002334 0.002613

Boldface indicates that we cannot reject the null hypothesis at 10% significance level.

127

The chi-square values at the 10% significance level for the independence tests confirm

that the null hypothesis of the first-order independence of exceptions is not rejected for

the three specified VaR models, i.e., implied volatility (VDAX)-GJR, FHS (GJR), and

RiskMetrics; however, this test could not be conducted for the implied volatility

(VDAX) model because there are no successive exceptions implying that the model is

correct. Finally, the conditional coverage test that needs to be tested in order to endorse

any VaR model is the joint null hypothesis that the VaR model has an adequate

frequency of independence exceptions. This test also provides important information

regarding the likelihood of the successive exceptions and the average number of days

between exceptions; this is the third and final step for a VaR model to be considered for

use. As can be observed, the chi-square values at the 10% significance level for the joint

null hypothesis tests verify that the three VaR models have an adequate frequency of

independence exceptions; as a result, the joint null hypothesis could not be rejected.

Similarly, for the implied volatility (VDAX) model, the conditional coverage test could

not be conducted because there are no successive exceptions implying that this model is

correct under the conditional coverage test.102

For comparison, the backtesting results for the VaR models at 97.5% and 95%

confidence levels are provided in panels B and C of Table 2.103 As can be seen, the chi-

square values for the null hypothesis of the unconditional coverage tests at the 10%

significance level also could not be rejected for three of the VaR models. However, in

the case of the RiskMetrics model, we found a higher unconditional coverage than

expected, leading to a rejection of the null hypothesis of correct unconditional coverage

at the 10% significance level. In comparison, the chi-square values at 10% significance

for the independence tests show that the null hypothesis of the independence of the

exceptions was not rejected for our VaR models. As can be observed, for the conditional

coverage test, the chi-square values at the 10% significance level for the joint null

hypothesis tests verify that the four VaR models have an adequate frequency of

independence exceptions; therefore, the joint null hypothesis could not be rejected.

However, the null hypothesis for the conditional coverage test was rejected at the 10%

significance level for the RiskMetrics model (at the 95% confidence level).

102 Chi-square critical value of 4.60 (for two degree of freedom) for 10% significance level is used to test the joint null hypothesis of the conditional coverage test. 103 The daily 97.5% and 95% VaR forecasts for each of the four specified VaR models are compared with corresponding daily returns in a time-series fashion in Figure 5, and 6, respectively.

128

In order to select a model, we can also use estimated quadratic scores for each VaR

model at different confidence levels such as 99%, 97.5% and 95%. However, the

quadratic scores are estimated using a loss function proposed by Dowd (2005), which

states that a lower minimum score level indicates a better VaR model. The quadratic

scores are presented in the last rows of each panel of Table 2. If we compare the

quadratic scores for each of the VaR models at the 99% confidence level, the rank of the

VaR models would be, from better to worse. implied volatility (VDAX), implied

volatility (VDAX)-GJR, FHS (GJR), and RiskMetrics. Alternatively, in a more

traditional way, one could also select a VaR model by evaluating it based on the number

of exceptions and the number of successive exceptions it has previously made. By these

criteria, the VaR models’ rank is similar to the rank we determined from the quadratic

score estimates. In addition, ranks for the VaR models at lower confidence levels

(97.5% and 95%) support the quadratic score estimates of the VaR models at the 99%

confidence level.

We conclude from the results presented in Table 2 that the information content of IV is

superior to that of HV, which is consistent with the earlier research on the information

content (e.g., Day and Lewis, 1992; Christensen and Prabhala, 1998; Fleming, 1998;

Dumas, Fleming and Whaley,1998; Blair et al., 2001; Ederington and Guan, 2002;

Poon and Granger, 2003; Mayhew and Stivers, 2003; Martens and Zein; 2004; and

Giot, 2005). The unconditional coverage, independence, and conditional coverage tests

and the quadratic scores suggest that the VDAX and a combination of the VDAX with

GJR-GARCH (1,1) VaR models always outperform filtered historical simulation (FHS)

and RiskMetrics models during our sample period from January 1, 1992 through May

29, 2009.

129

7 Conclusion

This essay examined the information content of the newly adopted VDAX (MFIV) index

for the daily VaR forecasts. The information content of the new VDAX was incorporated

into daily VaR forecasts and compared with the VaR forecasts from the FHS (GJR) and

the RiskMetrics models at various confidence levels (i.e., 99%, 97.5% and 95%), using

unconditional coverage, independence, and conditional coverage tests for backtesting

each of the VaR models. Furthermore, a quadratic score was estimated for each VaR

model for the period from January 1, 1992 through May 29, 2009. The backtesting

results showed that the new VDAX index contains significant information about actual

volatility in VaR models. The null hypotheses of independence and conditional

coverage backtests were never rejected for implied volatility, implied volatility plus

GJR, and FHS (GJR) VaR models. The number of VaR exceptions was not significantly

different from the set coverage rates. However, the null hypotheses of the RiskMetrics

model were rejected for lower confidence levels. It was also found that implied volatility

and implied volatility-GJR VaR models presented the fewest VaR exceptions and

clusters of exceptions, in contrast to the FHS (GJR) and RiskMetrics models. On the

other hand, the quadratic score for each model suggests the following ranking of VaR

models: implied volatility (VDAX), combined (implied volatility plus GJR), FHS, and

RiskMetrics. Our findings have implications for traders who hold long positions, risk

managers (internal), and regulators (external).

130

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135

Modeling Risk Factors Driving the EUR, USD, and GBP Swaption Volatilities

Ihsan Ullah Badshah



September 20, 2010

Abstract The purpose of this essay is twofold: First, we use principal component analysis to identify the common implied risk factors affecting implied volatility (IV) movements in the EUR, USD and GBP swaption markets. We then examine the dynamic interactions between the implied factors (i.e., factors in each of the EUR, USD, and GBP swaption IVs) by using techniques such as Granger causality and the generalized impulse response function (IRF). Second, we calibrate the string market model (SMM) for each of the swaption markets using multivariable nonlinear optimization in order to reproduce the swaption volatility matrix. We find that the first three implied factors explain 94 -97% of the variance in each of the EUR, USD and GBP swaption IVs. All of the significant factors present high degrees of correlation across swaption markets. Consequently, there are significant linkages across the factors and bi-directional causality is at work between the factors implied by EUR and USD swaption IVs. Furthermore, in innovation accounting investigations, the factors implied by the EUR and USD swaption IVs respond to each others’ shocks, and the factors implied by the GBP IVs respond to their shocks as well. However, a shock to the GBP factors does not affect their factors. On the other hand, calibration results show that the SMM can efficiently reproduce the whole swaption volatility matrix for each of the EUR, USD, and GBP markets. We obtain optimal solutions for the EUR, GBP and USD markets. Finally, we find similar characteristics between the EUR and the GBP swaption markets, but different characteristics in the USD swaption market. The implications of these results are important for the pricing of interest rate derivatives, Vega-hedging, risk management, and policy making. Keywords: calibration, implied volatility, linkages, principal component analysis, swaption JEL Classification: G12, G13, E4 The Author would like to thank Johan Knif, Mika Vaihekoski, George Skiadopoulos, Seppo Pynnonen, Gregory Koutmos, Kenneth Högholm, James Kolari, and Jussi Nikkinen for providing useful comments. The author would also like to thank to the participants of the Graduate School of Finance (GSF) Research workshops for their useful comments. The author acknowledges Evald & Hilda Nissi and HANKEN foundations for providing financial support.

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1 Introduction

The interest rate (IR) derivatives market has grown exponentially since the trading of

the first IR swap security gave birth to many complex (exotic) derivatives products

afterwards. The 1990s, in particular, saw phenomenal pace in the introduction of new

types of products. Besides the plain-vanilla derivatives, there were many other products

such as Bermudan swaptions, indexed-principal swaps knockout caps, ratchet caps,

callable inverse floaters, index accruals and many more (see e.g., Rebonato, 2002). In

fact, according to the International Swaps and Derivatives Association (ISDA), the IR

derivatives market is the largest derivatives market in the world. ISDA reported that

the notional amount of IR derivatives outstanding at the end of 2007 was 382 trillion

US dollars. Another detailed report was published by Bank of International Settlements

(BIS) at the end of 2007, showing that euro-denominated IR derivatives form a

substantial amount of this proportion (about $146 trillion), in addition to US dollar

(about $129 trillion), Japanese yen (about $53 trillion), and UK pound sterling (about

28 trillion). Thus, IR derivatives saw phenomenal growth in outstanding notional

amounts and consequently, huge development of different products and models to

support traders. Hence, this area has attracted many professionals from finance,

physics, mathematics and statistics.

However, IR markets present a high degree of correlation—a high dependence among

the market risk factors. When a few important sources of information are common to

market risk factors, there is a high degree of correlation among the risk factors. For

traders required to price and hedge huge portfolios, such portfolios might consist of

different IR securities and depend on hundreds of underlying risk factors. In order to

enhance efficiency and achieve parsimony, dimensionality reduction is necessary and

can be achieved through principal component analysis (PCA). PCA is a technique that

extracts the significant underlying risk factors from correlated IR securities (see e.g.,

London, 2004). Similarly, swaption implied volatilities (IVs) can be viewed as highly

correlated and can be explained by only two to three independent factors. The risk

factors implied by IVs could be used for a variety of purposes: vega-hedging, generating

implied volatility surface (IVS), model calibration, and risk management. For example,

plain-vanilla options are actively traded (i.e., in hundreds) with a variety of maturities

in order to estimate sensitivities of a portfolio with respect to each of these sources of

risk--the sensitivities can be consistently computed by using the significant principal

137

components (PCs).104 Second, the IVS can be projected onto the PCs. IVS are important

for traders in quoting options for thinly traded over-the-counter (OTC) market. Third,

the pricing model can be calibrated using only a few important PCs instead of a bunch

of underlying securities.105 Finally, the value-at-risk for the IR portfolios can be

estimated using the PCs. There is an extensive literature that identifies the common

PCs in term structure movements (see e.g., Litterman and Scheinkman, 1991; Knez,

Litterman, and Scheinkman, 1994; and a recent study on the swap curve dynamics by

Huang et al.,2008). However, besides Wadhwa (1999) we have not found any other

study identifying the PCs affecting volatility movements in the IR swaption markets.

Wadhwa (1999) studied USD LIBOR IVs movements, who found that four significant

PCs explained a large proportion of the total variation in the cap and swaption IVs

while using daily data from March 1992 to March 1999. Further, it was found that the

first PC was stable over time and never changed its shape, and the PCs beyond the first

PC were unstable. Bruggemann et al. (2008) modeled and analyzed the stochastic

properties of factors of IVs (i.e., implied from the options on the German stock market)

in a VAR framework. The VAR framework nicely described the dynamic linkages

between factors that determine the movements of the IVs. It was suggested that the

interaction between the first two factors (representing level and maturity slope) carries

useful risk management information. The relationship of the factors to movements in

economic variables was also investigated, and it was found that factor loadings are

linked to U.S. stock market returns and volatility. U.S. stock market returns and

volatility affect German stock market IV and the effect was found to be bi-directional.

On the other hand, let us consider the market models and their calibration procedures.

For instance, a trader of vanilla options (such as swaptions and caps) would expect his

model to price the required hedges (swaps and bonds) for each trade accurately and in

line with the vanilla options market. Equally, the exotic trader would expect his model

to price hedges (swaptions and caplets) in line with the market.106 A pricing model is

then calibrated to the vanilla market, facilitating to determine the price of the exotic

product and hedging positions of the exotic product.107 Thus, for accurate and efficient

pricing of exotic products, it is of paramount importance to find a parsimonious model

that accurately matches the market quoted prices. In this respect, Heath, Jarrow and

104 We use risk factors and PCs equivalently throughout this essay. 105 A good example is the LIBOR market model. 106 For detail on the calibration procedures, see Rebonato (2002). 107 Since quoted prices of the plain-vanilla products are available in the market, calibrating the model to the market prices allows us to extract information about the distribution of the underlying interest rates.

138

Morton (1992) (HJM) were the first to develop a model on the firm ground of

mathematical finance; this model is a framework for the continuously compounded

instantaneous forward rate (FR) models. However, this tractable model does not

guarantee that forward LIBOR rates have lognormal volatility structures. Therefore, the

HJM models conflict with the lognormal Black (1976) model commonly used in the

market. Later, Brace, Gatarek and Musiela (1997) (BGM) developed the LIBOR market

model (LMM) to overcome this shortcoming and were thereby consistent with both the

Black (1976) model and the HJM framework. LMM models the dynamics of observable

forward LIBOR rates directly, in contrast to instantaneous FRs. However, these models

share many of the same calibration issues and can be implemented with a small

number of factors. In practice, models require frequent recalibration, i.e., the constant

parameters of the model are often required to change in order to suit current market

conditions. The recalibration of the above models may ignore some hidden sources of

risks, because the models do not account for the dynamics of every FR. The string

market model (SMM) accounts for all hidden sources of risks in the market. The

stochastic string models were first developed by Kennedy (1994, 1997), who modeled

the evolution of the term structure of FRs as a stochastic string. Later, Goldstein

(2000), Longstaff and Schwartz (2001), Santa-Clara and Sornette (2001) and Longstaff

et al. (2001) developed similar models that allow for correlated strings of shocks to the

FR curve. The dynamics of each FR are modeled in the SMM model. Moreover, it

accounts for the correlations along each point of term structure. This model is also

called high dimensional (continuum of FRs). The calibration of the SMM model is more

straightforward than that of models with fewer factors (i.e., the LIBOR market model).

The parameters of the SMM model are directly estimated from the covariance matrix

and the parameterization of the SMM becomes more advantageous and parsimonious

than the LMM as the number of factors driving the term structure of FRs increases (see

e.g., Longstaff et al., 2001).

This essay has two purposes. First, we aim to examine the dynamics of swaption IVs for

each of the EUR, USD and GBP markets. Our goal is to characterize the number of

significant implied PCs required to explain the whole swaptions maturity surface and to

examine the dynamic linkages of the factor loadings implied from the three swaptions

markets. Second, we aim to calibrate the SMM model to the swaptions markets in a

discrete fashion, thereby reproducing the whole swaption volatility matrix for each the

139

EUR, USD, and GBP swaption markets.108 Thus, this study is related to that of Wadhwa

(1999) in terms of PCA and USD cap/swaptions, whereas the second part of the essay is

related to that of Longstaff et al. (2001) and Kerkhof and Pelsser (2002), who

calibrated the SMM to USD cap/swaption data. However, our study differs in several

important respects. First, we apply PCA to the EUR, USD and GBP swaption IVs,

whereas Wadhwa (1999) studied only the dynamics of USD cap/swaptions. Moreover,

we compare the significant PCs implied from the three markets and examine their

dynamic linkages. Second, we calibrate the SMM to the swaptions matrix for each of

the EUR, USD, and GBP swaption markets using a nonlinear multivariable

optimization, whereas the studies of Longstaff et al. (2001) and Kerkhof and Pelsser

(2002) only calibrated to the USD swaption data.

The main findings are that only three PCs explain 94-97% of the variations in each of

the EUR, USD, and GBP swaption IVs. The PC1s (which stand for the parallel shifts) are

found to be highly correlated across the three markets. The remaining PC2s and PC3s

are also correlated. As a result, we find considerable linkages across the swaption

markets. Bi-directional causality is at work between the EUR and USD factors. In

innovation accounting investigations, unit shocks to the EUR_PC1 and USD_PC1 have

significant influences on each other. Both influence GBP_PC1 as well, but not vice versa

(thus GBP_PC1 is found to be the least influential among the PC1s). Similar patterns

are found for the PC2s. On the other hand, after calibration of the SMM model to

swaption matrices, each of the three markets provides different results. The EUR

swaption optimization requires 1446 iterations (and 2000 function evaluations) to

obtain an optimal solution, the USD optimization requires 1433 iterations (and 2000

function evaluations), and the GBP optimization requires the least iterations among the

three optimization process: 1429 iterations (and 2000 function evaluations). Thereby

the smallest level of RMSE is observed for the EUR, followed by the GBP and USD

markets. Our results imply that the EUR and GBP swaption IVs contain identical

characteristics both for the PCA and calibration processes, but are somewhat different

from those of the USD market.

This essay is organized as follows. Section 2 discusses the swaptions market. Section 3

discusses the data. Section 4 examines factor loadings dynamics in a VAR framework.

108 Note that we use swaption volatility surface and swaption volatility matrix equivalently throughout this essay.

140

Section 5 calibrates the SMM model for each of the EUR, USD, and GBP swaption

markets. Section 6 summarizes and concludes.

2 The Swaptions Market

A swaption is an OTC-traded option that grants its owner the right to enter the

underlying swap contract, where the contract can be traded on a variety of IR swap

securities. There are two types of swaptions. In one type, the payer swaption, the owner

of the swaption is granted the right to enter into the IR swap, where he receives the

floating rate and pays the fixed rate. The second type of swaption is the receiver

swaption, which grants the owner of the swaption the right to enter into the IR swap

where he pays the floating rate and receive the fixed rate. Therefore, a swaption can be

viewed as call or put on coupon bonds.109 The market makers for the swaptions in the

major currencies (i.e., US dollar, euro, Japanese yen, and UK pound) are the big

investment and commercial banks. Whereas, the main participants in the swaption

markets are banks, firms, hedge funds, and other financial institutions. Banks and

firms normally use swaptions to manage IR risk arising from their borrowings or

businesses. For instance, if a bank holds a mortgage portfolio and wants to safeguard

against lower IRs (as lower IR will lead to early prepayment of the mortgages), the bank

may buy a receiver swaption. Likewise, a firm may want to manage the risk of higher

IRs by buying a payer swaption. A hedge fund that expects that the future IR will rise by

a certain percentage will sell a payer swaption, earning money in the form of swaption

premium. However, in practice, swaptions are quoted in terms of at-the-money (ATM)

IVs while using the Black (1976) model, the practitioner’s standard swaption model.

European swaptions are priced using the forward swap rate as an input to the Black

(1976) model. Jamshidian (1996) showed that the Black model is arbitrage-free under

the assumption of a lognormal swap rate.

� � * + ,KN(d2)FN(d1)eF

F/n)(111Payer rTtxn

��

��

� ��

(1)

� � * + ,KN(d1)d2)KN(eF

F/n)(111Receiver rTtxn

��

��

� ��

(2)

where

109 For more discussion see for instance, Longstaff et al. (2001).

141

,T�

/2)T(�ln(F/K)d2

1�

�

(3)

.T�dd 12 �� (4)

F is the forward swap rate, K is the strike rate of the swaption, t is the tenor of the swap,

� is the volatility of the forward swap rate, n is the yearly compounding of the swap

rate, and T is the time to expiration. Thus, swaption IVs can be obtained by inverting

the Black (1976) formula as shown above to obtain the IVs associated with the selected

pair of maturity and tenor.

3 Data

We use EUR-, USD- and GBP-denominated IR datasets for the analysis, as well as data

on money market rates and swap rates, and Black (1976)-IVs of swaptions.110 We have

weekly data on money market rates with maturities of 1, 3, 6, 9 and 12 months, and

data on EUR, USD, and GBP swap rates with maturities from 2 years up to 15 years.

The IR datasets are used to construct the FR curve and then the bootstrap technique is

used to arrive at the discount factors. This is essential for calibrating the SMM model

later on. On the other hand, the second type of datasets is the time series ATM Black-

IVs using weekly data backed out of swaptions for expiries of 3 months, 6 months and

1-8 years, and struck on the underlying swaps of 1-10 years maturities. The time series

data of swaption IVs for the EUR, USD and GBP markets from April 1, 2006 through

October 31, 2008.

110 Data are provided by ICAP, London, United Kingdom.

142

4 Factor Loading Dynamics

4.1 Factor Loadings

We apply PCA to reduce the swaptions maturity surface to only a few risk factors and

model the implied dynamics in a vector-autoregressive (VAR) framework (i.e., for the

EUR, USD, and GBP swaption markets). The rationale is to extract the independent

risk factors that explain most of the dynamics of the swaption IVs and to examine the

dynamic linkages across markets. Previously many researchers attempted to extract

those risk factors that explains most the dynamics in the volatility smirk (skew) or

surfaces for equity index options; for FTSE 100 index IV by Badshah (2008), and Cont

and da Fonseca (2002); for S&P 500 index IV, we consider Cont and da Fonseca

(2002), and Skiadopoulos et al. (1999); for DAX 30 index IV, Fengler et al. (2003),

whose findings confirmed that about 70-90% of the total variation in the surface or

smirk (skew) can be attributed to only three PCs. A closer study to ours is of

Bruggemann et al. (2008) who analyzed and modeled the stochastic properties of the

factors of IVs (i.e., obtained from options on German stock market) in a VAR

framework and found significant interactions between the factor loadings.

Thus, significant PCs are extracted for each of the EUR, USD, and GBP swaption IVs,

and their implied linkages across markets are examined in a VAR framework: 111 First,

weekly time-series data from April 1, 2006 to October 31, 2008 are assembled into

three groups: EUR, USD, and GPB. Then, the swaption IVs are incorporated as follows:

3M×1Y, 3M×2Y,…, 3M×10Y; 6M×1Y, 6M×2Y,…,6M×10Y; 9M×1Y, 9M×2Y,…,9M×10Y;

1Y×1Y, 1Y×2Y,…,1Y×10Y and so on up to 8Y×1Y, 8Y×2Y,…,8Y×10Y.112 Second, we

compute the first log-difference IVs across different expiries and time-to-maturities.

Finally, we extract the significant PCs from swaption IVs for each market.

PCA is a method of matrix decomposition into eigenvectors and eigenvalue matrices. It

is applied to a pool of ATM swaption IVs across different expiries and maturities,

effectively decomposing the covariance matrix as T�� , where the diagonal elements

of � are the eigenvalues and the columns � are the associated eigenvectors. The

choice of column labeling in � allows PC ordering such that 1e belongs to the largest

111 Bruggemann et al., 2008 used a similar VAR model for factor loading dynamics of IVs implied from the German option market. 112 M symbolizes month, while Y symbolizes year.

143

eigenvalue 1� , 2e belongs to second largest eigenvalue 2� and so forth. In highly

correlated IVs, the first eigenvalue would be much larger than the others.

Consequently, the first PC can explain much of the variation in the swaption IVs. If the

first three PCs explain most of the variations in the swaption IVs, then these PCs can

replace the original volatility variables without loss of much information (see e.g.,

Skiadopoulos et al.,1999; London, 2004). That is, the original volatility input data may

be written as a linear combination of the PCs, which reduces the dimensions of the

system.

Table 1 provides results for the PCA on the swaption IVs for the swaption markets

considered from April 1, 2006 through October 31, 2008. Each panel of Table 1

presents the cumulative proportion of variance explained by the first eight PCs.

Looking at the eigenvalues, we find that three PCs are significant, i.e., with eigenvalues

greater than 1, in EUR swaption IVs. The eigenvalues in the fourth PC in the USD and

GBP swaption IVs are also greater than 1. The proportions of the fourth PC that

contribute to the cumulative values are negligible, implying that only three PCs are

sufficient to capture much of the systematic variations for each of the three swaption

markets.113

The interpretations of the PCs are intuitively similar to those of Longstaff et al. (2001)

and Wadhwa (1999). The first PC generates parallel shifts in the term structure for

swaption IVs and captures about 86%, 88% and 80% for EUR, USD, and GBP,

respectively. The second PC that generates shifts in the slope of the term structure of

IVs captures an additional 9%, 6%, and 12% for EUR, USD, and GBP, respectively.

Finally, the third PC is the curvature factor, which generates movements in the term

structure of the IVs such that short-term and long-term IVs move in opposite directions

from the mid-term IVs and account for 1.5%, 2%, and 2% for EUR, USD, and GBP,

respectively. Thus, the first three PCs cumulatively explain about 96%, 97%, and 94% of

the variation in the IVs for each the EUR, USD and GBP swaption markets.

113 Here our results are consistent with the results of Wadhwa (1999), who found three significant PCs in the n .

144

Table 1. Principal Component Analysis of EUR, USD, and GBP swaption IVs

Panel A: EUR Swaption IVs

Number Eigenvalues Difference Proportion Cumulative Cumulative Prop.

1 86.09613 77.15441 0.8610 86.09613 0.8610

2 8.941718 7.387048 0.0894 95.03785 0.9504

3 1.554670 0.684989 0.0155 96.59252 0.9659

4 0.869682 0.350749 0.0087 97.46220 0.9746

5 0.518933 0.227666 0.0052 97.98113 0.9798

6 0.291267 0.014989 0.0029 98.27240 0.9827

7 0.276279 0.110757 0.0028 98.54868 0.9855

8 0.165522 0.038358 0.0017 98.71420 0.9871

Panel B: USD Swaption IVs


1 88.43159 82.02862 0.8843 88.43159 0.8843

2 6.402971 4.413318 0.0640 94.83456 0.9483

3 1.989653 0.828376 0.0199 96.82421 0.9682

4 1.161277 0.756448 0.0116 97.98549 0.9799

5 0.404829 0.150173 0.0040 98.39032 0.9839

6 0.254655 0.039919 0.0025 98.64498 0.9864

7 0.214736 0.050626 0.0021 98.85971 0.9886

8 0.164110 0.037694 0.0016 99.02382 0.9902

Panel C: GBP Swaption IVs


1 80.28617 67.94028 0.8029 80.28617 0.8029

2 12.34590 10.36036 0.1235 92.63207 0.9263

3 1.985536 0.916064 0.0199 94.61760 0.9462

4 1.069472 0.284469 0.0107 95.68708 0.9569

5 0.785003 0.235437 0.0079 96.47208 0.9647

6 0.549566 0.269289 0.0055 97.02165 0.9702

7 0.280277 0.034339 0.0028 97.30192 0.9730

8 0.245938 0.054895 0.0025 97.54786 0.9755

In sum Table 1 shows that the first three PCs cumulatively account for about 94-97% of

the total variation in each of the EUR, USD, and GBP swaption IVs. The parallel shifts

are evident in the USD, followed by the EUR and GBP swaption IVs, respectively. This

145

implies that the participants in the EUR swaption market are more concerned about

parallel shifts than about the other two underlying risk factors. The remaining two risk

factors are more important for the GBP and EUR market participants, respectively.

Figure 1 reports factor loadings for the first three PCs extracted from the time series

data for each the EUR, USD, and GBP swaption IVs from April 1, 2006 to October 31,

2008. The x-axis contains variables (i.e., IVs for a variety of maturities for instance

1M*1Y… 8Y*10Y), whereas the y-axis contains loadings for the three PCs.114 The PC1s

that correspond to parallel shifts in the swaption IVs are more stable and carry positive

weights throughout the sample period. PC1s are important for the shorter and medium

maturity swaptions, which are increasing, while the longer maturity swaptions remain

more stable and thus show a decreasing effect. Imply that the parallel shifts are

increasing and an important factor for the participants in the swaption markets during

the sample period is influential for all swaptions. In particular, short and medium

maturity swaptions are the most sensitive to parallel shifts. On the other hand, the

PC2s correspond to shifts in the slope of the swaption IVs. Here, the PC2s carry positive

weights for short maturity swaptions and gradually enter into negative territory,

moving downward as the term increases. This implies that the PC2s have generated

positive shifts in the term structure slope of shorter maturity swaption IVs and negative

shifts in the term structure slope of longer maturity swaption IVs. Finally, the PC3s are

viewed as the curvature factor that essentially influences swaption IVs. They carry

positive weights for the very short and very long maturity swaption IVs, and show

negative jumps for the medium term swaption IVs.

Figure 1 provides interesting results. EUR and GBP swaption IVs are affected by the

first three PCs and show somewhat similar characteristics, while the USD swaption IVs

are heavily affected by the first PC (i.e., parallel shifts are important for USD

swaptions) and to some extent by the second PC. Market participants in the USD

swaption market are concerned with short-term future changes in the interest rates,

whereas the additional two factors are essential for EUR and GBP markets.

114 About 100 variables are included in the principal component analysis.

146

.07

.08

.09

.10

.11

10 20 30 40 50 60 70 80 90 100

EUR_PC1 USD_PC1 GBP_PC1

-.12

-.08

-.04

.00

.04

.08

.12

.16

.20

.24

10 20 30 40 50 60 70 80 90 100


-.3

-.2

-.1

.0

.1

.2

.3

10 20 30 40 50 60 70 80 90 100


Figure 1. Factor loadings on the first three PCs for the EUR, USD, and GBP Swaption

IVs.

147

4.2 Correlation Analysis

Correlation analysis helps to quantify the extent to which important PCs are correlated

in and across markets. Table 2 presents the correlation matrix for the EUR, USD, and

GBP swaption PCs.

Table 2. Correlation matrix for the three PCs of the EUR, USD, and GBP swaption EUR USD GBP

PC1 PC2 PC3 PC1 PC2 PC3 PC1 PC2 PC3

EUR PC1 1

PC2 -0.770 [-11.9] (0.00)

1

PC3 -0.221 [-2.24] (0.03)

-0.001 [-0.01] (0.99)

1

USD PC1 0.969 [39.23] (0.00)

-0.690 [-9.43] (0.00)

-0.231 [-2.34] (0.02)

1

PC2 -0.760 [-11.5] (0.00)

0.993 [85.49] (0.00)

-0.005 [-0.05] (0.959)

-0.679 [-9.16] (0.00)

1

PC3 -0.414 [-4.50] (0.00)

0.0212 [0.20] (0.834)

0.916 [22.59] (0.000)

-0.441 [-4.86] (0.00)

-0.001 [-0.00] (0.99)

1

GBP PC1 0.969 [38.9] (0.00)

-0.721 [-10.29] (0.00)

-0.1975 [-1.99] 0.0488

0.952 [30.81] (0.00)

-0.698 [-9.65] (0.00)

-0.427 [-4.67] (0.00)

1

PC2 -0.721 [-10.2] (0.00)

0.991 [74.99] (0.00)

-0.0627 [-0.62] (0.53)

-0.642 [-8.28] (0.000)

0.992 [76.10] (0.00)

-0.052 [-0.51] (0.61)

-0.6677 [-8.88] (0.00)

1

PC3 -0.265 [-2.71] (0.00)

0.0551 [0.54] (0.58)

0.950 [30.21] (0.00)

-0.271 [-2.78] (0.00)

0.048 [0.47] (0.63)

0.8756 [17.94] (0.00)

-0.225 [-2.28] (0.02)

-0.001 [-0.01] (0.99)

1

In the square brackets t-statistics whereas in parenthesis probability values.

As can be seen from the correlation coefficients for the PC1s (which generate parallel

shifts in swaption IVs), the three swaption markets are highly correlated and

148

statistically significant. For example, the correlation between the PC1s for the EUR and

USD swaption IVs is positive, with the corresponding coefficient high at 0.97. A

similarly high correlation coefficient is found between the EUR and the GBP. A

correlation coefficient of 0.95 is observed between the USD and the GBP. Second, PC2s

that generate shifts in the slope of the swaption IVs are negative, correlations with the

PC1s ranging from a maximum of -0.72 to a minimum of -0.69. Third, the PC3

curvature factors in the swaption IVs. As can be observed, the PC3s are negative,

correlations with PC1s ranging from a maximum of -0.43 to a minimum of -0.23.

Finally, we found that PC2s and PC3s are uncorrelated for the three considered

swaption markets, and that the corresponding coefficients are not significant even at

the 5% level.

The conclusion drawn from the results in Table 2 is that that implied PC1s for the three

swaption markets are highly positively correlated. However, PC2s and PC3s are

negatively correlated with PC1s. The highest correlation is between the USD and the

EUR, whereas the USD and the GBP are the least correlated among the three swaption

markets.

149

4.3 VAR Model for Factor Loading Dynamics

The VAR model is used to investigate the dynamic interactions between the first two

PCs (i.e., PC1s, PC2s extracted from each of the EUR, USD, and GBP Swaption IVs).115

The VAR model was developed by Sims (1980) that successfully treats each endogenous

variable in a system as a function of the lagged values of all endogenous variables in the

dynamic simultaneous equation system. Hence, the VAR model for the factor loadings

dynamics can be represented mathematically as:

,uF��F t

L

1iitit ��

��

(5)

where ),z,z,z,z,z(zF GBPt2

GBPt1

USDt2

USDt1

EURt2

EURt1t �� is an 1m � vector of endogenous variables

representing the factors in the EUR, USD, and GBP swaption IVs; � L,3,2,1,i,�i ��

is an mm � matrix of coefficients; and tu is an 1m � vector of innovations that can be

contemporaneously correlated and uncorrelated with its own lagged values and with

other variables. Accordingly, the VAR model is estimated by the OLS estimation

method. However, for selecting a suitable number of lags for the dynamic VAR system,

we use a parsimonious Schwartz information criterion (SIC) that suggests two lags.

Consequently, analysis with Equation 5 is conducted using two lags.

To investigate the direction between PCs, we apply the Granger (1969) causality test,

which establishes the causal relationship between the PCs and hence confirms the lead-

lag relationship. The Granger causality test establishes that iPC is Granger-caused by

jPC if the information in the past and present values of jPC helps to improve the

forecasts of iPC . The results for the PCs are reported in Table 3 for lag 2 with

corresponding values of the F-tests. Bi-directional causality is at work between

EUR_PC1 and USD_PC1, and EUR_PC1 and USD_PC1 Granger-causes GBP_PC1, but

not vice-versa. On the other hand, there is significant bi-directional causality between

EUR_PC2 and USD_PC2, and they both influence GBP_PC2. Lastly, EUR_PC2 is

found to Granger-cause EUR_PC1 and USD_PC1 and similar results are found for

USD_PC2.

115 The first two PCs are selected because they are able to explain an average of 94% of the variation in each of the market IVs. We do not consider PC3s in our VAR framework. This also helps to maintain a reasonable number of parameters, increase efficiency, and achieve parsimony.

150

Table 3. Granger causality for implied PCs. Null Hypothesis 2 lags

F-Statistics P-Value

EUR_PC2 does not Granger Cause EUR_PC1 1.2394 0.294

EUR_PC1 does not Granger Cause EUR_PC2 1.5349 0.220

USD_PC1 does not Granger Cause EUR_PC1 8.5325*** 0.000

EUR_PC1 does not Granger Cause USD_PC1 6.5000*** 0.002

USD_PC2 does not Granger Cause EUR_PC1 0.2017 0.817

EUR_PC1 does not Granger Cause USD_PC2 0.5404 0.584

GBP_PC1 does not Granger Cause EUR_PC1 0.0425 0.958

EUR_PC1 does not Granger Cause GBP_PC1 3.3994** 0.037


EUR_PC1 does not Granger Cause GBP_PC2 1.0068 0.369

USD_PC1 does not Granger Cause EUR_PC2 1.6799 0.192

EUR_PC2 does not Granger Cause USD_PC1 7.1236*** 0.001

USD_PC2 does not Granger Cause EUR_PC2 2.8396* 0.063

EUR_PC2 does not Granger Cause USD_PC2 1.2269 0.297


EUR_PC2 does not Granger Cause GBP_PC1 0.4363 0.647


EUR_PC2 does not Granger Cause GBP_PC2 3.9015** 0.023

USD_PC2 does not Granger Cause USD_PC1 4.8363** 0.010

USD_PC1 does not Granger Cause USD_PC2 2.4546* 0.091

GBP_PC1 does not Granger Cause USD_PC1 1.8424 0.164

USD_PC1 does not Granger Cause GBP_PC1 6.2286*** 0.002


USD_PC1 does not Granger Cause GBP_PC2 0.8442 0.433


USD_PC2 does not Granger Cause GBP_PC1 0.0189 0.981


USD_PC2 does not Granger Cause GBP_PC2 10.381*** 0.000

GBP_PC2 does not Granger Cause GBP_PC1 0.1065 0.899

GBP_PC1 does not Granger Cause GBP_PC2 0.9315 0.397

***, **, and * Denote rejection of the null hypothesis at the 1% , 5%, and 10% significance levels respectively.

Granger causality in a VAR system using the F-test provides information about which

variables impact the future values of each of the variables in the VAR system. However,

F-test values do not provide the sign of the relationship, speed or persistence. The

151

VAR’s impulse response functions could provide information about this kind of

dynamic relationship (see, e.g., Brooks, 2002). An impulse response function (IRF)

measures the responses of the variables in the dynamic VAR system (in our case, the

first two factors of each the EUR, USD, and GBP swaption IVs) when a shock is given to

each factor. Thus, a one standard error shock is applied to the error of a factor, and the

effect on the dynamical VAR system over a specified period of time is recorded.

Pesaran and Shin (1998) proposed a generalized IRF, which is preferred since it does

not require orthogonalization of shocks and is invariant to the reordering of variables in

the VAR system. A mathematical illustration of IRFs is provided below, the moving

average representation of the Equation (5):

,u��F1i

itit �

��

(6)

where tF is an m-variate stochastic process; � is the mean of the process; and i� is

mm � MA matrices containing the responses to forecast errors tu that occurred i

periods ago. Hence, i� is computed through looping relationships. The IRF computes

the time profile of the effect of shocks on future values of the factor loadings in a

dynamic system at a given time. Here, the i� matrices are the dynamic multipliers of

the system, representing the model’s response to a unit shock in each of the factors. The

response of jF to a unit shock in kF is specified by the following sequence, called the

IRF:

,,�,�� jk,3jk,2jk,1 ��

(7)

where jk,i� is the jkth element of matrix i� . Since we know that ,�� )u�(u tt the

components of tu are contemporaneously correlated. Pesaran and Shin (1998) suggest

that whenever these correlations are high, separation of the response of jF to kF from

its response to other shocks that are correlated with ktu is unattainable and the

resulting values obtained from simulation will be misleading. Therefore, generalized

IRFs should be considered instead. The generalized IRF shocks only one element of tu ,

e.g.,�the kth element, and integrates out the effects of the error distribution of the other

shocks (i.e., the correlations among tu are taken into account). Assume that tu has a

multivariate normal distribution. For non-normal error distribution, stochastic-

simulation is required to obtain the conditional expectations )�u�(u kktt � (see, e.g.,

152

Koop et al., 1996; Pesaran and Shin,� 1998; Mills, 1999). The generalized IRF (in

combination with stochastic-simulations) ��:

,��e��)�..,,�(�)�u�(u k1

kkkk1

kkKk2k1kkktt��

(8)

Where � K1,,j,k� jk �� represents the elements of � and ke is a selection vector.

Hence, the response vector to a shock in variable k that occurred i periods ago is:

.��

�

e�

kk

k

kk

ki�

(9)

Finally, the generalized IRF that measures the impact of a shock of one standard error

in scaled form at time t on future values of f at time t+i as follows:

,.2,1,0,i,e��(i)� ki1/2

kkgk ��

(10)

Figure 2 translates the generalized IRFs for each EUR_PC1, EUR_PC2, USD_PC1,

USD_PC2, GBP_PC1, and GBP_PC2 to a unit shock in the innovations of each implied

PC (i.e., the same shock applied in each period starts at period 1 and ends at period 10).

For instance, the EUR_PC1 IRFs to a shock of one standard deviation (STD) in the

innovations of EUR_PC1, EUR_PC2, USD_PC1, USD_PC2,GBP_PC1, and GBP_PC2

are plotted in Figure 2, where the number of periods ahead is on the x-axis and the

IRFs in units are reported on the y-axis.116 A unit shock is applied to EUR_PC1,

USD_PC1 and GBP_PC1 in period 1 and the corresponding IRFs are traced, resulting in

an increase in EUR_PC1 equal to 0.001794 units, 0.001506 units, and 0.001402 units,

respectively. However, a shock in EUR_PC2, USD_PC2 and GBP_PC2 generates

decreases in the EUR_PC1 equal to -0.001240 units, -0.001323 units, and -0.000716

units, respectively. Similarly, the same shock is applied to each PC and we capture the

IRFs in each period ahead. The effects of the PC1s are positive and persistent for

EUR_PC1, though the pattern is decreasing and the effects die out after 10 periods. The

effects of PC2s on EUR_PC1 are negative, and the negative effects decrease and tend to

zero after 10 periods.

Figure 2 shows the IRFs of USD_PC1 to a shock of one STD in the innovations of each

PC considered. Likewise, a unit shock in each EUR_PC1, USD_PC1 and GBP_PC1

induces an increase in USD_PC1 equal to 0.001460 units, 0.001738 units, and

116 The standard deviations of the IRFs are estimated with 10,000 Monte Carlo runs.

153

0.001270 units, respectively. On the other hand, a shock in EUR_PC2, USD_PC2 and

GBP_PC2 induces a negative change in USD_PC1 equal to -0.001166 units, -0.001316

units, and -0.000856 units, respectively. If we look at the time profile of IRFs up to 10

periods, the effects of PC1s are positive and persistent on the USD_PC1, though

decreasing and tending towards zero after 10 periods. The effects of PC2s on USD_PC1

are negative and the negative effects are decreasing.

Figure 2 plots the IRFs of GBP_PC1 to a shock of one STD in the innovations of each

PC. A unit shock in each EUR_PC1, USD_PC1 and GBP_PC1 induces an increase in

GBP_PC1 equal to 0.001979 units, 0.001850 units, and 0.002533 units respectively.

However, a shock in EUR_PC2, USD_PC2 and GBP_PC2 induces a decrease in

GBP_PC1 equal to -0.001698 units, -0.001785 units, and -0.001322 units, respectively.

Similarly, the same shock in each PC provides IRFs for each period ahead. The effects

of the PC1s are positive and persistent on GBP_PC1, though they decrease and vanish

after 10 periods. The effects of PC2s on GBP_PC1 are negative and the negative effects

are decreasing.

The IRFs of EUR_PC2 to a unit shock in the innovations of each PCs are shown in

Figure 2. A unit shock is applied to each of EUR_PC2, USD_PC2 and GBP_PC2 that

induces a positive change in EUR_PC2 equal to 0.011322 units, 0.009673 units, and

0.008825 units respectively. However, a shock in EUR_PC1, USD_PC1 and GBP_PC1

induces a negative change in EUR_PC2 equal to -0.007828 units, -0.007597 units, and

-0.007592 units, respectively. We capture a time profile of IRFs and observe that the

effects of the PC2s are positive and persistent on EUR_PC2, though decreasing. The

effects of PC1s on EUR_PC2 are negative and decreasing. Second, we consider the IRFs

of USD_PC2 to a unit shock in the innovations of each PC. A shock in EUR_PC2,

USD_PC2 and GBP_PC2 induces a positive increase in USD_PC2 equal to 0.007032

units, 0.008231 units, and 0.006065 units, respectively. However, shocking EUR_PC1,

USD_PC1 and GBP_PC1 generates a decrease in USD_PC2 equal to -0.006070 units, -

0.006232 units, and -0.005802 units, respectively. A time profile of IRFs shows that

the effects of the PC2s are positive and persistent on USD_PC2, although decreasing.

The effects of PC1s on USD_PC2 are negative and decreasing. Finally, the IRFs of

GBP_PC2 to a unit shock in the innovations of each PC are plotted in Figure 2. A unit

shock in EUR_PC2, USD_PC2 and GBP_PC2 induces a positive change in GBP_PC2

equal to 0.011001 units, 0.010401 units, and 0.014114 units, respectively.

154

-.002

-.001

.000

.001

.002

.003

2 4 6 8 10

Res pons e of EUR_PC1 to EUR _PC1

-.002

-.001

.000

.001

.002

.003

2 4 6 8 10

Res pons e of EUR_PC1 to EUR _PC2

-.002

-.001

.000

.001

.002

.003

2 4 6 8 10

R es pons e of EUR_PC1 to USD_PC1

-.002

-.001

.000

.001

.002

.003

2 4 6 8 10

Res pons e of EUR_PC 1 to USD _PC2

-.002

-.001

.000

.001

.002

.003

2 4 6 8 10

Res pons e of EUR_PC 1 to GBP_PC 1

-.002

-.001

.000

.001

.002

.003

2 4 6 8 10

Res pons e of EUR _PC1 to GBP_PC2

-.02

-.01

.00

.01

.02

2 4 6 8 10

R es pons e of EUR_PC2 to EUR_PC1

-.02

-.01

.00

.01

.02

2 4 6 8 10

Res pons e of EUR_PC 2 to EUR _PC2

-.02

-.01

.00

.01

.02

2 4 6 8 10

Res pons e of EUR_PC2 to USD_PC 1

-.02

-.01

.00

.01

.02

2 4 6 8 10

Res pons e of EUR_PC2 to USD _PC2

-.02

-.01

.00

.01

.02

2 4 6 8 10

Res pons e of EUR_PC2 to GBP_PC1

-.02

-.01

.00

.01

.02

2 4 6 8 10

Res pons e of EUR _PC2 to GBP_PC2

-.002

-.001

.000

.001

.002

.003

2 4 6 8 10

Res pons e of USD_PC1 to EUR_PC 1

-.002

-.001

.000

.001

.002

.003

2 4 6 8 10

Res pons e of USD_PC1 to EUR _PC2

-.002

-.001

.000

.001

.002

.003

2 4 6 8 10

R es pons e of U SD_PC1 to USD_PC1

-.002

-.001

.000

.001

.002

.003

2 4 6 8 10

Res pons e of USD_PC 1 to USD _PC2

-.002

-.001

.000

.001

.002

.003

2 4 6 8 10

Res pons e of USD_PC 1 to GBP_PC 1

-.002

-.001

.000

.001

.002

.003

2 4 6 8 10

Res pons e of USD _PC1 to GBP_PC2

-.010

-.005

.000

.005

.010

2 4 6 8 10

R es pons e of U SD_PC2 to EUR_PC1

-.010

-.005

.000

.005

.010

2 4 6 8 10

Res pons e of USD_PC 2 to EUR _PC2

-.010

-.005

.000

.005

.010

2 4 6 8 10

Res pons e of USD_PC2 to USD_PC 1

-.010

-.005

.000

.005

.010

2 4 6 8 10

Res pons e of USD_PC2 to USD _PC2

-.010

-.005

.000

.005

.010

2 4 6 8 10

Res pons e of USD_PC2 to GBP_PC1

-.010

-.005

.000

.005

.010

2 4 6 8 10

Res pons e of USD _PC2 to GBP_PC2

-.004

-.002

.000

.002

.004

2 4 6 8 10

Res pons e of GBP_PC1 to EUR_PC 1

-.004

-.002

.000

.002

.004

2 4 6 8 10

Res pons e of GBP_PC1 to EU R_PC2

-.004

-.002

.000

.002

.004

2 4 6 8 10

Res pons e of GBP_PC1 to USD_PC1

-.004

-.002

.000

.002

.004

2 4 6 8 10

Res pons e of GBP_PC1 to USD_PC2

-.004

-.002

.000

.002

.004

2 4 6 8 10

Res pons e of GBP_PC1 to GBP_PC1

-.004

-.002

.000

.002

.004

2 4 6 8 10


-.02

-.01

.00

.01

.02

2 4 6 8 10

Res pons e of GBP_PC2 to EUR_PC1

-.02

-.01

.00

.01

.02

2 4 6 8 10

Res pons e of GBP_PC2 to EUR _PC2

-.02

-.01

.00

.01

.02

2 4 6 8 10

Res pons e of GBP_PC2 to USD_PC 1

-.02

-.01

.00

.01

.02

2 4 6 8 10

Res pons e of GBP_PC2 to U SD_PC2

-.02

-.01

.00

.01

.02

2 4 6 8 10


-.02

-.01

.00

.01

.02

2 4 6 8 10

Res pons e of GBP_PC 2 to GBP_PC2

Response to Generalized One S.D. Innovations ± 2 S.E.

Figure 2. Generalized IRFs of the first two PCs implied from Swaption IVs for each of

the EUR, USD, and GBP swaption market.

155

However, a shock in EUR_PC1, USD_PC1 and GBP_PC1 induces a decrease in

GBP_PC2 equal to -0.005632 units, -0.006951 units, and -0.007369 units,

respectively. The time profile of IRFs shows that the effects of the PC2s are positive and

persistent on GBP_PC2. However, the effects of PC1s on GBP_PC2 are negative and

decreasing.

We conclude from the IRFs plotted in Figure 2 that the PCs of EUR, USD and GBP

swaption IVs respond strongly to a unit shock in each PC of EUR and USD swaption

IVs. The effects are contemporaneous (period 1) and die out after 10 periods. However,

the PCs of both the EUR and the USD swaption IVs do not respond significantly to

shocks in any of the PCs implied from the GBP swaption IVs. Thus, the PCs of the GBP

market are found to be the least influential among the PCs. Nevertheless, the IRF

results presented in Figure 2 reconcile our results from the correlation analysis and the

Granger causality tests.

156

5 Calibration of a Discrete String Market Model

5.1 The String Market Model

We also model and reproduce the swaption volatility matrix on a particular day using

orthogonal PCs in an SMM framework. Santa-Clara and Sornette (2001), and Longstaff

et al. (2001) were the first to introduce the SMM model, also known as the high or

infinite dimensional model. Later, Kerkhof and Pelsser (2002) showed that the SMM

and the LMM are observationally equivalent. The mathematical illustration of the SMM

is as follows. Consider a finite number of forward rates (LIBOR forward rates) with an

increasing maturity structure. We define a set of dates or tenor structure

� ,M210 ,T,,T,TT �� (11)

where 0T represents the current time and 1M1 ,T,T �� forward tenor dates, and the

corresponding M forward rates are the .1,.....,Mkwhere),,TF(t,T 1kk �� Moreover,

the day count fractions can be defined k1kK TT �� , and are computed by the

maturity of LIBOR rate i.e., say three or six calendar months. On day t, the FR can be

represented by ),TF(t,T(t)F 1kkk �1 extracted as

k1kk1k

k

kk TT,1

)B(t,T)B(t,T

1(t)F ��

�

� �

��1 �

�

, (12)

where B(t,T) denotes the time t price of the discount bond maturing at time T. Thus,

the dynamics of M forward rates in SMM can be specified as follows

,M,1,........k(t),dW�(t)dt�(t)F(t)dF P

kkPk

k

k ��

(13)

where � k1i

Pk (t)W � are correlated Wiener processes under probability measure P with

* + .,M1,k,ldt,�(t),WWd klP

lP

k �� (14)

However, we are interested in finding the dynamics of (t)Fk under a measure 1kQ �

equivalent to measure P associated with numeraire ),tB( 1k�2 . To exclude arbitrage

157

opportunities in the above FR dynamics, we must have a drift (t)�1kQ

k�

equal to zero,

i.e.,

,M,1,........k(t),(t)dWF�(t)dF1kQ

ktkk ��

(15)

The change of measure does not affect volatility functions � M1kk� � and corresponding

correlations � M1k,lkl� � . Therefore, the volatility functions and correlations determine the

covariance matrix of the FRs. There are three widely used approaches for determining

the volatility functions and the covariance/correlation matrix of the FRs. First, they can

be obtained from historical data. Second, they can be obtained by calibrating the model

to the market prices of caps and swaptions. Third, a trader can explicitly specify

volatilities and correlations based on how he believes they will evolve. Here, the second

approach is selected by calibrating the SMM directly to the whole swaptions matrix.

For the SMM calibration, we select a separated approach (a calibration technique for

multifactor interest rate models) used widely in practice in the financial industry.

Separated calibration could be implemented in a straightforward way. For instance, the

SMM model could be calibrated to the whole swaption matrix. In fact, calibrating the

model to a swaption matrix is preferred over the caps because it is essentially important

in a situation where the prices of derivatives instruments are dependent on

covariance/correlation structures in addition to volatilities i.e., the values of exotic

instruments such as the Bermudan type of swaption, which are more dependent on

swaptions prices than on caps.117 We use a calibration approach proposed by Gatarek et

al. (2007). The technical details of separated calibration are provided in Appendix A.

The calibration of the SMM model to each of the EUR, USD and GBP swaption IVs are

provided in the following subsections.

5.2 Calibration of the SMM to the EUR swaption IVs

Because the EUR swaption market is relatively new, few studies have been devoted to

this market. This study attempts to fill the gap, calibrating the SMM model to the EUR

swaption IVs and reproducing the whole swaption matrix. Furthermore, the calibration

results obtained are compared with those of the EUR swaption IVs and GBP IVs. Thus,

a separated calibration technique is used in conjunction with the multivariable

optimization algorithm, where the target function is the calibration procedure that 117 See Gatarek et al. (2007).

158

minimizes the difference between the theoretical and market swaption volatilities.118

We have initial parameter values, and the time to maturity of swaptions is expressed in

years, i.e., [0.25, 0.5, 1, 2, 3, 4, 5, 6, 7, 8].119 Gatarek et al. (2007) argued that the

variance-covariance matrix must be positive definite. If this restriction is not to be

violated, then the algorithm must first remove eigenvectors associated with negative

eigenvalues.

Figure 3 provides results for the SMM calibration to the EUR market by using a

nonlinear multivariable optimization technique (i.e., fminsearch solver in Matlab). The

subplot at the top-right of Figure 3 shows the number of iterations on the x-axis and the

number of function evaluations on the y-axis. This plot requires 1446 iterations and

2000 function evaluations in order to reach the optimal solution. The target for the

optimization routine is the root mean squared error (RMSE) (i.e., the graph shows its

value ‘current function value’) between theoretical volatilities and market volatilities,

and we obtained RMSE=0.072506.

The subplot at the top-left shows the number of variables on the x-axis and the values

of optimized parameters on the y-axis. The optimized parameter values obtained for

our ten variables are [0.961, 1.681, 2.284, 2.084, 2.437, 1.977, 1.597, 1.21, 0.627, 0.399].

The subplot to the bottom-left shows a graphical representation for function values

during 1446 iterations against 2000 function evaluations. The y-axis function produces

a very high value (around 1800). By 1446 iterations, the RMSE level is reduced to

0.072506. When this value is smaller, we will have better results.

118 Mathematical illustration of the calibration procedure is provided in the Appendix A. 119 The approach is similar to the study of Longstaff et al. (2001).

159

Figure 3. Nonlinear multivariable optimization results for the EUR swaption market.

The corresponding theoretical volatilities (model volatilities), market volatilities and

their differences for the swaptions matrix on April 4, 2006 are reported in Table 4.120

Panel A of Table 4 provides final theoretical volatilities obtained by calibrating the

SMM to the EUR market data. Many intermediate steps are involved, as illustrated

mathematically in separated calibration procedure.121 Panel C of Table 4 reports

volatility differences between theoretical and market volatilities (i.e., model volatilities

minus market quoted volatilities). Here, the eigenvalues generate very small differences

between the theoretical and market swaption volatilities. The largest differences are

seen for swaptions with very short maturities, such as three, six months, and one year

maturities on one year underlying swaps. There are no significant differences for

maturities longer than one year in the whole swaption matrix. The whole swaption

matrix is reproduced more or less identically to the market swaption matrix. This

implies that SMM calibration using nonlinear multivariable optimization can be used in

practice for valuation of various types of interest rate and credit derivatives.

120 Figure 4 graphically translates the theoretical swaption market and difference volatilities in Table 4. 121 The variance-covariance and modified variance-covariance matrices are reported in Table 5 of Appendix B. Moreover, we had two negative eigenvalues in all ten eigenvalues during the optimization process for EUR swaptions.

160

Table 4. The EUR swaptions theoretical, market swaption volatilities and their Panel A : Theoretical volatilities

1Y 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y 10Y

3M 14.2168 14.7102 15.8130 16.1022 16.0104 15.6030 15.1019 14.7036 14.3034 13.9033

6M 15.3588 15.9081 16.0293 16.1319 16.0106 15.6063 15.1081 14.7069 14.4061 14.0000

1Y 16.5356 16.3159 16.2220 16.1055 15.8030 15.3046 14.9040 14.6035 14.3000 14.0000

2Y 16.5100 16.3046 16.0003 15.6002 15.2010 14.8010 14.5010 14.2000 13.9000 13.7000

3Y 16.3177 16.0016 15.5007 15.1018 14.7016 14.4014 14.1000 13.8000 13.6000 13.4000

4Y 16.1020 15.5007 15.1001 14.7001 14.4002 14.1000 13.8000 13.5000 13.3000 13.1000

5Y 15.6001 15.1008 14.7006 14.3005 14.0000 13.7000 13.5000 13.2000 13.0000 12.9000

6Y 16.0020 14.4010 14.2006 13.8000 13.5000 13.3000 13.1000 12.9000 12.7000 12.6000

7Y 14.4001 14.0001 13.7000 13.4000 13.1000 12.9000 12.7000 12.5000 12.4000 12.3000

8Y 17.1000 13.6000 13.3000 13.1000 12.9000 12.5000 12.4000 12.2000 12.1000 12.0000

Panel B: Market volatilities

1Y 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y 10Y

3M 14.0000 14.7000 15.8000 16.1000 16.0000 15.6000 15.1000 14.7000 14.3000 13.9000

6M 15.3000 15.8000 16.0000 16.1000 16.0000 15.6000 15.1000 14.7000 14.4000 14.0000

1Y 16.4000 16.3000 16.2000 16.1000 15.8000 15.3000 14.9000 14.6000 14.3000 14.0000

2Y 16.5000 16.3000 16.0000 15.6000 15.2000 14.8000 14.5000 14.2000 13.9000 13.7000

3Y 16.3000 16.0000 15.5000 15.1000 14.7000 14.4000 14.1000 13.8000 13.6000 13.4000

4Y 16.1000 15.5000 15.1000 14.7000 14.4000 14.1000 13.8000 13.5000 13.3000 13.1000

5Y 15.6000 15.1000 14.7000 14.3000 14.0000 13.7000 13.5000 13.2000 13.0000 12.9000

6Y 16.0000 14.4000 14.2000 13.8000 13.5000 13.3000 13.1000 12.9000 12.7000 12.6000

7Y 14.4000 14.0000 13.7000 13.4000 13.1000 12.9000 12.7000 12.5000 12.4000 12.3000

8Y 17.1000 13.6000 13.3000 13.1000 12.9000 12.5000 12.4000 12.2000 12.1000 12.0000

Panel C: Volatility difference between Theoretical and Market

1Y 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y 10Y

3M -0.2168 -0.0102 -0.0130 -0.0022 -0.0104 -0.0030 -0.0019 -0.0036 -0.0034 -0.0033

6M -0.0588 -0.1081 -0.0293 -0.0319 -0.0106 -0.0063 -0.0081 -0.0069 -0.0061 0.0000

1Y -0.1356 -0.0159 -0.0220 -0.0055 -0.0030 -0.0046 -0.0040 -0.0035 0.0000 0.0000

2Y -0.0100 -0.0046 -0.0003 -0.0002 -0.0010 -0.0010 -0.0010 0.0000 0.0000 0.0000

3Y -0.0177 -0.0016 -0.0007 -0.0018 -0.0016 -0.0014 0.0000 0.0000 0.0000 0.0000

4Y -0.0020 -0.0007 -0.0001 -0.0001 -0.0002 0.0000 0.0000 0.0000 0.0000 0.0000

5Y -0.0001 -0.0008 -0.0006 -0.0005 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

6Y -0.0020 -0.0010 -0.0006 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

7Y -0.0001 -0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

8Y -0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

161

0

2

4

6

8

10

02

46

81012

14

16

18

Swap Maturity

EUR - Theoretical Volatilities

Swaption Expiry

Vol

atility

0

2

4

6

8

10

02

46

81012

14

16

18

Swap Maturity

EUR - Market Volatilities

Swaption Expiry

Volat

ility

0

2

4

6

8

10

02

46

810

-0.4

-0.2

0

Swap Maturity

EUR - Vol Diff between Theoretical and Market

Swaption Expiry

Diff

eren

ce

Figure 4. EUR theoretical, market swaption volatilities and their differences.

162

5.3 Calibration of the SMM to the USD swaption IVs

The USD-denominated swaption market is the oldest and largest swaption market,

implying an abundance of research on this particular market relative to the EUR and

GBP swaption markets. Nonetheless, SMM is calibrated to the USD swaption matrix

(i.e., using separated calibration with multivariable nonlinear optimization technique),

so the whole swaption volatility matrix is reproduced for this market as well. For

calibration by optimization, we have initial parameters for [0.25, 0.5, 1, 2, 3, 4, 5, 6, 7,

8]. Optimization results are shown in Figure 5.

Figure 5: Nonlinear multivariable optimization results for the USD swaption market.

The subplot at the top-right of Figure 5 shows that the optimization requires 1433

iterations and 2000 function evaluations in order to produce an optimal solution for

the USD market, whereas for EUR it required 1446 iterations. The target for the

optimization routine is the RMSE level (its value is shown on graph as ‘current function

value’), which is 5.1568674. All ten optimized parameter values are reported in the

subplot at the top-left of Figure 5. The values are [1.063, 1.406, 1.768, 2.239, 2.327,

1.542, 0.971, 0.612, 0.311, 0.155]. The subplot at bottom left presents the function

values obtained according to the number of iterations and function evaluations (the

optimum function value obtained in 1433 iterations and 2000 function evaluations).

163

The function initially produces a very high function value (around 2300) and ultimately

minimizes at a final value of 5.1568674. If the RMSE level is compared with the EUR

RMSE level of 0.072506, then the EUR solution is found to be relatively optimal in

both, because the smaller RMSE level is the final solution.

Table 6 provides the theoretical, market swaption volatilities and their differences for

the USD swaptions matrix on April 4, 2006.122 Panel A of Table 6 presents the

theoretical volatility matrix obtained using the separated calibration procedure. During

the optimization process, there was one negative eigenvalue out of ten, though very

small in absolute value. Panel C of Table 6 provides the differences between the

theoretical and market volatilities. Like the EUR, here as well the eigenvalues generate

very small differences between the theoretical and market volatilities. The largest

differences are seen for the swaptions with the shorter maturities (three months, six

months, one year, and two years, with underlying one and two year swaps contracts).

However, there are no significant differences for maturities longer than two years. The

whole swaption matrix is reproduced with our calibration and is roughly similar to that

of the market quoted matrix. Compared to the EUR theoretical volatilities, we assert

that the SMM produced an optimal solution for the EUR swaption market rather than

for the USD swaption market. The differences between the theoretical and market

swaption volatilities are noticed more in shorter swaption maturities in the USD

theoretical volatility matrix, whereas this pattern is noticeable for maturities up to one

year in the EUR matrix. Beyond one year, the differences between theoretical and

market swaption volatilities are negligible.

122 Figure 6 graphically translates the theoretical, market swaption volatilities and their differences in Table 6.

164

Table 6. The USD swaptions theoretical, market swaption volatilities and their differences. Panel A : Theoretical volatilities

1Y 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y 10Y

3M 13.0115 13.5062 13.9192 14.1714 14.2297 14.1941 14.0561 13.9584 13.8512 13.7472

6M 14.0921 14.0248 14.4009 14.5008 14.7275 14.6138 14.4196 14.3186 14.1186 14.0000

1Y 15.1626 15.6954 15.6701 15.7485 15.6778 15.5728 15.3602 15.2527 15.0000 14.9000

2Y 16.7696 16.6266 16.6045 16.4484 16.3471 16.2386 16.0342 15.8000 15.7000 15.5000

3Y 16.9003 17.0561 16.8212 16.7237 16.5199 16.4180 16.2000 16.1000 15.9000 15.8000

4Y 17.1320 16.8283 16.6245 16.5179 16.3149 16.2000 16.0000 15.9000 15.7000 15.6000

5Y 16.8006 16.6012 16.4014 16.3017 16.1000 15.9000 15.8000 15.6000 15.5000 15.4000

6Y 16.4049 16.2027 16.0023 15.8000 15.6000 15.5000 15.3000 15.2000 15.1000 14.9000

7Y 16.1005 15.9005 15.6000 15.4000 15.2000 15.1000 14.9000 14.8000 14.7000 14.6000

8Y 15.7003 15.4000 15.2000 15.0000 14.8000 14.7000 14.5000 14.4000 14.3000 14.2000


1Y 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y 10Y

3M 11.5000 13.5000 13.9000 14.1000 14.2000 14.1000 14.0000 13.9000 13.8000 13.7000

6M 12.5000 13.9000 14.4000 14.5000 14.7000 14.6000 14.4000 14.3000 14.1000 14.0000

1Y 14.9000 15.4000 15.6000 15.6000 15.6000 15.5000 15.3000 15.2000 15.0000 14.9000

2Y 16.5000 16.6000 16.5000 16.4000 16.3000 16.2000 16.0000 15.8000 15.7000 15.5000

3Y 16.9000 17.0000 16.8000 16.7000 16.5000 16.4000 16.2000 16.1000 15.9000 15.8000

4Y 17.0000 16.8000 16.6000 16.5000 16.3000 16.2000 16.0000 15.9000 15.7000 15.6000

5Y 16.8000 16.6000 16.4000 16.3000 16.1000 15.9000 15.8000 15.6000 15.5000 15.4000

6Y 16.4000 16.2000 16.0000 15.8000 15.6000 15.5000 15.3000 15.2000 15.1000 14.9000

7Y 16.1000 15.9000 15.6000 15.4000 15.2000 15.1000 14.9000 14.8000 14.7000 14.6000

8Y 15.7000 15.4000 15.2000 15.0000 14.8000 14.7000 14.5000 14.4000 14.3000 14.2000


1Y 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y 10Y

3M -1.5115 -0.0062 -0.0192 -0.0714 -0.0297 -0.0941 -0.0561 -0.0584 -0.0512 -0.0472

6M -1.5921 -0.1248 -0.0009 -0.0008 -0.0275 -0.0138 -0.0196 -0.0186 -0.0186 0.0000

1Y -0.2626 -0.2954 -0.0701 -0.1485 -0.0778 -0.0728 -0.0602 -0.0527 0.0000 0.0000

2Y -0.2696 -0.0266 -0.1045 -0.0484 -0.0471 -0.0386 -0.0342 0.0000 0.0000 0.0000

3Y -0.0003 -0.0561 -0.0212 -0.0237 -0.0199 -0.0180 0.0000 0.0000 0.0000 0.0000

4Y -0.1320 -0.0283 -0.0245 -0.0179 -0.0149 0.0000 0.0000 0.0000 0.0000 0.0000

5Y -0.0006 -0.0012 -0.0014 -0.0017 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

6Y -0.0049 -0.0027 -0.0023 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

7Y -0.0005 -0.0005 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

8Y -0.0003 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

165

0

2

4

6

8

10

02

46

81012

14

16

18

Swap Maturity

USD - Theoretical Volatilities

Swaption Expiry

Vol

atility

0

2

4

6

8

10

02

46

81010

12

14

16

18

Swap Maturity

USD - Market Volatilities

Swaption Expiry

Vol

atility

0

2

4

6

8

10

02

46

810-2

-1.5

-1

-0.5

0

Swap Maturity

USD - Vol Diff between Theoretical and Market

Swaption Expiry

Diff

eren

ce

Figure 6. USD theoretical, market swaption volatilities and their differences

166

5.4 Calibration of the SMM to the GBP swaption IVs

The GBP-denominated swaption market is the smallest of the three, though it is older

than the EUR swaption market. More studies have been devoted to this market than to

the relatively new EUR swaption market. Our objective is thus to reproduce swaption

volatility matrix for the GBP swaption market. The calibration steps for the SMM are

similar. First of all, initial parameters are provided to the optimization routine, i.e., the

time to maturity of each swaption contract [0.25, 0.5, 1, 2, 3, 4, 5, 6, 7, 8]. The

optimization results are presented in Figure 7. The subplot at the top-right shows that

optimization requires 1429 iterations and 2000 function evaluations in order to

provide the optimal solution (this is more than the number of iterations required of the

EUR and USD markets). An RMSE level of 0.306703 is obtained for the GBP market, a

smaller level than the USD RMSE level though larger than the EUR RMSE level. The

corresponding optimized values are [1.501, 1.811, 1.97, 2.422, 2.55, 2.021, 1.56, 1.063,

0.546, 0.208]. The different function values are presented in the subplot at the bottom-

left and the optimum solution is found during 1429 iterations and 2000 function

evaluations. The function value reaches around 3300, translating into an optimal value

of 0.306703 after 1429 iterations.

Figure 7. Nonlinear multivariable optimization results for the GBP swaption market.

167

Table 8 reports the theoretical, market swaption volatilities and their differences for the

GBP swaption matrix on April 4, 2006.123 Panel A of Table 8 shows the theoretical

swaption volatilities obtained by calibrating SMM to the GBP swaption market. As with

the USD, one negative eigenvalue in ten eigenvalues is found to be negative, but very

small in absolute value. The corresponding differences between theoretical and market

volatilities are reported in Panel C of Table 8. These eigenvalues generate very small

differences between the theoretical and market swaption volatilities. The deviations are

observed for the swaptions of shorter maturities (three months, six months and one

year, with underlying contracts of one year lengths). However, there are no significant

differences for maturities longer than one year. The volatility patterns are similar to the

EUR theoretical volatilities, but different from the patterns of the USD theoretical

volatilities.

123 Figure 8 shows the theoretical, market swaption volatilities and their differences.

168

Table 8. The GBP theoretical, market swaption volatilities and their differences. Panel A: Theoretical volatilities

1Y 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y 10Y

3M 10.0083 10.4012 10.7066 11.0011 11.3000 11.5003 11.7006 12.0000 12.2001 12.4001

6M 10.9684 11.2259 11.3193 11.5106 11.7034 11.9011 12.0018 12.2010 12.4008 12.6000

1Y 12.4549 12.4056 12.3002 12.2007 12.3010 12.4001 12.5002 12.7001 12.8000 12.9000

2Y 13.5052 13.5024 13.3002 13.2000 13.1002 13.1000 13.1000 13.1000 13.1000 13.1000

3Y 14.1011 14.0000 13.8002 13.5000 13.4000 13.3000 13.3000 13.3000 13.3000 13.3000

4Y 14.4009 14.3008 14.0000 13.8001 13.6000 13.5000 13.5000 13.5000 13.5000 13.5000

5Y 14.5005 14.3000 14.1000 13.9000 13.8000 13.7000 13.6000 13.6000 13.6000 13.6000

6Y 14.6007 14.4001 14.2001 14.0000 13.8000 13.7000 13.7000 13.7000 13.7000 13.7000

7Y 14.7000 14.5000 14.3000 14.0000 13.8000 13.7000 13.8000 13.8000 13.8000 13.8000

8Y 14.7000 14.6000 14.3000 14.0000 13.9000 13.8000 13.8000 13.8000 13.8000 13.8000


1Y 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y 10Y

3M 9.6000 10.4000 10.7000 11.0000 11.3000 11.5000 11.7000 12.0000 12.2000 12.4000

6M 10.6000 11.2000 11.3000 11.5000 11.7000 11.9000 12.0000 12.2000 12.4000 12.6000

1Y 12.4000 12.4000 12.3000 12.2000 12.3000 12.4000 12.5000 12.7000 12.8000 12.9000

2Y 13.5000 13.5000 13.3000 13.2000 13.1000 13.1000 13.1000 13.1000 13.1000 13.1000

3Y 14.1000 14.0000 13.8000 13.5000 13.4000 13.3000 13.3000 13.3000 13.3000 13.3000

4Y 14.4000 14.3000 14.0000 13.8000 13.6000 13.5000 13.5000 13.5000 13.5000 13.5000

5Y 14.5000 14.3000 14.1000 13.9000 13.8000 13.7000 13.6000 13.6000 13.6000 13.6000

6Y 14.6000 14.4000 14.2000 14.0000 13.8000 13.7000 13.7000 13.7000 13.7000 13.7000

7Y 14.7000 14.5000 14.3000 14.0000 13.8000 13.7000 13.8000 13.8000 13.8000 13.8000

8Y 14.7000 14.6000 14.3000 14.0000 13.9000 13.8000 13.8000 13.8000 13.8000 13.8000


1Y 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y 10Y

3M -0.4083 -0.0012 -0.0066 -0.0011 -0.0000 -0.0003 -0.0006 -0.0000 -0.0001 -0.0001

6M -0.3684 -0.0259 -0.0193 -0.0106 -0.0034 -0.0011 -0.0018 -0.0010 -0.0008 0.0000

1Y -0.0459 -0.0056 -0.0002 -0.0007 -0.0010 -0.0001 -0.0002 -0.0001 0.0000 0.0000

2Y -0.0052 -0.0024 -0.0002 -0.0000 -0.0000 -0.0000 -0.0000 0.0000 0.0000 0.0000

3Y -0.0011 -0.0000 -0.0002 -0.0000 -0.0000 -0.0000 0.0000 0.0000 0.0000 0.0000

4Y -0.0009 -0.0008 -0.0000 -0.0001 -0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

5Y -0.0005 -0.0000 -0.0000 -0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

6Y -0.0007 -0.0001 -0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

7Y -0.0000 -0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

8Y -0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

169

0

2

4

6

8

10

02

46

81010

12

14

16

Swap Maturity

GBP - Theoretical Volatilities

Swaption Expiry

Vol

atility

0

2

4

6

8

10

02

46

8108

10

12

14

16

Swap Maturity

GBP - Market Volatilities

Swaption Expiry

Vol

atility

0

2

4

6

8

10

02

46

810

-0.8

-0.6

-0.4

-0.2

0

Swap Maturity

GBP - Diff between Theoretical and Market

Swaption Expiry

Diff

eren

ce

Figure 8. GBP theoretical, market swaption volatilities and their differences.

170

6 Conclusion

We estimate and extract the important implied risk factors affecting IV movements in

each of the EUR, USD and GBP swaption markets. We then provide intuitive

interpretations of the observed, implied risk factors. Moreover, we examine the

dynamic interactions between the risk factors across markets using techniques such as

the Granger causality test and the impulse response function. In the second part of the

study, we calibrate the string market model to swaption IVs by using multivariable

nonlinear optimization to reproduce the swaption matrices for the EUR, USD, and GBP

swaption markets.

First, we applied PCA to each of the EUR, USD, and GBP swaption IVs to discover the

important risk factors. We found that three risk factors explain about 94 -97% of the

variance in each of the EUR, USD, GBP swaption IVs. Second, the significant implied

factors present high correlations across swaption markets; consequently, there are

strong linkages across the three markets. Bi-directional causality is at work between the

implied factors from each of the EUR and USD swaption markets. The factors from

EUR and USD swaption markets Granger-cause the factors from the GBP swaption

market, but not vice-versa. Furthermore, in innovation accounting investigations,

shocks to both factors implied by the EUR and USD IVs are found to be influential for

the factor implied from the GBP IVs. However, a shock to the GBP factors does not

affect the factors noticed in the other two markets. Finally, there are many similar

characteristics between EUR and GBP markets, in contrast to the USD swaption

market. The whole swaption matrix for the EUR, USD and GBP markets is reproduced

using the SMM model. The least differences are observed between the theoretical and

market volatilities for EUR, GBP, and USD, respectively. Here too, similar

characteristics are found between EUR and GBP markets.

The identification of risk factors is important in practice. These factors can be used for

hedging of the portfolio position, particularly vega-hedging, generating smooth implied

volatility surfaces, risk management, and for model calibration purposes. First, as in

the interest rate market, we find securities with a variety of maturities; therefore, it is

almost impossible for a trader to vega-hedge portfolios against each and every

individual risk. Hence, in the easiest way to account for most of the risks, a trader uses

the major risk factors instead of hedging against every underlying risk. Second, we find

a limited number of OTC-traded swaptions in the swaption markets. When a trader

171

needs to quote options with different maturities, he is generally restricted. This could

be solved by a swaption volatility surface and thereby we could use the risk factors in

order to generate swaption volatility surface. A trader can easily quote new options by

taking estimates from the volatility surface. Third, by employing these factors, a trader

can manage the overall downside risk for his portfolio i.e., value-at-risk for a portfolio

consisting of interest rate derivatives. Finally, market models could be easily calibrated

to only few important factors i.e., LIBOR market models.

The SMM is an important pricing and hedging tool. First, this model enriches the LMM

by calibrating to the whole swaption matrix, whereas the LMM is calibrated to only few

risk factors. Thus, the SMM calibration accounts for all independent risk factors in a

parsimonious fashion. Second, once a whole swaption matrix is reproduced, the pricing

and hedging of exotic derivatives such as Bermudan swaptions could be done by using

these plain-vanilla products. Finally, in conjunction with SMM, multivariable nonlinear

optimization could be used for increased accuracy and efficiency in the pricing and

hedging activities of exotic interest rate derivatives.

172

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174

Appendix A: Separated Calibration Procedure We use the separated calibration procedure proposed by Gatarek et al. (2007).124 The calibration steps are as follows. First, we form a matrix of market quoted swaption IVs,

imM� , where m represents swaption maturity and M the underlying swap maturity. For

example, i1,2� is the swaption IV for a swaption maturing at 1T with the underlying

swap period from .),T(T 21 Second, we consider other market data such as LIBOR rates,

FRA and IRS and create a vector of discount factors B as

� � � � � � � �� ,0,TB,,0,TB,0,TB,0,TBB M321 �� Then the covariance matrix of FRs, i

mmC , is estimated through the following iterative process

,)dt,T(t,T)�,T(t,T�CiT

0k1k

instl1l

instikl � ��

Where ki � and li � and ),T(t,T� l1l

inst� is the stochastic instantaneous volatility of

the FRs ),T(t,TF l1ll � , while assuming that

.0�,C�C kli

ikl ��

Then the initial values i� are supplied, given that i� values are positive and particularly

important for obtaining on the diagonal. Hence, the diagonal parameters are

.k2

1kki

0,kkk �),T(t,T�C ��

To determine the non-diagonal elements of the matrix C , the parameters (t)F k

i,j are

required to be estimated as

)B(0,T)B(0,T)B(0,T)B(0,T(0)F jik1kk

i,j �� Using the above equation, the complete matrix of parameters F is estimated iteratively. This is a three-dimension or three-array matrix: the first array elements are indexed with i, called the outer loop; the second array elements are indexed with j, called the second loop; and the final array is indexed with k, called the inner most loop. Consequently, the off-diagonal parameters of matrix C are estimated using the estimated parameters of matrix F . With the help of the following iterative process, the off-diagonal parameters are

124 Using similar notations as Gatarek et al. (2007).

175

.(0)(0)FF2�(0)F(0)C2F(0)F(0)F�C nkn

1kknk

n

1ki

nk,n1k,n

1kkn

lkN11li

ik,n

n

1kl

2knk1k,n

�

��

��

��

�

��

�� 3

The whole matrix is computed using this iterative process, starting as ..,m1,........k �

and .2,........kn �� . Finally, we obtain the matrix C . If negative eigenvalues are found, then we remove the associated eigenvectors and compute a new PCA-approximated

matrix PCAC . Finally, the theoretical volatility matrix is obtained by modifying the initial covariance matrix C as

PCAkli

PCAkl C�C

i

�

and ��

��

��

��

��

n

1ki

lkN

PCA11,li

ik,n

n

1klk

2knk, (0)F(0)CF��

The SMM model uses kk � � . The theoretical volatilities are then

��

��

��

��

��

n

1ki

lkN

PCA11,li

ik,n

n

1kl

2kn (0)F(0)CF�

RMSE is computed between market and theoretical swaptions volatilities

� �2m

1i,j

Marketlij

lTheoriticaij ��RMSE �

�

��

Afterward, the unconstrained nonlinear optimization method is used to obtain an optimal solution. For this purpose, we use the Matlab optimization solver called “fminsearch”, which minimizes unconstrained multidimensional objective function illustrated above. The fminsearch optimization solver uses the Nelder-Mead (1965) “simplex” algorithm, which finds the minimum of a scalar-valued nonlinear function of many variables given initial values. Suppose we have � �0fun,xfminsearchx � , which starts at the initial

point 0x and finds a local minimum x of the function described in fun. The 0x can be a

scalar, a vector or a matrix.

176

Appendix B: Variance-Covariance and Modified-Variance-Covariance Matrices

Table 5. EUR swaption market VarianceCovariance Matrix 0.0513 0.0517 0.0643 0.0502 0.0128 -0.0509 -0.0807 -0.0605 -0.1020 -0.1742 0.0517 0.0703 0.0644 0.0474 0.0196 -0.0266 -0.0874 -0.1079 -0.1354 -0.1625 0.0643 0.0644 0.0994 0.0806 0.0322 -0.0131 -0.0573 -0.1398 -0.1568 -0.1919 0.0502 0.0474 0.0806 0.1312 0.0916 0.0156 -0.0357 -0.0975 -0.1423 -0.1831 0.0128 0.0196 0.0322 0.0916 0.1631 0.1038 0.0004 -0.0862 -0.1602 -0.2249 -0.0509 -0.0266 -0.0131 0.0156 0.1038 0.2576 0.1646 0.0364 -0.0831 -0.1461 -0.0807 -0.0874 -0.0573 -0.0357 0.0004 0.1646 0.3687 0.2034 0.1031 -0.1621 -0.0605 -0.1079 -0.1398 -0.0975 -0.0862 0.0364 0.2034 0.6058 0.1201 -0.3845 -0.1020 -0.1354 -0.1568 -0.1423 -0.1602 -0.0831 0.1031 0.1201 1.1148 0.1529 -0.1742 -0.1625 -0.1919 -0.1831 -0.2249 -0.1461 -0.1621 -0.3845 0.1529 2.8122 Modified_VarianceCovariance Matrix 0.0529 0.0509 0.0627 0.0507 0.0121 -0.0506 -0.0807 -0.0610 -0.1021 -0.1743 0.0509 0.0708 0.0654 0.0471 0.0200 -0.0268 -0.0874 -0.1076 -0.1353 -0.1624 0.0627 0.0654 0.1011 0.0801 0.0330 -0.0134 -0.0574 -0.1393 -0.1566 -0.1918 0.0507 0.0471 0.0801 0.1313 0.0913 0.0157 -0.0357 -0.0976 -0.1424 -0.1831 0.0121 0.0200 0.0330 0.0913 0.1635 0.1036 0.0004 -0.0860 -0.1601 -0.2249 -0.0506 -0.0268 -0.0134 0.0157 0.1036 0.2576 0.1646 0.0363 -0.0831 -0.1461 -0.0807 -0.0874 -0.0574 -0.0357 0.0004 0.1646 0.3687 0.2034 0.1031 -0.1621 -0.0610 -0.1076 -0.1393 -0.0976 -0.0860 0.0363 0.2034 0.6059 0.1201 -0.3844 -0.1021 -0.1353 -0.1566 -0.1424 -0.1601 -0.0831 0.1031 0.1201 1.1148 0.1529 -0.1743 -0.1624 -0.1918 -0.1831 -0.2249 -0.1461 -0.1621 -0.3844 0.1529 2.8122

177

Table 7. USD swaption market VarianceCovariance Matrix 0.0626 0.0993 0.0736 0.0485 0.0297 -0.0637 -0.0728 -0.0641 -0.1280 -0.0762 0.0993 0.0840 0.1034 0.0897 0.0425 0.0350 -0.0548 -0.1255 -0.0757 -0.2384 0.0736 0.1034 0.1254 0.1149 0.0503 -0.0272 -0.0874 -0.1775 -0.3091 -0.2477 0.0485 0.0897 0.1149 0.1817 0.1258 -0.0155 -0.1056 -0.2271 -0.2627 -0.4904 0.0297 0.0425 0.0503 0.1258 0.2441 0.1434 -0.0703 -0.2387 -0.4642 -0.5499 -0.0637 0.0350 -0.0272 -0.0155 0.1434 0.4632 0.2493 -0.1464 -0.4661 -0.8917 -0.0728 -0.0548 -0.0874 -0.1056 -0.0703 0.2493 0.8581 0.4954 -0.1692 -0.6951 -0.0641 -0.1255 -0.1775 - 0.2271 -0.2387 -0.1464 0.4954 1.5091 0.5427 -1.2504 -0.1280 -0.0757 -0.3091 -0.2627 -0.4642 -0.4661 -0.1692 0.5427 3.3371 1.2359 -0.0762 -0.2384 -0.2477 -0.4904 -0.5499 -0.8917 -0.6951 -1.2504 1.2359 7.2584 Modified_VarianceCovariance Matrix 0.0801 0.0793 0.0825 0.0588 0.0293 -0.0525 -0.0738 -0.0601 -0.1261 -0.0741 0.0793 0.1067 0.0934 0.0781 0.0429 0.0222 -0.0536 -0.1301 -0.0778 -0.2408 0.0825 0.0934 0.1299 0.1200 0.0501 -0.0215 -0.0879 -0.1755 -0.3082 -0.2467 0.0588 0.0781 0.1200 0.1877 0.1256 -0.0089 -0.1062 -0.2247 -0.2617 -0.4892 0.0293 0.0429 0.0501 0.1256 0.2441 0.1432 -0.0703 -0.2388 -0.4642 -0.5500 -0.0525 0.0222 -0.0215 -0.0089 0.1432 0.4704 0.2486 -0.1439 -0.4649 -0.8903 -0.0738 -0.0536 -0.0879 -0.1062 -0.0703 0.2486 0.8582 0.4952 -0.1693 -0.6952 -0.0601 -0.1301 -0.1755 -0.2247 -0.2388 -0.1439 0.4952 1.5100 0.5432 -1.2500 -0.1261 -0.0778 -0.3082 -0.2617 -0.4642 -0.4649 -0.1693 0.5432 3.3373 1.2361 -0.0741 -0.2408 -0.2467 -0.4892 -0.5500 -0.8903 -0.6952 -1.2500 1.2361 7.2586 Table 9. GBP swaption market VarianceCovariance Matrix 0.0308 0.0335 0.0189 0.0165 -0.0021 -0.0303 -0.0573 -0.0252 -0.0937 -0.1220 0.0335 0.0467 0.0419 0.0174 -0.0051 -0.0182 -0.0514 -0.0993 -0.0913 -0.1627 0.0189 0.0419 0.0781 0.0605 0.0029 -0.0573 -0.0848 -0.0994 -0.1461 -0.0977 0.0165 0.0174 0.0605 0.1129 0.0700 -0.0264 -0.0865 -0.1014 -0.1537 -0.1732 -0.0021 -0.0051 0.0029 0.0700 0.1562 0.1031 -0.0069 -0.1019 -0.1561 -0.2080 -0.0303 -0.0182 -0.0573 -0.0264 0.1031 0.2569 0.1772 0.0071 -0.1248 -0.2490 -0.0573 -0.0514 -0.0848 -0.0865 -0.0069 0.1772 0.4046 0.2393 -0.0607 -0.3296 -0.0252 -0.0993 -0.0994 -0.1014 -0.1019 0.0071 0.2393 0.7023 0.2328 -0.6249 -0.0937 -0.0913 -0.1461 -0.1537 -0.1561 -0.1248 -0.0607 0.2328 1.5846 0.0080 -0.1220 -0.1627 -0.0977 -0.1732 -0.2080 -0.2490 -0.3296 -0.6249 0.0080 4.6734 Modified_VarianceCovariance Matrix 0.0335 0.0305 0.0203 0.0160 -0.0024 -0.0300 -0.0571 -0.0256 -0.0935 -0.1221 0.0305 0.0500 0.0404 0.0179 -0.0049 -0.0185 -0.0517 -0.0988 -0.0915 -0.1626 0.0203 0.0404 0.0788 0.0602 0.0028 -0.0571 -0.0847 -0.0997 -0.1460 -0.0978 0.0160 0.0179 0.0602 0.1130 0.0701 -0.0265 -0.0866 -0.1013 -0.1537 -0.1732 -0.0024 -0.0049 0.0028 0.0701 0.1562 0.1031 -0.0070 -0.1019 -0.1561 -0.2080 -0.0300 -0.0185 -0.0571 -0.0265 0.1031 0.2569 0.1772 0.0071 -0.1247 -0.2490 -0.0571 -0.0517 -0.0847 -0.0866 -0.0070 0.1772 0.4046 0.2393 -0.0607 -0.3296 -0.0256 -0.0988 -0.0997 -0.1013 -0.1019 0.0071 0.2393 0.7023 0.2327 -0.6249 -0.0935 -0.0915 -0.1460 -0.1537 -0.1561 -0.1247 -0.0607 0.2327 1.5846 0.0080 -0.1221 -0.1626 -0.0978 -0.1732 -0.2080 -0.2490 -0.3296 -0.6249 0.0080 4.6734

EKONOMI OCH SAMHÄLLE Skrifter utgivna vid Svenska handelshögskolan

ECONOMICS AND SOCIETY

Publications of the Hanken School of Economics

185. MARIA SUOKANNAS: Den anonyma seniorkonsumenten identifieras. Om identitetsskapande processer i en marknadsföringskontext. Helsingfors 2008.

186. RIIKKA SARALA: The Impact of Cultural Factors on Post-Acquisition Integration.

Domestic and Foreign Acquisitions of Finnish Companies in 1993-2004. Helsinki 2008.

187. INGMAR BJÖRKMAN et al. (Eds.): Innovation, Leadership, and Entrepreneurship.

A Festschrift in Honour of Professor Martin Lindell on his 60th Birthday. Helsinki 2008.

188. JOACIM TÅG: Essays on Platforms. Business Strategies, Regulation and Policy in

Telecommunications, Media and Technology Industries. Helsinki 2008. 189. HENRIK TÖTTERMAN: From Creative Ideas to New Emerging Ventures.

Entrepreneurial Processes Among Finnish Design Entrepreneurs. Helsinki 2008. 190. ANNIKA RAVALD: Hur uppkommer värde för kunden? Helsingfors 2008. 191. TOM LAHTI: Angel Investing in Finland: An Analysis Based on Agency Theory and

the Incomplete Contracting Theory. Helsinki 2008. 192. SYED MUJAHID HUSSAIN: Intraday Dynamics of International Equity Markets.

Helsinki 2009. 193. TEEMU TALLBERG: The Gendered Social Organisation of Defence. Two Ethno-

graphic Case Studies in the Finnish Defence Forces. Helsinki 2009. 194. JONAS HOLMQVIST: Language Influence in Services. Perceived Importance of

Native Language Use in Service Encounters. Helsinki 2009. 195. ENSIO ERÄ-ESKO: Beskattningsrätt och skattskyldighet för kyrkan i Finland.

Steuerrecht und Versteuerung der Kirche in Finnland. Mit einer deutschen Zusammenfassung. Helsingfors 2009.

196. PIA BJÖRKWALL: Nyttighetsmodeller - ett ändamålsenligt innovationsskydd?

Helsingfors 2009. 197. ARTO THURLIN: Essays on Market Microstructure. Price Discovery and Informed

Trading. Helsinki 2009. 198. PETER NYBERG: Essays on Risk and Return. Helsinki 2009. 199. YANQING JIANG: Growth and Convergence: The Case of China. Helsinki 2009. 200. HANNA WESTMAN: Corporate Governance in European Banks. Essays on Bank

Ownership. Helsinki 2009.

201. CATHARINA von KOSKULL: Use of Customer Information. An Ethnography in Service Development. Helsinki 2009.

202. RITVA HÖYKINPURO: Service Firms’ Action upon Negative Incidents in High

Touch Services: A Narrative Study. Helsinki 2009. 203. SUVI NENONEN: Customer Asset Management in Action. Using Customer

Portfolios for Allocating Resources Across Business-to-Business Relationships for Improved Shareholder Value. Helsinki 2009.

204. CAMILLA STEINBY: Multidimensionality of Actors in Business Networks. The

Influence of Social Action in Pharmacy Networks in Finland. Helsinki 2009. 205. JENNIE SUMELIUS: Developing and Integrating HRM Practices in MNC

Subsidiaries in China. Helsinki 2009. 206. SHERAZ AHMED: Essays on Corporate Governance and the Quality of Disclosed

Earnings – Across Transitional Europe. Helsinki 2009. 207. ANNE HOLMA: Adaptation in Triadic Business Relationship Settings. A Study in

Corporate Travel Management. Helsinki 2009. 208. MICHAL KEMPA: Monetary Policy Implementation in the Interbank Market.

Helsinki 2009. 209. SUSANNA SLOTTE-KOCK: Multiple Perspectives on Networks. Conceptual

Development, Application and Integration in an Entrepreneurial Context. Helsinki 2009.

210. ANNA TALASMÄKI: The Evolving Roles of the Human Resource Function.

Understanding Role Changes in the Context of Large-Scale Mergers. Helsinki 2009. 211. MIKAEL M. VAINIONPÄÄ: Tiering Effects in Third Party Logistics: A First-Tier

Buyer Perspective. Helsinki 2010. 212. ABDIRASHID A. ISMAIL: Somali State Failure. Players, Incentives and

Institutions. Helsinki 2010. 213. ANU HELKKULA: Service Experience in an Innovation Context. Helsinki 2010. 214. OLLE SAMUELSON: IT-innovationer i svenska bygg- och fastighetssektorn. En

studie av förekomst och utveckling av IT under ett decennium. Helsingfors 2010. 215. JOANNA BETH SINCLAIR: A Story about a Message That was a Story. Message

Form and Its Implications to Knowledge Flow. Helsinki 2010. 216. TANJA VILÉN: Being in Between. An Ethnographic Study of Opera and Dialogical

Identity Construction. Helsinki 2010. 217. ASHIM KUMAR KAR: Sustainability and Mission Drift in Microfinance. Empirical

Studies on Mutual Exclusion of Double Bottom Lines. Helsinki 2010. 218. HERTTA NIEMI: Managing in the “Golden Cage”. An Ethnographic Study of Work,

Management and Gender in Parliamentary Administration. Helsinki 2010. 219. LINDA GERKMAN: Topics in Spatial Econometrics. With Applications to House

Prices. Helsinki 2010.

Modeling and Forecasting Implied Volatility

Documents