Electronic copy available at: http://ssrn.com/abstract=1663803 1 Using Volatility Instruments as Extreme Downside Hedges Bernard Lee * Sim Kee Boon Institute for Financial Economics Singapore Management University 90 Stamford Road Singapore 178903 Phone: +65 6828-1990 Fax: +65 6828-1922 [email protected]Yueh-Neng Lin Department of Finance National Chung Hsing University 250 Kuo-Kuang Road Taichung, Taiwan Phone: +886 (4) 2284-7043 Fax: +886 (4) 2285-6015 [email protected]Abstract “Long volatility” is thought to be an effective hedge against a long equity portfolio, especially during periods of extreme market volatility. This study examines using volatility futures and variance futures as extreme downside hedges, and compares their effectiveness against traditional “long volatility” hedging instruments such as out-of-the-money put options. Our results show that CBOE VIX and variance futures are more effective extreme downside hedges than out-of-the-money put options on the S&P 500 index, especially when reasonable actual and/or estimated costs of rolling contracts have taken into account. In particular, using 1-month rolling as well as 3-month rolling VIX futures presents a cost-effective choice as hedging instruments for extreme downside risk protection as well as for upside preservation. Keywords: VIX futures; Variance futures; CBOE VIX Term Structure; S&P 500 puts; Rolling cost Classification code: G12, G13, G14 * Corresponding author: Bernard Lee, Sim Kee Boon Institute for Financial Economics, Singapore Management University, 90 Stamford Road, Singapore 178903, Phone: +(65) 6828-1990, Fax: +(65) 6828-1922, Email: [email protected]. The authors would like to thank Srikanthan Natarajan for his able research assistance, and the MAS-FGIP scheme for generous financial support. The work of Yueh-Neng Lin is supported by a grant from the National Science Council, Taiwan.
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Using Volatility Instruments As Extreme Downside Hedges-August 23, 2010
“Long volatility” is thought to be an effective hedge against a long equity portfolio, especially during periods of extreme market volatility. This study examines using volatility futures and variance futures as extreme downside hedges, and compares their effectiveness against traditional “long volatility” hedging instruments such as out-of-the-money put options. Our results show that CBOE VIX and variance futures are more effective extreme downside hedges than out-of-the-money put options on the S&P 500 index, especially when reasonable actual and/or estimated costs of rolling contracts have taken into account. In particular, using 1-month rolling as well as 3-month rolling VIX futures presents a cost-effective choice as hedging instruments for extreme downside risk protection as well as for upside preservation.
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Electronic copy available at: http://ssrn.com/abstract=1663803
1
Using Volatility Instruments as Extreme Downside Hedges
Bernard Lee∗∗∗∗
Sim Kee Boon Institute for Financial Economics Singapore Management University
∗ Corresponding author: Bernard Lee, Sim Kee Boon Institute for Financial Economics, Singapore Management University, 90 Stamford Road, Singapore 178903, Phone: +(65) 6828-1990, Fax: +(65) 6828-1922, Email: [email protected]. The authors would like to thank Srikanthan Natarajan for his able research assistance, and the MAS-FGIP scheme for generous financial support. The work of Yueh-Neng Lin is supported by a grant from the National Science Council, Taiwan.
Electronic copy available at: http://ssrn.com/abstract=1663803
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1. Introduction
Index funds and exchange traded funds (ETFs) replicate the performance of the
S&P 500 index (SPX). In recent years, they are extremely popular among global
investors, resulting in the need to hedge both the price risk and the volatility risk of
index-linked investment vehicles, particularly during periods of extreme market shocks.
Volatility and variance swaps have been popular in the OTC equity derivatives market
for about a decade. The Chicago Board Options Exchange (CBOE) successively
launched three-month variance futures (VT) on May 18, 2004 and twelve-month
variance futures (VA) on March 23, 2006. The CBOE variance futures contracts can
generate the same volatility exposures on the SPX as OTC variance swaps, with the
additional benefits associated with exchange-traded products. VT is the first
exchange-traded contract in the U.S. to isolate pure realized variance exposure.
Outside the U.S., variance futures on FTSE 100, CAC 40 and AEX indices were
launched on September 15, 2006 on LIFFE Bclear. The CBOE also successively
launched Volatility Index (VIX) futures on March 26, 2004 and VIX options on
February 24, 2006. The trading volume and open interests of VIX options and VIX
futures have since grown significantly, reflecting their acceptance and growing
importance.
Electronic copy available at: http://ssrn.com/abstract=1663803
3
Futures and options on VIX allow investors to buy or sell the VIX (which
measures the SPX’s implied volatility over 30 days), whereas VT (VA) allows
investors to trade the difference between implied and realized variance of the SPX over
three (twelve) months. The distinction between variance futures and variance swaps is
minimal, as the information contained in them is virtually identical. There are several
problems associated with using the CBOE variance futures for empirical analysis: First,
they are illiquid. The VT and VA contracts are far less liquid than the VIX futures
(Huang and Zhang, 2010). For example, on March 2, 2010, the trading volume of the
VIX futures was 13,864 contracts, which was 4,621 times greater than 3 (2) contracts
traded on the VT (VA). Second, VT has a maturity of three months and VA of twelve
months. This means that there have been only 18 non-overlapping VT observations
during the June 2004 to December 2009 period since the contract’s inception. Given
the high volatility of returns on variance futures, this is not enough data to determine if
the mean return is statistically significant.
Where data may be too sparse to be credible, this study uses the so-called “VIX
squared” (which corresponds to the 1-month S&P500 variance swap rate) if necessary.
Since the VIX goes back to the 1990s, this would give us about 240 non-overlapping
monthly variance swap returns, making it possible to establish a variance risk premium
4
with more meaningful precision. Given that the trading volume in variance futures is
quite low, the study reconciles characteristic patterns of CBOE VIX Term Structure
(hereafter as �������) with VT, which behaves like a derivative instrument with
observable VIX being its underlying asset.
From an investor’s point of view, it seems attractive that the negative correlation
between volatility and stock index returns is particularly pronounced in stock market
downturns, thereby offering protection against stock market losses when it is needed
most. Empirical studies, however, indicate that this kind of downside or crash
protection might be expensive because of its constant negative carry, and that
practically it may be impossible to time the market to pay for protection only during a
significant market downturn. Driessen and Maenhout (2007) show that using data on
SPX options, constant relative risk aversion investors find it always optimal to short
out-of-the-money puts and at-the-money straddles. The option positions are
economically and statistically significant and robust after correcting for transaction
costs, margin requirements, and Peso problems. Hafner and Wallmeier (2008) analyze
the implications of optimal investments in volatility. Egloff, Leippold and Wu (2010)
have an extensive analysis of how variance swaps or volatility futures fit into optimal
portfolios in dynamic context that takes into account how variance swaps, in addition
5
to improving Sharpe ratios, improve the ability of the investor to hedge time-variations
in investment opportunities. Moran and Dash (2007) discuss the benefits of a long
exposure to VIX futures and VIX call options. Szado (2009) analyzes the
diversification impacts of a long VIX exposure during the 2008 financial crisis. His
results show that, while long volatility exposure may result in negative returns in the
long term, it may provide significant protection in downturns. In particular, investable
VIX products such as VIX futures and VIX options could have been used to provide
diversification during the crisis of 2008. Additionally, his results suggest that, dollar
for dollar, VIX calls could have provided a more efficient means of diversification than
provided by SPX puts.
This study begins by using volatility futures and variance swaps as extreme
downside hedges. We apply hedging techniques as they are actually used in real-life
trading that takes into account the costs of rolling contracts - we approach this
analysis from the perspective of real-life trading practices in order to come up with
realistic estimates on true hedging costs. Practically speaking, what will be the
reasonable notional amount that investors can hedge without significantly widening the
bid-ask spreads on hedging instruments, thereby making volatility and variance futures
ineffective extreme downside hedges? For large trades, an investor often needs a dealer
6
who is willing to take the other side of the trade on the exchange because of the lack of
liquidity, while the dealers are simply replicating their VIX or variance futures
exposures with options positions. Therefore, in theory it is hard to see how the VIX
and variance futures will be more efficient than the underlying options market (since
dealers require profit margins, unless there is significant native volume in the VIX
future or variance futures markets). Even if the VIX or variance futures are in fact
more effective than the underlying SPX options as hedging instruments, we want to
understand the degree at which increased transaction costs may negate the hedging
effectiveness of VIX and variance futures per unit of hedging cost. To analyze the
effectiveness of using VIX or variance futures during the crisis, we use a long SPX
portfolio and compare various hedging strategies using: (i) VIX futures; (ii) variance
futures, and (iii) out-of-the-money (OTM) SPX put options.
The remainder of the paper is organized as follows. Section 2 describes the
hedging strategies implemented. Section 3 provides an analysis of the hedging results.
Section 4 concludes.
2. Methodology
This section will provide an in-depth discussion of the methodologies used in
this paper: (i) the hedging schemes and the estimation of bid/ask spreads; (ii) the
7
rolling methodology for the VIX futures; (iii) the rolling methodology for the variance
futures and the creation of synthetic 1-month variance futures data; and (iv) the rolling
methodology for OTM put options on SPX.
2.1 Hedging Schemes
Kuruc and Lee (1998) describe the generalized delta-gamma hedging algorithm.
In principle, this algorithm can be expanded to delta-gamma-vega by naming vega risk
factors, but the authors are not aware of any simple and practical solution to the vega
mismatch problem: i.e. simply adding vegas corresponding to implied volatilities with
different moneyness and maturities does not provide meaningful solutions.
The generalized delta solution using minimal Value-at-Risk (VaR) objective
function is defined as follows: Let our objective function be ����� = √���Θ�� ,
where �� = � ����� ����� ⋯ ������� is the vector of “delta equivalent cashflow” positions of
our portfolio � as measured against a m-dimensional vector of nominated risk factors
�� = ��� �� ⋯ � !�, with �" = #"√∆%&�"�"', and #" is the volatility of the log changes
in the j-th risk factor multiplied by a scalar dependent on the confidence level, and Θ
is the correlation matrix for the nominated risk factors in �� = ��� �� ⋯ � !�. The
corresponding variance/covariance matrix is given by
Σ = diag"&#"√Δ%'Θ diag"&#"√Δ%' . In the case of delta Value-at-Risk, the
8
.-dimension vector ℎ0 representing the hedging solution can be obtained from solving
min344� #�&�� + �ℎ4�' , where #�∙� denotes the variance of the hedged portfolio
&�� + �ℎ4�' and � is a � × .-matrix with its 8-th column being the corresponding
� -dimensional delta equivalent cashflow mapping vector for the 8 -th hedging
instrument. Its closed-form solution, in the absence of any hedging constraints, is
ℎ0 = −�:���:��!;�:���:���
where : is the Cholesky decomposition of Σ, or Σ = :�:.
The more general delta-gamma solution can be obtained by using a modified
objective function ���� = <���Θ�� + �� %��=�>Θ>Θ� , where (using ?@" as the
high (low) futures price over the two days % and % + 1 . The high-low spread
estimator in Eq. (1) is derived based on the assumptions that (i) futures trade
continuously and that (ii) the value of the futures does not change while the market is
closed. French and Roll (1986) and Harris (1986) have shown that stock prices often
move significantly over non-trading periods, which will cause the high-low spread to
be underestimated accordingly. To correct for overnight returns, we decrease (increase)
both the high and low prices for day % + 1 by the amount of the overnight increase
(decrease) when calculating spreads if the day % + 1 low (high) is above (below) the
day % close. Further, if the observed two-day variance is large enough, the high-low
spread estimate will be negative. We adjust for those overnight returns using the
difference between the day % closing price and the day % + 1 opening price. In cases
with large overnight price changes, or when the total return over the two consecutive
days is large relative to the intraday volatility during volatile periods, the high-low
16
spread estimate will be still negative. As a practical work-around in the analysis to
follow, we set all negative two-day spreads to the difference between the day % − 1
close and the day % open, divided by the day % daily settlement price.1 The opening
ask (closing bid) for each day is thus calculated as the observed opening price (closing
price) multiplied by one plus (minus) half the assumed bid-ask spread � . By
construction, high-low spread estimates are always non-negative.
Table 1 provides summary statistics for estimated bid-ask spreads ¬�|, based
on the high-low spread estimators (Corwin and Schultz, 2010) for daily settlement
prices of VIX futures and variance futures. Actual bid-ask spreads ¬��UJVd® of VIX
futures, variance futures, and 10% OTM SPX puts are also reported in Table 1. Bid,
ask and midpoints of spot and forward �������s are utilized to construct the bid-ask
spread estimators ¬�¯°±�c� of synthetic 1-month variance futures. The spread
estimates for the period of the Financial Crisis triggered by the bankruptcy of Lehman
Brothers are also separately tabulated in Panel B of Table 1.
To reconcile the differences in multipliers across alternate contracts, the unit of
bid-ask spreads in Table 1 is expressed by the US dollars.2 The mean (median)
1 If the day % opening price equals day % − 1 closing price, the study uses the high-low spread estimate from the previous day. 2 The contract size of VIX futures is $1,000 times the VIX. The contract multiplier for the VT contract is $50 per variance point. One point of SPX options equals $100.
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monthly- and quarterly-rolling bid-ask spreads ¬�| for VIX futures are $337.41
($157.76) and $262.94 ($127.05), respectively, both of which are greater than their
actual bid-ask spreads $109.70 ($90.00) and $128.49 ($100.00). The distribution of
monthly-rolling spreads ¬��UJVd® of VIX futures are on average greater and less
volatile in magnitude than their quarterly-rolling spreads during the 2008 Financial
Crisis. Noticeably, the bid-ask spreads of 10% OTM SPX puts are increasing
significantly in the 2008 crisis period, which are on average greater than those of VIX
futures and VT. Panel B of Table 1 demonstrates higher bid-ask spreads for all hedging
instruments during the financial crisis period triggered by the Lehman Brothers
Bankruptcy. ¬�| consistently overestimates ¬��UJVd® , but the extent of the
overestimation is not hugely unreasonable for this type of models and seems fairly
consistent, which gives rise to the possibility for practitioners to recalibrate the
estimates to observed bid-ask spreads.
[Table 1 about here]
2.2 VIX Futures
In order to test the various hedging strategies, our study uses the daily settlement
prices on VIX futures from June 10, 2004 to October 14, 20093 (translating to 1,347
3 This study chooses to roll at the fifth business day prior to the expiration date for monthly and quarterly rolling strategies of VIX futures and VT to avoid liquidity problems with the last week of
18
trading days, spanning 64 monthly expirations and 21 quarterly expirations)4 for the
1-month rolls and from July 19, 2004 to September 9, 2009 for the 3-month rolls. Price
data on the VIX futures are obtained from the transaction records provided by the
Chicago Futures Exchange (CFE).
According to the product specifications published by the CFE5 , the final
settlement date for the VIX futures is the Wednesday which is 30 days before the third
Friday of the calendar month immediately following the month in which the contract
expires. The study uses the following algorithms to roll monthly VIX futures contracts
five business days before the expiration date, in order to avoid well-known liquidity
problems in the last week of trading. More specifically, on the first day of constructing
a new return series, we want to take long positions on the second-nearby monthly VIX
contracts at the opening of the market. Since the ask prices at the opening of the
market is not available to our study, the study uses the opening prices plus half of the
trading. The CBOE started trading VT on May 18, 2004, the maturity date of June-matured VT and SPX puts is June 18, 2004. Thus, monthly and quarterly rolling of VT begins on ²³.� 14, 2004 that is the Monday after the fifth business day before the expiration date. The last monthly (quarterly) rolling ceases on ´=%µ¶�� 9, 2009 (~�F%��¶�� 11, 2009) for VT, which is the second Friday, a week before their expiration on October 16, 2009 (September 18, 2009). Similarly, since the final settlement date for June-matured VIX futures is June 16, 2004, monthly and quarterly rolling of VIX futures starts on ²³.� 10, 2004 that is the Thursday, one week prior to its maturity. Their last monthly (quarterly) rolling cease on ´=%µ¶�� 14, 2009 (~�F%��¶�� 9, 2009), that is the Wednesday, a week before its expiration on October 21, 2009 (September 16, 2009). 4 VIX futures with contract months of December 2004, April 2005, July 2005 and September 2005 are not available from the Chicago Futures Exchange website. This study uses their second nearby contracts for monthly and quarterly empirical analyses. 5 See http://cfe.cboe.com/Products/Spec_VIX.aspx.
19
bid-ask spreads. The bid-ask spreads are estimated either using actual market bids and
asks or based on the methodology of Corwin and Schultz (2010), who derived a
bid-ask spread estimator as a function of high-low ratios over one-day and two-day
intervals. The daily cumulative payoffs are calculated using daily settlement prices.
The contracts are then closed at their synthetic bid prices, or the closing prices minus
half of the bid-ask spreads on the Wednesdays the week before maturity. On the next
day, we buy back the second-nearby contract at the synthetic ask prices, and so on.6
For the quarterly series, we apply the same algorithm except by using quarterly instead
of monthly rolls unless the next quarterly contract is still not actively traded; in which
case, we will use the next contract available.
Since an investor does not pay upfront cash for the futures, his mark-to-market at
the end of the day is the market value of his futures contract plus the cash balance of
any financing required. The act of finally closing the futures in itself should create cash
receivable/payable. The daily P&L should be computed based on a combination of the
change in market values of the assets and in the balance of cash borrrowed to finance
any final settlement. For the purpose of this calculation, the potential need to finance
one’s margin requirements is ignored. We then initiate a new contract on the next day
6 The bid-ask spread is used to estimate total hedging costs. However, the transaction costs are not used for the purpose of computing an effective hedge solution.
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to maintain the hedge. If the futures contracts close in the money, one should receive
the exercise value of the contracts as cash, or pay cash if the contracts close out of the
money. Any interest charges on a negative balance or interest accruals on a positive
balance from the current period also become part of the P&L for the next period. The
cumulative P&L below can be used as our mark to market of the futures contracts:
i³�³C�%8·� ¸&« % + C∆%� of VIX futures in the first rolling month
= ¹$1000 × »·8T»³%UV % + C∆%�, C = 0,1,2, … , E� − 1=��ℎ % + C∆%�, C = E� { where E� ≡ �� − %�/∆% is the number of trading days between the current day %
and position closing day �� ; »·8T»³%UV % + C∆%� = ·8T»³%¾cJJ®c% + C∆%� −·8T»³%ªbc�%� is the cumulative value of the futures contract at daily settlement on
day % + C∆% , taking the difference between the daily settle futures price,
·8T»³%¾cJJ®c% + C∆%�, and the day-% futures price at the initiation of the contract,
·8T»³%ªbc�%�.
On day �� we close out the first VIX futures and keep any resulting net
cashflow in a cash account. Since the contract size of VIX futures is $1,000 multiplied
by VIX points, the day-�� cash account is
=��ℎ��� = $1000 × »·8T»³%UV ���
= $1000 × l·8T»³%U®ª¾c��� − ·8T»³%ªbc�%�w
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The cumulative P&L for VIX futures initiated on day (�� + ∆%) in the second
rolling month depends on whether interest charges from the first period become part of
the P&L for the second period:
i³�³C�%8·� ¸&« �� + ¿∆%� of VIX futures in the second rolling month
where E� ≡ �� − �� + ∆%��/∆% ; »·8T»³%UV �� + ¿∆%� = ·8T»³%¾cJJ®c�� +¿∆%� − ·8T»³%ªbc��� + ∆%� is the cumulative value of the contract on day �� + ¿∆%
with the opening price, ·8T»³%ªbc��� + ∆%�, of the second VIX futures initiated on
day �� + ∆%. The cash balance account, =��ℎ �� + ¿∆%�, is given by
=��ℎ �� + ¿∆%� in the second rolling month
= =��ℎ �� + ¿ − 1�∆%� × �À��MÁÂJ�×ÂJ
where �� + ¿∆%� is the continuously compounded zero-coupon interest rate on day
�� + ¿∆%. Similar cumulative P&L calculations are used for subsequent periods.
Typically investors gain exposure to the SPX Index by trading ETF on the SPX.7
Depositary receipts on the SPX, or “SPDRs,” represent ownership in unit trusts
designed to replicate the underlying index. As such, SPDRs closely if not perfectly
replicate movements in the underlying stock index. One of the most popular SPDRs,
7 An ETF represents fractional ownership in an investment trust, or unit trusts, patterned after an underlying index, and is a mutual fund that is traded much like any other fund. Unlike most mutual funds, ETFs can be bought or sold throughout the trading day, not just at the closing price of the day.
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the SPY, are valued at approximately 1/10th the value of the Index.8 SPDRs typically
tend to be transacted in 100-lot (or “round-lot”) increments, like most other equities.9
Further, the contract size of VIX futures is $1,000 times the index value of the VIX. In
order to compute the number of VIX futures contracts required for a 100-lot unit of
SPX ETF, we apply the appropriate multipliers for adjusting unit size and unit dollar
values in the hedged portfolio.
In this study, we assume that a typical investor holds the long asset already, but it
will be rare for any fully-invested portfolio to set aside surplus cash to pay for the cost
of hedging. The total amount realized for the asset, when the profit or loss on the hedge
is taken into account, is denoted by mark-to-market (E�E ), so that for, Ã =0,1,2, … , E = � − %�/∆%,
E�E % + ÃΔ%� = $10 × ~¸�% + ÃΔ%�
+ℎ × �$1000 × »·8T»³%UV % + ÃΔ%�! + =��ℎ% + ÃΔ%�
The corresponding cumulative P&L is given by
E�EUV W&X% + ÃΔ%� = E�E % + ÃΔ%� − E�E %�
8 A single SPDR was quoted at $78.18, or approximately 1/10th the value of the S&P 500 at 778.12, on March 17, 2009. 9 If a single unit of SPDRs was valued at $78.18 on March 17, 2009, it implies that a 100-lot unit of SPDRs was valued at $7,818 on that day.
where »·8T»³%UV % + ÃΔ%� is the cumulative value of the futures contract on day
% + ÃΔ% »µ� ∀à ; =��ℎ% + ÃΔ%� is the cash balance account; E�E%� = $10 ×~¸�ªbc�%�; and ~¸�UV W&X% + ÃΔ%� = ~¸�U®ª¾c% + ÃΔ%� − ~¸�ªbc�%�.
When hedging is used, the hedger chooses a value for the hedge ratio ℎ that
minimizes an objective function of the value of the hedged portfolio, such as its
variance. It is important to use the percentages in the cumulative P&L as input, i.e.,
E�EUV W&X% + ÃΔ%�/E�EUV W&X% + Ã − 1�Δ%� − 1 , because doing so avoids
unstable and even non-sensical numerical values when there are massive market
shocks in the market, and also because that is the most natural quantity to hedge
against as seen from the investor’s perspective. Figure 1 presents the cumulative dollar
P&L on the SPX and the cumulative dollar P&Ls on the VIX futures and the bank cash
balance account for 1- and 3-month rolls. Note that the red line of the lower
right-hand-corner graph in each panel is the sum of the security asset and cash balance
accounts represented by the red lines in the upper half of each panel. We expect that an
effective hedging instrument will appear like a crude mirror image of the P&L
represented by the SPX portfolio. That is roughly the case for both panels in Figure 1.
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[Figure 1 about here]
2.3 Variance Futures
Our study uses the daily VT futures prices from June 14, 2004 to October 09,
2009 for the 1-month rolls and from June 14, 2004 to September 11, 2009 for the
3-month rolls. In the following, we describe the algorithms for the rolling strategies of
variance futures at 5 business days before the expiration date. Where 3-month VT data
may be too sparse to be credible, this study performs monthly rolls based on synthetic
1-month VT, replicated from using ������� observations. Monthly rolling occurs
over the period from June 14, 2004 to October 9, 2009, whereas the 3-month rolling
strategy is performed over the period from June 14, 2004 to September 11, 2009.
2.3.1 Algorithm for monthly rolls of synthetic 1-month variance futures
VT contracts are forward starting three-month variance swaps. Once a futures
contract becomes the front-quarter contract, it enters the three-month window during
which realized variance is calculated. Because VT is based on the realized variance of
the SPX, the price of the front-month contract can be stated as two distinct components:
the realized variance (Å:) and the implied forward variance (�Å:). Å: indicates
the realized variance of the SPX corresponding to the front-quarter VT contract. �Å:
represents the future variance of the SPX that is implied by the daily settlement price
25
of the front-quarter VT contract.
Using martingale pricing theory with respective to a risk-neutral probability
measure Q, the time-t VT price in terms of variance points is the annualized forward
integrated variance, ÆJ̄ ��� = �a� nJÇ&È�;a�,�' for f�=3 months = 1/4 year. The value
of a forward-starting VT contract is composed of 100% implied forward variance
(�Å:�;a�,�), as given by
ÆJ̄ �,p¾�� = �a� nJÇ&È�;a�,�' = �Å:�;a�,� (2)
where 0 < % < � − f� < �. The analytical pricing formula for front-month VT is
given by
ÆJ̄ �,p �� = �a� nJÇ&È�;a�,�'
= Ê1 − �;Ja� Ë Å:�;a�,J + Ê�;Ja� Ë �Å:J,�, (3)
where 0 < � − f� < % < �. The formula to calculate the annualized realized variance
(Å:) is as follows10
−×= ∑
−
=
1
1
2 )1/(252aN
i
ei NRRUG (4)
where @ = ln @̧M�/ @̧� is daily return of the S&P 500 from @̧ to @̧M�; @̧M� is the
final value of the S&P500 used to calculate the daily return; and @̧ is the initial value
10 See http://cfe.cboe.com/education/VT_info.aspx for the details. Our Å: in Eq.(3) multiplying 10,000 is the Å: data available in the Chicago Futures Exchange website.
26
of the S&P 500 used to calculate the daily return. This definition is identical to the
settlement price of a variance swap with N prices mapping to 1−N returns. aN
is the actual number of days in the observation period, and eN is the expected number
of days in the period. The actual and expected number of days may differ if a market
disruption event results to the closure of relevant exchanges, such as September 11,
2001.
Because the �ͳ��� µ» ��� (denoted by ���J,�� ) is defined as the variance
swap rate, we are able to evaluate ���J,�� by computing the conditional expectation
under the risk-neutral measure Q, as follows
���J,�� ≡ ��;J nJÇ&ÈJ,�' (5)
Based on Eqs. (2)−(5), the �Å: portion of a front-quarter VT contract can be
replicated by ���J,�� extracted from ������� with identical days to maturity. In
other words, we can synthesize the front-quarter VT with the following:
The following steps are used to construct the monthly rolling of VT. On day t,
we take a long position of the synthetic forward-starting 1-month variance futures at
the ask (where available) or synthetic ask price. For forward-starting contracts, the
daily cumulative payoffs are calculated using midpoints of »����;a�,�� %� for
% < � − f� based on Eq. (7), while for synthetic front-month contracts, we use
Å:�;a�,J and midpoints of ���J,�� for � − f� ≤ % < � based on Eq. (6). The
contracts are then closed at their bid prices (where available) or synthetic bid prices on
the second Friday of the contract month. On the next day, we buy back the next
synthetic forward-starting 1-month variance futures at the ask price, and so on. The
primary reason to roll the synthetic 1-month variance futures one week before
expiration (on the third Friday of the contract month) is to ensure consistency with
28
other rolling strategies used in this study. Figure 2 represents the �Fµ% and »µ����Ð
surfaces of ������� midpoints with the axes representing the trading date and day
to maturity over the period from June 14, 2004 to October 9, 2009.
[Figure 2 about here]
Given the growth of the futures and option markets on VIX, the CBOE has
calculated daily historical values for ������� dating back to 1992. ������� is a
representation of implied volatility of SPX options, and its calculation involves
applying the VIX formula to specific SPX options to construct a term structure for
fairly-valued variance. The generalized VIX formula has been modified to reflect
business days to expiration. As a result, investors will be able to use ������� to
track the movement of the SPX option implied volatility in the listed contract months.
������� of various maturities allows one to infer a complete initial term structure of
�Å: that is contemporaneous with the prices of variance futures of various maturities.
Figure 3 represents the price errors (PEs) between market VT and synthetic VT
constructed from daily returns of the SPX and ������� across the market close
dates and VT maturities over the period from June 18, 2004 to October 21, 2009.
[Figure 3 about here]
Detailed results, as tabulated in Table 2, give the summary statistics for the PEs.
29
The median Midpoint value is −1.4238, with a standard deviation of 40.18 and t value
of −1.29, which is not significantly different from zero at the 95% confidence interval.
By analyzing of the distribution of the PEs on synthetic VT, the synthetic VT contracts
were within 1.5 VIX-point PEs in 96.37% out of the 1,323 trading days for �������
bid quotes, 99.02% for ������� midpoints and 94.18% for ������� ask quotes.
This suggests that the synthetic VT contracts as computed from SPX options are
generally consistent with the VT traders’ thinking.
We observe pricing errors of up to 5.82% from ������� ask quotes, which
can be caused by the inaccuracy of synthetic Å: calculation from the lack of S&P
500 Special Opening Quotation (“SOQ”) data.11 In addition, for simplicity, Ñc, or the
number of expected S&P 500 values needed to calculate daily returns during the
three-month period, is approximated by Ñd, the actual number of S&P 500 values used
to calculate daily returns during the three-month period. The discrepancies between
market VT and synthetic VT could also be primarily a function of relative liquidity in
these two markets, given that VIX futures would keep trading because it is a lot more
liquid — at least the VIX futures while we may observe “stale” quotes in VT.
11 For purposes of calculating the settlement value, CFE calculates the three-month realized variance from a series of values of the S&P 500 beginning with the Special Opening Quotation (“SOQ”) of the S&P 500 on the first day of the three-month period, and ending with the S&P 500 SOQ on the last day of the three-month period. All other values in the series are closing values of the S&P 500.
30
[Table 2 about here]
Panel A of Figure 4 presents the cumulative gain and loss of a monthly rolling
strategy of long synthetic 1-month variance futures. We do not observe any “rough
mirror image” resemblance between the red line and the blue line in the lower
right-hand-corner graph prior to the Financial Crisis, but such is generally the case
after the Financial Crisis.
[Figure 4 about here]
2.3.2 Algorithm for quarterly rolls of 3-month variance futures
The 3-month rolls of VT are rolled at the fifth business day before the expiration
day. In other words, we roll on the second Friday of the contract month (i.e., 5 trading
days ahead) to avoid any liquidity issues due to contract expiration. The following
steps are used to construct quarterly rolling of VT. On day t, we take a second nearby
VT contract at the ask (where available) or synthetic ask price. Since the ask prices at
the opening of the market is not available to our study, we use the opening prices plus
half of the bid-ask spreads. The bid-ask spread is estimated based on the methodology
of Corwin and Schultz (2010) who derived a bid-ask spread estimator as a function of
high-low ratios over one-day and two-day intervals. The daily cumulative payoffs are
calculated using daily settlement prices. The contracts are then closed at their bid
31
prices (where available) or synthetic bid prices on the second Friday of the contract
month. On the next day, we buy back the next second-nearby VT at the synthetic ask
price, and so on.
Panel B of Figure 4 presents the cumulative gain and loss of a 3-month rolling
strategy of long VT. Price data on the VT come from the transaction records provided
by the CFE. Once again, we do not observe any “rough mirror image”
resemblancebetween the red line and the blue line in the lower right-hand-corner graph
prior to the Financial Crisis, but such is generally the case after the Financial Crisis.
2.4 Out-of-the-Money SPX Put Options
The monthly series of out-of-the-money (OTM) SPX put options are created by
purchasing 10% OTM SPX puts monthly one month prior to their expiration. Given
good liquidity relative to the volatility derivatives market and the significant bid/ask in
the options market (e.g., 37.32 −75.72% easily), we will just let any purchased options
expire instead of trying to roll them forward. This is consistent with real-world
practice.
The quarterly series of out-of-the-money (OTM) SPX put options are created by
purchasing 10% OTM SPX puts three months prior to their expiration. The option is
rolled up (by paying additional premium) whenever a bullish market move results in
32
the contract becoming 20% out-of-the-money. On the other hand, the option is rolled
down (i.e. monetizing earned premia) whenever a bearish market move results in the
contract turning at the money. Whenever an option is rolled up or rolled down, the new
option purchased will be one with its expiration closest to being three months out.
This study accounts for the option premium in SPX put options primarily by
using the “burn rate” (daily theta) of the premium.12 Although an investor pays
upfront cash for the premium, his mark-to-market at the end of the day is his negative
cash position paid plus the value of his option. The act of purchasing the option in
itself should not create any big P&L shock unless there is a major shock to the
underlying. The interest charges, while small initially, can become quite significant
over time, thus the need to account for it as part of the cost in running this strategy. In
general, one is expected to maintain a negative cash balance until the option strategy
generates enough profits to cover the outstanding debt. In other words, the P&L should
be computed based on a combination of money borrowed to finance the option and the
option itself. In a sense, one is not expected see any negative value representing the
12 Suppose that the investor has a securities account. He has to account for both the asset and liability columns when computing his P&L. On day % he buys an option: the cash account is “−¸³%%�” while the asset account is “+¸³%%�”. If he sells the option right away, the net account on day % is back at 0 P&L. On day % + 1, if the underlying price has not changed, the cash account still remains at “−¸³% %�” and asset account at “¸³% % + 1� = ¸³% %� − µ.� Ð�Ò µ» %ℎ�%�”. Thus, .�% ¸&« on day % + 1 is equal to “µ.� Ð�Ò µ» %ℎ�%�”. If the option expires worthless, his cumulative P&L become “−¸³%%� −8.%����%” ONLY at the expiration day. In other words, while he has already paid upfront cash for the option on day %, the full negative P&L for the option premium usually does not manifest itself until the expiration day.
33
entire option premium, unless the option expires at less than the original premium paid
plus interest cost, or unless the option position has lost most of its intrinsic value.13
Suppose the put option is marked to market at regular intervals of length ∆%. As
described above, at time % we short an instantaneously maturing risk-free bond ¬%�
to raise cash, and then go long on the put option ¸³%%�, such that the net P&L at time
% is zero. In other words, the combined position is a self-financed portfolio: The
investor borrows cash in order to finance the purchase of the option, such that
¬%� = ¸³%%� . Accordingly, interest based on a deterministic continuously
compounded rate %� should be paid when money is borrowed to purchase the
option. At time % + ∆%, the mark-to-market (E�E) is given as follows:
E�E% + ∆%� = ¸³%% + ∆%� − ¬%��ÀJM∆J�∆J
where ¬%� = ¸³%%�. We shall repeat the net P&L calculation at time % + 2∆%, and
for C = 1, … , E ≡ � − %�/∆% and � is the option maturity date. The study uses the
formula above to estimate the cumulative P&L of a mechanical rolling strategy of
13 Some researchers treat the option premium as a negative P&L because “money is paid” upfront. They always take a large P&L shock when the option is paid. Technically, that is incorrect because one can buy the option in the morning and sell it in the afternoon. Thus, no P&L changes if the price of the option stays the same.
34
buying one option and rolling it forward every month. The study then runs statistics on
it to estimate the hedge ratio by minimizing residual hedge (or any other alternative
objective functions).
Since bids and asks right before expiration often do not reflect actual tradable
values, it is more reliable to use the exercise value of the option at expiration date.
Once a settlement price is published on a specific contract month, the movement of
that put no longer reflects changes in the value of the underlying index; it is going into
“settlement mode”. Accordingly, we initiate a new contract on its expiration day to
maintain the hedge. Typically, execution traders will be given at least 1 trading session
to “build” a new position. To reflect real-world conditions, our study initiates a new
10% OTM put contract on its subsequent trading day. If the option expires in the
money, one should include the exercise value into the P&L, i.e., one receives cash into
the cash account if the option expires in the money. If the cash infusion is big enough
to create a positive cash balance, there will be a positive profit count interest earned.
By contrast, there is no value left in the option if it expires out of the money. Any
interest surpluses (charges) from the current period also become part of the positive
(negative) P&L for the next period. The cumulative P&L below can be used as our
mark to market of the option:
35
E�E¾bNbVJ% + C∆%� in the first rolling month
= ÔÕÖ 0 , »µ� C = 0 ¸³%% + C∆%� − ¸³%%� Ó �ÀJM@∆J�×∆J®
@^� , »µ� C = 1,2, … , E ≡ �� − %�/∆%{
where E ≡ �� − %�/∆% , and ¸³%��� = &�� − ~��×�'M with the final index
settlement value ~��� at expiration day �� and the strike price ��.
The cumulative P&L for the put option on its first trading day (�� + ∆%) of the
second rolling month will depend on whether interest charges (surpluses) from the first
period will become part of the negative (positive) P&L for the second period:
E�E¾bNbVJ�� + ∆%�
= − ¸³%%� Ó �ÀJM@∆J�×∆JK@^� − &�� − ~��×�'M§ �À��M∆J�×∆J
The market-to-market of the put option on its 2nd, 3rd,…, and E day during the
second rolling month is given, for C = 2,3, … , E ≡ �� − ���/∆%, by
E�E¾bNbVJ�� + C∆%� =
¸³%�� + ¿∆%� − ¸³%�� + Δ%� Ó �À��M@∆J�×∆J®@^�− ¸³%%� Ó �ÀJM@∆J�×∆JK
@^� − &�� − ~��×�'M§ × �À��M∆J�×∆J Ó �À��M@∆J�×∆J®
@^�
and similar MTM calculations apply to any subsequent periods.
Figure 5 presents the cumulative P&Ls of both the 1-month (Panel A) and
36
3-month (Panel B) rolling strategies using OTM SPX puts. Before the Financial Crisis,
we observe a straight line representing the negative carry of any long-option strategy,
with the line become steeper as we approach the Financial Crisis consistent with a
steady increase in implied volatility toward the Crisis. Once we have reached the
Financial Crisis, we observe the “rough mirror image” resemblance between the red
line and the blue line in the lower right-hand-corner graph, as in the case of VT futures.
The upside for the 3-month roll is less impressive than that for the 1-month roll. This is
not surprising since markets often recover after major shocks, translating into fewer
opportunities to lock in profits with the 3-month roll.
[Figure 5 about here]
3. Hedging Performance
This section will discuss the empirical results from: (i) the hedging schemes as
applied to the VIX futures; (ii) the hedging schemes as applied to the variance futures;
and (iii) the hedging schemes as applied to the 10% OTM SPX puts.
3.1 1-Month VIX Futures
The study conducts the empirical hedging analysis based on the five different
hedging methodologies as described above, by using VIX futures as a hedge to a
100-lot unit of long SPX ETF. The rebalancing, done every month, takes place five
37
business days prior to the expiration of VIX futures to avoid well-known liquidity
problems in the last week of trading of futures contracts. The study focuses on a
one-month out-of-sample hedging horizon, using data for the period October 14, 2004
through October 14, 2009. Hedge effectiveness is measured based on the magnitude of
percentage drawdown reduction from before the hedge to after the hedge:
NA NA NA NA ¸�.�C ¬. «�ℎ��. ¬�µ%ℎ��� ¬�.ß�³F%=Ò ~�F%��¶�� 15, 2008 − Z�=��¶�� 31, 2008� Ñ ¬��UJVd® 76 NA 76 76 76 76
¬�| 76 NA NA 76 76 NA
¬�¯°±�c� NA 76 NA NA NA NA
E ¬��UJVd® 254.61
(0.52) NA
397.08 (23.40)
246.18
(0.58) 8,914.67
(5.16) 439.19 (25.93)
¬�| 1,348.14
(2.84) NA NA
953.41
(2.19) 10,603.05
(6.44) NA
¬�¯°±�c� NA 26,887.52
(16.69) NA NA NA NA
EÐ. ¬��UJVd® 230.00
(0.45) NA
400.00 (16.28)
190.00
(0.45) 8,750.00
(4.97) 410.00 (15.03)
¬�| 1,025.15
(2.54) NA NA
625.00
(2.00) 7,475.00
(4.30) NA
¬�¯°±�c� NA 19,984.64
(15.50) NA NA NA NA
E�T ¬��UJVd® 990.00
(1.75) NA
1,470.00 (200.00)
1,350.00
(2.55) 25,000.00
(11.61) 1,500.00 (200.00)
¬�| 4,922.70
(8.39) NA NA
4,284.70
(6.47) 43,875.00
(37.29) NA
¬�¯°±�c� NA 97,974.12
(56.44) NA NA NA NA
E8. ¬��UJVd® 10.00
(0.02) NA
10.00 (4.38)
10.00
(0.02) 250.00 (0.73)
10.00 (5.67)
¬�| 10.00
(0.02) NA NA
13.29
(0.04) 0.00
(0.00) NA
¬�¯°±�c� NA 5,214.95
(2.76) NA NA NA NA
~n ¬��UJVd® 196.49
(0.38) NA
304.15 (27.60)
218.56
(0.51) 4,989.16
(2.15) 297.89 (36.57)
¬�| 1,107.31
(2.23) NA NA
871.19
(1.71) 10,188.15
(6.24) NA
¬�¯°±�c� NA 19,371.58
(9.19) NA NA NA NA
~ß��.��� ¬��UJVd® 1.10
(0.97) NA
1.14 (4.35)
2.03
(1.61) 0.62
(0.53) 1.02
(3.63)
¬�| 0.96
(0.73) NA NA
1.19
(0.53) 1.24
(2.07) NA
¬�¯°±�c� NA 1.71
(1.30) NA NA NA NA
�³�%µ�8� ¬��UJVd® 4.29
(3.53) NA
4.45 (25.79)
9.88
(5.86) 3.93
(3.22) 4.76
(16.21)
¬�| 3.53
(2.68) NA NA
4.39
(2.20) 3.96
(9.72) NA
¬�¯°±�c� NA 5.69
(6.04) NA NA NA NA
54
Table 2 Summary Statistics of Price Differences between Real-Market and
Synthetic 3-Month Variance Futures (VT)
Summary statistics for price differences between market VT and synthetic VT that is calculated from daily
returns of the S&P 500 index and the CBOE VIX Term Structure Bids, Asks and Midpoints. The sample
covers the period from June 14, 2004 to September 11, 2009. Defining ÆJ¾à�J3cJ@U ¯��� = Ê1 −�;Ja� Ë Å:�;a�à�J3cJ@U + Ê�;Ja� Ë ���J,�,Ø@Ù� , these are given as the bid pricing error: ¸nØ@Ù = ÆJKd�ecJ ¯��� −ÆJ¾à�J3cJ@U ¯���, where ���J,�,Ø@Ù is the bid quotation of CBOE VIX Term Structure with comparable
days to expiration and Å:�;a�à�J3cJ@U is realized variance implicit in the daily returns of the S&P 500
index. Similarly, ¸n @Ù and ¸nd¾e are calculated from midpoints ���J,� @Ù and ask quotes ���J,�d¾e ,
respectively. The unit of ¸n is the annualized variance point multiplied by 10000. For example, on
March 4, 2005, the front-month VT contract had 10 business days remaining until settlement. The Å:
reported by CFE that evening was 94.97 and the VT daily settlement price was 99.50. Using the above
formula, we can calculate the implied forward variance (�Å:) for the remaining ten days. �Å:=123.06.
Taking the square root of the �Å:, one finds the futures price is implying an annualized S&P 500 return
standard deviation or volatility of 11.09% over the next ten days. On March 4, 2005, the bid, midpoint and
ask quotes of front-month CBOE VIX Term Structure are 10.96, 11.51 and 12.04, respectively, which give
us an estimate of 120.122, 132.480 and 144.962 for �Å:.
Table 3 Descriptive Statistics on Hedging The first row of each panel gives the descriptive statistics on the unhedged SPX ETF. The remaining rows of each panel give the descriptive statistics on the hedged
P&L under the five different models used, ranked by the maximum drawdown reduction starting from the most effective hedging model. Panels A, B, C and D provide
summary statistics on hedging with 1-month roll VIX futures, 3-month roll VIX futures, 1-month roll VT, and 1-month roll SPX puts, respectively.