Using VIX futures to hedge forward implied volatility risk YuehNeng Lin, Anchor Y. Lin PII: DOI: Reference:
S10590560(15)001835 doi: 10.1016/j.iref.2015.10.033 REVECO 1168
To appear in:
International Review of Economics and Finance
Please cite this article as: Lin, Y.N. & Lin, A.Y., Using VIX futures to hedge forward implied volatility risk, International Review of Economics and Finance (2015), doi: 10.1016/j.iref.2015.10.033
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Using VIX futures to hedge forward implied volatility risk
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YuehNeng Lina, and Anchor Y. Lina
ABSTRACT
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The fair value of VIX futures is derived by pricing the forward 30day volatility which underlies the volatility risk of S&P 500 in the 30 days after the futures expiration. While forward implied volatility can also be traded with forwardstart strangles, this study
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demonstrates that VIX futures could offer more effective volatilityrisk hedge for an
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investor who has a short position on the S&P 500 futures call option. In particular, the
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deltaveganeutral hedging strategy incorporating stochastic volatility on average
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outperforms in outofsample hedging. Adding price jumps further enhances the hedging performance during the crash period.
Keywords: VIX futures; Forward implied volatility; Forwardstart strangles; Stochastic volatility; Price jumps Classification code: G12, G13, G14
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Department of Finance, National Chung Hsing University, Taiwan. Corresponding author: YuehNeng Lin, Department of Finance, National Chung Hsing University, 250, KuoKuang Road, Taichung, Taiwan, Phone: (886) 4 22857043, Fax: (886) 4 22856015, Email:
[email protected]
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1. Introduction
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The U.S. stock market crashed on October 19, 1987, when the Dow Jones Industrials
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Average lost 22.6% of its market value and the S&P 500 dropped 20.4% in one day. The 1987 crash brought volatility products to the attention of academics and practitioners. As
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the boomingcrash cycle of financial markets becomes often and makes markets highly uncertain, effectively managing volatility risk has become urgent for market participants.
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Other than volatility and variance swaps that have been popular in the OTC equity derivatives market for about a decade, hedging volatility risk has been studied through log
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contract (Neuberger, 1994), straddle options (Carr & Madan, 1998), straddles (Brenner, Ou,
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& Zhang, 2006), vanilla and volatility options (Psychoyios & Skiadopoulos, 2006), and the
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Chicago Board Options Exchange (CBOE) Volatility Index (VIX) (GonzalezPerez, 2015). The CBOE launched the VIX Futures on March 26, 2004, successively followed by the ThreeMonth S&P 500 Variance Futures on May 18, 2004, the TwelveMonth S&P 500 Variance Futures on March 23, 2006, the VIX Option on February 24, 2006, and the new S&P 500 Variance Futures on December 10, 2012. These were the first of an entire family of volatility products to be traded on exchanges. The VIX uses all outofthemoney calls and puts written on the S&P 500 Index (SPX) with valid quotes. Atthemoney call and put
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options on SPX are also included with their prices averaged. It attempts to gauge the
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expected volatility over the next 30 days. The introduction of VIX futures has been a major
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financial innovation that facilitates to a great extent the hedging of volatility risk. Moran and Dash (2007) and Szado (2009) discuss the benefits of using VIX futures as equity
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downside hedge. Fig. 1 indicates the explosive growth of the trading volume, open interests and block trader order execution of VIX futures in recent years, clearly reflecting a demand
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for a tradable vehicle which can be used to hedge or to implement a view on volatility. [Fig. 1 about here]
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The fair value of VIX futures is derived by pricing the forward 30day volatility
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which underlies the volatility risk of S&P 500 in the 30 days after the futures expiration.
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The most straightforward application of VIX futures is to manage the forward implied volatility. While forward implied volatility can also be traded with forwardstart straddles or by unwinding deltahedged option positions, VIX futures offer a cleaner and less costly exposure which does not need to be adjusted when the market moves. Using VIX futures as forward volatility risk hedge is an important issue that has not yet been concluded in the literature. This study addresses this issue by using VIX futures to hedge forward volatility risk of a short position on the S&P 500 futures call option from the perspective of an
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investor who faces stochastic volatility and price jumps.1 To conduct this hedging exercise,
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the study proposes a deltaveganeutral strategy to manage the underlying price risk and
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volatility risk and to examine to what extent the risk factors of stochastic volatility and jumps in the S&P 500 prices influence the hedging effectiveness of VIX futures. For the
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purpose of comparison, the hedging effectiveness of forwardstart strangles is also investigated. Forwardstart strangles consist of two wide strangles with different maturities,
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so that the changes of the underlying stock price will not affect the payoff of the portfolio. In other words, the forwardstart strangles can manage forward volatility risk without
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exposure to delta and gamma risk (Rebonato, 1999).
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Recent empirical work on index options identifies factors such as stochastic volatility,
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jumps in prices and jumps in volatility. The results in the literature regarding these issues are mixed. For example, tests using option data disagree over the importance of price jumps: Bakshi, Cao, and Chen (1997) find substantial benefits from including jumps in prices, whereas Bates (2000) and others find that such benefits are economically small, if not negligible. Pan (2002) finds that pricing errors decrease when price jumps are added for 1
A short position on the S&P 500 futures call option is considered because its vega risk consists of (i) spot volatility randomness over the period from current time and the option maturity, and (ii) forward volatility changes between the option maturity and the underlying futures expiry. Since VIX futures settle to the forward implied volatility of S&P 500, they, at least in theory, should provide an effective hedge on the vega risk of S&P 500 futures options.
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certain strikematurity combinations, but increase for others. Eraker (2004) finds that
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adding jumps in returns and volatility decreases errors by only 1%. Bates (2000) finds a
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10% decrease, but it falls to around 2% when timeseries consistency is imposed. Broadie, Chernov, and Johannes (2007) find strong evidence for jumps in prices and modest evidence
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for jumps in volatility for the cross section of SPX futures option prices from 1987 to 2003. Furthermore, while studies using the time series of returns unanimously support jumps in
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prices, they disagree with respect to the importance of jumps in volatility. In terms of empirical index option hedging, Bakshi, Cao, and Chen (1997) find that once the stochastic
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volatility is modeled, the hedging performance may be improved by incorporating neither
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price jumps, nor stochastic interest rates into the SPX option pricing framework. Bakshi and
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Kapadia (2003) use Heston’s (1993) stochasticvolatility option pricing model to construct a deltahedged strategy for a long position on SPX call options. They find that the volatility risk is priced and the price jump affects the hedging efficiency. Lin, Lin, and Chen (2012) point out the presence of a negative volatility risk premium in the individual LIFFE equity options based on a deltaneutral strategy. Vishnevskaya (2004) follows the structure of Bakshi and Kapadia (2003) and constructs a deltavegahedged portfolio consisting of the underlying stock, another option and the moneymarket fund, to hedge a long position on
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the SPX call option. His result suggests the existence of some other sources of risk. Guided
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by previous studies, this study considers both stochastic volatility and price jumps, denoted
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the “SVJ” model.2 The setup contains the competing stochasticvolatility (SV) futures option formula as special cases.
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Since SPX futures option contracts are Americanstyle, it is important to take into account the extra value accruing from the ability to exercise the options prior to maturity.
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One can follow such a nonparametric approach as in AïtSahalia and Lo (1998) and Broadie, Detemple, Ghysels, and Torrés (2000) to price American options. Closedform option
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pricing formulas, however, make it possible to derive hedge ratios analytically. For options
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with early exercise potential, this study therefore computes a quadratic approximation for
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evaluating American futures options. The approximation is based on the one developed by MacMillan
(1987),
examined
by
BaroneAdesi
and
Whaley
(1987)
for
the
constantvolatility process, extended by Bates (1991) for the jumpdiffusion process, and modified by Bates (1996) for the currency futures options under the SV and SVJ processes. In conducting the hedging exercise, the study considers a perfect hedge may not be
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Our results could be generalized by introducing timevarying jump intensity, jumps in the volatility, or a more complicated correlation structure for the state variables. While such generalization would add realism, this study is aimed to examine the hedging performance of the VIX futures in the presence of the most robust sources of incomplete market risk, i.e., stochastic volatility and price jumps.
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practically feasible in the presence of stochastic jump sizes (e.g. for the SVJ model). This
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difficulty is seen from the existing work by Cox and Ross (1976), Merton (1976), Bates
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(1996), and Bakshi, Cao, and Chen (1997). For this reason, whenever jump risk is present, this study follows the literature and only aims for a partial hedge in which diffusion risks
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(delta and vega) are completely neutralized but jump risk is left uncontrolled for. The overall impact on hedging effectiveness of not controlling for jump risk can be small or
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large, depending on whether the hedge is frequently rebalanced or not. This study examines hedging performance based on both daily and weekly rebalancing frequencies. Our findings
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reveal that VIX futures generally outperform forwardstart strangles over the outofsample
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hedging period, August 2006June 2009. These diagnostics document the extraordinary
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significance in our hedging exercise that occurs following the stock market crash on September 15, 2008. Based on empirical analyses, this study concludes that using VIX futures as vega hedge for a short position on a SPX futures call option on average provides superior hedging effectiveness than forwardstart strangles. In addition, adding price jumps into the SPX price process further improve hedging performance during the crash period. The rest of this paper proceeds as follows. Next section illustrates hedging strategies. Pricing models for calculating delta and vega hedge ratios are presented in Section 3.
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Section 4 summarizes data and model parameter estimation procedure. Section 5 presents
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outofsample hedging results. Section 7 finally concludes.
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summary statistics of parameter estimates and insample pricing errors. Section 6 analyzes
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2. Hedging strategies
Consider a timet short position on the T1 matured call option written on T2
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matured SPX futures as the unhedged portfolio, i.e., Ct (F ) for t T1 T2 . This study
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then constructs two hedging strategies (HS1 and HS2) to hedge the position.
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Hedging Strategy 1 (HS1): The hedged portfolio ( ) consists of a short position on a SPX
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futures call, N1, t shares of underlying SPX futures (F), and N 2, t shares of forwardstart strangles. That is, t Ct ( F ) N1,t Ft N 2,T INSTt , where INSTt is forwardstart strangles that comprise a short position on a T1 matured strangle and a long position on a
T2 matured strangle. A short position on a T1 matured strangle is constructed by shorting sell a T1 matured SPX call with strike K2, and shorting sell a T1 matured SPX put with strike K1. A long position on a T2matured strangle is constructed by long a T2matured SPX call with strike K2, and long a T2matured SPX put with strike K1. That is, forwardstart
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strangles at time t ( INSTt ) is given by
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INSTt ct ( S , T1 , K 2 ) pt ( S , T1 , K1 ) ct ( S , T2 , K 2 ) pt ( S , T2 , K1 )
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where ct ( S , T1, K 2 ) and ct ( S , T2 , K 2 ) are K 2 strike SPX call options with maturities T1
maturities T1 and T2 , respectively.
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and T2 , respectively. pt ( S , T1, K1 ) and pt ( S , T2 , K1 ) are K1 strike SPX put options with
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Hedging Strategy 2 (HS2): The hedged portfolio ( ) consists of a short position on a SPX futures call, N1, t shares of underlying SPX futures, and N 2, t shares of the VIX futures.
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That is, t Ct ( F ) N1,t Ft N 2,t INSTt , where INSTt is the timet price of the VIX
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futures with expiry T1 , FtVIX (T1 ) .
Further, add the constraints of deltaneutral and veganeutral by t INSTt Ct ( F ) N1,t N 2,t 0 Ft Ft Ft
(1)
t INSTt Ct ( F ) N 2,t 0 t t t
(2)
where t is the instantaneous variance of SPX. The shares of hedging assets are computed as
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Ct ( F ) INSTt N 2,t Ft Ft
(3)
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N1,t
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C ( F ) INSTt / N 2,t t t t
for
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The illustrations of Ct ( F ) / Ft , INSTt / Ft , Ct ( F ) / t and INSTt / t
(4)
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alternate models are provided in the following section.
Next, this study couples these two hedging schemes with the SV and the SVJ option models to construct four hedging strategies: HS1SV, HS1SVJ, HS2SV and HS2SVJ.
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Assuming that there are no arbitrage opportunities, the hedged portfolio t should earn the
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t is expressed as
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riskfree rate of return. In other words, the change in the value of this hedged portfolio over
(5)
t t N1,t [ Ft t (T2 ) Ft (T2 )] N 2,t [ INSTt t INSTt ] [Ct t ( F ) Ct ( F )] . The
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where
t t t t t
hedging error as a function of rebalancing frequency
is defined as the additional profit
(loss) over the riskfree rate of return and it can be written as HEt (t t ) t t t (ert 1) N1,t [ Ft t (T2 ) Ft (T2 )] N 2,t [ INSTt t INSTt ] [Ct t ( F ) Ct ( F )] (ert 1) [ N1,t Ft (T2 ) N 2,t INSTt Ct ( F )]
Finally, compute the average absolute hedging error:
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(6)
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(7)
(8)
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, and the average dollarvalue hedging error:
where
and T1 is the expiry of the unhedged SPX futures call option.
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Measurements of hedging performances are based on the framework proposed in previous papers, for instance, Bakshi, Cao, and Chen (1997) and Psychoyios and Skiadopoulos
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(2006).
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3. Empirical pricing models
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Hedging strategies are constructed using the data of SPX, SPX futures, SPX options,
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SPX futures calls, VIX, and VIX futures. Therefore, their fair value and related greeks are required for further empirical analyses. The most general process considered in this paper is the jumpdiffusion and stochastic volatility (SVJ) process of Bates (1996) and Bakshi, Cao, and Chen (1997). This general process contains stochastic volatility (SV) of Heston (1993) as a special case.
3.1 The SVJ process for SPX prices
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Contingent claims are priced as if investors were riskneutral and the SPX price is
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assumed to follow a jumpdiffusion with stochastic volatility (SVJ) under the riskneutral
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probability measure Q:
dSt (b J J ) St t St dS ,t J t St dNt
(9)
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where b is the cost of carry coefficient (0 for futures options and r for stock options with a cash dividend yield ). J t is the percentage jump size with mean J . The jumps
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in the asset logprice are assumed to be normally distributed, i.e., ln( 1 J t ) ~ N ( J , J2 ) . Satisfying the noarbitrage condition, J exp( J J2 / 2) 1 . N t is the jump frequency
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following a Poisson process with mean J . The instantaneous variance t of the SPX
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follows a riskneutral meanreverting square root process (10)
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d t ( t )dt t d ,t
where is the speed of meanreverting adjustment of t ; / is the longrun mean of t ; is the variation coefficient of t ; and S , t and , t are two correlated Brownian motions with the correlation coefficient dt corr (dS ,t , d ,t ) .
3.2 Fair value to SPX options SPX options are Europeanstyle. Bakshi, Cao, and Chen (1997) provide the timet
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value of SPX call and put options with strike K and maturity T for the SVJ model:
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ct ( S , T , K ) St e (T t )1 Ke r (T t ) 2
(12)
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pt ( S , T , K ) St e (T t ) (1 1) Ke r (T t ) ( 2 1)
(11)
where 1 and 2 are riskneutral probabilities that are recovered from inverting the
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characteristic functions f1 and f 2 , respectively,
i ln K f j (t , T t; St , r , t ; ) 1 1 e j (T t , K ; St , r , t ) Re d 2 0 i
(13)
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for j = 1, 2. The characteristic functions f1 and f 2 for the SVJ model are given in
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equations (A12) and (A13) of Bakshi, Cao, and Chen (1997). Delta and vega of the
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European SPX options are given in equation (13) of Bakshi, Cao, and Chen (1997). Finally,
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delta and vega of the forwardstart strangles can be calculated straightforward.
3.3 Fair value to SPX futures call options Since SPX futures call options are Americanstyle, it is important, in principle, to take into account the extra value accruing from the ability to exercise the options prior to maturity. Referred to Bates (1996), the futures call option is
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if Ft (T2 ) / K yc*
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(14)
if Ft (T2 ) / K yc*
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q2 Ft (T2 ) / K * C ( F , T1 , K ) KA2 * Ct ( F , T1 , K ) t y c Ft (T2 ) K
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where Ct* ( F , T1 , K ) e r (T1 t ) [ Ft (T2 )1 K2 ] is the timet price of a Europeanstyle futures call option with strike K and expiry T1 . Ft (T2 ) is the timet futures price with
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maturity T2 . For j=1, 2, j are the relevant tail probabilities for evaluating futures call options, which are the same to j in equation (13) except replacing K with Ke ( r ) (T
2
T1 )
.
equation of q given as follows
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A2 ( yc* / q2 )[1 Ct* ( yc* , T1 ,1) / Ft ] . For the SVJ process, q2 is the positive root to the
2 r J [(1 J ) q e q ( q 1) J / 2 1] 0 q r (T1 t ) 1 e
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1 2 1 q J J 2 2
(15)
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is the expected average variance over the lifetime of the option conditional on no jumps, [1 e (T t ) ] t . The critical futures price/strike price ratio yc* 1 i.e., (T1 t )
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1
above which the futures call is exercised immediately is given implicitly by
y* C * ( y* , T ,1) yc* 1 Ct* ( yc* , T1,1) c 1 t c 1 Ft q2
(16)
The closed form solutions to the parameters q2 and y c* are provided for given model parameters and for given maturity T1 . Since linear homogeneity in underlying asset and strike holds for European options, by Euler theorem the following equations sustain:
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1 * Ct ( F , T1 , K ) e r (T1 t ) ( yc 1 2 ) K
Ct* ( yc , T1 ,1) of y c* in equation (16) Ft
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where yc Ft (T2 ) / K . Replacing Ct* ( yc , T1 ,1) and
(18)
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Ct* ( yc , T1 ,1) 1 Ct* ( F , T1 , K ) 1 r (T1 t ) e 1 Ft K Ft K
(17)
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Ct* ( yc , T1 ,1)
with equations (17) and (18), the solution to y c* is given by
1 e r (T1 t )2 1 1 1 1e r (T1 t )1 q2 q2 K
(19)
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yc*
Thus, the solution to A2 is computed as
(20)
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[1 e r (T1t ) 2 ] 1 r (T1t ) A2 1 e 1 1 r (T1t ) K 1 q 2 1 K q2 e
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Finally, the calculation for delta and vega of the futures call option is straightforward.
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3.4 Fair value to VIX futures From Lin (2007), the timet fair price of the VIX futures expiring at T1 under the SVJ model is given by
FtVIX (T1 ) 2
1
0
(a 0 T1 t t a 0 T1 t b 0 )
a 0 0
2
(CT1 t t DT1 t )
1 8 2 (a 0 T1 t t a 0 T1 t b 0 ) 0
3/ 2
(21)
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, b 0
0 (VIX t2 2 ) b a 0
0
(1 T t ) , 1
2 2 ( 0 a ) , CT t ( T t T2 t ) , DT t 2 (1 T t ) 2 , 2 0
1
1
.
1
1
1
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t
(1 e 0 )
T
a 0
, 2 2J ( J J ) , T1t e (T1t ) , T1 t
0 30 / 365
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where
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Pricing formulas and related greeks for the SV model are obtained as a special case of the general model with price jumps restricted to zero, i.e., J t dN t 0 and thus J = J = J
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=0.
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4. Data and estimation procedure
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The sample period spans from July 3, 2006, through June 30, 2009. 3 Spot VIX, spot
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SPX, daily midpoints of the last bid and last ask quotations for SPX options, and daily settlement prices for VIX futures are obtained from the CBOE. Intradaily data of SPX futures and SPX futures call options from CME are used to avoid imperfect synchronization of closing prices with the underlying index price. Midpoints of their bid and ask quotes are used. The analysis uses the last quote for a particular futures call option in terms of exercise price, maturity, and type for all trading days in the sample. The averages of U.S. Treasury
3
The period after June 30, 2009 is excluded due to tick data availability of historical intraday data for the SPX futures and SPX futures call options.
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bill bid and ask discounts, collected from Datastream, with maturity closest to an option’s
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maturity are used as riskfree interest rates. Since SPX option contracts are Europeanstyle,
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index levels are adjusted for dividends by the subtraction of the present value of future cash dividend payments (from S&P Corporation) before each option’s expiration date.
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The contracts that are selected for empirical analyses are described as follows: First, the selected SPX futures contracts expire in March, June, September and December. Second,
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the SPX futures call options that expire in February, May, August and November are selected as the unhedged portfolio. Third, the forwardstart strangles involve two strangles.
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This study uses the SPX options contracts that expire in February, May, August and
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November to construct a shortterm strangle, and that expire in March, June, September,
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and December for another longterm strangle. Finally, the VIX futures that expire in February, May, August and November are selected as the hedging instrument. Several exclusion filters are applied to construct the option price data. First, as options with less than six days to expiration may induce liquidityrelated biases, they are excluded from the sample. Second, to mitigate the impact of price discreteness on option valuation, option prices lower than 3/8 are excluded (Bakshi, Cao, & Chen, 1997).4 Finally, 4
Minimum tick for SPX options trading below 3.00 is 0.05 ($5.00) and for all other series, 0.10 ($10.00).
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option prices that violate the upper and lower boundaries are not included in the sample.
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Based on these criteria, 29.90% and 10.50% of the original selected SPX options and SPX
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futures call options are eliminated, respectively. A total of 24,761 records of joint SPX futures call options (20,839 records), SPX options (3,200 observations), and VIX futures
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(722 records) are used for our empirical work.
Table 1 reports descriptive properties of the SPX futures call options, along with the
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underlying SPX futures, VIX futures, VIX, the strikes of selected SPX options, and the total number of futures call options, for each moneynessmaturity category. Moneyness is
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defined as the SPX futures price divided by the SPX futures call option’s strike price, i.e.,
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Ft (T2 ) / K . Out of 20,839 SPX futures call option observations, about 50% is
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outofthemoney (OTM) and 35% is atthemoney (ATM). By maturity, 6,221 transactions are under 30 days to maturity, 8,613 are between 30 and 60 days to maturity, and 6,005 are between 30 and 121 days to maturity. The average futures call price ranges from 5.57 points for shortterm (<30 days) deep outofthemoney (DOTM) call options to 118.56 points for mediumterm (30－60 days) deep inthemoney (DITM) call options. The underlying SPX Tick size for SPX futures options is 0.05 ($12.50) for premium less than or equal to 5.00 and for all other premium, 0.10 ($25.00). Following Bakshi, Cao, and Chen (1997), the SPX option prices lower than 3/8 that corresponds to a 7.5 ticksize interval are excluded. Indeed, this study advocates discarding all SPX futures call options for this reason. That is, the SPX futures call option prices lower than 3/8 that also corresponds to a 7.5 ticksize interval are excluded.
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futures price ranges from 947.85 points for shortterm DOTM calls to 1,347.89 points for
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mediumterm ATM calls.
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The average VIX futures prices range from 21.02 corresponding to the mediumterm ATM calls to 46.23 for the shortterm DOTM calls. The VIX varies from 20.68 (for
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mediumterm ATM call options) to 50.87 (for shortterm DOTM call options). The price of a VIX futures contract to VIX is an analogy to what a 30day forward interest rate is to a
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30day spot interest rate. The price of a VIX futures contract can be lower, equal to or higher than VIX, depending on whether the market expects volatility to be lower, equal to
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or higher in the 30day forward period (covered by the VIX futures) than in the 30day spot
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period (covered by VIX). Owing to the tendency of VIX to have a negative correlation with
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the SPX, the VIX futures prices have traded at a discount to the VIX before contract settlement especially during the incrash period, September 15, 2008 to December 31, 2008. A total of 800 records of simultaneous strikes K1 and K 2 of SPX options are used for the construction of the forwardstart strangles over the sample period, consisting of 722 trading days. Average option quotes of K1 range from 599.89 for DOTM shortterm futures calls to 1,100.31 for ATM shortterm futures calls, whereas average option quotes of
K 2 range 1,066.82 for shortterm DOTM futures calls to 1,464.49 for mediumterm ATM
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futures calls.
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[Table 1 about here]
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This study uses SPX options to construct the forwardstart strangles that expire in the February quarterly cycle as T1 strangle and that expire in the March quarterly cycle as T2
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strangle. Therefore, the pair of ( T1 , T2 ) data must be FebruaryMarch, MayJune, AugustSeptember and NovemberDecember. The strike K1 ( K 2 ) is the minimum
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(maximum) strike of the SPX put (call) option, which expire in the February and March quarterly cycles simultaneously. To enhance the integrity of the study, less heavily traded
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SPX options are eliminated here. On the basis of an analysis of patterns of trading volumes
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on SPX options, four pairs of maturitystrike combinations are selected to form
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forwardstart strangles on each trading day: (T1 , K1 ) and (T2 , K1 ) of SPX puts, and
(T1, K2 ) and (T2 , K 2 ) of SPX calls. There are in total 3,200 SPX option observations, consisting of 1,600 SPX calls and 1,600 SPX puts. Those option observations compose of 800 pairs of calls with strike K 2 and puts with strike K1 . The selected strikes K1 of the SPX puts are on average 938.03 index points, whereas the strikes K 2 of the SPX calls are on average 1,377.21. Table 2 reports the average point of those SPX options for each moneynessmaturity category over the period, 3 July 2006 to 30 June 2009. Moneyness is
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defined as the current SPX value divided by the SPX option’s strike, S/K. The average call
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prices range from 1.3716 points for shortterm OTM call options to 59.4300 points for
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longterm nearthemoney call options. The average put prices range from 0.4125 points for shortterm OTM put options to 66.8500 points for longterm nearthemoney put options.
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[Table 2 about here]
The vector of structural parameters for alternate processes is backed out by
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minimizing the sum of the squared pricing errors between model and market prices over the period, July 3, 2006 to May 14, 2009.5 The minimization6 is given by NT N t
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min [Cn Cn* ()]2
(22)
t 1 n 1
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where NT is the number of trading days in the estimation sample, N t is the number of
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SPX futures call options,7 SPX options and VIX futures on day t, and Cn and C n* are the observed and model prices, respectively.8 The parameters of the SV and the SVJ models are 5
May 14, 2009 is the last trading day of Maymatured SPX futures call options in 2009. The rich set of options and VIX futures data provides a challenging test of our empirical results. Using the model parameters estimated from the joint data from several markets in minimizing squared deviations between model and market prices, the study finds that the parameter estimates are comparable to the estimates obtained in prior studies (Bates, 1996; Bakshi, Cao, & Chen, 1997; Bates, 2000; Pan, 2002; Eraker, 2004). 7 Only 6.76 percent and 7.95 percent of the SPX futures call options are inthemoney (ITM) and deepinthemoney (DITM), respectively. There is no observation with maturity greater than 122 days to expiration. 8 Since SPX futures options are American, the study follows Broadie, Chernov, and Johannes (2007) procedure to account for the early exercise feature. Our calibration procedure begins by converting American market prices to equivalent European market prices, and estimates model parameters based on European prices. This allows us to use computationally efficient European pricing routines that render the largescale calibration procedure feasible. Using the observed price ( ) the study computes an American binomialtree implied volatility, that is, a volatility value such that , where denotes the binomialtree 6
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estimated separately each month and thus are assumed to be constant over a month.
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The assumption that the structural parameters are constant over a month is justified by an
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appeal to parameter stability (Bates, 1996; Eraker, 2004; Zhang & Zhu, 2006). The estimation period is chosen because the settlement day of the SPX futures option is the third
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Friday of contract months. Hence, the month is defined as the period from the third Friday of the prior calendar month to the third Thursday of this calendar month. Table 3 reports the
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monthly average of each estimated parameter series and the monthly averaged insample
[Table 3 about here]
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mean squared errors for the SV and SVJ models, respectively.
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5. Riskneutral parameter estimates and insample pricing fit Table 3 reports summary statistics for instantaneous volatility and riskneutral parameter estimates along with insample mean squared errors from the SV and SVJ models. In Q4 2008, a series of bank and insurance company failures triggered a financial crisis that effectively halted global credit markets and required unprecedented government intervention. To examine role of VIX futures in the crash dump, other than the entire sample American option price. The study then estimates that an equivalent European option would trade in the market at a price , where denotes the Black (1976) European option price.
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period, the study divides the data into four subperiods: (i) the period prior to the Lehman
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Brothers bankruptcy: July 3, 2006 to August 14, 2008, and (ii) the 2008 crash fear periods:
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August 15, 2008 to December 18, 2008, August 15, 2008 to June 18, 2009, and December 19, 2008 to June 18, 2009.9
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Modelspecific estimates of implicit distributions indicate extremely turbulent conditions in the market over September 2008‒ December 2008, with somewhat quieter
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conditions over January 2009‒ June 2009 following the end of the Lehman Brothers bankruptcy. Indeed, Table 3 finds parameter estimates of the full sample period, July
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2006‒ June 2009, diverge from estimates for a September 2008‒ June 2009 subsample. The
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estimate of jumpfrequency intensity
that captures outliers indicates the major shocks
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that affected the markets over September 2008‒ December 2008. The possibility of SPX price jumps occurs with an average size uncertainty
of 0.13, 0.12, and 0.11 (with jump size
estimated at 0.35, 0.30, and 0.49) for the July 2006‒ August 2008,
September 2008‒ December 2008, and January 2009‒ June 2009 periods, respectively. The riskneutral meanreverting volatility process is controlled by three parameters:
9
The data split point, for example, August 14, 2008 and December 18, 2008, is chosen because the estimation month is defined as the period from the third Friday of the prior calendar month to the third Thursday of this calendar month.
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, and the
. For instance, on the basis of mean values of
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variation coefficient,
, to which variance reverts, the speed of reversion,
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the longterm mean,
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{5.77, 0.35, 0.77} for the SV model and {3.93, 0.59, 1.80} for the SVJ model over the crash period, volatility took 30 and 44 trading days to revert halfway toward a longterm mean of
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34.58% for the SV model and 58.62% for the SVJ model, respectively.10 The mean speed of reversion value of 5.77 and 3.19 for the SV model over the crash periods (Panels C and D)
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is higher than the values of 0.86 (Panel B) and 1.47 (Panel E) over the pre and postcrashrelaxation periods. However, it is close to the value of 5.02 found by the SVJ
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model over the postcrashrelaxation period. The average longterm mean variance
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of 34.36% for the SVJ model over the crash period compares with values of 3.91% in the
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precrash period and 13.33% in the postcrashrelaxation period. The variation coefficient, , determines how fattailed the distribution is and thereby the relative values of deep OTM options versus nearthemoney options. The mean value of 1.80 found in the SVJ model over the crash period compares with values of 0.22 and 0.46 over the pre and postcrashrelaxation periods. The consistently negative estimates of
indicate that
implied volatility and index returns are negatively correlated. This means the implied 10
The halflife of volatility shocks for the squareroot process was calculated as
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(2)/
252 trading days.
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for the SVJ model reached 0.91 in the
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traders is negatively skewed. The mean values of
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distribution perceived by SPX option traders, SPX futures option traders and VIX futures
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crash period and 0.97 in the postcrashrelaxation period. An explanation for these extreme values is that a relatively continuous bear market in the SPX starting from September 15,
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2008 caused investors to perceive the underlying distribution as being heavily left skewed. The SV model captures this effect by attaching a higher value to
. The mean value of
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of 0.57 for the SV model in the precrash period compares with more extreme values of 0.75 in the crash period and 0.77 in the postcrashrelaxation period. The implied , estimated by alternate models are able to capture the substantial 2008
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volatilities,
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crash. In particular, the SVJ model adds a pricejump component that directly captures the
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2008 outliers. To address whether jumps in volatility significantly impact the results of our study, this study calculates skewness and kurtosis statistics of modelimplied instantaneous variance increments. As shown in the Table 3, the data, however, are extremely nonnormal, and thus the square root diffusion specification has little chance to fit the observed data. Therefore, our conclusions regarding the misspecification of the square root volatility process hold for the set of parameters in the SV and SVJ models in Table 3. Table 3 also gives the
values for the parameter estimates of the index price process. These results
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indicate that all parameter estimates are significantly different from zero at the 90%
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confidence interval.
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The parameter estimates in Table 3 are interesting in light of estimates obtained in prior studies. Bakshi, Cao, and Chen (1997) estimate the jump frequency for the SVJ model
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as a 0.59 annualized jump probability, and the jumpsize parameter and 7%, respectively. Their estimate of
and
and
are 5.37%
are 2.03 and 0.38. Pan (2002), Bates
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(2000) and Eraker (2004) assume that the jump frequency depends on the spot volatility. The average jump intensity point estimates in Pan (2002) are in the range [0.07, 0.3%]
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across different model specifications, whereas Eraker (2004) indicates two to three jumps in
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a stretch of 1,000 trading days. Interestingly, Bates (2000) obtains quite different results,
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with an average jump intensity of 0.005. Bates (2000) also reports a jump size mean ranging from 5.4% to 9.5% and standard deviations of about 10% to 11%. Hence, Bates’ estimates imply more frequent and more severe crashes than the parameter estimates reported in Bakshi, Cao, and Chen (1997) and Eraker (2004). Our estimates are in the same ballpark as those reported in Bates (1996). Finally, the fact that allowing pricing jumps to occur enhances the SV model’ fit is illustrated by the model’s average of the squared pricing errors between the market price
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). The insample mean squared errors are
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and the model price in an average month (
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considerably and consistently smaller for more complicated models. However, the resulting for the various specifications is significant under the crash period,
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increase in
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suggesting that this jump risk assessment fails to capture the substantial 2008 crash.11
6. Hedging results
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The need for a perfect deltaneutral hedge can arise in situations where not only is the underlying price risk present, but also are volatility and jump risks in our specifications. In
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conducting this exercise, whenever jump risk is present (for example, for the SVJ model),
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the study follows Merton (1976) and Bakshi, Cao, and Chen (1997) to only aim for a partial
for.
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hedge in which diffusion risks are completely neutralized but jump risk is left uncontrolled
To examine the hedging effectiveness, consider a short position on the SPX futures call option at time
and establish the hedge using two hedging schemes under two SPX
price processes. The hedger will need a position in futures (to control for price risk), and
shares of the underlying SPX
units of either forwardstart strangles or VIX
11
Bates (2006) found the stochasticintensity jump model fits S&P returns better than the constantintensity specification, when jumps are drawn from a finiteactivity normal distribution or mixture of normals.
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. The study uses the previous month’s structural
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futures to control for volatility risk
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parameters and the current day’s (t) observations of SPX, SPX futures, SPX futures call
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options, SPX options, VIX futures and U.S. Treasurybill rates to construct the hedged portfolio. Next, this study calculates the hedging error as of day
if the rebalancing takes place every five days. These
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rebalanced daily or as of day
if the hedge is
steps are repeated for each futures call option that expires in February quarterly cycle and
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every trading day in the sample. Finally, the average absolute and the average dollar hedging errors for each moneynessmaturity category are then reported for each model in
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Tables 4 and 5.
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In Table 4, under the SV model, the hedging errors of HS1 are from 0.26 points
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(DOTM shortterm of oneday revision in the postcrashrelaxation period) to 42.55 points (ATM mediumterm of fiveday revision in the crash period), whereas the hedging errors of HS2 range from 0.12 points (ATM shortterm of fiveday revision in the postcrashrelaxation period) to 48.80 points (ATM mediumterm of fiveday revision in the crash period). For the SVJ model, HS1 has hedging errors from 0.22 points (OTM shortterm of oneday revision in the postcrashrelaxation period) to 148.83 points (ITM longterm of oneday revision in the postcrashrelaxation period), whereas HS2 has
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hedging errors from 0.22 points (OTM shortterm of oneday revision in the
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postcrashrelaxation period) to 51.04 points (ATM mediumterm of fiveday revision in the
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crash period).
When the hedge revision frequency changes from daily to once every five days, the
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hedging errors increase, except for the SV shortterm options in the postcrashrelaxation period. Though HS1 outperforms HS2 in some cases within shortterm categories under
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fiveday rebalancing, the HS1 performs poorly in daily rebalancing. The findings shed a light on the instability issue of using forwardstart strangles as forward volatility hedge.
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Improvement by the HS2 strategy is more evidence than the improvement by the HS1
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strategy when the same pricing model is used. The finding indicates a consequence of using
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a better hedging scheme, HS2. Further, improvement by the SVJ model is more obvious in the crash period (Panel C). It represents the random price jump feature commonly exists in the SPX price process, especially in the incrash periods. In particular, those SVJ models coupled with HS2 have virtually smaller hedging errors. Noticeably, for the futures call options with medium and longterm maturities in the other periods, however, adding price jumps to the SV model does not improve its hedging performance. Thus, except for shortterm options, the result in the noncrash period seems to be consistent with those of
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Bakshi, Cao, and Chen (1997) and Bakshi and Kapadia (2003), which show the hedging
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superiority of the SV model relative to the SVJ model. Note that the parameter of
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jumpfrequency intensity J in Bakshi, Cao, and Chen (1997) is 0.59, i.e. one year and half for a price jump to occur, which is comparable to 0.60 of our empirical work only in
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the precrash period. Based on daily or every fiveday rebalancing, Bakshi, Cao, and Chen (1997) conclude that the reason for the SV model dominates the SVJ model in terms of hedging performance is the chance for a price jump to occur is small in the daily or fiveday
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rebalancing period. The estimated parameter J in the crash (postcrashrelaxation) period,
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however, is 9.97 (1.02) larger than that of Bakshi et al. (1997). For this reason, whenever
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jump risk is present, the overall impact on shortterm hedging effectiveness of not allowing
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for jump risk is significant. Therefore, Bakshi, Cao, and Chen’s (1997) reason does not completely hold for our empirical results. Another possible reason is that the SPX futures call options used in this study are Americanstyle, while SPX options are Europeanstyle for prior research. Since the traders with Americanstyle options positions have earlyexercise choice and thus can take caution to prevent any loss from the potential jump events than the ones with Europeanstyle options. Thus, Americanstyle option buyers (sellers) may even favor (hate) volatility risk than the ones with Europeanstyle options. In addition, given the
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possibility of price jumps, the specification of SVJ can provide more accurate parameter
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estimates than SV (Bates, 1996). Thus, the deltaveganeutral strategy could be constructed
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in a more effective way under SVJ than SV for the shortterm options. It is thus not surprising for our hedging results for some shortterm options showing that the SVJ model
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outperforms the SV model in the crash periods.
Further, the HS1 (HS2) hedge of the SVJ model results in poor hedging performance
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comparable to that of the SV model in some moneynessmaturity categories, which suggests that model misspecification may have a significant effect on hedging when using the HS1
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(HS2) strategy. About the maturity impact on hedging performance, there is no pattern
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showing the difference between hedging schemes decreases with maturity and increases
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with maturity, as discovered by Psychoyios and Skiadopoulos (2006) for ITM and OTM target options. In terms of the moneyness effect, our results show that the absolute hedging errors of HS1 and HS2 strategies have the tendency to increase in a Ushape with moneyness. In contrast, Psychoyios and Skiadopoulos (2006) find that the options perform best for ATM and worse for ITM, and the difference between hedging schemes is minimized for ATM and maximized for ITM. In sum, the results indicate that the forwardstart strangles are in general a less efficient instrument to hedge forward volatility risk than VIX
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futures. A paired ttest with pvalue in Table 4 supports the significance of the superiority of
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[Table 4 about here]
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including VIX futures in the hedging strategy under the SV and SVJ models, respectively.
Theoretically, if a portfolio is perfectly hedged, it should earn the riskfree rate of
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interest, and the average hedging errors should be close to zero. In this study, the hedging error is defined as the changes in the value of the hedged portfolio minus the riskfree return.
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Table 5 reports the average hedging errors. If the figure is greater (less) than zero, it means that the strategy gets more (less) profits than riskfree return.
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During the incrash period, the dollar hedging errors are relatively sensitive to
CE
revision frequency. Another pattern to note from Table 5 is that the HS1 and HS2’s dollar
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hedging errors are positive for mediumterm options in the crashrelated periods (Panels C and D), indicating that both HS1 and HS2 exhibit a systematic hedging bias. The remaining dollar hedging errors are more random and can take either sign. As shown in the full sample period in Table 5, dollar hedging errors are lower for the HS2 strategy than for the HS1 strategy, indicating that VIX futures are a superior hedge instrument relative to the forwardstart strangles. [Table 5 about here]
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7. Conclusion
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This study has extended a parsimonious American futures call option pricing model
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that admits stochastic volatility and random price jumps (SVJ). It is shown that this closedform pricing formula is practically implementable, leads to useful analytical hedge
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ratios, and contains the stochasticvolatility (SV) option formula as a special case. This last feature has made it relatively straightforward to study the relative empirical hedging
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performance of the two models. In particular, the study considers whether and by how much VIX futures improve option hedging from the perspective of an investor who has a short
PT
position on SPX futures call options. For the purpose of comparison, forwardstart strangles
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are constructed for managing forward volatility. This study then couples two hedging
AC
schemes with two SPX price processes to hedge the short position. On the one hand, the SPX futures and forwardstart strangles (HS1) are used to construct two hedging strategies (HS1SV and HS1SVJ). On the other hand, SPX futures and VIX futures (HS2) are used to construct the other two hedging strategies (HS2SV and HS2SVJ). There are some interesting empirical findings. First, HS2 in general dominates HS1, suggesting that VIX futures are a more efficient instrument to hedge forward volatility risk than forwardstart strangles. This finding contributes to the existing literature documented
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by Psychoyios and Skiadopoulos (2006), who point out that volatility options are less
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powerful instruments to hedge a short position on a European call option than plainvanilla
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options. Second, gauged by the absolute hedging errors, while the SVJ model performs slightly better during the crash period, the SV model coupled with HS2 achieves better
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performance for the remaining cases. Overall, our results support the claim that VIX futures are a better alternative to the traditional forwardstart strangles for managing the implied
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forward volatility risk of a short position on the SPX futures call option. In particular, VIX futures not only perform better in a stochasticvolatility and/or pricejump economy but also
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are practically implementable.
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The contributions of this paper are threefold. First, closedform solutions to American
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futures call options under two SPX price processes are derived. Second, the concept of forward volatility risk applied to VIX futures and forwardstart strangles is introduced. Third, this study derives the hedging ratios of VIX futures and the forwardstart strangles that are convenient to practical participants for risk management purposes.
Acknowledgements We thank Carl R. Chen (the editor), MinTeh Yu (the visiting editor), Thomas Chiang (the
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visiting editor), and one anonymous reviewer for their constructive comments, which helped us to improve the manuscript. YuehNeng Lin gratefully acknowledges research support from
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Taiwan National Science Council.
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Table 1 Sample properties of S&P 500 futures call options
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The average points of futures call options ($250 per point), along with the average points of underlying futures ($250 per point), the average points of VIX futures ($1,000 per point), the average points of VIX, the average strikes (K1 and K2) of selected SPX options and the total number of futures call options, are presented in each moneynessmaturity category. Contract months of both futures call options and VIX futures are in the February Quarterly Cycle, whereas contract months of SPX futures are in the March Quarterly Cycle. K1 and K2 are selected strikes of SPX put and call options to construct forwardstart strangles, both of which expire on the maturity dates of the SPX futures call and the SPX futures simultaneously. The study divides the futures call option data and contemporaneous futures levels, VIX futures, VIX, K 1 and K2 into several categories according to either moneyness or term to expiration. Define Ft(T2)/K as moneyness of a SPX futures call. A SPX futures call is then said to be deep outofthemoney (DOTM) if its Ft(T2)/K<0.94; outofthemoney (OTM) if Ft(T2)/K [0.94,0.97); atthemoney (ATM) if Ft(T2)/K [0.97,1.03); inthemoney (ITM) if Ft(T2)/K [1.03,1.06); and deep inthemoney (DITM) if Ft(T2)/K≧1.06. By the term to expiration, an option contract can be classified as (i) shortterm (<30 days); (ii) mediumterm (30－60 days); and (iii) longterm (>60 days). The data period is from July 3, 2006 to June 30, 2009. Days to Maturity Moneyness <30 3060 >60 Subtotal SPX futures call 5.5696 6.8933 8.4643 SPX futures 947.8534 1078.1347 1167.9128 VIX futures 46.2295 34.2556 30.4921 DOTM <0.94 VIX 50.8742 38.7459 32.2216 K1 599.8907 720.3157 793.8094 K2 1066.8182 1255.1761 1340.5847 Observation 869 2,867 3,053 6,789 SPX futures call 7.7356 12.3137 17.0321 SPX futures 1231.1835 1323.0393 1328.2712 VIX futures 29.7906 23.3573 21.4009 OTM 0.940.97 VIX 30.5768 23.3098 20.9756 K1 972.7829 1005.5325 1002.6713 K2 1345.6457 1456.6037 1464.8980 Observation 875 1,615 1,226 3,716 SPX futures call 13.8811 22.9812 33.3377 SPX futures 1311.4333 1345.6390 1320.7551 VIX futures 23.3768 21.3924 21.7445 ATM1 0.971.00 VIX 23.6443 20.9470 21.1742 K1 1100.3112 1046.4877 998.3112 K2 1400.5785 1464.4924 1458.7431 Observation 1,478 1,704 903 4,085 SPX futures call 33.7082 44.3218 55.6945 SPX futures 1306.8449 1347.8856 1305.5544 VIX futures 23.2381 21.0243 22.9012 ATM2 1.001.03 VIX 23.3762 20.6828 22.1135 K1 1100.1285 1050.9091 974.7164 K2 1394.2252 1463.4335 1445.6805 Observation 1,323 1,331 529 3,183 SPX futures call 62.0779 72.8389 84.7926 SPX futures 1225.0017 1292.5961 1217.1777 VIX futures 28.9174 25.3501 27.0426 ITM 1.031.06 VIX 29.3179 25.5828 27.1678 K1 987.9844 960.6031 850.1330 K2 1328.0622 1421.9455 1364.6277 Observation 707 514 188 1,409 SPX futures call 109.8420 118.5584 116.0519 SPX futures 1060.5809 1070.8000 1041.5613 DITM ≧1.06 VIX futures 35.8835 34.9171 35.6560 VIX 36.5378 36.1897 36.7596 K1 806.5996 695.8935 634.9057
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K2 Observation
1165.8359 1213.4278 1156.9811 969 582 106 6,221 8,613 6,005
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Subtotal
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1,657 20,839
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Table 2 Sample characteristics of S&P 500 index options.
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Outofthemoney and near atthemoney SPX call and put options are selected in this study to construct forwardstart strangles, both of which expire on the maturity dates of the SPX futures call and the SPX futures simultaneously. The average points of the SPX options ($100 per point) and the total number of the SPX options are presented in each moneynessmaturity category. Contract months of SPX options are in the February and March quarterly Cycles. This study classifies those observations into shortterm (<30 days), mediumterm (30–60 days), and longterm (>60 days). Moneyness is defined as S/K where S is the price of the SPX and K is the strike price of SPX options. DOTM (DITM), OTM (ITM), ATM1 (ATM2), ATM2 (ATM1), ITM (OTM) and DITM (DOTM) for calls (puts) are defined by Moneyness <0.94, 0.94–0.97, 0.97–1.00, 1.00–1.03, 1.03–1.06, and≧1.06, respectively. The data period is from July 3, 2006 to June 30, 2009.
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<30 1.5607 105 1.3716 74 4.9698 24 203
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Moneyness DOTM Option price Observation OTM Option price Observation ATM1 Option price Observation Subtotal
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Days to Expiration Call 30–60 >60 3.8948 8.2840 293 812 6.2070 15.9739 146 110 16.5250 59.4300 26 10 465 932
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Subtotal
1,210 330
60 1,600
<30 0.5031 192 0.4125 8 12.5417 3 203
Put 30–60 >60 2.1075 3.0217 452 930 8.4417 53.9000 9 1 31.3625 66.8500 4 1 465 932
Subtotal 1,574 18 8 1,600
ACCEPTED MANUSCRIPT Table 3 Implied parameter estimation.
SC
RI P
T
The insample mean squared pricing errors ( ) and estimated structural parameters reported in Panel A are their averages over 36 nonoverlapping estimation months from the whole sample period, July 3, 2006 to June 18, 2009, with a total of 20,839 SPX futures calls, 722 VIX futures, and 3,200 SPX options. Panel B presents estimation results that obtained preceding the Lehman Brothers bankruptcy on September 15, 2008. Panels C, D and E report parameter estimates and insample mean squared errors that occurred covering the stock market crash on September 15, 2008. The study divides the post’08 crash fears in the S&P 500 market into three periods: August 15, 2008 to December 18, 2008, August 15, 2008 to June 18, 2009, and December 19, 2008 to June 18, 2009. The instantaneous variance of the S&P 500 index, , is replaced by a function of VIX and structural parameters, given by
where 0 30 / 365 , 2 2J ( J J ) , b ( 0 a ) / , a (1 e ) / ,and . The figures within parentheses are the statistics of parameter estimates. The symbols of ***, ** and * indicate significance of statistics at the 1%, 5% and 10% levels, respectively. and are the skewness and kurtosis of the modelimplied instantaneous variance increments, defined as, and , respectively. performs a JarqueBera test of the null hypothesis that the modelimplied instantaneous variance comes from a normal distribution, against the alternative that it does not come from a normal distribution at a 95% confidence interval. 0
Longrun mean of ν (
)
CE
Total variation of ν ( )
)
PT
Adjustment speed of ν (
ED
A. July 3, 2006－June 18, 2009 Insample pricing error ( )
Correlation between diffusion Brownian motions ( )
0
0
MA NU
0
SV
SVJ
8.2329 1.5071*** (4.90) 0.1838*** (6.26) 0.7548*** (12.54) 0.6254*** (12.90)
26.1041
4.1949 3.8388*** (8.30) 0.0886*** (3.91) 0.4363*** (3.18) 0.7844*** (13.22) 1.7078** (1.95) 0.1248*** (6.35) 0.3698*** (6.51) 27.0852
17.1793
14.2989
1.1374
1.0819
0.0010
0.0010
7.1789 0.8602*** (4.62) 0.1718*** (5.95) 0.7294*** (11.30) 0.5723*** (10.15)
3.6659 3.5516*** (7.05) 0.0391*** (5.80) 0.2197*** (4.98) 0.7211*** (9.26) 0.5961**
AC
Mean jump intensity ( )
Mean of price jumpsize innovations ( ) Variance of price jumpsize innovations ( Implied volatility (%) (

)
)
of a JarqueBera test
B. July 3, 2006－August 14, 2008 Insample pricing error ( ) Adjustment speed of ν ( Longrun mean of ν (
) )
Total variation of ν ( ) Correlation between diffusion Brownian motions ( ) Mean jump intensity ( )
42
ACCEPTED MANUSCRIPT
18.2347
(2.34) 0.1301*** (5.17) 0.3520*** (6.20) 19.2580
8.7814
9.8201
1.0871
1.1694
0.0010
0.0010
12.6901 5.7651*** (7.80) 0.1196* (1.99) 0.7664*** (8.50) 0.7467** (4.37)
54.4009
5.6520 3.9328* (2.20) 0.3436** (2.36) 1.8034* (1.72) 0.9127*** (10.46) 9.9650* (1.64) 0.1201** (2.74) 0.3043* (1.65) 52.1165
4.4195
4.2566
0.3995
0.4310
0.0174
0.0223
10.4888 3.1893*** (3.98) 0.2150** (2.80) 0.8210*** (5.80) 0.7633*** (9.09)
46.5644
5.3299 4.5854*** (4.42) 0.2174*** (3.32) 0.9996** (2.23) 0.9491*** (25.85) 4.5983* (1.54) 0.1112*** (3.92) 0.4161*** (2.84) 47.4362
7.5112
6.7361
0.7064
0.6857
0.0010
0.0010
8.7177 1.4721**
5.1039 5.0204***
Mean of price jumpsize innovations ( ) Variance of price jumpsize innovations (

T
)
RI P
Implied volatility (%) (
)
of a JarqueBera test
)
Longrun mean of ν (
)
Total variation of ν ( )
MA NU
Adjustment speed of ν (
SC
C. August 15, 2008－December 18, 2008 Insample pricing error ( )
Correlation between diffusion Brownian motions ( ) Mean jump intensity ( )
ED
Mean of price jumpsize innovations ( ) Variance of price jumpsize innovations (
of a JarqueBera test
CE

)
PT
Implied volatility (%) (
)
D. August 15, 2008－June 18, 2009 Insample pricing error ( )
AC
Adjustment speed of ν ( Longrun mean of ν (
)
)
Total variation of ν ( ) Correlation between diffusion Brownian motions ( ) Mean jump intensity ( ) Mean of price jumpsize innovations ( ) Variance of price jumpsize innovations ( Implied volatility (%) (
)
)
of a JarqueBera test E. December 19, 2008－June 18, 2009 Insample pricing error ( ) Adjustment speed of ν ( )
43
ACCEPTED MANUSCRIPT
Longrun mean of ν (
(3.06) 0.2787** (2.33) 0.8574*** (3.62) 0.7743*** (7.97)
)
Total variation of ν ( )
Mean of price jumpsize innovations ( )
7.2650
0.7536 0.0010
0.5738
SC
)
CE
PT
ED
of a JarqueBera test
AC

)
MA NU
Implied volatility (%) (
7.1635
RI P
Mean jump intensity ( )
Variance of price jumpsize innovations (
41.3401
T
Correlation between diffusion Brownian motions ( )
(3.68) 0.1333*** (4.63) 0.4636** (3.28) 0.9734*** (36.61) 1.0205* (1.68) 0.1053** (2.61) 0.4906** (2.86) 44.316
44
0.0010
ACCEPTED MANUSCRIPT
Table 4 Absolute hedging errors.
M
l 1
 HEt (l 1) t (t lt )e rt ( M l )  / M where
and
is the maturity date of SPX
RI P
T
The figures in this table denote the average points of absolute hedging errors ($250 per point):
futures call options. The hedging error between time t and time t t is defined as HE t (t t ) . The matured SPX futures,
SC
instrument portfolio of hedging scheme 1 (HS1) consists of N 1, t shares of
, and N 2 , t shares of forwardstart strangle portfolios. The forwardstart strangle portfolio
MA NU
consists of a short position on a T1 matured strangle and a long position on a T2 matured strangle. The instrument portfolio of hedging scheme 2 (HS2) consists of N 1, t shares of and N 2 , t shares of the VIX futures with expiry T1 , moneynessmaturity categories. Define
matured SPX futures,
. This study classifies hedging errors into
as moneyness of a SPX futures call. A SPX futures
call is then said to be deep outofthemoney (DOTM) if its
[0.97,1.03); inthemoney (ITM) if
ED
[0.94,0.97); atthemoney (ATM) if
<0.94; outofthemoney (OTM) if
≧1.06. By the term to expiration, a SPX
[1.03,1.06); and deep inthemoney (DITM) if
PT
futures call option contract can be classified as (i) shortterm (<30 days); (ii) mediumterm (30–60 days); and (iii) longterm (>60 days). The outofsample hedging period is from 21 July 2006 to 18 June 2009.
E(
CE
The pvalues under the two null hypotheses of “H0: E( )” and “H0: E(
) q E(
) q E(
) vs. H1: E(
) vs. H1: E(
) > E(
)>
)” are to test whether
AC
the difference in the absolute hedging errors of the two hedging strategies (HS1 and HS2) is statistically significant under the SV and SVJ models, respectively. The symbols of
**
and * indicate significance of
statistics at the 5% and 10% levels, respectively. 1Day Revision DaystoExpiration <30 30 60
Hedging Moneyness Scheme Model A. Jul 21, 2006 Jun 18, 2009 HS1 SV SVJ DOTM HS2 SV SVJ
5.8329 8.4933 5.7488 8.3319
6.2008 31.6892 4.6044 5.0073
H0:E(
) q E(
)
0.9217
0.0000**
0.0000**
H0:E(
) q E(
)
0.8986
0.0000**
SV SVJ
4.7340 4.8790
NA NA
OTM
HS1
45
>60
5Day Revision DaystoExpiration <30 30 60 >60
5.7913 16.4219 8.3815 14.0761 1.8824 13.4179 2.1983 11.2527 pvalue
8.6271 22.7737 8.3442 9.2326
10.5529 9.3129 3.1552 3.6305
0.1254
0.5089*
0.0000**
0.0002**
0.0614*
0.0000**
0.0000**
5.2275 8.0138
14.2134 15.6147
NA NA
6.7398 9.6305
ACCEPTED MANUSCRIPT HS2
SV SVJ
2.4048 3.1536
NA NA
2.2145 6.6859 3.0531 9.3376 pvalue
NA NA
3.6940 4.9829
) q E(
)
0.0138**
NA
0.0000**
0.0032**
NA
0.0000**
H0:E(
) q E(
)
0.1928
NA
0.0000**
0.0276**
NA
0.0000**
SV SVJ SV SVJ
4.8706 5.3093 3.6236 2.2060
6.2623 32.1829 5.0806 5.6141
7.8156 8.6025 11.2080 10.6070 3.8468 3.6578 4.8969 4.7077 pvalue
12.9177 22.8385 12.5941 13.2571
10.5497 14.0824 6.7227 8.3758
HS2
) q E(
)
0.1412
0.0171**
H0:E(
) q E(
)
0.0006**
0.0125**
HS1 ATM2
HS2
0.0000**
0.0067**
0.7463
0.0000**
0.0000**
0.0074**
0.0075**
0.0001**
MA NU
H0:E(
RI P
ATM1
SC
HS1
T
H0:E(
SV SVJ SV SVJ
6.8883 6.4570 6.8828 4.4955
8.9601 23.2452 7.5894 8.1042
8.6977 12.9543 5.4217 6.9022 pvalue
6.1710 8.4305 7.9739 7.6883
19.1191 24.8081 18.6367 19.2646
12.5169 18.1361 10.0803 12.5638
) q E(
)
0.9967
0.0615*
0.0000**
0.4081
0.7381
0.0267**
H0:E(
) q E(
)
0.1472
0.0773*
0.0026**
0.6847
0.0835*
0.0107**
9.1297 7.9787 9.0078 7.8946
9.4940 27.1869 8.8718 9.2986
7.1300 16.9370 39.4481 16.8191 5.1574 16.1473 6.3767 15.5056 pvalue
21.3883 26.4431 20.7272 20.9980
11.8076 27.8737 11.0048 13.0975
0.9318
0.6205
0.0028**
0.8273
0.7771
0.4609
0.9548
0.2554
0.0528*
0.6561
0.3348
0.0397**
16.9423 14.8874 16.1436 14.7896
12.1066 94.6282 12.1011 12.0533
9.9524 34.3362 20.2989 32.0880 8.0175 34.0936 13.9179 32.0985 pvalue
28.9465 57.8329 28.2546 28.0366
16.1561 30.6637 12.9926 24.1756
ITM
HS2
) q E(
H0:E(
) q E(
)
CE
H0:E(
SV SVJ SV SVJ
PT
HS1
)
SV SVJ SV SVJ
DITM
AC
HS1
ED
H0:E(
H0:E(
) q E(
)
0.6389
0.9962
0.1363
0.9551
0.7437
0.0655*
H0:E(
) q E(
)
0.9513
0.0091**
0.0118**
0.9978
0.0041**
0.0867*
B. Jul 21, 2006 Aug 14, 2008 HS1 SV SVJ DOTM HS2 SV SVJ
NA NA NA NA
2.1962 2.1832 1.9477 2.0341
NA NA NA NA
2.3741 2.1979 2.3704 2.1301
1.5551 1.3644 1.4860 1.0437
H0:E(
) q E(
)
NA
0.3362
0.0000**
NA
0.9912
0.3691
H0:E(
) q E(
)
NA
0.6719
0.9637
NA
0.8835
0.0028**
SV SVJ SV
NA NA NA
NA NA NA
3.2045 2.1027 1.3354
NA NA NA
NA NA NA
3.8217 3.0037 2.5328
HS2
HS1 OTM HS2
46
1.1342 1.0408 0.8792 1.0371 pvalue
ACCEPTED MANUSCRIPT SVJ
NA
NA
1.6922 pvalue
NA
NA
2.9046
) q E(
)
NA
NA
0.0000**
NA
NA
0.0000**
H0:E(
) q E(
)
NA
NA
0.0039**
NA
NA
0.5875
SV SVJ SV SVJ
NA NA NA NA
4.2869 4.6295 3.4436 3.8990
NA NA NA NA
6.3865 7.2811 6.2879 6.8936
8.0379 7.6641 5.9108 6.9447
0.0000**
0.8359
0.0001**
0.0000**
NA
0.4781
0.1514
NA NA NA NA
11.2765 12.1989 11.1193 11.6040
9.8152 10.2419 9.2718 10.1104
ATM1
HS2
)
NA
0.0358**
H0:E(
) q E(
)
NA
0.0991*
SV SVJ SV SVJ
NA NA NA NA
6.9181 7.2796 5.6824 6.2526
SC
) q E(
NA
H0:E(
) q E(
)
NA
0.0450**
0.0003**
NA
0.8586
0.4455
H0:E(
) q E(
)
NA
0.1190
0.0243**
NA
0.5219
0.8729
SV SVJ SV SVJ
ED
H0:E(
5.9348 5.3591 2.8907 3.3706 pvalue
RI P
HS1
T
H0:E(
NA NA NA NA
6.9824 7.0991 6.6085 6.7266
3.6398 3.9125 3.2016 3.7983 pvalue
NA NA NA NA
13.7744 13.8075 12.6659 13.2618
6.0485 6.6600 5.5578 6.6144
)
NA
0.7403
0.2604
NA
0.5691
0.4941
)
NA
0.7613
0.8021
NA
0.7891
0.9532
SV SVJ SV SVJ
NA NA NA NA
13.5854 12.7503 12.7643 11.5203
6.4183 8.4154 5.6090 7.6518 pvalue
NA NA NA NA
13.8552 13.1806 12.0081 12.5431
12.4903 13.0566 11.8338 12.2788
HS2
HS1 ITM
HS2
) q E(
H0:E(
) q E(
CE
H0:E(
HS1 HS2
H0:E(
AC
DITM
6.4690 6.6270 4.3589 5.1346 pvalue
MA NU
ATM2
PT
HS1
) q E(
)
NA
0.7793
0.5607
NA
0.6578
0.8547
H0:E(
) q E(
)
NA
0.6398
0.7419
NA
0.8503
0.8277
7.6480 11.4905 7.5772 11.4174
10.5635 11.2614 8.5305 9.6118
10.9136 22.4222 3.3582 18.6405 3.0728 18.0134 2.9827 14.7426 pvalue
14.0124 15.7957 13.5323 15.4117
21.6642 5.1409 4.4312 4.2620
C. Aug 15, 2008 Dec 18, 2008 HS1 SV SVJ DOTM HS2 SV SVJ H0:E(
) q E(
)
0.9398
0.0033**
0.0000**
0.0000**
0.4306
0.0004**
H0:E(
) q E(
)
0.9540
0.0297**
0.0876*
0.0000**
0.5927
0.0306**
SV SVJ SV
5.8033 6.2498 2.5001
NA NA NA
10.3276 8.6948 6.3684
18.1836 18.8846 8.5513
NA NA NA
14.0134 9.9149 7.8971
HS1 OTM HS2
47
ACCEPTED MANUSCRIPT SVJ
4.0152
NA
7.3196 10.7651 pvalue
NA
8.5085
) q E(
)
0.0032**
NA
0.0163**
0.0001**
NA
0.0224**
H0:E(
) q E(
)
0.1369
NA
0.0571*
0.0112**
NA
0.1318
SV SVJ SV SVJ
5.6013 6.4635 3.7156 2.5303
14.7696 16.1270 11.4663 12.9278
31.2404 32.5086 30.5141 31.7426
18.1844 12.9225 10.0351 11.0667
HS2
) q E(
)
0.0611*
0.1044
H0:E(
) q E(
)
0.0000**
0.1279
SV SVJ SV SVJ
8.3341 8.2593 7.8299 5.5400
17.9943 19.0578 15.8407 17.0484
HS1 ATM2
HS2
0.0387**
0.0020**
0.8372
0.0294**
0.1118
0.0066**
0.8530
0.1526
8.0280 9.2510 8.0043 8.2282
42.5501 44.0385 41.7969 43.0365
18.6694 16.9034 12.2722 14.2951
14.6713 14.6458 10.6875 12.1188 pvalue
MA NU
H0:E(
RI P
ATM1
13.7353 10.6191 12.1018 11.9816 8.9570 4.5807 10.3001 4.5126 pvalue
SC
HS1
T
H0:E(
) q E(
)
0.7514
0.4752
0.2184
0.9919
0.8678
0.2148
H0:E(
) q E(
)
0.0775*
0.5139
0.1209
0.6846
0.8447
0.2079
SV SVJ SV SVJ
10.4907 9.4448 10.0307 8.3360
14.7204 11.0775 14.9294 9.4768 10.8168 10.2878 12.0713 8.9498 pvalue
37.7847 38.1067 36.9593 37.1115
23.6318 23.3064 22.7341 23.2022
ITM
HS2
) q E(
H0:E(
) q E(
HS1
0.7493
0.6571
0.1025
0.8322
0.9130
0.8174
)
0.4537
0.3891
0.0370**
0.8790
0.9095
0.9613
16.5189 15.5831 15.8198 14.6301
12.1581 13.2079 11.7373 12.3661
11.3606 27.0118 23.5030 24.7545 10.6019 26.4372 17.2753 23.0175 pvalue
29.5314 28.8187 29.3436 28.2613
16.9069 28.7006 13.2451 26.6083
HS2
SV SVJ SV SVJ
H0:E(
AC
DITM
)
CE
H0:E(
16.0566 17.9491 13.9291 13.5954
PT
HS1
ED
H0:E(
) q E(
)
0.6797
0.8936
0.6954
0.8885
0.9714
0.1706
H0:E(
) q E(
)
0.5232
0.8013
0.0974*
0.6411
0.9243
0.6692
D. Aug 15, 2008 Jun 18, 2009 HS1 SV SVJ DOTM HS2 SV SVJ
6.4329 8.4933 5.7488 8.3319
6.8937 36.8297 5.0641 5.5045
9.2821 16.4219 13.8840 14.0761 2.6343 13.4179 2.9187 11.2527 pvalue
9.7092 26.3464 9.2914 10.4098
17.2975 15.2710 4.4064 4.8423
H0:E(
) q E(
)
0.4246
0.0000**
0.0000**
0.1254
0.3793
0.0000**
H0:E(
) q E(
)
0.8986
0.0000**
0.0002**
0.0614*
0.0000**
0.0000**
SV SVJ SV SVJ
4.7340 4.8790 2.4048 3.1536
NA NA NA NA
10.6376 23.8219 4.5656 6.6926
14.2134 15.6147 6.6859 9.3376
NA NA NA NA
14.5439 27.3526 6.7992 9.4712
HS1 OTM
HS2
48
ACCEPTED MANUSCRIPT pvalue 0.0138
**
NA
0.0000**
0.0032**
NA
0.0000**
0.0000**
0.0276**
NA
0.0000**
23.3288 54.9344 22.3978 26.3855
17.1645 30.9849 8.8608 12.1448
0.0067**
0.6612
0.0002**
0.0074**
0.0056**
0.0001**
) q E(
)
H0:E(
) q E(
)
0.1928
NA
SV SVJ SV SVJ
4.8706 5.3093 3.6236 2.2060
10.3376 89.0275 8.4577 9.1524
12.7689 8.6025 26.6114 10.6070 6.3648 3.6578 8.9163 4.7077 pvalue 0.0000**
HS2
) q E(
)
0.1412
0.0931*
H0:E(
) q E(
)
0.0006**
0.0135**
SV SVJ SV SVJ
6.8883 6.4570 6.8828 4.4955
13.0222 55.0048 11.3828 11.7874
HS1 ATM2
HS2
0.0008**
13.6708 27.0726 7.7933 10.8463 pvalue
6.8840 8.4305 6.8839 7.6883
31.7308 49.8909 30.5907 34.5034
18.5452 35.7508 11.8843 16.6995
MA NU
H0:E(
RI P
ATM1
SC
HS1
T
H0:E(
) q E(
)
0.9967
0.3295
0.0036**
0.9999
0.7045
0.0336**
H0:E(
) q E(
)
0.1472
0.0904*
0.0089**
0.6847
0.0771*
0.0041**
SV SVJ SV SVJ
9.1297 7.9787 8.4078 7.8946
12.0504 47.6335 11.1755 10.8986
11.3279 82.1890 7.5098 9.4780 pvalue
8.9370 8.8191 8.8368 8.5056
29.1382 39.3043 28.9324 31.8723
18.7345 53.3887 16.3535 19.6924
ITM
HS2
) q E(
H0:E(
) q E( HS1 HS2
AC
DITM
)
0.6128
0.6883
0.0036**
0.9779
0.9550
0.2555
)
0.9548
0.2479
0.0530*
0.9152
0.4956
0.0318**
14.9423 13.8874 14.1436 12.7896
11.9002 106.0531 10.9146 11.8486
10.7869 23.3362 23.1047 22.0880 8.3501 22.0936 14.9251 21.4985 pvalue
31.0522 64.0635 30.9634 29.9194
17.0217 34.8209 12.5579 25.3318
CE
H0:E(
PT
HS1
ED
H0:E(
SV SVJ SV SVJ
) q E(
)
0.6389
0.4289
0.1180
0.7733
0.9684
0.0221**
) q E(
)
0.4935
0.0089**
0.0063**
0.8761
0.0036**
0.0343**
E. Dec 19, 2008 Jun 18, 2009 HS1 SV SVJ DOTM HS2 SV SVJ
0.2552 0.2698 0.2466 0.2621
3.7353 58.8359 2.0805 1.9695
6.6704 30.7344 1.9325 2.8162 pvalue
0.7291 2.1383 0.6988 2.1252
6.0054 35.4272 4.7806 5.2441
10.3069 31.4880 4.3666 5.7714
H0:E(
) q E(
)
0.9326
0.0000**
0.0000**
0.8830
0.0209**
0.0000**
H0:E(
) q E(
)
0.8949
0.0001**
0.0002**
0.9759
0.0000**
0.0000**
SV SVJ SV SVJ
1.0983 0.2186 1.0807 0.2040
NA NA NA NA
11.0911 45.9416 1.9294 5.7757
0.7148 4.4971 0.3437 4.4841
NA NA NA NA
15.3196 52.8511 5.1939 10.8790
H0:E( H0:E(
HS1 OTM
HS2
49
ACCEPTED MANUSCRIPT pvalue ) q E(
)
0.9384
NA
0.0000**
0.0000**
NA
0.0000**
H0:E(
) q E(
)
0.9156
NA
0.0000**
0.9387
NA
0.0000**
SV SVJ SV SVJ
2.1305 0.9810 1.2787 0.9799
6.8348 146.6424 6.0798 6.1686
11.4281 46.7435 2.7680 6.9963 pvalue
1.0402 5.4522 0.1968 5.4394
17.0761 72.6580 16.6123 17.4098
15.7494 56.0465 7.2314 13.6406
0.0001**
0.7691
0.0000**
0.0014**
0.9604
0.0024**
0.0002**
12.2774 44.3815 3.7622 9.0738 pvalue
1.3427 6.2974 0.1209 6.2847
22.0154 55.1462 19.9361 19.6574
18.3723 62.0025 11.3440 20.0485
HS2
) q E(
)
0.0123**
0.4059
H0:E(
) q E(
)
0.9950
0.0149**
SV SVJ SV SVJ
3.1292 1.7709 3.0204 1.7598
8.5574 87.2837 7.3797 7.0633
HS1 ATM2
HS2
0.0000**
MA NU
H0:E(
RI P
ATM1
SC
HS1
T
H0:E(
) q E(
)
0.6607
0.3206
0.0000**
0.0000**
0.3043
0.0017**
H0:E(
) q E(
)
0.9529
0.0984*
0.0159**
0.8164
0.0259**
0.0060**
SV SVJ SV SVJ
4.0260 2.4809 3.4469 2.0897
9.4581 66.8410 9.3937 9.1536
0.9104 6.3525 0.6204 6.3399
23.5434 40.0792 22.7974 24.5999
13.8821 83.1949 10.0316 15.2241
ITM
HS2
0.0270**
0.9700
0.0000**
0.0900*
0.7756
0.0014**
)
0.0377**
0.2729
0.0578*
0.7338
0.3678
0.0293**
SV SVJ SV SVJ
5.2200 3.4308 4.8072 3.0395
11.7884 146.2860 11.6914 10.7576
9.8307 22.4410 4.5971 11.0082 pvalue
0.6702 5.6443 0.1415 5.6319
31.7112 79.3362 30.7986 29.7712
17.2130 45.0214 11.4124 23.2043
) q E(
)
0.3106
0.9327
0.0416**
0.2203
0.6921
0.0364**
) q E(
)
0.1931
0.0085**
0.0231**
0.9605
0.0027**
0.0125**
H0:E(
) q E(
CE
) q E(
HS1
H0:E(
HS2
AC
H0:E(
7.9665 148.8315 4.2333 6.9084 pvalue
)
H0:E(
DITM
PT
HS1
ED
H0:E(
50
ACCEPTED MANUSCRIPT
Table 5 Average hedging errors.
M
l 1
HEt (l 1) t (t lt )e rt ( M l ) / M where
and
is the maturity date of SPX futures
RI P
T
The figures in this table denote the average points of hedging errors ($250 per point):
call options. The hedging error between time t and time t t is defined as HE t (t t ) . The matured SPX futures,
SC
instrument portfolio of hedging scheme 1 (HS1) consists of N 1, t shares of
, and N 2 , t shares of forwardstart strangles. The forwardstart strangles consist of a short
MA NU
position on a T1 matured strangle and a long position on a T2 matured strangle. The instrument portfolio of hedging scheme 2 (HS2) consists of N 1, t shares of shares of the VIX futures with expiry T1 , moneynessmaturity categories. Define
matured SPX futures, and N 2 , t
. This study classifies hedging errors into
as moneyness of a SPX futures call. A SPX futures
call is then said to be deep outofthemoney (DOTM) if its
[0.97,1.03); inthemoney (ITM) if
ED
[0.94,0.97); atthemoney (ATM) if
<0.94; outofthemoney (OTM) if
≧1.06. By the term to expiration, a SPX
[1.03,1.06); and deep inthemoney (DITM) if
PT
futures call option contract can be classified as (i) shortterm (<30 days); (ii) mediumterm (30–60 days);
CE
and (iii) longterm (>60 days). The outofsample hedging period is from 21 July 2006 to 18 June 2009.
AC
Hedging Moneyness Scheme Model A. Jul 21, 2006 Jun 18, 2009 HS1 SV SVJ DOTM HS2 SV SVJ HS1 SV SVJ OTM HS2 SV SVJ HS1 SV SVJ ATM1 HS2 SV SVJ HS1 SV SVJ ATM2 HS2 SV SVJ HS1 SV SVJ ITM HS2 SV SVJ
1Day Revision DaystoExpiration <30
30 60
>60
2.8390 5.2072 2.7281 5.1884 1.1157 2.0840 0.8611 2.0223 1.5552 0.6932 1.1694 0.6374 5.1354 4.9462 4.4271 3.5067 7.4345 6.9341 7.1144 6.8463
2.8762 25.9343 1.9852 1.7105 NA NA NA NA 0.6738 24.8450 0.6507 0.6552 1.5561 14.0600 1.4691 1.2689 4.0729 20.0595 3.4119 2.6260
1.8114 0.7847 0.0012 0.2691 0.2503 1.3175 0.0666 0.0174 1.2684 0.0859 0.0384 0.0237 0.1142 0.1406 0.1111 0.1402 1.9169 1.6727 1.4529 1.4544
51
5Day Revision DaystoExpiration <30 30 60 13.7181 11.8536 11.2294 9.5215 7.5231 8.3460 2.4801 4.6683 3.5010 4.3550 2.6266 0.4759 4.2973 4.7267 3.9269 3.3109 8.9370 8.4496 8.8368 8.3006
4.6776 9.5738 4.5018 4.9837 NA NA NA NA 4.2204 10.5435 4.1317 5.0167 7.7341 10.6113 7.0446 7.8498 13.3552 15.1941 12.8081 11.9149
>60 0.4372 1.2054 1.0195 0.3605 0.6831 2.3168 0.5634 0.5864 0.3892 1.3111 0.3834 0.4709 1.0545 1.4848 1.0498 1.4157 4.5900 0.3254 4.5067 0.1530
ACCEPTED MANUSCRIPT
6.4425 86.6284 5.7767 5.3072
7.7711 2.3076 3.4147 2.1691
23.1700 22.0880 23.1611 21.3961
14.7266 36.9230 14.6212 13.7740
11.4645 8.1373 7.2423 6.7845
NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
0.8411 1.0124 0.1183 1.0083 NA NA NA NA 1.9793 2.7606 1.4356 2.0058 2.1785 3.2895 1.4021 2.2925 0.0048 1.7478 0.0035 1.6630 3.5868 0.0080 0.3598 0.0036
0.1770 0.1320 0.0392 0.1041 2.0308 0.9101 0.0744 0.1318 3.8498 3.2230 0.4293 0.4943 2.4401 2.5936 0.1110 0.0735 0.6201 1.1181 0.4451 0.7377 2.0003 2.1317 2.0026 2.0478
NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
0.7835 0.7573 0.6927 0.7560 NA NA NA NA 1.5468 2.3004 0.7233 2.0327 1.3849 2.6603 0.5328 2.5582 3.6893 1.4969 3.6595 0.9506 1.9752 2.3968 1.9111 1.4656
0.3253 0.1128 0.2880 0.1008 2.1418 1.1026 0.2438 0.3454 3.9888 3.4000 0.7642 0.7010 1.7701 1.8573 0.6547 0.7992 0.2854 0.5566 0.2247 0.3106 1.6812 0.5475 0.5541 0.5134
3.8282 7.2995 3.5487 7.2171 1.1208 2.6408 0.5024 2.0759 2.5380 1.1397 2.5223 1.0714 8.3141 7.5297 7.8299 5.5400 10.4907 9.4448 10.0307 8.3360 16.5189 15.5831 15.8198 14.6301
4.5523 5.0446 3.6732 3.9902 NA NA NA NA 8.4063 8.8251 8.1388 8.5812 12.6079 13.1475 12.4412 13.1332 11.2037 11.8665 11.1040 11.8868 5.5772 5.4481 5.5494 5.5132
19.2037 15.5682 16.0579 12.3495 9.9460 12.1234 3.1084 7.3601 4.7120 6.9702 3.3795 0.8478 5.4337 4.1225 5.0043 4.7057 11.0775 9.0088 10.2254 8.4235 26.9973 24.7545 26.1183 22.5741
11.2566 11.9120 11.2338 11.4042 NA NA NA NA 31.2404 32.1716 30.5141 31.7426 42.5501 44.0385 41.7969 43.0365 37.7847 38.1067 36.9593 37.1115 29.5314 28.8187 29.3436 28.2613
2.7077 1.1593 2.2417 0.9752 5.8195 5.6768 5.8166 3.8669 6.9820 8.5429 6.8260 4.2104 5.4160 10.7723 5.1376 5.5466 14.8300 18.6708 13.9976 18.3482 15.2319 20.1900 9.5881 2.2296
CE
AC
RI P
SC
MA NU 52
T
13.4856 12.9300 13.2439 12.8297
ED
SV SVJ DITM HS2 SV SVJ B. Jul 21, 2006 Aug 14, 2008 HS1 SV SVJ DOTM HS2 SV SVJ HS1 SV SVJ OTM HS2 SV SVJ HS1 SV SVJ ATM1 HS2 SV SVJ HS1 SV SVJ ATM2 HS2 SV SVJ HS1 SV SVJ ITM HS2 SV SVJ HS1 SV SVJ DITM HS2 SV SVJ C. Aug 15, 2008 Dec 18, 2008 HS1 SV SVJ DOTM HS2 SV SVJ HS1 SV SVJ OTM HS2 SV SVJ HS1 SV SVJ ATM1 HS2 SV SVJ HS1 SV SVJ ATM2 HS2 SV SVJ HS1 SV SVJ ITM HS2 SV SVJ HS1 SV SVJ DITM HS2 SV SVJ
PT
HS1
2.1467 0.3448 0.0512 0.5503 0.9383 1.5313 0.7628 0.3355 2.4975 3.2675 1.4064 0.8982 1.2458 5.5048 1.2159 2.2863 4.1159 7.1031 4.0811 6.2648 9.2562 14.0350 4.4799 4.5303
3.2284 30.5973 2.3082 2.2588 NA NA NA NA 6.1472 81.7971 5.8738 6.1450 8.9851 48.5724 8.0774 8.3533 8.2234 42.2563 7.7128 8.0095 6.8409 98.7172 6.5325 6.1571
3.3019 1.2740 0.0273 0.3927 4.5112 7.2749 0.4437 0.4165 5.5297 8.8002 0.9907 0.8522 5.0756 5.3328 1.5006 1.2976 4.9683 2.3397 3.7358 2.0911 9.1337 2.3492 3.7245 2.3339
13.7181 11.8536 11.2294 9.5215 7.5231 8.3460 2.4801 4.6683 3.5010 4.3550 2.6266 0.4759 4.2973 4.7267 4.2269 4.3109 8.9370 8.4496 8.8368 8.3006 23.1700 22.0880 22.0936 21.4985
5.3514 11.3616 5.2474 5.9804 NA NA NA NA 16.1184 37.0414 16.0486 19.5599 25.8740 37.0119 25.0965 28.5540 23.1937 29.1358 23.1503 25.0100 17.0571 42.4095 16.6738 15.9005
0.5210 2.0245 0.3430 0.4877 8.2378 11.4613 3.8244 3.0784 9.0904 13.7178 5.2224 3.5571 7.3574 8.9422 4.8719 3.1144 10.4540 1.3862 10.3394 1.2989 14.5683 10.1879 9.0832 7.2971
0.2520 0.2650 0.2466 0.2559 1.0983 0.1910 1.0807 0.2001 2.1305 0.9810 1.2787 0.9799 3.1292 1.7709 3.0204 1.7598 4.0260 2.4809 3.4469 2.0897 5.2200 3.4308 4.8072 3.0395
2.0888 52.5901 1.1334 0.7687 NA NA NA NA 4.3617 139.4685 3.2934 3.4293 5.7319 80.3826 3.2608 3.0734 6.2950 61.9202 5.5185 5.5006 7.3885 139.1338 6.5252 6.0027
5.1512 2.7616 0.0108 0.1405 9.7357 15.6734 0.0230 0.5349 9.7368 16.4767 0.4140 0.7883 10.4100 5.0932 0.0864 0.0795 5.8128 11.6959 1.5111 1.9372 8.9296 17.1273 2.4656 5.3399
0.6289 2.1383 0.3988 2.1252 0.7148 4.4971 0.3437 4.4841 1.0402 5.4522 0.1968 5.4394 1.3427 6.2974 0.0484 6.2847 0.9104 6.3525 0.6204 6.3399 0.4315 5.6443 0.1415 5.6319
0.2689 10.8879 0.0622 0.4091 NA NA NA NA 4.1671 40.8901 4.1453 5.1897 10.8996 30.7023 9.5083 8.3656 13.7524 23.3311 14.7798 13.2973 11.6515 48.2989 12.7502 9.6774
5.6898 3.4095 0.0956 0.2928 11.7739 19.9199 0.7359 1.9254 12.0159 20.8979 1.6098 2.6505 10.0615 6.3932 1.0695 0.2733 6.1182 15.7397 2.7653 4.3236 13.4623 6.4823 8.2415 6.4429
ED
CE
AC
RI P
MA NU 53
T
2.8390 5.2072 2.7281 5.1884 1.1157 2.0840 0.8611 2.0223 1.5552 0.6932 1.1694 0.6374 5.1354 4.9462 4.4271 3.5067 7.4345 6.9341 7.1144 6.8463 13.4856 12.9300 13.1436 12.7896
PT
D. Aug 15, 2008 Jun 18, 2009 HS1 SV SVJ DOTM HS2 SV SVJ HS1 SV SVJ OTM HS2 SV SVJ HS1 SV SVJ ATM1 HS2 SV SVJ HS1 SV SVJ ATM2 HS2 SV SVJ HS1 SV SVJ ITM HS2 SV SVJ HS1 SV SVJ DITM HS2 SV SVJ E. Dec 19, 2008 Jun 18, 2009 HS1 SV SVJ DOTM HS2 SV SVJ HS1 SV SVJ OTM HS2 SV SVJ HS1 SV SVJ ATM1 HS2 SV SVJ HS1 SV SVJ ATM2 HS2 SV SVJ HS1 SV SVJ ITM HS2 SV SVJ HS1 SV SVJ DITM HS2 SV SVJ
SC
ACCEPTED MANUSCRIPT
AC
CE
PT
ED
MA NU
SC
RI P
T
ACCEPTED MANUSCRIPT
Fig. 1. Average daily volume, average daily open interest, and average daily block trader order execution of VIX futures across years 2004 –2013.
54