Using Volatility Instruments As Extreme Downside Hedges-August 23, 2010

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

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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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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.


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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

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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

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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

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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

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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

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.-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

Kronecker delta notation):

>@" = #@#"∆% A B��BC.�@ BC.�"D = #@#"∆% A�@�" B��B�@ B�" + ?@"�" B�B�"D

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

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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 (E�E) value of the hedged portfolio,

Fℎ��%�, or

min3 G |Fℎ��%�|�IJI�

where Fℎ��%� = K�KJ�K�KJ;�� − 1 = �J�M3NOP� Q&SJ��J;��M3NOP� Q&SJ;�� − 1 ; �%� is the

day-% mark-to-market of the unhedged portfolio; and TUV W&X%� = T%� −

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T0� is the cumulative P&L of the hedging instrument on day %, assuming

T0� = 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, Fℎ��%�, or

min3 G &Fℎ��%�'��IJI�

3. Minimize the peak-to-trough maximum drawdown of the mark-to-market value

of the hedged portfolio:

min3 E�TZZ�; \�%� + ℎTUV W&X%�]J^_� �

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 E�E%� = �%� + ℎTUV W&X%� be the dollar

mark-to-market value of the brokerage account representing the hedged

portfolio at the end of the period �0, %!, and E�E_IaIJbcde = ��T_IaIJ�E�Ef�! be the maximum dollar mark-to-market value in the �0, %! period. The

drawdown at any time %, denoted ZZ%�, is defined as

ZZ%� = E�E_IaIJbcde − E�E%�

The Maximum Drawdown (E�TZZ ) from time 0 up to time � is the

maximum of the drawdown over the history of the E�E. Formally,

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E�TZZ�; \E�E%�]J^_� � = max_IJI�\ZZ%�]

Alternatively, E�TZZ% is also used to describe the percentage drop from the

peak to the trough as measured from the peak. Using E�TZZ as a measure of

risk, the optimization procedure will find an estimate of the hedge ratio ℎ that

can achieve the lowest possible E�TZZ.

4. Minimize the Expected Shortfall or Conditional Value-at-Risk at 95%

computed from the daily P&L of the hedged portfolio, or equivalently,

min3 i��&�W&X%� + ℎ TW&X%�' with �W&X%� = �%� − �% − 1� and

TW&X%� = T%� − T% − 1� using the Cornish-Fisher expansion (Lee and Lee,

2004), where

i��jk%∙� = −lm∙� + #∙�n&oUp,�;qrs > 95%'w

= −m∙� − #∙�nxyyyyz{ o|�;q� + 16 &o|�;q�� − 1'~∙�

+ 124 &o|�;q�� − 3o|�;q�'�∙�− 136 &2o|�;q�� − 5o|�;q�'~∙��

� s > 95%��

with o|�;q� being the critical value for probability 1 − s with standard

normal distribution (e.g. o|�;q� = −1.64 �% s = 95%), while m, #, ~ and

� following the standard definitions of mean, volatility, skewness and excess

kurtosis, respectively, as computed from the daily P&L of the hedged portfolio.

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Practically, the expected value of the tail of oUp,�;q at and above 95%

estimated numerically by using the discrete average of o|�;q� taken at 95.5%,

96.5%, 97.5%, 98.5% and 99.5% (e.g. o|�;q� = −1.70 at s = 95.5% ,

o|�;q� = −1.81 at s = 96.5% , o|�;q� = −1.96 at s = 97.5% ,

o|�;q� = −2.17 at s = 98.5%, and o|�;q� = −2.58 at s = 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:

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�~ ≡ ∑ �@�@@o�;#� + 12 ∑ �@o@M#@��@ o�;#� − 12 o�;#�

where

�@ = excess return rate of the 8-th asset of the portfolio �; given that our study

uses a single hedging instrument, the 8-th asset is the portfolio itself, which is

calculated as the percentage changes in the mark-to-market value of the hedged

portfolio, Fℎ��%�

�@ = 8-th position of the portfolio �

o�M = dN&�O�&��',_'�� where o|M is critical value for probability s and

o�; = @�&�O�&��',_'�� where o|; is critical value for probability 1 − s

(e.g. o|M = 2.33 at 1%, o|; = −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

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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

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and elegant. Using the high and low prices from two consecutive days, the high-low

spread estimate of � is given as follows

� = �c�;��Mc� (1)

where � = ��;��;�√� − < ��;�√� , > ≡ ∑ C. ¡¢£�¤¥X£�¤¥ ¦§��"^_ , ¨ = C. ¡¢£,£��¥X£,£��¥ ¦§�

, ©Jª («Jª )

is the observed high (low) futures price for day %, and ©J,JM�ª («J,JM�ª ) 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

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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

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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.

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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

= $1000 × »·8T»³%UV �� + ¿∆%� + =��ℎ�� + ¿∆%� , »µ� ¿ = 1,2, … , E�

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.

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 (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.

23

= $10 × ~¸�UV W&X% + ÃΔ%� + ℎ × �$1000 × »·8T»³%UV % + ÃΔ%�!+ =��ℎ% + ÃΔ%�

where »·8T»³%UV % + ÃΔ%� is the cumulative value of the futures contract on day

% + ÃΔ% »µ� ∀Ã ; =��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.

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 (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 �Í³�� 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:

ÆJ̄ �,p �� = Ê1 − �;Ja� Ë × Å:�;a�,J + Ê�;Ja� Ë × ��J,�� (6)

where f� 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 f� > � − %.

Since the market price of a forward-starting VT future is completely attributable

27

to �Å:, this study takes the initial forward �� (denoted »��) curve implicit in

�� to synthesize the forward-starting VT price, i.e., for ∀% ∈ �0, � − f�!, ÆJ̄ �,p¾�� = »��;a�,�� %� (7)

Specifically, the following equation uses historic �� observations to compute

a time series history of forward ��.

»��;a�,�� %� = �a� l��J,�� × � − %� − ��J,�;a�� × � − f� − %�w (8)

where 0 < % < � − f� < �.

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.


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.


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


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.


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

so on. Their mark-to-markets are

E�E% + C∆%� = ¸³%% + C∆%� − ¸³%%� Ó �ÀJM@∆J�×∆J®@^�

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.


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:

E�TZZ%�; E�EØcpª�c 3cÙÚc� − E�TZZ%�; E�EdpJc� 3cÙÚc�

where E�EØcpª�c 3cÙÚc = $10 × ~¸� ; E�EdpJc� 3cÙÚc = $10 × ~¸� + ℎ ×$1000 × »·8T»³%UV W&X + =��ℎ; and E�TZZ%�;∙� 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:

E�TZZ%�; \E�E]J^_� � = max_IJI� ÛE�E_IaIJbcde − E�E%�E�E_IaIJbcde Ü

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 (E�E) time series are starting at

$0 (or at the ³.ℎ�ÐÝ�Ð value of 100-lot SPX ETF) at the beginning of the empirical

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 i�� 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

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.



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.


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 ³.ℎ�ÐÝ�Ð value of one 100-lot

unit of SPX ETF ($0) at the beginning of the empirical analysis, from September 8,

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

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

42

results are plotted as shown in Figure 8.


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 ³.ℎ�ÐÝ�Ð 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

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.

44


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 ³.ℎ�ÐÝ�Ð 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 .µ% 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

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

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

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.


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

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

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.

50

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diversification during the 2008 financial crisis, Journal of Alternative Investments

12, 68−85.

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 ¬�¯°±�c� of synthetic 1-month VT are constructed from CBOE VIX

Term Structure. Actual bid and ask prices ¬��UJVd® 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 ¬8Ð − ��ß ~F��Ð ($) VIX Futures Synthetic

1-month VT 10% OTM

SPX Puts VIX Futures VT

10% OTM SPX Puts ¸�.�C �. Æ³CC ~��FC� Ñ ¬��UJVd® 1,347 NA 1,342 1,322 1,323 1,323

¬�| 1,347 NA NA 1,322 1,323 NA

¬�¯°±�c� NA 1,342 NA NA NA NA

E ¬��UJVd® $ 109.70

(0.60%) NA

$ 59.09 (75.72 %)

$ 128.49

(0.65%) $ 2,070.97

(8.35%) $ 99.07

(37.32%)

¬�| 337.41

(1.34) NA NA

262.94

(1.05) 1,596.83

(5.24) NA

¬�¯°±�c� NA $ 3,898.52

(14.62%) NA NA NA NA

EÐ. ¬��UJVd® 90.00

(0.48) NA

30.00 (53.06)

100.00

(0.56) 750.00 (7.25)

55.00 (18.75)

¬�| 157.76

(0.95) NA NA

127.05

(0.72) 425.00 (3.72)

NA

¬�¯°±�c� NA 1,517.05

(14.39) NA NA NA NA

E�T ¬��UJVd® 990.00

(2.46) NA

1,470.00 (200.00)

1,650.00

(11.10) 25,000.00

(43.28) 1,500.00 (200.00)

¬�| 4,922.70

(16.48) NA NA

4,284.70

(9.51) 43,875.00

(140.71) NA

¬�¯°±�c� NA 97,974.12

(56.44) NA NA NA NA

E8. ¬��UJVd® 10.00

(0.02) NA

5.00 (1.74)

10.00

(0.02) 50.00 (0.35)

5.00 (1.60)

¬�| 0.12

(0.00) NA NA

0.09

(0.00) 0.00

(0.00) NA

¬�¯°±�c� NA 197.19 (2.41)

NA NA NA NA

~�Z ¬��UJVd® 85.70

(0.42) NA

117.20 (63.66)

107.03

(0.50) 3,053.77

(5.08) 137.70 (47.03)

¬�| 506.63

(1.34) NA NA

393.34

(1.07) 3,841.03

(6.35) NA

¬�¯°±�c� NA 7,960.01

(5.27) NA NA NA NA

~ß��.�� ¬��UJVd® 3.00

(0.99) NA

5.84 (0.91)

5.03

(7.56) 2.66

(2.01) 3.82

(2.44)

¬�| 3.71

(2.89) NA NA

3.78

(2.31) 5.54

(9.20) NA

53

¬�¯°±�c� NA 5.79

(0.79) NA NA NA NA

�³�%µ�8� ¬��UJVd® 19.69

(3.46) NA

47.36 (2.50)

52.52

(144.57) 11.71 (8.51)

25.14 (8.36)

¬�| 21.82

(19.84) NA NA

23.08

(10.41) 41.75

(174.40) NA

¬�¯°±�c� NA 47.90 (6.25)

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:.

~Ò.%ℎ�%8= ��µ�� ¸nØ@Ù ¸n @Ù ¸nd¾e

Sample size 1323 1323 1323 E 24.7233 −3.8346 −32.3176 EÐ. 8.5006 −1.4238 −12.8784

Maximum 542.3729 310.0932 105.2920

Minimum −114.3666 −494.9200 −1125.9654 ~n 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

55

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.

N M Mdn Max Min SE Skewness Kurtosis %MaxDD MaxDD

Reduction ¸�.�C �. ©�ÐÝ8.Ý �8%ℎ 1 − Eµ.%ℎ µCC �� Æ³%³��

Unhedged Daily P&L of ETF 1,286 -0.1886 10.4000 1,041.3000 -1,068.5000 158.5964 -0.3951 9.8015 56.7754 NA

Hedged P&L of CVaR (99%) 1,286 0.1064 0.7892 578.4022 -676.4387 108.2907 0.1230 8.0682 30.2757 26.4997

Hedged P&L of CVaR (95%) 1,286 -0.0962 -1.5148 578.3888 -612.0395 102.8994 0.3253 9.2858 31.5135 25.2619

Hedged P&L of Squared Residuals 1,286 0.7446 -3.4703 710.0810 -504.6844 103.3491 1.1553 13.4364 34.4279 22.3475

Hedged P&L of ASR 1,286 -0.8495 -3.9042 615.9115 -563.8233 102.4816 0.5676 10.7006 34.6108 22.1645

Hedged P&L of Absolute Residuals 1,286 0.0314 -4.5413 745.4857 -522.8739 107.8800 1.2038 13.7347 36.7407 20.0347

Hedged P&L of Maximum Drawdown 1,286 -2.1053 -5.7246 738.0256 -645.1556 108.8194 0.7424 12.1729 37.5413 19.2341 ¸�.�C ¬. ©�ÐÝ8.Ý �8%ℎ 3 − Eµ.%ℎ µCC �� Æ³%³��


Hedged P&L of Absolute Residuals 1,261 1.9462 -0.8721 782.3203 -571.4506 108.5573 0.4204 11.3838 29.2318 27.5436

Hedged P&L of Squared Residuals 1,261 1.6406 0.1705 1,076.2284 -590.4172 118.2364 1.0349 15.8422 29.8988 26.8765

Hedged P&L of CVaR (95%) 1,261 0.8558 0.1447 911.8982 -643.0565 112.6528 0.4959 13.9038 31.6962 25.0792

Hedged P&L of ASR 1,261 0.6987 -0.2771 1,039.5360 -713.6413 116.8077 0.6805 16.1108 32.0416 24.7338

Hedged P&L of CVaR (99%) 1,261 0.7469 1.3222 1,037.7292 -712.5766 119.3144 0.5603 15.7663 32.1667 24.6087

Hedged P&L of Maximum Drawdown 1,261 1.7532 0.4049 1,087.0567 -739.8727 120.3740 0.8829 17.0031 33.3421 23.4333 ¸�.�C i. ©�ÐÝ8.Ý �8%ℎ 1 − Eµ.%ℎ µCC ��


Hedged P&L of CVaR (99%) 1,280 2.0736 6.8298 5,574.7937 -1,693.6304 241.5745 10.9839 249.3082 62.4046 -5.6293

Hedged P&L of CVaR (95%) 1,280 1.3339 5.8185 5,358.2763 -1,693.1674 237.7148 10.3652 230.5780 65.6095 -8.8341

Hedged P&L of Squared Residuals 1,280 3.1297 5.4710 7,765.6996 -1,801.5350 293.8598 16.0251 409.5746 72.5345 -15.7591

Hedged P&L of ASR 1,280 0.9630 4.5467 6,745.0399 -1,836.2260 267.5257 13.5350 336.2760 74.7476 -17.9722

Hedged P&L of Absolute Residuals 1,280 1.6028 5.2071 8,058.8990 -1,945.6195 299.4175 16.2921 430.7170 82.1645 -25.3891

Hedged P&L of Maximum Drawdown 1,280 1.3942 4.9793 8,555.2542 -2,013.0657 312.3023 17.0915 460.2193 85.1590 -28.3836 ¸�.�C Z. ©�ÐÝ8.Ý �8%ℎ 1 − Eµ.%ℎ µCC ~¸� ¸³%�


Hedged P&L of Maximum Drawdown 1,280 5.7609 3.7184 4,352.7777 -4,993.1594 281.1385 0.0627 145.0942 53.1397 3.6357

56

Hedged P&L of CVaR (99%) 1,280 -6.0883 -8.9577 4,354.0381 -4,997.0452 271.4551 -0.1946 161.0495 88.4597 -31.6844

Hedged P&L of Squared Residuals 1,280 2.9678 -4.3139 5,230.0272 -6,002.5866 330.8145 0.1095 158.7258 88.9501 -32.1747

Hedged P&L of CVaR (95%) 1,280 -4.5555 -8.6922 4,354.0284 -4,997.0823 281.5294 0.1179 144.7715 89.3658 -32.5904

Hedged P&L of Absolute Residuals 1,280 4.3372 -4.9899 6,104.1945 -7,006.3837 383.9240 0.1252 163.0461 95.0837 -38.3083

Hedged P&L of ASR 1,280 0.1249 -4.7818 4,352.1665 -4,995.4903 280.9029 0.0760 145.8005 98.5676 -41.7923

57

Panel A. 1-month roll of VIX futures

Panel B. 3-month roll of VIX futures

Figure 1 Cumulative dollar P&Ls of the ETF, VIX futures and cash balance account. The 1-month

roll strategy in Panel A covers the period from June 10, 2004 to October 14, 2009, while the 3-month

roll strategy in Panel B is from June 10, 2004 to September 9, 2009. SPX ETFs are valued at

approximately 1/10th the value of the SPX and typically tend to be transacted in 100-lot (or “round-lot”)

increments. Since the study tries to figure out how many contracts of VIX futures are required to hedge a

100-lot unit of SPX ETF, a multiplier for adjusting unit size and unit dollar value is applied in the

hedged portfolio. The contract size of VIX futures is $1,000 times the VIX. E�Eá@NpVJ is the

accumulation of the security asset and cash balance accounts of VIX futures. The fourth subplot of each

panel plots trading dates versus cumulative P&L of ETF with y-axis labeling on the left and plots trading

dates versus cumulative P&L of E�Eá@NpVJ with y-axis labeling on the right.

58

Panel A. âãäå æçè Evolution

Panel B. éäêëìêí æçè Evolution

Figure 2. æçèîïêð evolution. Panel A of this chart represents the CBOE VIX Term Structure

midpoints as of the market close dates and days to VT maturities over the period from June 14, 2004 to

October 9, 2009. The round markers indicate actual observations. Panel B represents the initial forward

VIX as of the market close dates and days to nearby VT maturities. The round markers indicate actual

forward VIX observations.

59

Panel A. Evolution of price differences between market VT and synthetic VT ñääòóôõ ìöä÷ï åøï ùú âûêüìýï

ñääòóôõ öïþäë åøï ùú âûêüìýï

Panel B. Histogram of price differences between market VT and synthetic VT

Figure 3. Price differences between market VT and synthetic VT. Panel A represents the price

differences between market VT and synthetic VT constructed from daily returns of the SPX and CBOE

VIX Term Structure midpoints across the market close dates and VT maturities over the period from

June 18, 2004 to October 21, 2009. The round markers indicate actual observations. Panel B shows

histograms of those pricing errors across bid, midpoint and ask quotes of CBOE VIX Term Structure.

60

Panel A. 1-month roll of synthetic VT

Panel B. 3-month roll of VT

Figure 4 Cumulative dollar P&Ls of the ETF, VT and cash balance account. The 1-month roll

strategy in Panel A covers the period from June 14, 2004 to October 9, 2009, while the 3-month roll

strategy in Panel B is from June 14, 2004 to September 11, 2009. SPX ETFs are valued at approximately

1/10th the value of the SPX and typically tend to be transacted in 100-lot (or “round-lot”) increments.

Since the study tries to figure out how many contracts of VT are required to hedge a 100-lot unit of SPX

ETF, a multiplier for adjusting unit size and unit dollar value is applied in the hedged portfolio. The

contract multiplier for the VT contract is $50 per variance point. E�E¯� is the accumulation of the

security asset and cash accounts of VT. The fourth subplot of each panel plots trading dates versus

cumulative P&L of ETF with y-axis labeling on the left and plots trading dates versus cumulative P&L

of E�E¯� with y-axis labeling on the right.

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Panel A. 1-month roll of SPX puts

Panel B. 3-month roll of SPX puts

Figure 5 Cumulative dollar P&Ls of the ETF, SPX puts and cash balance account. The 1-month

roll strategy in Panel A covers the period from June 21, 2004 to October 16, 2009, while the 3-month

roll strategy in Panel B is from June 21, 2004 to September 18, 2009. SPX ETFs are valued at

approximately 1/10th the value of the SPX and typically tend to be transacted in 100-lot (or “round-lot”)

increments. Since the study tries to figure out how many contracts of SPX puts are required to hedge a

100-lot unit of SPX ETF, a multiplier for adjusting unit size and unit dollar value is applied in the

hedged portfolio. One point of SPX options equals $100. E�E¾bNbVJ is the accumulation of the

security asset and cash accounts of SPX puts. The fourth subplot of each panel plots trading dates versus

cumulative P&L of ETF with y-axis labeling on the left and plots trading dates versus cumulative P&L

of E�E¾bNbVJ with y-axis labeling on the right.

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Figure 6. VIX Futures 1-month rolling contracts (September 8, 2004 − October 14, 2009). The

ETF line is the unhedged MTM of holding one 100-lot unit of the S&P 500 ETF in dollars. The APE

line is the hedged MTM by minimizing the sum of absolute residuals. The SPE line is the hedged MTM

by minimizing the sum of squared residuals. MaxDD is the hedged MTM by minimizing maximum

drawdown. The CVaR(95%) and CVaR(99%) lines represent the hedged MTM by minimizing the

conditional Value-at-Risk computed using the Cornish-Fisher expansion at 95% and 99%, respectively.

Finally, the ASR line is the hedged MTM by maximizing the Alternative Sharpe Ratio of the hedged

portfolio.

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Figure 7. VIX futures 3-month rolling contracts (September 8, 2004 to September 9, 2009). The ETF

line is the unhedged MTM of holding one 100-lot unit of the S&P500 ETF in dollars. The APE line is the

hedged MTM by minimizing the sum of absolute residuals. The SPE line is the hedged MTM by

minimizing the sum of squared residuals. MaxDD is the hedged MTM by minimizing maximum




portfolio.

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Figure 8. VT futures 1-month rolling contracts (September 10, 2004 to October 9, 2009). The ETF

line is the unhedged MTM of holding one 100-lot unit of the S&P500 ETF in dollars. The APE line is the

hedged MTM by minimizing the sum of absolute residuals. The SPE line is the hedged MTM by

minimizing the sum of squared residuals. MaxDD is the hedged MTM by minimizing maximum




portfolio.

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Figure 9. 10% OTM SPX put options 1-month rolling contracts (September 17, 2004 to October 16,

2009). The ETF line is the unhedged MTM of holding one 100-lot unit of the S&P500 ETF in dollars.

The APE line is the hedged MTM by minimizing the sum of absolute residuals. The SPE line is the

hedged MTM by minimizing the sum of squared residuals. MaxDD is the hedged MTM by minimizing

maximum drawdown. The CVaR(95%) and CVaR(99%) lines represent the hedged MTM by minimizing

the conditional Value-at-Risk computed using the Cornish-Fisher expansion at 95% and 99%,

respectively. Finally, the ASR line is the hedged MTM by maximizing the Alternative Sharpe Ratio of the

hedged portfolio.

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Figure 10. Comparison of different hedging instruments under the squared residuals hedging

model. The ETF line is the unhedged MTM of holding one 100-lot unit of the S&P500 ETF in dollars.

The 1-month rolling SPXPUT line is the hedged MTM by using 1-month rolling SPX puts. The 1-month

rolling VIX futures line is the hedged MTM by using 1-month rolling VIX futures. The 3-month rolling

VIX futures line is the hedged MTM by using 3-month rolling VIX futures. Finally, the 1-month rolling

VT line is the hedged MTM by using 1-month rolling variance futures.

Using Volatility Instruments As Extreme Downside Hedges-August 23, 2010

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