“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.

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
bernardlee@smu.edu.sg
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
ynlin@dragon.nchu.edu.tw
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: bernardlee@smu.edu.sg. 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.
1
Electronic copy available at: http://ssrn.com/abstract=1663803

2. 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.
2
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
3
Electronic copy available at: http://ssrn.com/abstract=1663803

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
4

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
5

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
6

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
7

8. -dimension vector ℎ representing the hedging solution can be obtained from solving
min + ℎ , where ∙ denotes the variance of the hedged portfolio
+ ℎ and is a × -matrix with its -th column being the corresponding
-dimensional delta equivalent cashflow mapping vector for the -th hedging
instrument. Its closed-form solution, in the absence of any hedging constraints, is
ℎ=−
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
Kronecker delta notation):
= ∆ = ∆ +
In general, the hedging objective functions can be made arbitrarily complex, but
doing so often results in non-unique boundary value solutions. Hedging problems in
the real world usually have constraints: for instance, it is not uncommon for hedgers to
be disallowed from “shorting” put options or other volatility instruments by their risk
and compliance departments.
No matter how simple or complex the hedging methodology, hedging is almost
always an optimization problem with different objective functions. To the best of the
8

9. authors’ knowledge, the formulation by Kuruc and Lee (1998) was one of the few
original hedging formulations that do not assume a high degree of resemblance
between the portfolio and the chosen hedging instruments, as it is often the case when
hedging problems are posed as linear regression problems. Moreover, the general
vector and matrix framework can still apply by customizing different objective
functions for the specific hedging problem. For example, the value of a large fixed
income portfolio is usually expressed in terms of deltas in order to consolidate a
complex portfolio of many different bonds with relevant yield curve factors. For a
simple portfolio of one or two assets, one can simply use the asset itself as the risk
factor. In our case, we can simplify the problem by saying that a portfolio of a 100-lot
unit of SPX ETF has an exposure to the dollar S&P 500 factor. These are the five
objective functions used in our study when applied to a single hedging instrument:
1. Minimum absolute residual hedge by minimizing the sum of absolute
percentage changes in the mark-to-market ( ) value of the hedged portfolio,
ℎ , or
min ℎ
ℎ = −1= &
−1 ;
&
where is the
day- mark-to-market of the unhedged portfolio; and & = −
9

10. 0 is the cumulative P&L of the hedging instrument on day , assuming
0 = 0 in general.
2. Minimum variance hedge by minimizing the sum of squared percentage
changes in the mark-to-market value of the hedged portfolio, ℎ , or
min ℎ
3. Minimize the peak-to-trough maximum drawdown of the mark-to-market value
of the hedged portfolio:
min ; +ℎ &
The drawdown is the measure of the decline from a historical peak in some
time series, typically representing the historical mark-to-market value of a
financial asset. Let = +ℎ & be the dollar
mark-to-market value of the brokerage account representing the hedged
portfolio at the end of the period 0, , and =
be the maximum dollar mark-to-market value in the 0, period. The
drawdown at any time , denoted , is defined as
= −
The Maximum Drawdown ( ) from time 0 up to time is the
maximum of the drawdown over the history of the . Formally,
10

11. ; = max
Alternatively, % is also used to describe the percentage drop from the
peak to the trough as measured from the peak. Using as a measure of
risk, the optimization procedure will find an estimate of the hedge ratio ℎ that
can achieve the lowest possible .
4. Minimize the Expected Shortfall or Conditional Value-at-Risk at 95%
computed from the daily P&L of the hedged portfolio, or equivalently,
min & +ℎ & with & = − −1 and
& = − − 1 using the Cornish-Fisher expansion (Lee and Lee,
2004), where
% ∙ =− ∙ + ∙ , > 95%
1
+ −1 ∙
6
1
=− ∙ − ∙ + −3 ∙ > 95%
24
1
− 2 −5 ∙
36
with being the critical value for probability 1 − with standard
normal distribution (e.g. = −1.64 = 95%), while , , and
following the standard definitions of mean, volatility, skewness and excess
kurtosis, respectively, as computed from the daily P&L of the hedged portfolio.
11

12. Practically, the expected value of the tail of , at and above 95%
estimated numerically by using the discrete average of taken at 95.5%,
96.5%, 97.5%, 98.5% and 99.5% (e.g. = −1.70 at = 95.5% ,
= −1.81 at = 96.5% , = −1.96 at = 97.5% ,
= −2.17 at = 98.5%, and = −2.58 at = 99.5%; their
average is −2.04).
5. All of the above are risk measures. It is not uncommon that minimizing risk
measures will result in minimizing both the downside and the upside of the
profit and loss stream. Since the specific problem is essentially one of
combining 2 long assets, we will explore the possibility of maximizing an
Omega-function-like measure (Omega functions are essentially functions based
on ratios of a measure of upside cumulants to a measure of downside cumulants)
known as the Alternative Sharpe Ratio (Lee and Lee, 2004). This is a more
“balanced” approach of optimal hedging from the perspective of not only
minimizing risk (which also tends to minimize returns) but also achieving an
optimal balance between “upside moments” and “downside moments”, and is
generally consistent with real-world practice in that traders tend to underhedge.
The objective function to maximize is defined as:
12

13. 1 1
≡ + −
2 2
where
= excess return rate of the -th asset of the portfolio ; given that our study
uses a single hedging instrument, the -th asset is the portfolio itself, which is
calculated as the percentage changes in the mark-to-market value of the hedged
portfolio, ℎ
= -th position of the portfolio
=
,
where is critical value for probability and
= is critical value for probability 1 −
,
where
(e.g. = 2.33 at 1%, = −2.33 at 99%)
Consistent with real-world hedging practice, we use a minimum of 60 daily data
points prior to the hedging date to compute the hedging ratio. For objective functions
=
.
(1) and (2), all the residuals are exponentially weighted, based on ,
so that the 252nd data point only contains a 5% weight, starting from a 100% weight
for the first data point corresponding to the first date. At each rolling or hedge
rebalancing date, the hedging ratio is recomputed.
An important factor affecting hedge effectiveness is the transaction cost
assumption. Studies examining transaction costs in futures markets are scarce. One
13

14. reason can be the simple lack of data: unlike in the cash equity markets, not every
quote or every transaction is reported on a ticker tape. Manaster and Mann (1996)
analyze transaction data from the Chicago Mercantile Exchange (CME). They find that
futures dealers actively manage their inventory levels throughout the day. Studies by
Kempf and Korn (1999) and Hasbrouck (2004) demonstrate similar effects and, in
addition, show that signed order flow does result in significant price impacts. Their
results show that the price impact function is non-decreasing in trade size for most
contracts. Kempf and Korn (1999) suggest that price impacts should be a nonlinear
function of trade size rather than linear at least for the DAX futures. However, results
are certainly not as firm and convincing as they are for cash equities and other markets,
since the number of studies in futures markets is fairly low. Corwin and Schultz (2010)
use daily high and low prices of NYSE, Amex and Nasdaq listed stocks to estimate the
bid-ask spreads. Price pressure from large orders may result in execution at daily high
or low prices. Similarly, a continuation of buy or sell orders in a shallow market will
often lead to executions at high or low prices. The high and low prices can capture
these transient price effects in addition to the bid-ask spread. Consequently, Corwin
and Schultz’s (2010) high-low spread estimator captures liquidity in a broader sense
than just the bid-ask spread. This study adopts their methodology, which is both simple
14

15. and elegant. Using the high and low prices from two consecutive days, the high-low
spread estimate of is given as follows
= (1)
where = − , ≡ , = ,
, ( )
,
is the observed high (low) futures price for day , and , ( , ) is the observed
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
15

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 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 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.
16

17. 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 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 , 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
17

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 CFE 5 , 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
14, 2004 that is 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
9, 2009 ( 11, 2009) for VT, which is the second Friday, a week before
Monday after the fifth business day before the expiration date. The last monthly (quarterly) rolling
ceases on
their expiration on October 16, 2009 (September 18, 2009). Similarly, since the final settlement date for
10, 2004 that is the Thursday, one week prior to its maturity. Their last monthly (quarterly) rolling
June-matured VIX futures is June 16, 2004, monthly and quarterly rolling of VIX futures starts on
cease on 14, 2009 ( 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.
18

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.
19

20. 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:
& + ∆ of VIX futures in the first rolling month
$1000 × + ∆ , = 0,1,2, … , −1
=
ℎ + ∆ , =
where ≡ − /∆ is the number of trading days between the current day
and position closing day ; + ∆ = + ∆ −
is the cumulative value of the futures contract at daily settlement on
day + ∆ , taking the difference between the daily settle futures price,
+ ∆ , and the day- futures price at the initiation of the contract,
.
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 ×
= $1000 × −
20

21. 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:
& + ∆ of VIX futures in the second rolling month
= $1000 × + ∆ + ℎ + ∆ , = 1,2, … ,
where ≡ − +∆ /∆ ; + ∆ = +
∆ − +∆ is the cumulative value of the contract on day + ∆
with the opening price, + ∆ , of the second VIX futures initiated on
day + ∆ . The cash balance account, ℎ + ∆ , is given by
ℎ + ∆ in the second rolling month
= ℎ + −1 ∆ × ×
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.
21

22. 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 ( ), so that for, =
0,1,2, … , = − /∆ ,
+ Δ = $10 × + Δ
+ℎ × $1000 × + Δ + ℎ + Δ
The corresponding cumulative P&L is given by
& + Δ = + Δ −
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.
22

23. = $10 × & + Δ + ℎ × $1000 × + Δ
+ ℎ + Δ
where + Δ is the cumulative value of the futures contract on day
+ Δ ∀ ; ℎ + Δ is the cash balance account; = $10 ×
; and & + Δ = + Δ − .
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.,
& + Δ / & + −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.
23

24. [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 ( ) and the implied forward variance ( ). indicates
the realized variance of the SPX corresponding to the front-quarter VT contract.
represents the future variance of the SPX that is implied by the daily settlement price
24

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, = , for =3 months = 1/4 year. The value
of a forward-starting VT contract is composed of 100% implied forward variance
( , ), as given by
= =
,
, , (2)
where 0 < < − < . The analytical pricing formula for front-month VT is
given by
=
,
,
= 1− , + , , (3)
where 0 < − < < . The formula to calculate the annualized realized variance
( ) is as follows10
 N a −1 
RUG = 252 ×  ∑ Ri2 /( N e − 1) 
  (4)
 i =1 
where = ln / is daily return of the S&P 500 from to ; 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 in Eq.(3) multiplying
10,000 is the data available in the Chicago Futures Exchange website.
25

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 N − 1 returns. N a
is the actual number of days in the observation period, and Ne 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 (denoted by , ) is defined as the variance
swap rate, we are able to evaluate , by computing the conditional expectation
under the risk-neutral measure Q, as follows
, ≡ , (5)
Based on Eqs. (2)−(5), the portion of a front-quarter VT contract can be
replicated by , extracted from with identical days to maturity. In
other words, we can synthesize the front-quarter VT with the following:
= 1− × + ×
,
, , (6)
where is the total number of business days in the original term to expiration of the
VT contract, while − is the number of business days left until the final expiration
of the VT contract, and > − .
Since the market price of a forward-starting VT future is completely attributable
26

27. to , this study takes the initial forward (denoted ) curve implicit in
to synthesize the forward-starting VT price, i.e., for ∀ ∈ 0, − ,
=
,
, (7)
Specifically, the following equation uses historic observations to compute
a time series history of forward .
, = , × − − , × − − (8)
where 0 < < − < .
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 , for
< − based on Eq. (7), while for synthetic front-month contracts, we use
, and midpoints of , for − ≤ < 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
27

28. other rolling strategies used in this study. Figure 2 represents the and
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
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.
28

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 calculation from the lack of S&P
500 Special Opening Quotation (“SOQ”) data.11 In addition, for simplicity, , or the
number of expected S&P 500 values needed to calculate daily returns during the
three-month period, is approximated by , 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.
29

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
30

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
31

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 “− ” while
P&L. On day + 1, if the underlying price has not changed, the cash account still remains at “−
the asset account is “+ ”. If he sells the option right away, the net account on day is back at 0
+1 = − ℎ ”. Thus, & on day + 1 is
”
ℎ ”. If the option expires worthless, his cumulative P&L become “− −
and asset account at “
equal to “
” 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.
32

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 , 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
= . 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 ( ) is given as follows:
+∆ = +∆ − ∆ ∆
where = . We shall repeat the net P&L calculation at time + 2∆ , and
so on. Their mark-to-markets are
+ ∆ = + ∆ − ∆ ×∆
for = 1, … , ≡ − /∆ 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.
33

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:
34

35. + ∆ in the first rolling month
0 , =0
= + ∆ − ∆ ×∆
, = 1,2, … , ≡ − /∆
where ≡ − /∆ , and = − 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:
+∆
=− ∆ ×∆
− − ∆ ×∆
The market-to-market of the put option on its 2nd, 3rd,…, and day during the
second rolling month is given, for = 2,3, … , ≡ − /∆ , by
+ ∆ =
+ ∆ − +Δ ∆ ×∆
− ∆ ×∆
− −
× ∆ ×∆ ∆ ×∆
and similar MTM calculations apply to any subsequent periods.
Figure 5 presents the cumulative P&Ls of both the 1-month (Panel A) and
35

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
36

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:
% ; − % ;
where = $10 × ; = $10 × +ℎ×
$1000 × & + ℎ; and % ;∙ is defined as the maximum
sustained percentage decline (peak to trough) for period 0, , which provides an
intuitive and well-understood empirical measure of the loss arising from potential
extreme events. We use the percentage Maximum Drawdown in this case, which is
calculated as the percentage drop from the peak to the trough as measured from the
peak:
−
% ; = max
The graphical results are plotted in Figure 6. Descriptive statistics on both the
unhedged and hedged profits and losses (P&Ls) are also reported in Panel A of Table 3.
Note the following technical details: First, all P&L ( ) time series are starting at
$0 (or at the ℎ value of 100-lot SPX ETF) at the beginning of the empirical
37

38. analysis, from September 8, 2004 to October 14, 2009, covering a period of extreme
volatility due to the bankruptcy of Lehman Brothers. The in-sample data period starts
from June 10, 2004, allowing the use of roughly three months of data to estimate the
first out-of-sample hedging ratio. Second, summary statistics are computed based on
raw daily P&Ls, without any time scaling. Third, maximum drawdown is computed
based on the percentage drop from the peak to the trough as measured from the peak.
Fourth, in this specific analysis, we have computed the Cornish-Fisher at 95%
and 99%, but have noticed minimal differences among the two choices. One may
conclude from the statistical results that:
1. Minimizing Cornish-Fisher CVaR is effective in minimizing maximum drawdown.
“Second-order techniques” such as minimizing absolute residuals and squared
residuals have also shown reasonable performance as extreme downside hedges,
but they also produce some of the most impressive upsides after the bankruptcy of
Lehman Brothers.
2. Minimizing Maximum Drawdown is ineffective. This is not surprising considering
the “look back” nature of maximum drawdown as a measure. It is generally
believed that, under this method, one can only compute the correct hedge ratio
after a significant drawdown has already happened. By then, the hedge is put on
38

39. only when it is no longer needed, and when the hedging instrument is “recoiling”
in its P&L.
3. In theory, the Alternative Sharpe Ratio should perform well as an objective
function for extreme downside hedges, given its attempt to balance both the left
and the right tails of the return distribution. In this case, we can observe a modest
reduction in drawdown without any significant improvement of the upside.
[Table 3 about here]
[Figure 6 about here]
3.2 3-Month VIX Futures
We conduct the empirical hedging analysis based on the five different hedging
methodologies as described above, by using VIX futures as a hedge to one 100-lot unit
of long SPX ETF. The rebalancing, done every three months, is done one week before
the VIX futures roll dates. The graphical results are plotted in Figure 7.
[Figure 7 about here]
The study has also computed the standard descriptive statistics on both the
unhedged and hedged P&Ls in Panel B of Table 3. Note the following technical details:
First, all MTM (P&L) time series are starting at the ℎ value of one 100-lot
unit of SPX ETF ($0) at the beginning of the empirical analysis, from September 8,
39

40. 2004 to September 9, 2009, covering a period of extreme volatility due to the
bankruptcy of Lehman Brothers. The in-sample data period starts from June 10, 2004,
allowing the use of roughly three months of data to estimate the first out-of-sample
hedging ratio. Note that while the SPX ETF starts on September 8, 2004 to ensure a
consistent starting date as in the case of 1-month rolling VIX futures, the 3-month
rolling VIX futures time series actually rolls off on September 09, 2009, resulting in a
shorter time series than that for the 1-month rolling VIX futures time series. Second,
summary statistics are computed based on raw daily P&L, without any time scaling.
Third, maximum drawdown is computed based on the percentage drop from the peak
to the trough as measured from the peak. Fourth, in this specific analysis, we have
computed the Cornish-Fisher CVaR at 95% and 99%, but have noticed minimal
differences between the two choices. One may conclude from the statistical results
that:
1. “Second-order techniques” such as minimizing absolute residuals and squared
residuals have performed well as extreme downside hedges and also in perserving
upside.
2. Minimizing Cornish-Fisher CVaR is more effective than Minimizing Maximum
Drawdown in this case, but all of them perform worse than second-order
40

41. techniques.
3. The Alternative Sharpe Ratio continues to be an reliable average performer —
giving neither surprisingly good nor poor results.
While there is a difference between the 1-month and the 3-month rolling time
series, the difference is not dramatic between the two choices. In practice, rolling every
3-month saves on transaction costs, but the longer-dated futures are also known to be
less responsive to shocks in the spot VIX index, making them less desirable as hedging
instruments. For real-life traders, the choice of an appropriate tenor is likely to be
determined by the actual liquidity and transaction costs available at the time of
execution — any mechanical comparison between the 1-month and the 3-month rolling
time series may seem irrelevant. For the rest of this paper, we will focus on comparing
different hedging instruments based on their 1-month rolling time series, which tends
to be where liquidity can be found in real-life trading.
3.3 1-Month Variance Futures
This section conducts the empirical hedging analysis based on the five different
hedging methodologies as described above, by using VT futures as a hedge to one
100-lot unit of long S&P500 ETF. The algorithm for creating the monthly rolls of
synthetic 1-month variance futures has been described in Section 2.3.1. The graphical
41

42. results are plotted as shown in Figure 8.
[Figure 8 about here]
Their standard descriptive statistics on both the unhedged and hedged profits and
losses (P&Ls) are reported in Panel C of Table 3. Note the following technical details:
First, all MTM (P&L) time series are starting at the ℎ value of one 100-lot
unit of SPX ETF ($0) at the beginning of the empirical analysis, from September 10,
2004 to October 9, 2009, covering a period of extreme volatility due to the bankruptcy
of Lehman Brothers. The in-sample data period starts from June 14, 2004, allowing the
use of roughly three months of data to estimate the first out-of-sample hedging ratio.
Second, summary statistics are computed based on raw daily P&Ls, without any time
scaling. Third, maximum drawdown is computed based on the percentage drop from
the peak to the trough as measured from the peak. Fourth, in this specific analysis, we
have computed the Cornish-Fisher CVaR at 95% and 99%, but have noticed minimal
differences between the two choices. One may conclude from the statistical results
that:
1. “Second-order techniques” such as minimizing absolute residuals and squared
residuals are not reasonably effective as extreme downside hedges in this case.
2. Minimizing the Cornish-Fisher CVaR is more effective than minimizing the
42

43. Maximum Drawdown; in addition, minimizing the Cornish-Fisher CVaR offers a
significant improvement over second-order techniques.
3. Once again, the Alternative Sharpe Ratio continues to be an average performer —
giving neither surprisingly good nor poor results.
The problem with using the VT is that its implied negative carry costs can be
very expensive. This implied negative carry is caused by the burn rate on the premia of
options used to replicate the variance futures. Because of VT is the square of the VIX,
when the upside of VT kicks in, it does make an dramatic improvement to the portfolio.
In fact, the consistent negative carry (running at roughly 20% per year before the
Financial Crisis) has resulted in an increase in maximum drawdown for all hedging
models. Such a large negative carry will likely deter any real-life traders from using
such an instrument for hedging.
3.4 1-Month Out-of-Money SPX Put Options
This section conducts the out-of-sample hedging analysis based on the five
different hedging methodologies as described above, by using 10% OTM SPX puts as
a hedge to one 100-lot unit of long S&P500 ETF. The algorithm for creating the
monthly rolls of 1-month SPX puts has been described in Section 2.4. The graphical
results are plotted in Figure 9.
43

44. [Figure 9 about here]
The standard descriptive statistics on both the unhedged and hedged profits and
losses (P&Ls) are presented in Panel D of Table 3. Note the following technical details:
First, all P&L (MTM) time series are starting at $0 (at the ℎ value of one
100-lot unit of SPX ETF) at the beginning of the empirical analysis, from September
17, 2004 to October 16, 2009, covering a period of extreme volatility due to the
bankruptcy of Lehman Brothers. The in-sample data period starts from June 21, 2004,
allowing the use of roughly three months of data to estimate the first out-of-sample
hedging ratio. Second, descriptive statistics are computed based on raw daily P&Ls,
without any time scaling. Third, maximum drawdown is computed based on the
percentage drop from the peak to the trough as measured from the peak. Fourth, in this
specific analysis, we have computed the Cornish-Fisher CVaR at 95% and 99%, but
have noticed minimal differences between the two choices. One may conclude from
the statistical results that:
1. “Second order techniques” such as minimizing absolute residuals and squared
residuals are effective in controlling the drawdown, but they have produced
some very interesting upsides for the portfolio.
2. In this case, minimizing the Maximum Drawdown is ineffective. Minimizing
44

45. Cornish-Fisher CVaR is not effective in controlling the drawdown (the drawdown
actually increases), but has produced interesting upsides for the portfolio.
3. Once again, the Alternative Sharpe Ratio is a poor performer for controlling the
drawdown in this case — but it has produced average results for maintaining the
upside.
Monthly 10% OTM SPX puts go into the money about once every decade. Very
often, they become costly propositions as extreme downside hedges when market
volatility shoots up (thereby increasing the costs of the options), but the net return to
the hedger (even after the option goes into money) is still too small to cover the
cumulative option premia over time. Because of the steep premia of OTM puts, many
hedgers either (i) underhedge with a smaller than suitable notional amount or (ii) use
options further out of the money, lowering the payoff when the option goes into the
money. In this case, the observed burn rate is roughly 20% per year before the
Financial Crisis, which is likely deter to any real-life traders from using such an
instrument for hedging.
3.5 Overall Comparison on Choices of Hedging Instruments
Figure 10 presents a comparison of the results from using different hedge
instruments under the hedging model of minimizing squared residuals. The “squared
45

46. residual” model is chosen because (i) it has produced reasonably consistent
performance under almost all cases and (ii) it is a widely-accepted hedging models
among practitioners. Some key observations are as follows: First, as noted earlier, the
negative carries from using VT and 10% OTM SPX puts are way too high for them to
be deployed as practical hedging solutions. Interesting enough, it is more costly to use
10% OTM SPX puts than to use VT, but without any substantial improvements to the
upside despite the higher costs involved. This is surprising considering that VT can be
created from a series of SPX options in theory, and 10% OTM puts are among the
cheapest liquid options available. This suggests a possible market anomaly since it
seems unusual that using the synthesized product is cheaper than using the raw
materials, unless the “smile” of the volatility curve is so steep that it becomes
cost-ineffective to use way out-of-the-money put options. Second, using 3-month VIX
futures has a lower negative carry than using 1-month VIX futures. This is consistent
with our expectations given that the more frequent rolling of 1-month VIX futures will
increase transaction costs. Interestingly, the upsides achieved by both methods are
reasonably close. In other words, the market is reasonably “efficient” in that any
increase in transaction costs more or less offsets the lack of responsiveness (relative to
the spot VIX) by using the 3-month VIX. Real-life traders have developed the correct
46

47. market intuition by making the choice of tenor primarily based on the liquidity
available at the time of execution. Third, the overall winner in our analysis is the use of
3-month VIX futures, which has provided both extreme downside protection as well as
upside preservation. The pragmatic issue faced by real-world hedgers is whether they
can execute any such hedging trades with the extended tenor in reasonable size. That is
an empirical question that can only be satisfactorily answered by placing large trades
directly in the VIX futures market.
[Figure 10 about here]
4. Conclusions
This paper attempts to address whether “long volatility” is an effective hedge
against a long equity portfolio, especially during periods of extreme market volatility.
Our study examines using volatility futures and variance futures as extreme downside
hedges, and compares their effectiveness against traditional “long volatility” hedging
instruments such as 10% OMT put options on the SPX. In each case, out-of-sample
hedging ratios are calculated based on five reasonable choices of objective functions,
and the hypothetical performance of the hedge is computed again a long portfolio
consisted of a 100-lot unit of SPX ETF.
Our empirical results show that the CBOE VIX and variance futures are more
47

48. effective extreme downside hedges than out-of-the-money put options on the S&P 500
index, especially when the empirical analysis has taken reasonable actual and/or
estimated costs of rolling contracts 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.
The overall winner in our analysis is the use of 3-month VIX futures. Empirical
evidence also suggests that the pros and cons of using liquid VIX futures contracts
with different tenors more or less offset one another: i.e. real-life traders have
developed the correct market intuition by making the choice of tenor primarily based
on the liquidity available at the time of execution. The pragmatic issue faced by
real-world hedgers is whether they can execute any such hedging trades with the
extended tenor in reasonable size, which is an empirical question that can only be
satisfactorily answered by placing large trades directly in the VIX futures market.
By replicating hedging techniques used by real-life traders with bid/ask-level
market data, our findings suggest the following conclusions and recommendations:
First, using volatility instruments as extreme downside hedges, especially when
combined with the appropriate hedging techniques, can be a viable alternative to
buying a series of out-of-the-money put options on SPX. Second, the
48

49. volatility/variance market may be more efficient than generally believed. Third, there
is a business case supporting that volatility/variance instruments can be made more
widely available as extreme downside hedging instruments. In Asia ex-Japan, this can
be accomplished by (i) creating daily benchmark volatility indices on the Hang Seng
Index (HSI) and the Straits Times Index (STI), in a fashion similar to the Volatility
Index Japan (VXJ), the Volatility Index of TAIFEX index options, and the Volatility
Index of the KOSPI 200 (VKOSPI); and (ii) creating exchange-traded
volatility/variance instruments once such reference indices gain acceptance by the
OTC market.
49

50. References
Corwin, Shane A., and Paul Schultz, 2010, A simple way to estimate bid-ask spreads
from daily high and low prices, Working Paper, Mendoza College of Business,
University of Notre Dame.
Driessen, Joost, and Pascal Maenhout, 2007, An empirical portfolio perspective on
option pricing anomalies, Review of Finance 11, 561–603.
Egloff, Daniel, Markus Leippold, and Liuren Wu, 2010, Variance risk dynamics,
variance risk premia, and optimal variance swap investments, Journal of
Financial and Quantitative Analysis, forthcoming.
French, Kenneth, and Richard Roll, 1986, Stock return variances: The arrival of
information and the reaction of traders, Journal of Financial Economics 17,
5−26.
Hafner, Reinhold, and Martin Wallmeier, 2008, Optimal investments in volatility,
Financial Markets and Portfolio Management 22, 147−167.
Harris, Lawrence, 1986, A transaction data study of weekly and intradaily patterns in
stock returns, Journal of Financial Economics 16, 99−117.
Hasbrouck, Joel, 2004, Liquidity in the futures pits: Inferring market dynamics from
incomplete data, Journal of Financial and Quantitative Analysis 39 (2), 305−326.
50

51. Huang, Yuqin, and Jin E. Zhang, 2010, The CBOE S&P 500 three-month variance
futures, Journal of Futures Markets 30, 48–70.
Kempf, Alexander and Olaf Korn, 1999, Market depth and order size, Journal of
Financial Markets 2, 29–48.
Kuruc, Alvin, and Bernard Lee, 1998, How to Trim Your Hedges, Risk 11 (12), 46−49.
Lee, Bernard, and Youngju Lee, 2004, The alternative sharpe ratio, in: Schachter, B.
(Ed.), Intelligent Hedge Fund Investing (ed. B.), Risk Books, London, pp. 143 –
177.
Manaster, Steven and Steven C. Mann, 1996, Life in the pits: Competitive market
making and inventory control, Review of Financial Studies 9, 953−975.
Moran, Matthew T., and Srikant Dash, 2007. VIX futures and options: Pricing and
using volatility products to manage downside risk and improve efficiency in
equity portfolios. Journal of Trading 2, 96−105.
Szado, Edward, 2009, VIX futures and options — A case study of portfolio
diversification during the 2008 financial crisis, Journal of Alternative Investments
12, 68−85.
51

52. Table 1 Distribution of Bid-Ask Spreads
This table 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 the 3-month
VT. Daily bid and ask estimators of synthetic 1-month VT are constructed from CBOE VIX
Term Structure. Actual bid and ask prices for VIX futures, the 3-month VT, and 10% OTM
SPX puts are obtained from Bloomberg and CBOE, respectively. Monthly-rolling daily spread estimates
are calculated for the full sample period from June 2004 through October 2009, while quarterly-rolling
daily spreads are computed for the full sample period from June 2004 through September 2009. The
spread estimates for the period of the Financial Crisis triggered by the bankruptcy of Lehman Brothers
are also separately tabulated in Panel B. The unit of bid-ask spreads is the dollar premium quote. 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 $100. The figure in parentheses is a spread of %,
of which actual/estimated bid-ask spreads equal % times daily settlement prices of VIX futures and VT
or the midpoints of SPX puts.
−
Monthly Rolling Quarterly Rolling
Synthetic 10% OTM 10% OTM
.
($) VIX Futures VIX Futures VT
1-month VT SPX Puts SPX Puts
1,347 NA 1,342 1,322 1,323 1,323
1,347 NA NA 1,322 1,323 NA
NA 1,342 NA NA NA NA
$ 109.70 $ 59.09 $ 128.49 $ 2,070.97 $ 99.07
NA (75.72 %) (8.35%) (37.32%)
(0.60%) (0.65%)
337.41 262.94 1,596.83
NA NA (5.24) NA
(1.34) (1.05)
$ 3,898.52
NA (14.62%) NA NA NA NA
90.00 30.00 100.00 750.00 55.00
NA (53.06) (7.25) (18.75)
(0.48) (0.56)
157.76 127.05 425.00
NA NA (3.72) NA
(0.95) (0.72)
1,517.05 NA NA NA
NA (14.39) NA
990.00 1,470.00 1,650.00 25,000.00 1,500.00
NA
(2.46) (200.00) (11.10) (43.28) (200.00)
4,922.70 4,284.70 43,875.00
NA NA NA
(16.48) (9.51) (140.71)
97,974.12 NA NA NA
NA (56.44) NA
10.00 5.00 10.00 50.00 5.00
NA (1.74) (0.35) (1.60)
(0.02) (0.02)
0.12 0.09 0.00
NA NA (0.00) NA
(0.00) (0.00)
197.19 NA NA NA
NA (2.41) NA
85.70 117.20 107.03 3,053.77 137.70
NA (63.66) (5.08) (47.03)
(0.42) (0.50)
506.63 393.34 3,841.03
NA NA (6.35) NA
(1.34) (1.07)
7,960.01
NA (5.27) NA NA NA NA
3.00 5.84 5.03 2.66 3.82
NA (0.91) (2.01) (2.44)
(0.99) (7.56)
3.71 3.78 5.54
NA NA (9.20) NA
(2.89) (2.31)
52

53. 5.79 NA NA NA
NA (0.79) NA
19.69 47.36 52.52 11.71 25.14
NA (2.50) (8.51) (8.36)
(3.46) (144.57)
21.82 23.08 41.75
NA NA (174.40) NA
(19.84) (10.41)
47.90
. ℎ ℎ 15, 2008 − 31, 2008
NA (6.25) NA NA NA NA
76 NA 76 76 76 76
76 NA NA 76 76 NA
NA 76 NA NA NA NA
254.61 397.08 246.18 8,914.67 439.19
NA (23.40) (5.16) (25.93)
(0.52) (0.58)
1,348.14 953.41 10,603.05
NA NA (6.44) NA
(2.84) (2.19)
26,887.52 NA NA NA
NA (16.69) NA
230.00 400.00 190.00 8,750.00 410.00
NA (16.28) (4.97) (15.03)
(0.45) (0.45)
1,025.15 625.00 7,475.00
NA NA (4.30) NA
(2.54) (2.00)
19,984.64
NA (15.50) NA NA NA NA
990.00 1,470.00 1,350.00 25,000.00 1,500.00
NA (200.00) (11.61) (200.00)
(1.75) (2.55)
4,922.70 4,284.70 43,875.00
NA NA (37.29) NA
(8.39) (6.47)
97,974.12 NA NA NA
NA (56.44) NA
10.00 10.00 10.00 250.00 10.00
NA
(0.02) (4.38) (0.02) (0.73) (5.67)
10.00 13.29 0.00
NA NA NA
(0.02) (0.04) (0.00)
5,214.95 NA NA NA
NA (2.76) NA
196.49 304.15 218.56 4,989.16 297.89
NA (27.60) (2.15) (36.57)
(0.38) (0.51)
1,107.31 871.19 10,188.15
NA NA (6.24) NA
(2.23) (1.71)
19,371.58 NA NA NA
NA (9.19) NA
1.10 1.14 2.03 0.62 1.02
NA (4.35) (0.53) (3.63)
(0.97) (1.61)
0.96 1.19 1.24
NA NA (2.07) NA
(0.73) (0.53)
1.71
NA (1.30) NA NA NA NA
4.29 4.45 9.88 3.93 4.76
NA (25.79) (3.22) (16.21)
(3.53) (5.86)
3.53 4.39 3.96
NA NA (9.72) NA
(2.68) (2.20)
5.69
NA (6.04) NA NA NA NA
53

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 = 1−
+ , , , these are given as the bid pricing error: = −
, where , , is the bid quotation of CBOE VIX Term Structure with comparable
days to expiration and is realized variance implicit in the daily returns of the S&P 500
index. Similarly, and are calculated from midpoints , and ask quotes , ,
respectively. The unit of 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
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 ( ) for the remaining ten days. =123.06.
Taking the square root of the , 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 .
ℎ
Sample size 1323 1323 1323
24.7233 −3.8346 −32.3176
8.5006 −1.4238 −12.8784
Maximum 542.3729 310.0932 105.2920
Minimum −114.3666 −494.9200 −1125.9654
62.4415 40.1845 71.5199
Q1 2.9785 −8.5093 −26.9828
Q3 26.7968 3.5121 −4.8220
Skewness 4.1088 −1.8077 −5.3837
Kurtosis 27.2165 29.2939 55.1695
54