Using VIX futures to hedge forward implied volatility risk

Using VIX futures to hedge forward implied volatility risk

    Using VIX futures to hedge forward implied volatility risk Yueh-Neng Lin, Anchor Y. Lin PII: DOI: Reference: S1059-0560(15)00183-5 d...

805KB Sizes 3 Downloads 71 Views

    Using VIX futures to hedge forward implied volatility risk Yueh-Neng Lin, Anchor Y. Lin PII: DOI: Reference:

S1059-0560(15)00183-5 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

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT

RI P

T

Using VIX futures to hedge forward implied volatility risk

SC

Yueh-Neng Lina, and Anchor Y. Lina

ABSTRACT

MA NU

The fair value of VIX futures is derived by pricing the forward 30-day 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 forward-start strangles, this study

ED

demonstrates that VIX futures could offer more effective volatility-risk hedge for an

PT

investor who has a short position on the S&P 500 futures call option. In particular, the

CE

delta-vega-neutral hedging strategy incorporating stochastic volatility on average

AC

outperforms in out-of-sample hedging. Adding price jumps further enhances the hedging performance during the crash period.

Keywords: VIX futures; Forward implied volatility; Forward-start strangles; Stochastic volatility; Price jumps Classification code: G12, G13, G14

a

Department of Finance, National Chung Hsing University, Taiwan. Corresponding author: Yueh-Neng Lin, Department of Finance, National Chung Hsing University, 250, Kuo-Kuang Road, Taichung, Taiwan, Phone: (886) 4 2285-7043, Fax: (886) 4 2285-6015, Email: [email protected]

ACCEPTED MANUSCRIPT

T

1. Introduction

RI P

The U.S. stock market crashed on October 19, 1987, when the Dow Jones Industrials

SC

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

MA NU

the booming-crash cycle of financial markets becomes often and makes markets highly uncertain, effectively managing volatility risk has become urgent for market participants.

ED

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

PT

contract (Neuberger, 1994), straddle options (Carr & Madan, 1998), straddles (Brenner, Ou,

CE

& Zhang, 2006), vanilla and volatility options (Psychoyios & Skiadopoulos, 2006), and the

AC

Chicago Board Options Exchange (CBOE) Volatility Index (VIX) (Gonzalez-Perez, 2015). The CBOE launched the VIX Futures on March 26, 2004, successively followed by the Three-Month S&P 500 Variance Futures on May 18, 2004, the Twelve-Month 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 out-of-the-money calls and puts written on the S&P 500 Index (SPX) with valid quotes. At-the-money call and put

1

ACCEPTED MANUSCRIPT

T

options on SPX are also included with their prices averaged. It attempts to gauge the

RI P

expected volatility over the next 30 days. The introduction of VIX futures has been a major

SC

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

MA NU

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

ED

for a tradable vehicle which can be used to hedge or to implement a view on volatility. [Fig. 1 about here]

PT

The fair value of VIX futures is derived by pricing the forward 30-day volatility

CE

which underlies the volatility risk of S&P 500 in the 30 days after the futures expiration.

AC

The most straightforward application of VIX futures is to manage the forward implied volatility. While forward implied volatility can also be traded with forward-start straddles or by unwinding delta-hedged 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

2

ACCEPTED MANUSCRIPT

T

investor who faces stochastic volatility and price jumps.1 To conduct this hedging exercise,

RI P

the study proposes a delta-vega-neutral strategy to manage the underlying price risk and

SC

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

MA NU

purpose of comparison, the hedging effectiveness of forward-start strangles is also investigated. Forward-start strangles consist of two wide strangles with different maturities,

ED

so that the changes of the underlying stock price will not affect the payoff of the portfolio. In other words, the forward-start strangles can manage forward volatility risk without

PT

exposure to delta and gamma risk (Rebonato, 1999).

CE

Recent empirical work on index options identifies factors such as stochastic volatility,

AC

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.

3

ACCEPTED MANUSCRIPT

T

certain strike-maturity combinations, but increase for others. Eraker (2004) finds that

RI P

adding jumps in returns and volatility decreases errors by only 1%. Bates (2000) finds a

SC

10% decrease, but it falls to around 2% when time-series consistency is imposed. Broadie, Chernov, and Johannes (2007) find strong evidence for jumps in prices and modest evidence

MA NU

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

ED

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

PT

volatility is modeled, the hedging performance may be improved by incorporating neither

CE

price jumps, nor stochastic interest rates into the SPX option pricing framework. Bakshi and

AC

Kapadia (2003) use Heston’s (1993) stochastic-volatility option pricing model to construct a delta-hedged 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 delta-neutral strategy. Vishnevskaya (2004) follows the structure of Bakshi and Kapadia (2003) and constructs a delta-vega-hedged portfolio consisting of the underlying stock, another option and the money-market fund, to hedge a long position on

4

ACCEPTED MANUSCRIPT

T

the SPX call option. His result suggests the existence of some other sources of risk. Guided

RI P

by previous studies, this study considers both stochastic volatility and price jumps, denoted

SC

the “SVJ” model.2 The setup contains the competing stochastic-volatility (SV) futures option formula as special cases.

MA NU

Since SPX futures option contracts are American-style, it is important to take into account the extra value accruing from the ability to exercise the options prior to maturity.

ED

One can follow such a nonparametric approach as in Aït-Sahalia and Lo (1998) and Broadie, Detemple, Ghysels, and Torrés (2000) to price American options. Closed-form option

PT

pricing formulas, however, make it possible to derive hedge ratios analytically. For options

CE

with early exercise potential, this study therefore computes a quadratic approximation for

AC

evaluating American futures options. The approximation is based on the one developed by MacMillan

(1987),

examined

by

Barone-Adesi

and

Whaley

(1987)

for

the

constant-volatility process, extended by Bates (1991) for the jump-diffusion 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

2

Our results could be generalized by introducing time-varying 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.

5

ACCEPTED MANUSCRIPT

T

practically feasible in the presence of stochastic jump sizes (e.g. for the SVJ model). This

RI P

difficulty is seen from the existing work by Cox and Ross (1976), Merton (1976), Bates

SC

(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

MA NU

(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

ED

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

PT

reveal that VIX futures generally outperform forward-start strangles over the out-of-sample

CE

hedging period, August 2006June 2009. These diagnostics document the extraordinary

AC

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 forward-start 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.

6

ACCEPTED MANUSCRIPT

T

Section 4 summarizes data and model parameter estimation procedure. Section 5 presents

SC

out-of-sample hedging results. Section 7 finally concludes.

RI P

summary statistics of parameter estimates and in-sample pricing errors. Section 6 analyzes

MA NU

2. Hedging strategies

Consider a time-t short position on the T1 -matured call option written on T2

ED

-matured SPX futures as the unhedged portfolio, i.e.,  Ct (F ) for t  T1  T2 . This study

PT

then constructs two hedging strategies (HS1 and HS2) to hedge the position.

CE

Hedging Strategy 1 (HS1): The hedged portfolio (  ) consists of a short position on a SPX

AC

futures call, N1, t shares of underlying SPX futures (F), and N 2, t shares of forward-start strangles. That is,  t  Ct ( F )  N1,t  Ft  N 2,T  INSTt , where INSTt is forward-start 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 T2-matured strangle is constructed by long a T2-matured SPX call with strike K2, and long a T2-matured SPX put with strike K1. That is, forward-start

7

ACCEPTED MANUSCRIPT

T

strangles at time t ( INSTt ) is given by

RI P

INSTt  ct ( S , T1 , K 2 )  pt ( S , T1 , K1 )  ct ( S , T2 , K 2 )  pt ( S , T2 , K1 )

SC

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.

MA NU

and T2 , respectively. pt ( S , T1, K1 ) and pt ( S , T2 , K1 ) are K1 -strike SPX put options with

ED

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.

PT

That is,  t  Ct ( F )  N1,t  Ft  N 2,t  INSTt , where INSTt is the time-t price of the VIX

AC

CE

futures with expiry T1 , FtVIX (T1 ) .

Further, add the constraints of delta-neutral and vega-neutral 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

8

ACCEPTED MANUSCRIPT

Ct ( F ) INSTt  N 2,t Ft Ft

(3)

T

N1,t 

RI P

 C ( F )   INSTt   /   N 2,t   t     t t    

for

SC

The illustrations of Ct ( F ) / Ft , INSTt / Ft , Ct ( F ) /  t and INSTt /  t

(4)

MA NU

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: HS1-SV, HS1-SVJ, HS2-SV and HS2-SVJ.

ED

Assuming that there are no arbitrage opportunities, the hedged portfolio  t should earn the

CE

t is expressed as

PT

risk-free 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

AC

where

 t  t   t  t   t

hedging error as a function of rebalancing frequency

is defined as the additional profit

(loss) over the risk-free rate of return and it can be written as HEt (t  t )   t  t   t (ert  1)  N1,t [ Ft  t (T2 )  Ft (T2 )]  N 2,t [ INSTt  t  INSTt ]  [Ct  t ( F )  Ct ( F )]  (ert  1) [ N1,t Ft (T2 )  N 2,t INSTt  Ct ( F )]

Finally, compute the average absolute hedging error:

9

(6)

ACCEPTED MANUSCRIPT

T

(7)

(8)

SC

RI P

, and the average dollar-value hedging error:

where

and T1 is the expiry of the unhedged SPX futures call option.

MA NU

Measurements of hedging performances are based on the framework proposed in previous papers, for instance, Bakshi, Cao, and Chen (1997) and Psychoyios and Skiadopoulos

ED

(2006).

PT

3. Empirical pricing models

CE

Hedging strategies are constructed using the data of SPX, SPX futures, SPX options,

AC

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 jump-diffusion 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

10

ACCEPTED MANUSCRIPT

T

Contingent claims are priced as if investors were risk-neutral and the SPX price is

RI P

assumed to follow a jump-diffusion with stochastic volatility (SVJ) under the risk-neutral

SC

probability measure Q:

dSt  (b  J  J ) St   t St dS ,t  J t St dNt

(9)

MA NU

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

ED

in the asset log-price are assumed to be normally distributed, i.e., ln( 1  J t ) ~ N ( J , J2 ) . Satisfying the no-arbitrage condition,  J  exp( J   J2 / 2)  1 . N t is the jump frequency

PT

following a Poisson process with mean J . The instantaneous variance  t of the SPX

CE

follows a risk-neutral mean-reverting square root process (10)

AC

d t  (   t )dt    t d ,t

where  is the speed of mean-reverting adjustment of  t ;  /  is the long-run 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 (dS ,t , d ,t ) .

3.2 Fair value to SPX options SPX options are European-style. Bakshi, Cao, and Chen (1997) provide the time-t

11

ACCEPTED MANUSCRIPT

T

value of SPX call and put options with strike K and maturity T for the SVJ model:

RI P

ct ( S , T , K )  St e  (T  t )1  Ke r (T  t ) 2

(12)

SC

pt ( S , T , K )  St e  (T  t ) (1  1)  Ke r (T  t ) ( 2  1)

(11)

where 1 and  2 are risk-neutral probabilities that are recovered from inverting the

MA NU

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)

ED

for j = 1, 2. The characteristic functions f1 and f 2 for the SVJ model are given in

PT

equations (A12) and (A13) of Bakshi, Cao, and Chen (1997). Delta and vega of the

CE

European SPX options are given in equation (13) of Bakshi, Cao, and Chen (1997). Finally,

AC

delta and vega of the forward-start strangles can be calculated straightforward.

3.3 Fair value to SPX futures call options Since SPX futures call options are American-style, 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

12

ACCEPTED MANUSCRIPT

if Ft (T2 ) / K  yc*

T

(14)

if Ft (T2 ) / K  yc*

RI P

q2   Ft (T2 ) / K  *  C ( F , T1 , K )  KA2  * Ct ( F , T1 , K )   t y c     Ft (T2 )  K

SC

where Ct* ( F , T1 , K )  e  r (T1  t ) [ Ft (T2 )1  K2 ] is the time-t price of a European-style futures call option with strike K and expiry T1 . Ft (T2 ) is the time-t futures price with

MA NU

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

ED

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

PT

1 2  1  q    J  J   2 2 

(15)

CE

 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 )

AC

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:

13

ACCEPTED MANUSCRIPT

1 * Ct ( F , T1 , K )  e  r (T1  t ) ( yc 1  2 ) K

Ct* ( yc , T1 ,1) of y c* in equation (16) Ft

SC

where yc  Ft (T2 ) / K . Replacing Ct* ( yc , T1 ,1) and

(18)

RI P

Ct* ( yc , T1 ,1) 1 Ct* ( F , T1 , K ) 1  r (T1 t )   e 1 Ft K Ft K

(17)

T

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     1e  r (T1  t )1 q2  q2 K 

(19)

MA NU

yc* 

Thus, the solution to A2 is computed as

(20)

ED

[1  e  r (T1t ) 2 ]  1 r (T1t )   A2  1 e 1   1  r (T1t )    K  1   q 2  1   K  q2  e    

CE

PT

Finally, the calculation for delta and vega of the futures call option is straightforward.

AC

3.4 Fair value to VIX futures From Lin (2007), the time-t 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)

14

ACCEPTED MANUSCRIPT



, 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

SC

t 

(1  e   0 )

T

a 0 

,  2  2J (  J   J ) , T1t  e (T1t ) , T1 t 

 0  30 / 365

RI P

where

MA NU

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

ED

=0.

PT

4. Data and estimation procedure

CE

The sample period spans from July 3, 2006, through June 30, 2009. 3 Spot VIX, spot

AC

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.

15

ACCEPTED MANUSCRIPT

T

bill bid and ask discounts, collected from Datastream, with maturity closest to an option’s

RI P

maturity are used as risk-free interest rates. Since SPX option contracts are European-style,

SC

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.

MA NU

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,

ED

the SPX futures call options that expire in February, May, August and November are selected as the unhedged portfolio. Third, the forward-start strangles involve two strangles.

PT

This study uses the SPX options contracts that expire in February, May, August and

CE

November to construct a short-term strangle, and that expire in March, June, September,

AC

and December for another long-term 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 liquidity-related 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).

16

ACCEPTED MANUSCRIPT

T

option prices that violate the upper and lower boundaries are not included in the sample.

RI P

Based on these criteria, 29.90% and 10.50% of the original selected SPX options and SPX

SC

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

MA NU

(722 records) are used for our empirical work.

Table 1 reports descriptive properties of the SPX futures call options, along with the

ED

underlying SPX futures, VIX futures, VIX, the strikes of selected SPX options, and the total number of futures call options, for each moneyness-maturity category. Moneyness is

PT

defined as the SPX futures price divided by the SPX futures call option’s strike price, i.e.,

CE

Ft (T2 ) / K . Out of 20,839 SPX futures call option observations, about 50% is

AC

out-of-the-money (OTM) and 35% is at-the-money (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 short-term (<30 days) deep out-of-the-money (DOTM) call options to 118.56 points for medium-term (30-60 days) deep in-the-money (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 tick-size 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 tick-size interval are excluded.

17

ACCEPTED MANUSCRIPT

T

futures price ranges from 947.85 points for short-term DOTM calls to 1,347.89 points for

RI P

medium-term ATM calls.

SC

The average VIX futures prices range from 21.02 corresponding to the medium-term ATM calls to 46.23 for the short-term DOTM calls. The VIX varies from 20.68 (for

MA NU

medium-term ATM call options) to 50.87 (for short-term DOTM call options). The price of a VIX futures contract to VIX is an analogy to what a 30-day forward interest rate is to a

ED

30-day 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

PT

or higher in the 30-day forward period (covered by the VIX futures) than in the 30-day spot

CE

period (covered by VIX). Owing to the tendency of VIX to have a negative correlation with

AC

the SPX, the VIX futures prices have traded at a discount to the VIX before contract settlement especially during the in-crash 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 forward-start strangles over the sample period, consisting of 722 trading days. Average option quotes of K1 range from 599.89 for DOTM short-term futures calls to 1,100.31 for ATM short-term futures calls, whereas average option quotes of

K 2 range 1,066.82 for short-term DOTM futures calls to 1,464.49 for medium-term ATM

18

ACCEPTED MANUSCRIPT

T

futures calls.

RI P

[Table 1 about here]

SC

This study uses SPX options to construct the forward-start strangles that expire in the February quarterly cycle as T1 -strangle and that expire in the March quarterly cycle as T2

MA NU

-strangle. Therefore, the pair of ( T1 , T2 ) data must be FebruaryMarch, MayJune, AugustSeptember and NovemberDecember. The strike K1 ( K 2 ) is the minimum

ED

(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

PT

SPX options are eliminated here. On the basis of an analysis of patterns of trading volumes

CE

on SPX options, four pairs of maturity-strike combinations are selected to form

AC

forward-start 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 moneyness-maturity category over the period, 3 July 2006 to 30 June 2009. Moneyness is

19

ACCEPTED MANUSCRIPT

T

defined as the current SPX value divided by the SPX option’s strike, S/K. The average call

RI P

prices range from 1.3716 points for short-term OTM call options to 59.4300 points for

SC

long-term near-the-money call options. The average put prices range from 0.4125 points for short-term OTM put options to 66.8500 points for long-term near-the-money put options.

MA NU

[Table 2 about here]

The vector of structural parameters  for alternate processes is backed out by

ED

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

PT

min [Cn  Cn* ()]2 

(22)

t 1 n 1

CE

where NT is the number of trading days in the estimation sample, N t is the number of

AC

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 May-matured 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 in-the-money (ITM) and deep-in-the-money (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 large-scale calibration procedure feasible. Using the observed price ( ) the study computes an American binomial-tree implied volatility, that is, a volatility value such that , where denotes the binomial-tree 6

20

ACCEPTED MANUSCRIPT

T

estimated separately each month and thus  are assumed to be constant over a month.

RI P

The assumption that the structural parameters are constant over a month is justified by an

SC

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

MA NU

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

ED

monthly average of each estimated parameter series and the monthly averaged in-sample

[Table 3 about here]

CE

PT

mean squared errors for the SV and SVJ models, respectively.

AC

5. Risk-neutral parameter estimates and in-sample pricing fit Table 3 reports summary statistics for instantaneous volatility and risk-neutral parameter estimates along with in-sample 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.

21

ACCEPTED MANUSCRIPT

T

period, the study divides the data into four sub-periods: (i) the period prior to the Lehman

RI P

Brothers bankruptcy: July 3, 2006 to August 14, 2008, and (ii) the 2008 crash fear periods:

SC

August 15, 2008 to December 18, 2008, August 15, 2008 to June 18, 2009, and December 19, 2008 to June 18, 2009.9

MA NU

Model-specific estimates of implicit distributions indicate extremely turbulent conditions in the market over September 2008‒ December 2008, with somewhat quieter

ED

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

PT

2006‒ June 2009, diverge from estimates for a September 2008‒ June 2009 subsample. The

CE

estimate of jump-frequency intensity

that captures outliers indicates the major shocks

AC

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 risk-neutral mean-reverting 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.

22

ACCEPTED MANUSCRIPT

, and the

. For instance, on the basis of mean values of

RI P

variation coefficient,

, to which variance reverts, the speed of reversion,

T

the long-term mean,

SC

{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 long-term mean of

MA NU

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)

ED

is higher than the values of 0.86 (Panel B) and 1.47 (Panel E) over the pre- and post-crash-relaxation periods. However, it is close to the value of 5.02 found by the SVJ

PT

model over the post-crash-relaxation period. The average long-term mean variance

CE

of 34.36% for the SVJ model over the crash period compares with values of 3.91% in the

AC

pre-crash period and 13.33% in the post-crash-relaxation period. The variation coefficient, , determines how fat-tailed the distribution is and thereby the relative values of deep OTM options versus near-the-money 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 post-crash-relaxation periods. The consistently negative estimates of

indicate that

implied volatility and index returns are negatively correlated. This means the implied 10

The half-life of volatility shocks for the square-root process was calculated as

23

(2)/

252 trading days.

ACCEPTED MANUSCRIPT

for the SVJ model reached -0.91 in the

RI P

traders is negatively skewed. The mean values of

T

distribution perceived by SPX option traders, SPX futures option traders and VIX futures

SC

crash period and -0.97 in the post-crash-relaxation period. An explanation for these extreme values is that a relatively continuous bear market in the SPX starting from September 15,

MA NU

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

ED

of -0.57 for the SV model in the pre-crash period compares with more extreme values of -0.75 in the crash period and -0.77 in the post-crash-relaxation period. The implied , estimated by alternate models are able to capture the substantial 2008

PT

volatilities,

CE

crash. In particular, the SVJ model adds a price-jump component that directly captures the

AC

2008 outliers. To address whether jumps in volatility significantly impact the results of our study, this study calculates skewness and kurtosis statistics of model-implied 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

24

ACCEPTED MANUSCRIPT

T

indicate that all parameter estimates are significantly different from zero at the 90%

RI P

confidence interval.

SC

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

MA NU

as a 0.59 annualized jump probability, and the jump-size parameter and -7%, respectively. Their estimate of

and

and

are -5.37%

are 2.03 and 0.38. Pan (2002), Bates

ED

(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%]

PT

across different model specifications, whereas Eraker (2004) indicates two to three jumps in

CE

a stretch of 1,000 trading days. Interestingly, Bates (2000) obtains quite different results,

AC

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

25

ACCEPTED MANUSCRIPT

). The in-sample mean squared errors are

T

and the model price in an average month (

RI P

considerably and consistently smaller for more complicated models. However, the resulting for the various specifications is significant under the crash period,

SC

increase in

MA NU

suggesting that this jump risk assessment fails to capture the substantial 2008 crash.11

6. Hedging results

ED

The need for a perfect delta-neutral 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

PT

conducting this exercise, whenever jump risk is present (for example, for the SVJ model),

CE

the study follows Merton (1976) and Bakshi, Cao, and Chen (1997) to only aim for a partial

for.

AC

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 forward-start strangles or VIX

11

Bates (2006) found the stochastic-intensity jump model fits S&P returns better than the constant-intensity specification, when jumps are drawn from a finite-activity normal distribution or mixture of normals.

26

ACCEPTED MANUSCRIPT

. The study uses the previous month’s structural

T

futures to control for volatility risk

RI P

parameters and the current day’s (t) observations of SPX, SPX futures, SPX futures call

SC

options, SPX options, VIX futures and U.S. Treasury-bill 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

MA NU

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

ED

every trading day in the sample. Finally, the average absolute and the average dollar hedging errors for each moneyness-maturity category are then reported for each model in

PT

Tables 4 and 5.

CE

In Table 4, under the SV model, the hedging errors of HS1 are from 0.26 points

AC

(DOTM short-term of one-day revision in the post-crash-relaxation period) to 42.55 points (ATM medium-term of five-day revision in the crash period), whereas the hedging errors of HS2 range from 0.12 points (ATM short-term of five-day revision in the post-crash-relaxation period) to 48.80 points (ATM medium-term of five-day revision in the crash period). For the SVJ model, HS1 has hedging errors from 0.22 points (OTM short-term of one-day revision in the post-crash-relaxation period) to 148.83 points (ITM long-term of one-day revision in the post-crash-relaxation period), whereas HS2 has

27

ACCEPTED MANUSCRIPT

T

hedging errors from 0.22 points (OTM short-term of one-day revision in the

RI P

post-crash-relaxation period) to 51.04 points (ATM medium-term of five-day revision in the

SC

crash period).

When the hedge revision frequency changes from daily to once every five days, the

MA NU

hedging errors increase, except for the SV short-term options in the post-crash-relaxation period. Though HS1 outperforms HS2 in some cases within short-term categories under

ED

five-day rebalancing, the HS1 performs poorly in daily rebalancing. The findings shed a light on the instability issue of using forward-start strangles as forward volatility hedge.

PT

Improvement by the HS2 strategy is more evidence than the improvement by the HS1

CE

strategy when the same pricing model is used. The finding indicates a consequence of using

AC

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 in-crash periods. In particular, those SVJ models coupled with HS2 have virtually smaller hedging errors. Noticeably, for the futures call options with medium- and long-term maturities in the other periods, however, adding price jumps to the SV model does not improve its hedging performance. Thus, except for short-term options, the result in the non-crash period seems to be consistent with those of

28

ACCEPTED MANUSCRIPT

T

Bakshi, Cao, and Chen (1997) and Bakshi and Kapadia (2003), which show the hedging

RI P

superiority of the SV model relative to the SVJ model. Note that the parameter of

SC

jump-frequency 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

MA NU

the pre-crash period. Based on daily or every five-day 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 five-day

ED

rebalancing period. The estimated parameter J in the crash (post-crash-relaxation) period,

PT

however, is 9.97 (1.02) larger than that of Bakshi et al. (1997). For this reason, whenever

CE

jump risk is present, the overall impact on short-term hedging effectiveness of not allowing

AC

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 American-style, while SPX options are European-style for prior research. Since the traders with American-style options positions have early-exercise choice and thus can take caution to prevent any loss from the potential jump events than the ones with European-style options. Thus, American-style option buyers (sellers) may even favor (hate) volatility risk than the ones with European-style options. In addition, given the

29

ACCEPTED MANUSCRIPT

T

possibility of price jumps, the specification of SVJ can provide more accurate parameter

RI P

estimates than SV (Bates, 1996). Thus, the delta-vega-neutral strategy could be constructed

SC

in a more effective way under SVJ than SV for the short-term options. It is thus not surprising for our hedging results for some short-term options showing that the SVJ model

MA NU

outperforms the SV model in the crash periods.

Further, the HS1 (HS2) hedge of the SVJ model results in poor hedging performance

ED

comparable to that of the SV model in some moneyness-maturity categories, which suggests that model misspecification may have a significant effect on hedging when using the HS1

PT

(HS2) strategy. About the maturity impact on hedging performance, there is no pattern

CE

showing the difference between hedging schemes decreases with maturity and increases

AC

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 U-shape 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 forward-start strangles are in general a less efficient instrument to hedge forward volatility risk than VIX

30

ACCEPTED MANUSCRIPT

T

futures. A paired t-test with p-value in Table 4 supports the significance of the superiority of

SC

[Table 4 about here]

RI P

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 risk-free rate of

MA NU

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 risk-free return.

ED

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 risk-free return.

PT

During the in-crash 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

AC

hedging errors are positive for medium-term options in the crash-related 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 forward-start strangles. [Table 5 about here]

31

ACCEPTED MANUSCRIPT

T

7. Conclusion

RI P

This study has extended a parsimonious American futures call option pricing model

SC

that admits stochastic volatility and random price jumps (SVJ). It is shown that this closed-form pricing formula is practically implementable, leads to useful analytical hedge

MA NU

ratios, and contains the stochastic-volatility (SV) option formula as a special case. This last feature has made it relatively straightforward to study the relative empirical hedging

ED

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, forward-start strangles

CE

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 forward-start strangles (HS1) are used to construct two hedging strategies (HS1SV and HS1SVJ). On the other hand, SPX futures and VIX futures (HS2) are used to construct the other two hedging strategies (HS2SV and HS2SVJ). 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 forward-start strangles. This finding contributes to the existing literature documented

32

ACCEPTED MANUSCRIPT

T

by Psychoyios and Skiadopoulos (2006), who point out that volatility options are less

RI P

powerful instruments to hedge a short position on a European call option than plain-vanilla

SC

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

MA NU

performance for the remaining cases. Overall, our results support the claim that VIX futures are a better alternative to the traditional forward-start strangles for managing the implied

ED

forward volatility risk of a short position on the SPX futures call option. In particular, VIX futures not only perform better in a stochastic-volatility and/or price-jump economy but also

PT

are practically implementable.

CE

The contributions of this paper are threefold. First, closed-form solutions to American

AC

futures call options under two SPX price processes are derived. Second, the concept of forward volatility risk applied to VIX futures and forward-start strangles is introduced. Third, this study derives the hedging ratios of VIX futures and the forward-start strangles that are convenient to practical participants for risk management purposes.

Acknowledgements We thank Carl R. Chen (the editor), Min-Teh Yu (the visiting editor), Thomas Chiang (the

33

ACCEPTED MANUSCRIPT

visiting editor), and one anonymous reviewer for their constructive comments, which helped us to improve the manuscript. Yueh-Neng Lin gratefully acknowledges research support from

AC

CE

PT

ED

MA NU

SC

RI P

T

Taiwan National Science Council.

34

ACCEPTED MANUSCRIPT

T

References

RI P

Aït-Sahalia, Y., & Lo, A. W. (1998). Nonparametric estimation of state-price densities

SC

implicit in financial asset prices. Journal of Finance, 53, 499-548. Bakshi, G., Cao, C., & Chen, Z. (1997). Empirical performance of alternative option pricing

MA NU

models. Journal of Finance, 52 (5), 2003-2049.

Bakshi, G., & Kapadia, N. (2003). Delta-hedged gains and the negative market volatility

ED

risk premium. Review of Financial Studies, 16, 527-566. Barone-Adesi, G., & Whaley, R. (1987). Efficient analytic approximation of American

PT

option values. Journal of Finance, 42, 301-320.

CE

Bates, D.S. (1991). The crash of ’87: Was it expected? The evidence from options markets.

AC

Journal of Finance, 46, 1009-1044. Bates, D.S. (1996). Jumps and stochastic volatility: Exchange rate processes implicit in Deutsche Mark options. Review of Financial Studies, 9, 69-107. Bates, D.S. (2000). Post-’87 crash fears in the S&P 500 futures option market. Journal of Econometrics, 94, 181-238. Bates, D.S. (2006). Maximum likelihood estimation of latent affine processes. Review of Financial Studies, 19, 909-965.

35

ACCEPTED MANUSCRIPT

T

Black, F. (1976). The pricing of commodity contracts. Journal of Financial Economics, 3,

RI P

167-179.

SC

Brenner, M., Ou, E. Y., & Zhang, J. E. (2006). Hedging volatility risk. Journal of Banking & Finance, 30, 811-821.

MA NU

Broadie, M., Chernov, M., & Johannes, M. (2007). Model specification and risk premia: Evidence from futures options. Journal of Finance, 62, 1453-1490.

ED

Broadie, M., Detemple, J., Ghysels, E., & Torrés, O. (2000). American options with stochastic dividends and volatility: A nonparametric investigation. Journal of

PT

Econometrics, 94, 53-92.

CE

Carr, P., & Madan, D. (1998). Towards a theory of volatility trading. In: Jarrow, R. (Ed.),

AC

Volatility, (pp. 417-427). Risk Publications. Cox, J., & Ross, S. (1976). The valuation of options for alternative stochastic processes. Journal of Financial Economics, 3, 145-166. Eraker, B. (2004). Do equity prices and volatility jump? Reconciling evidence from spot and option prices. Journal of Finance, 59, 1367-1403. Gonzalez-Perez, M. T. (2015). Model-free volatility indexes in the financial literature: A review. International Review of Economics & Finance, forthcoming.

36

ACCEPTED MANUSCRIPT

T

Heston, S. (1993). A closed-form solution for options with stochastic volatility with

RI P

applications to bond and currency options. Review of Financial Studies, 6 (2),

SC

327-343.

Lin, B. H., Lin, Y. N., & Chen, Y. J. (2012). Volatility risk premium decomposition of

MA NU

LIFFE equity options. International Review of Economics & Finance, 24, 315-326. Lin, Y. N. (2007). Pricing VIX futures: Evidence from integrated physical and risk-neutral

ED

probability measures. Journal of Futures Markets, 27, 1175-1217. MacMillan, L. W. (1987). Analytic approximation for the American put options. Advances

PT

in Futures and Options Research, 1A, 119-139.

CE

Merton, R. (1976). Option pricing when the underlying stock returns are discontinuous.

AC

Journal of Financial Economics, 4, 125-144. Moran, M. T., & Dash, S. (2007). VIX futures and options: Pricing and using volatility products to manage downside risk and improve efficiency in equity portfolios. Journal of Trading, 2, 96-105. Neuberger, A. (1994). The log contract. Journal of portfolio management, 20 (2), 74-80. Pan, J. (2002). The jump-risk premia implicit in options: Evidence from an integrated time-series study. Journal of Financial Economics, 63, 3-50.

37

ACCEPTED MANUSCRIPT

T

Psychoyios, D., & Skiadopoulos, G. (2006). Volatility options: Hedging effectiveness,

RI P

pricing, and model error. Journal of Futures Markets, 26 (1), 1-31.

Interest-Rate Options. New York: John Wiley.

SC

Rebonato, R. (1999). Volatility and Correlation: In the Pricing of Equity, FX, and

MA NU

Szado, E. (2009). VIX futures and options: A case study of portfolio diversification during the 2008 financial crisis. Journal of Alternative Investments, 12, 68-85.

ED

Vishnevskaya, A. (2004). Essays in option pricing: Vega-hedged gains and the volatility risk premium. Working Paper, University of Virginia.

AC

CE

PT

Zhang, J. E., & Zhu, Y. (2006). VIX futures. Journal of Futures Markets, 26, 521-531.

38

ACCEPTED MANUSCRIPT

Table 1 Sample properties of S&P 500 futures call options

AC

CE

PT

ED

MA NU

SC

RI P

T

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 moneyness-maturity 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 forward-start 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 out-of-the-money (DOTM) if its Ft(T2)/K<0.94; out-of-the-money (OTM) if Ft(T2)/K [0.94,0.97); at-the-money (ATM) if Ft(T2)/K [0.97,1.03); in-the-money (ITM) if Ft(T2)/K [1.03,1.06); and deep in-the-money (DITM) if Ft(T2)/K≧1.06. By the term to expiration, an option contract can be classified as (i) short-term (<30 days); (ii) medium-term (30-60 days); and (iii) long-term (>60 days). The data period is from July 3, 2006 to June 30, 2009. Days to Maturity Moneyness <30 3060 >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 0.940.97 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 0.971.00 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 1.001.03 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 1.031.06 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

39

ACCEPTED MANUSCRIPT

K2 Observation

1165.8359 1213.4278 1156.9811 969 582 106 6,221 8,613 6,005

AC

CE

PT

ED

MA NU

SC

RI P

T

Subtotal

40

1,657 20,839

ACCEPTED MANUSCRIPT

Table 2 Sample characteristics of S&P 500 index options.

SC

RI P

T

Out-of-the-money and near at-the-money SPX call and put options are selected in this study to construct forward-start 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 moneyness-maturity category. Contract months of SPX options are in the February and March quarterly Cycles. This study classifies those observations into short-term (<30 days), medium-term (30–60 days), and long-term (>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.

ED

<30 1.5607 105 1.3716 74 4.9698 24 203

AC

CE

PT

Moneyness DOTM Option price Observation OTM Option price Observation ATM1 Option price Observation Subtotal

MA NU

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

41

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 in-sample 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 in-sample 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  2J (  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 model-implied instantaneous variance increments, defined as, and , respectively. performs a Jarque-Bera test of the null hypothesis that the model-implied 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

Long-run mean of ν (

)

CE

Total variation of ν ( )

)

PT

Adjustment speed of ν (

ED

A. July 3, 2006-June 18, 2009 In-sample 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 jump-size innovations ( ) Variance of price jump-size innovations ( Implied volatility (%) (

-

)

)

of a Jarque-Bera test

B. July 3, 2006-August 14, 2008 In-sample pricing error ( ) Adjustment speed of ν ( Long-run 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 jump-size innovations ( ) Variance of price jump-size innovations (

-

T

)

RI P

Implied volatility (%) (

)

of a Jarque-Bera test

)

Long-run mean of ν (

)

Total variation of ν ( )

MA NU

Adjustment speed of ν (

SC

C. August 15, 2008-December 18, 2008 In-sample pricing error ( )

Correlation between diffusion Brownian motions ( ) Mean jump intensity ( )

ED

Mean of price jump-size innovations ( ) Variance of price jump-size innovations (

of a Jarque-Bera test

CE

-

)

PT

Implied volatility (%) (

)

D. August 15, 2008-June 18, 2009 In-sample pricing error ( )

AC

Adjustment speed of ν ( Long-run mean of ν (

)

)

Total variation of ν ( ) Correlation between diffusion Brownian motions ( ) Mean jump intensity ( ) Mean of price jump-size innovations ( ) Variance of price jump-size innovations ( Implied volatility (%) (

)

)

of a Jarque-Bera test E. December 19, 2008-June 18, 2009 In-sample pricing error ( ) Adjustment speed of ν ( )

43

ACCEPTED MANUSCRIPT

Long-run mean of ν (

(3.06) 0.2787** (2.33) 0.8574*** (3.62) -0.7743*** (-7.97)

)

Total variation of ν ( )

Mean of price jump-size innovations ( )

7.2650

0.7536 0.0010

0.5738

SC

)

CE

PT

ED

of a Jarque-Bera test

AC

-

)

MA NU

Implied volatility (%) (

7.1635

RI P

Mean jump intensity ( )

Variance of price jump-size 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  lt )e rt ( 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 forward-start strangle portfolios. The forward-start 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 , moneyness-maturity 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 out-of-the-money (DOTM) if its

[0.97,1.03); in-the-money (ITM) if

ED

[0.94,0.97); at-the-money (ATM) if

<0.94; out-of-the-money (OTM) if

≧1.06. By the term to expiration, a SPX

[1.03,1.06); and deep in-the-money (DITM) if

PT

futures call option contract can be classified as (i) short-term (<30 days); (ii) medium-term (30–60 days); and (iii) long-term (>60 days). The out-of-sample hedging period is from 21 July 2006 to 18 June 2009.

E(

CE

The p-values 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. 1-Day Revision Days-to-Expiration <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

5-Day Revision Days-to-Expiration <30 30 60 >60

5.7913 16.4219 8.3815 14.0761 1.8824 13.4179 2.1983 11.2527 p-value

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 p-value

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 p-value

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 p-value

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 p-value

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 p-value

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 p-value

ACCEPTED MANUSCRIPT SVJ

NA

NA

1.6922 p-value

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 p-value

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 p-value

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 p-value

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 p-value

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 p-value

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 p-value

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 p-value

MA NU

H0:E(

RI P

ATM1

13.7353 10.6191 12.1018 11.9816 8.9570 4.5807 10.3001 4.5126 p-value

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 p-value

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 p-value

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 p-value

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 p-value 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 p-value 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 p-value

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 p-value

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 p-value

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 p-value

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 p-value ) 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 p-value

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 p-value

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 p-value

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 p-value

)

H0:E(

DITM

PT

HS1

ED

H0:E(

50

ACCEPTED MANUSCRIPT

Table 5 Average hedging errors.

M

l 1

HEt (l 1) t (t  lt )e rt ( 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 forward-start strangles. The forward-start 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 , moneyness-maturity 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 out-of-the-money (DOTM) if its

[0.97,1.03); in-the-money (ITM) if

ED

[0.94,0.97); at-the-money (ATM) if

<0.94; out-of-the-money (OTM) if

≧1.06. By the term to expiration, a SPX

[1.03,1.06); and deep in-the-money (DITM) if

PT

futures call option contract can be classified as (i) short-term (<30 days); (ii) medium-term (30–60 days);

CE

and (iii) long-term (>60 days). The out-of-sample 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

1-Day Revision Days-to-Expiration <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

5-Day Revision Days-to-Expiration <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