TILBURG SCHOOL OF ECONOMICS AND MANAGEMENT Over the underlying relationship between Volatilities of three major U.S. Market Equity Indexes and their respective CBOE Volatility Indexes Supervisor: Prof. Juan Carlos Rodríguez Second Reader: Prof. L.B.D Raes Candidate: Luca Ribichini ANR: 914427 Dow Jones Industrial Average (DJIA)--CBOE DJIA Volatility Index (VXD) Nasdaq 100 Index (NDX)--CBOE Nasdaq 100 Volatility Index (VXN) Russell 2000 Index (RUT)--CBOE Russell 2000 Volatility Index (RVX) M. SC. FINANCE THESIS ACADEMIC Y EAR 2014 - 2015
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Over the underlying relationship between Volatilities of three major U.S. Market Equity Indexes and their respective CBOE Volatility Indexes
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TILBURG SCHOOL OF ECONOMICS AND MANAGEMENT
Over the underlying relationship between
Volatilities of three major U.S. Market Equity
Indexes and their respective CBOE Volatility
Indexes
Supervisor:
Prof. Juan Carlos Rodríguez
Second Reader:
Prof. L.B.D Raes
Candidate:
Luca Ribichini
ANR: 914427
Dow Jones Industrial Average (DJIA)--CBOE DJIA Volatility Index (VXD)
Nasdaq 100 Index (NDX)--CBOE Nasdaq 100 Volatility Index (VXN)
Russell 2000 Index (RUT)--CBOE Russell 2000 Volatility Index (RVX)
M.SC. FINANCE THESIS
ACADEMIC YEAR
2014 - 2015
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Abstract
In this study, I analyze the underlying relationship between the Dow Jones Industrial
Average (DJIA), Nasdaq 100 (NDX), Russell 2000 (RUT) group (henceforth: MEX) volatilities
and their respective CBOE volatility indexes group (henceforth: VOX). I examine their directional
influence by performing a set of linear regression models to assess the VOX forecast power over
its respective MEX volatility. I show that the commonly accepted view of VOX as predictor of
MEX future 22-day volatility is misleading. Indeed, this view does not reflect nor the true nor the
best relationship between these two measures. I systematically find superior R^2 and correlation
coefficients, when I start to calculate MEX volatility into the past from actual dates of VOX. Thus,
instead of predicting future volatility of its respective set of equity indexes, VOX is surprisingly
predicted in an opposite way by that set. These results are always strongly and statistically
significant. I also examine VOX and MEX future 22-day volatility empirical distributions,
comparing them against their theoretical (Normal) ones, and I find high levels of skewness and
kurtosis of empirical distributions. From an academic standpoint, it is interesting to compare these
findings to those already found about the VIX, since they would complete the analysis over the
full set of CBOE major market volatility indexes. Moreover, they would lead to an identification
of common patterns and reactions of the CBOE volatility indexes. From a market standpoint, this
work enhances the investor awareness and improve her interpretation of this relatively new set of
financial tools. Indeed, providing new empirical results and assessing recurrent patterns over each
single index, it would lead financial participants to use these volatility indexes not just as sentiment
indicators or source of hedging. It might be useful for investors who want to take a directional
view over this set of indexes, since the market (CBOE) offers options and futures over them.
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To my grandfather
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1. Introduction
“Volatility forecasting” is a very peculiar topic of finance literature. Indeed, volatility is a
fundamental component of each portfolio trading and hedging strategy. In finance theory, there
are two main approaches to volatility forecasting: one that uses time-series data and the other that
uses option prices (implied volatility) according to the Black-Scholes model. The popularity of the
last approach has hugely grown through the recent years and it has led to some specific volatility
indexes. The most famous one is the VIX, better known as fear index. This index, provided by the
Chicago Board Options Exchange (CBOE), is supposed to estimate over the next 30-day period
the expected volatility of S&P 500 index (SPX), by averaging the weighted prices of the S&P 500
index puts and calls over a wide range of strike prices. It is largely used by traders to have a better
understanding of investor sentiment, and thus possible reversals in the market. Its large use among
investors has induced academics to question about the VIX reliability as good predictor of S&P
500 future one-month volatility. Indeed, several studies have been conducted over the forecasting
power of the VIX and they have led to contradictory results. Poon and Granger (2003) concluded
that VIX construction is a good tool for model-based forecasting; Becker and Clemens (2007)
instead, rejected the notion that VIX contains any information for SPX volatility forecasting. Two
years later, they corrected previous conclusions (Becker and Clemens, 2009) and after having
examined the forecast performance of VIX, they concluded that VIX could not simply be viewed
as a combination of various measures in model based forecasting either. Vodenska and Chambers
(2013), in alternative, directly undertook a statistical analysis between VIX and SPX volatility
over a 20 years period, finding a reversal forecast power between these two indexes.
Due to the VIX great success, CBOE has extended through the years its set of volatility indexes,
using the same VIX methodology, to other U.S major market equity ones (Dow Jones Industrial
Average, Nasdaq 100, Russell 2000). This new set of volatility indexes is the one of interest of
this study. Along the paper, I refer to this set (CBOE volatility indexes group) as “VOX” and to
its respective equity set (U.S major market equity indexes group) as “MEX”. These two sets are
respectively summarized in the first and second column of Table 1, in the next page.
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Table 1
VOX MEX
CBOE DJIA Volatility Index (VXD) Dow Jones Industrial Average Index (DJIA)
CBOE Nasdaq 100 Volatility Index (VXN) Nasdaq 100 Index (NDX)
CBOE Russell 2000 Volatility Index (RVX) Russell 2000 Index (RUT)
Despite the wide range of volatility indexes, academics have always been concerned about the
forecasting power of the VIX over the S&P 500 future volatility. Curiously, there are no specific
studies specifically concerning RVX and VXD forecasting power. Regarding VXN instead, Simon
(2003) conducted a study from 1995 to 2002 concluding that implied volatilities from options on
the Nasdaq 100 index reflect the stochastic properties of the index itself, but they also show
behaviors that appear to be more closely related to investor sentiment. On the other side, Corrado
and Miller (2005) stated that the implied volatilities (VXN) appear to provide high quality
forecasts of NDX future volatility. Notwithstanding the limited research, CBOE provides these
indexes and they are relevant for investors who actually do hedge, diversify, invest in related ETFs
or take a directional view on volatility in these markets. This lack of studies is odd, since these
indexes represent a wider and more specific range of market segments than just the S&P 500 one.
This study is intended to extend the academic’s question about VIX reliability to the remaining
CBOE volatility indexes (VOX), by statistically examining the underlying relationship between
these two classes of indexes. Moreover, this work might give investors a deeper insight over the
whole CBOE volatility indexes set, and thus over the VIX methodology reliability. To accomplish
this task, I follow the same research methodology involved in the Vodenska and Chambers paper
(2013). Therefore, I linearly regress each index inside MEX against its respective one inside VOX,
I test for different volatility periods, calculation-starting points and volatility regimes and I finally
examine distributions of each index.
This paper is structured as follows. In section 2, I define research methodology. In section 3, I
introduce each index inside VOX and MEX. In section 4, I explain data building. In section 5, I
perform all regression combinations and I discuss my results. In section 6, I analyze high and
normal volatility regimes. In section 7, I examine MEX future 22-day volatility and VOX
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empirical and theoretical distributions. In section 8, I discuss conclusions. In section 9, I insert
references. The final part is devoted to graphs and figures I use to refer to, along the whole paper.
In Appendix, I provide graphs of the best theoretical fitting distributions of both VOX and MEX.
2. Research methodology
I follow the same research methodology of Vodenska and Chambers (2013), so I directly
undertake a statistical analysis between VOX and MEX.
I start examining the daily VOX and MEX returns for the maximum period available 1
according to each VOX (and corresponding MEX), using data from CBOE and Yahoo-Finance
databases. I run a set of linear regressions to detect whether, and to what extent, VOX predicts
MEX volatility for different periods. I first linearly regress MEX future 22-trading day (henceforth:
day) volatility2 against VOX3 to analyze the VOX forecast power over MEX future4 one-month
volatility. This first set of regression corresponds to my reference model. Secondly, I perform
regression analysis of MEX future 22-day volatility against VOX, this time including VOX past
22-day volatility as additional independent variable. I do so in order to catch any incremental
explanatory information from the simple model with just VOX as independent variable. Thirdly, I
regress different MEX volatility periods (6, 11, 33-day volatility windows) against VOX. Fourthly,
based on 22-day volatility, I shift the starting calculation point of MEX future volatility into the
past and into the future (+/- 11, 22, 33 days), in order to find the best relationship between MEX
22-day future volatility and VOX. Then, I shift again the starting point to calculate the MEX future
volatility, this time just into the past (- 11, 22, 33 days), and I combine this shift with different
MEX volatility periods (6, 11, 33-day volatility windows) against VOX.
In addition, I provide summary graphs and tables of R^2 and correlation coefficients for each
combination of MEX volatility period and volatility calculation starting point. Furthermore, I use
estimated regression parameters of each MEX volatility period to plot estimated MEX future
1 From year of each index introduction to 2014 or 2015 2 “22‐trading day volatility” is basically the same as one‐month or 30‐calendar‐day volatility, ignoring weekends and
holidays 3 When I do not specify the period, I mean “present” 4 When I use words “past” and “future”, they are intended in regard of VOX date
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volatilities against their real ones, both for the full and the out-of-the-sample (1 year: 2014-2015)
estimation periods. I also add scatter plots with estimated linear regression interpolations, for each
MEX volatility period. In each out-of-the-sample extrapolation, estimated versus real data, I
always test for mean and variance similarity of the two data series.
Then, I divide MEX sample periods in two regimes: high and normal. Where “high” stands for
higher than two standard deviations from the mean and “normal” stands for lower than two
standard deviations from the mean. I apply this sorting to empirically show that during normal
volatility regimes VOX tends to overestimate MEX future 22-day volatility and to underestimate
it during high volatility regimes. Therefore, I report tables with main statistics and percentages of
overestimation (for normal regimes) and underestimation (for high regimes) within the same
periods. Finally, I perform a distribution analysis for the full sample of both MEX future 22-day
volatility and VOX. I analyze their empirical distributions against their respective Normal ones,
through graphical comparisons and normality tests for each index. I always test the null hypothesis
of normality. I also provide specific graphs of tail distributions. In Appendix, I provide graphs and
estimated parameters of the best fitting theoretical distributions for each index.
3. VOX and MEX: an overview
3.1. CBOE Volatility Indexes (VOX)
The first volatility index, introduced by the Chicago Board of Exchange, was the VIX. It stands
for “Volatility Index”, even if it just refers to S&P 500 volatility. Created by Robert E. Whaley in
1993, it was originally designed to measure the market’s expectation of 30-day volatility implied
by the at-the-money S&P 100 Index (OEX) option prices. In 2003, CBOE and Goldman Sachs
modified the VIX calculation to set up the index on call/put options over the S&P 500 Index (SPX).
Due to the VIX great success, the CBOE volatility index supply expanded and it currently
embodies twenty-nine volatility indexes. They are designed to measure the expected volatility of
six different security classes: stock indexes, interest rates, currency futures, ETFs, single stocks
and VVIX (volatility of VIX). The “stock indexes” class is the one of interest in this study, because
it includes Dow Jones Industrial Average Volatility Index (VXD), Nasdaq 100 Volatility Index
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(VXN) and Russell 2000 Volatility Index (RVX). They all share the same VIX methodology
construction and they were introduced respectively in 1997, 2001 and 2004. Market participants
commonly intend these volatility indexes as measures of MEX market expectations of near-term
volatility, conveyed by listed option prices. As the CBOE revised white paper (2015) suggests:
“they are volatility indexes comprised of options rather than stocks, with the price of each option
reflecting the market’s expectation of future volatility”.
VOX calculation is performed on a real-time basis5 throughout each trading day, by averaging
the weighted prices of MEX put/call options over a large range of strike prices. Like conventional
indexes, VOX calculation employs rules to select component options and formulas to calculate
index values.
Here below I report the VOX calculation steps:
1. Option selection. The option selection criteria is settled with the calculation of the current
"forward index level (F)", which is based on the options strike price at which the absolute
difference between call and put prices is the smallest. Then, by taking the nearest strike
price below the forward index level (F) for both the near-term and next-term options, you
determine the strike price K0. Selection6 is given by taking MEX out-of-the-money put and
calls options with strike prices respectively < K0 and > K0, and MEX at-the-money put and
calls options with strike price K0.
2. Near-term and next-term volatility calculation. You just apply inputs in VOX equation
(1) below.
(1) Near‐term 𝐨𝐩𝐭𝐢𝐨𝐧𝐬: 𝜎12 =
2
𝑇1 ∑
∆𝐾𝑖𝐾𝑖2
𝑖
𝑒𝑅1𝑇1𝑄(𝐾𝑖) − 1
𝑇1 [𝐹1𝐾0− 1]
2
(1) Next‐term 𝐨𝐩𝐭𝐢𝐨𝐧𝐬: 𝜎22 =
2
𝑇2 ∑
∆𝐾𝑖𝐾𝑖2
𝑖
𝑒𝑅2𝑇2𝑄(𝐾𝑖) − 1
𝑇2 [𝐹2𝐾0− 1]
2
5 Minutes and even seconds 6 Excluding any put and call options that have a bid price equal to zero (i.e., no bid)
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With:
σ : 𝑉𝑂𝑋
100 → 𝑉𝑂𝑋 = σ * 100
T : Time to expiration
F : Forward index level
𝑲𝟎 : First strike price below F
𝑲𝒊 : Strike price of the ith out-of-the-money option; a call if 𝐾𝑖> 𝐾0; and a put if
𝐾𝑖< 𝐾0; both put and call if 𝐾𝑖= 𝐾0
∆𝑲𝒊 : Interval between strike prices - half the difference between the strike on
either side of 𝐾𝑖
𝑸(𝑲𝒊) : The midpoint of the bid-ask spread for each option with strike 𝐾𝑖
3. VOX index level. You multiply by 100 the square root of the 30-day weighted average of
𝜎12 and 𝜎2
2. The process is described by VOX equation (1.1) below.
(𝟏. 𝟏) 𝑽𝑶𝑿 = 100 × √[𝑇1 𝜎12 (𝑁𝑇2 − 𝑁30
𝑁𝑇2 − 𝑁𝑇1) + 𝑇2 𝜎2
2 (𝑁30 − 𝑁𝑇1𝑁𝑇2 − 𝑁𝑇1
)] × 𝑁365𝑁30
With:
𝑵𝑻𝟏 : number of minutes to settlement of the near-term options
𝑵𝑻𝟐 : number of minutes to settlement of the next-term options
𝑵𝟑𝟎 : number of minutes in a 30 days
𝑵𝟑𝟔𝟓 : number of minutes in a 365-day year
3.2. Market Equity Indexes (MEX)
Dow Jones Industrial Average (DJIA)
Dow Jones Industrial Average index is a price-weighted average of the highest 30 stock price
companies traded on the New York Stock Exchange and the Nasdaq. S&P Dow Jones Indices,
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controlled by McGraw-Hill Financial, provides it. This index represents the most famous of the
Dow Averages, being the second oldest U.S. market index (The Wall Street Journal started
publishing it every day from Oct. 7, 1896). In 1916, DJIA included up to twenty stocks, then
thirty in 1928 and it currently holds the same number. It is expected to track the performance of
the U.S industrial sector in an overall way.
Investing in DJIA is quite easy, since a great variety of financial securities is provided and
largely traded: ETFs, futures and options contracts. Indeed, stock DJIA components are very
liquid and widely held by both individual and institutional investors. This gives the index a
considerable “timeliness”, which means that the index is based at any point in time on very recent
transactions. Its calculation is given by the sum of all the thirty stock prices divided the “Dow
Divisor”. This last term refers to a number, provided and periodically updated by S&P Dow Jones
Indices, which is committed to keep the index historical continuity by accounting for stock splits,
spinoffs and changes among the DJIA stock components. Formula is described in the next page
by equation (2).
(𝟐) 𝑫𝑱𝑰𝑨 = ∑ 𝑆𝑡𝑜𝑐𝑘 𝑝𝑟𝑖𝑐𝑒𝑖30𝑖
𝐷𝑜𝑤 𝐷𝑖𝑣𝑖𝑠𝑜𝑟
Nasdaq 100 (NDX)
Nasdaq 100 is a modified capitalization-weighted index made by the one-hundred largest
non-financial companies listed on the Nasdaq stock exchange. It was introduced on January 31,
1985 by the Nasdaq and it was first limited to U.S companies. Then, after 1998, also foreign
companies started to be admitted but they had to respect stringent restrictions. NDX stock
components belong to Industrial, Technology, Retail, Telecommunication, Biotechnology,
Health Care, Transportation, Media and Service sectors.
NDX derivatives market is a very deep one, with high trading volumes at the exchange. The
same is for its main ETF that in August 2012 was the third most actively traded exchange-traded
product in the world. Nasdaq rebalances this index just once a year. They do so by reviewing
NDX constituents, making out-of-the-index evaluations, provisions and ranking appropriate
companies. Its basic structure is given by the formula of a modified capitalization-weighted
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method. The term “modified” means that the largest stocks are stopped to a maximum weight
percentage of the total stock index, and the surplus weight it is equally reallocated among the
stocks under that percentage. Formula is described below by equation (3).
(𝟑) 𝑵𝑫𝑿 = ∑ 𝑃𝑖,1𝑖 ∗ 𝑄𝑖,0∑ 𝑃𝑖,0𝑖 ∗ 𝑄𝑖,0
With:
𝑷𝒊,𝟎= Price at base time 0 of the component stock
𝑷𝒊,𝟏= Price at current time 1 of the component stock
𝑸𝒊,𝟎 = Quantity at base time 0 of the component stock
Russell 2000 (RVX)
Russell 2000 is a free-float capitalization-weighted index, composed by (approximately) the
two thousands companies with the smallest market capitalization of the Russell 3000 index.
Introduced by Russell Investments researchers in 1984, it reached a wide success through the
years being commonly used by mutual funds as a benchmark for small-cap stocks in the U.S and
as measure of the small-caps total performance to the one of mid-caps. Indeed, RVX is generally
recognized as the most objective barometer of global small-caps, representing around the 10%
of Russell 3000 total market capitalization. RVX stock constituents are clustered in the Financial
Services, Consumer Discretionary, Producer Durables, Technology and Health Care sectors.
RVX composition is annually adjusted to avoid larger stocks misrepresent performance and
characteristics of the true small-cap block. Moreover, being a free-float adjusted index, it just
includes those stocks that are tradable by the general public of investors (excluding government,
large corporate and large private holdings). Formula is described below by equation (4).
(𝟒) 𝑹𝑽𝑿 = ∑ 𝑃𝑖,1𝑖 ∗ 𝑄𝑖,0∑ 𝑃𝑖,0𝑖 ∗ 𝑄𝑖,0
With:
𝑷𝒊,𝟎= Price at base time 0 of the component stock
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𝑷𝒊,𝟏= Price at current time 1 of the component stock
𝑸𝒊,𝟎 = “Free-float” adjusted quantity at base time 0 of the component stock
4. Data Building
I collect data from CBOE and Yahoo-finance databases. I start data-collection from the first
date available for each index contained in VOX and I begin from there to collect observations of
corresponding MEX dates7. I do so to have the same corresponding number of observations for
each couple of indexes. The final date of each couple of indexes is the same and it is around mid-
2014 (to leave one year for out-of-the-sample estimations)8. The purpose of having updated and
longer as possible samples is to maximize the statistical accuracy of my results9. Starting and final
dates for each index are reported below in Table 1.1.
Table 1.1
VOX MEX Periods
DJIA Volatility Index (VXD) Dow Jones Industrial
Average Index (DJIA) 10/07/1997 – mid-2014
Nasdaq 100 Volatility Index
(VXN) Nasdaq 100 Index (NDX) 02/01/2001 – mid-2014
Russell 2000 Volatility Index
(RVX) Russell 2000 Index (RUT) 01/02/2004 – mid-2014
From daily-adjusted closing prices of each MEX index at time t, I start to calculate daily rate
of returns. Therefore, I compute the natural logarithm for each day and I multiply it by10 100. The
formula is described in the next page by equation (5).
7 I collect data in this way, since indexes inside MEX are far older than their corresponding ones inside VOX 8 When the out-of-the-sample estimations is meaningless, I extend the final date to 2015. 9 I ignore holidays and weekends 10 This way I will have the exact type of numbers in my time-series
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(𝟓) 𝑴𝑬𝑿𝒕 = 𝑙𝑛 (𝑀𝐸𝑋𝑡𝑀𝐸𝑋𝑡−1
) × 100
Since I am interested in MEX future n-day volatility, I need to calculate and annualize it11. The
process is described below by equation (6).
(𝟔) 𝑴𝑬𝑿. 𝑽𝒕;𝒔;𝒏 =
{
√∑[𝑀𝐸𝑋𝑡+𝑠;𝑖 − (∑
𝑀𝐸𝑋𝑡+𝑠;𝑖𝑛 + 1
𝑛
𝑖=0
)]
2𝑛
𝑖=0}
× √252
With:
t = current date
n = future trading days from the current date, so the volatility period length
s = number of trading days that must be subtracted or added (±) to the current date
t in order to shift the volatility calculation starting point (if 0 it is just the current
date t)
The method I employ to estimate linear regression parameters is the Ordinary Least Square
(OLS). Representative equation is given below by equation (a).
(𝐚) 𝒀 = 𝜶 + 𝜷 ∗ 𝑿 + 𝜺
5. Over VOX and MEX relationship: regression analysis
Volatility indexes represent a measure of market expectations over the next-future volatility of
their respective set of equity indexes. The key driver of this kind of index is the implied volatility12
of options taken into calculation. The underlying assumption in using VOX, as MEX volatility
forecast tool, is that investor sentiment has some predictive power over future short-term market
movements. This is a reasonable assumption, since participants translate their beliefs in market
11 I do so because, according to CBOE methodology, VOX indexes are annualized 12 According to the Black-Scholes Formula
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transactions thus in market movements. Therefore, I expect a strong relationship between VOX
and MEX future short-term volatility. This relationship should bring out a good forecasting power
of VOX, or at least good enough to justify its large use by market participants. I perform a set of
OLS linear regression models with different MEX volatility periods and calculation starting points.
I do so in order to statistically catch the underlying relationship between VOX and MEX future
volatility and to assess which is the best one.
5.1. VOX and MEX future 22-day volatility: reference model
I directly start with an investigation over VOX and MEX future 22-day volatility relationship,
since according to CBOE this should be the “unique” one13. Its reliability is the main driver of the
market participants’ use. I postulate linear regression Model 1 as my reference model, with MEX
future 22-day (n =22) volatility as time interval and s = 0 (volatility calculation starting point the
same VOX day). I put MEX future volatility and VOX respectively as my dependent and
independent variable. Formula is described below by equation (1.a).
𝐌𝐨𝐝𝐞𝐥 𝟏 ∶
(𝟏. 𝐚) 𝑴𝑬𝑿. 𝑽𝒕;𝟎;𝟐𝟐 = 𝛼 + 𝛽 ∗ 𝑽𝑶𝑿𝒕 + 𝜀𝑡
Estimated parameters and t-stat coefficients (in brackets) summary are listed below.
Dow Jones Industrial Average, from 10/07/1997 to 06/01/2014, (4176 observations)
I summarize R^2 and correlation coefficients, for each couple of indexes and combinations, in
Tables from 9 to 11 below and in the next page.
Tables 9 – Dow Jones Industrial Average
DJIA (R^2)
n = 6 n = 11 n = 22 n = 33
s = -33 43,17% 53,22% 65,85% 78,03%
s = -22 50,92% 61,30% 76,61% 81,63%
s = -11 59,81% 71,28% 76,70% 81,62%
s = 0 55,98% 62,60% 56,60% 53,66%
DJIA (Corr)
n = 6 n = 11 n = 22 n = 33
s = -33 65,71% 72,95% 81,15% 88,34%
s = -22 71,36% 78,29% 87,53% 90,35%
s = -11 77,34% 84,43% 87,58% 90,34%
s = 0 74,82% 79,12% 75,23% 73,26%
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Tables 10 – Nasdaq 100
Tables 11 – Russell 2000
I graphically summarize R^2 and correlation coefficients, including those of Model 1 (s =0 and
n =22), respectively in set of Figures 5.1 and Figures 5.2.
I extended periods to 07/2015, because for these types of variables an out-of-the-sample
extrapolation would not be feasible. I notice that all coefficients are significant, at any level of
confidence. Moreover, as highlighted from graphs, there is a clear evidence of how the best
relationship is described by combination: s=-22 and n=33. This means that VOX best reflects the
past 1 month + future ½ month MEX volatility, showing the highest R^2 and correlation
coefficients. It is evident a path that shows how, fixing the volatility period, estimation goodness
rises up by shortening the volatility calculation period up to 11 days before. It is worth to notice
that VOX slightly better reflects past 33-day volatility (s=-33; n=33) rather than 22-day one (s=-
22; n=22).
NDX (R^2)
n = 6 n = 11 n = 22 n = 33
s = -33 56,66% 66,92% 77,91% 86,42%
s = -22 62,56% 73,27% 85,16% 88,63%
s = -11 69,36% 79,94% 84,76% 83,50%
s = 0 66,98% 72,96% 73,30% 71,42%
NDX (Corr)
n = 6 n = 11 n = 22 n = 33
s = -33 75,27% 81,80% 88,27% 92,96%
s = -22 79,10% 85,60% 92,28% 94,14%
s = -11 83,28% 89,41% 92,06% 91,38%
s = 0 81,84% 85,41% 85,61% 84,51%
RUT (Corr)
n = 6 n = 11 n = 22 n = 33
s = -33 70,89% 78,54% 86,37% 92,66%
s = -22 75,87% 82,96% 92,06% 94,20%
s = -11 82,00% 89,25% 92,13% 89,93%
s = 0 80,04% 82,55% 80,87% 78,67%
RUT (R^2)
n = 6 n = 11 n = 22 n = 33
s = -33 50,25% 61,69% 74,59% 85,86%
s = -22 57,56% 68,82% 84,74% 88,73%
s = -11 67,24% 79,65% 84,88% 80,88%
s = 0 64,07% 68,15% 65,40% 61,90%
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6. VOX and MEX future 22-day volatility: high and normal volatility
periods
From the analysis performed over the reference model (Model 1), I see a VOX tendency to
overestimate and underestimate MEX future n=22 volatility respectively during normal and high
volatility periods of MEX. In order to check this finding, I sort my datasets into two different
regimes15: high-volatility regime and normal-volatility regime. This sorting is based on MEX
volatility levels during the full sample. With “high-volatility”, I mean greater than two standard
deviations from the mean, so16: 𝑯.𝑽 𝑖𝑓 𝑎𝑛𝑑 𝑜𝑛𝑙𝑦 𝑖𝑓 > 𝝁 + 𝟐 ∗ 𝝈
I perform the same regressions of Model 1, this time using just high-volatility observations for
each index as datasets. Estimated parameters and t-stat coefficients (in brackets) are shown below:
Dow Jones Industrial Average, (212 observations) :
𝐇𝐢𝐠𝐡 𝐩𝐞𝐫𝐢𝐨𝐝𝐬: 𝑫𝑱𝑰𝑨. 𝑽𝒕;𝟎;𝟐𝟐 = 29.29 + 0.50 ∗ 𝑽𝑿𝑫𝒕
(12.72) (8.41)
Nasdaq 100, (232 observations) :
𝐇𝐢𝐠𝐡 𝐩𝐞𝐫𝐢𝐨𝐝𝐬: 𝑵𝑫𝑿. 𝑽𝒕;𝟎;𝟐𝟐 = 57.80 + 0.07 ∗ 𝑽𝑿𝑵𝒕
(16.69) (1.15)
Russell 2000, (99 observations) :
𝐇𝐢𝐠𝐡 𝐩𝐞𝐫𝐢𝐨𝐝𝐬: 𝑹𝑼𝑻. 𝑽𝒕;𝟎;𝟐𝟐 = 57.47 + 0.17 ∗ 𝑹𝑽𝑿𝒕
(10.74) (1.80)
I provide R^2 and correlation coefficients summary in Table 12, below.
Table 12
Index R^2 Correlation
Dow Jones Industrial Average 25.22% 50.22%
Nasdaq 100 0.58% 7.62%
Russell 2000 3.25% 18.03%
15 Since it is meaningless to perform an out-of-the-sample extrapolation, I use data updated to 06/2015 16 Of course, “with normal-volatility” I mean all the other. So: 𝑵.𝑽 𝑖𝑓 𝑎𝑛𝑑 𝑜𝑛𝑙𝑦 𝑖𝑓 < 𝝁 + 𝟐 ∗ 𝝈
31
Compared to Model 1, I notice a huge change in sign, magnitude and significance of estimation
coefficients. Indeed, there is basically no predictive power for Nasdaq 100 and Russell 2000 as it
is shown by R^2 and correlations. For Dow Jones, I observe a greater predictive power (even if
poor in an absolute value) and this because volatilities are not as “high” as for the other two indexes.
Anyway, these results are misleading if taken as pure results. Indeed, datasets used are composed
of comprised volatility periods with few observations and often very far one each other.
Notwithstanding limitations, it is useful to highlight the big difference from previous findings.
I catch these differences comparing descriptive statistics for both VOX and MEX future 22-day
volatility. Moreover, I indicate the length of each regime and I show correlation within each period.
In addition, I empirically test the hypothesis that VOX tends to overestimate and underestimate
MEX future 22-day volatility during respectively normal and high volatility periods. I provide the
realization percentages17 of this hypothesis.
Realization percentages for each period are given by formulas of equations (8) and (9) below.