A remark on Lin and Chang's paper ‘Consistent modeling of S&P 500 and VIX derivatives’

Journal of Economic Dynamics & Control 36 (2012) 708–715

Contents lists available at SciVerse ScienceDirect

Journal of Economic Dynamics & Control journal homepage: www.elsevier.com/locate/jedc

A remark on Lin and Chang’s paper ‘Consistent modeling of S&P 500 and VIX derivatives’ Jun Cheng a,b, Meriton Ibraimi c, Markus Leippold c,n, Jin E. Zhang d,e a

Postdoctoral Research Station, Research Center, Shanghai Stock Exchange, Shanghai 200120, China School of Management and Engineering, Nanjing University, Nanjing 210093, China University of Zurich, Department of Banking and Finance, Plattenstrasse 14, 8032 Zurich, Switzerland d School of Economics and Finance, The University of Hong Kong, Pokfulam Road, Hong Kong e Department of Accountancy and Finance, School of Business, University of Otago, Dunedin 9054, New Zealand b c

a r t i c l e in f o

abstract

Article history: Received 1 November 2011 Received in revised form 29 November 2011 Accepted 14 December 2011 Available online 18 January 2012

Lin and Chang (2009, 2010) establish a VIX futures and option pricing theory when modeling S&P 500 index by using a stochastic volatility process with asset return and volatility jumps. In this note, we prove that Lin and Chang’s formula is not an exact solution of their pricing equation. More generally, we show that the characteristic function of their pricing equation cannot be exponentially affine, as proposed by them. Furthermore, their formula cannot serve as a reasonable approximation. Using the (Heston, 1993) model as a special case, we demonstrate that Lin and Chang formula misprices VIX futures and options in general and the error can become substantially large. & 2012 Elsevier B.V. All rights reserved.

JEL classification: G13 Keywords: VIX option pricing Affine jump diffusion Characteristic function

1. Introduction VIX options have become very successful exchange-listed products for volatility trading. The bid-ask spread of VIX options market is large due to the fact that a commonly accepted VIX option pricing model is not available yet. Hence, developing a tractable VIX option pricing model is important for the healthy growth of the new market. Yet, as the VIX index is directly linked to the implied volatility of the S&P 500 index and hence to index options, a VIX option pricing model needs to provide enough flexibility to jointly price in a consistent manner options on the S&P 500 as well as on the VIX index. The first attempt to express the price of VIX futures was made in Zhang and Zhu (2006), where the stochastic volatility model of Heston (1993) is used to describe S&P 500. They developed a simple theoretical model for VIX futures prices and tested the model using the actual futures price on one particular day. Dotsis et al. (2007) studied the continuous-time models of the volatility indices. Zhu and Zhang (2007) further derived a no-arbitrage pricing model for VIX futures using the time-dependent long-term mean level in the volatility model. Lin (2007) incorporates simultaneous jumps in both asset return and volatility processes. Sepp (2008) used the square root stochastic variance model with jumps in the variance process to describe the evolution of S&P 500 volatility, and showed how to price and hedge VIX futures and VIX

n

Corresponding author. Tel.: þ 41 44 634 50 69. E-mail addresses: [email protected] (J. Cheng), [email protected] (M. Ibraimi), [email protected] (M. Leippold), [email protected] (J.E. Zhang). 0165-1889/$ - see front matter & 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.jedc.2012.01.002

J. Cheng et al. / Journal of Economic Dynamics & Control 36 (2012) 708–715

709

options in this model. Albanese et al. (2009) studied volatility derivatives by using spectral methods. Zhang and Huang (2010) studied the CBOE S&P 500 three-month variance futures market, and showed a linear dependence between the price of fixed time-to-maturity variance futures and the VIX by using a simple mean-reverting stochastic model for the S&P 500 index. Lu and Zhu (2010) studied the variance term structure using VIX futures market. Zhang et al. (2010) studied VIX futures market by using a stochastic volatility model with stochastic long-term mean level. Some other recent studies about the VIX and its derivatives include Chen et al. (2011), Dupoyet et al. (2011), Hilal et al. (2011), Konstantinidi and Skiadopoulos (2011), Cont and Kokholm (2011), Shu and Zhang (2012), and Zhu and Lian (2012) among others. Carr and Lee (2009) provided an interesting review on volatility derivatives market. Lin and Chang (2009, 2010) establish a VIX futures and option pricing theory when modeling S&P 500 index by using a stochastic volatility process with asset return and volatility jumps. Hence, their model seems to suggest a pricing framework which is both tractable and flexible enough to consistently price index options and options on the VIX. However, we show that Lin and Chang’s (2009, 2010) formula published in both papers is not an exact solution of their pricing equation. More generally, we formally prove that the characteristic function of their pricing equation cannot be exponentially affine, as proposed by them. One could still argue that their formula provides a reasonable approximation for an option pricing formula that, given their general setup, does not allow for a closed-form solution. However, by using a reduced-form specification of their model, we find that their formula can also not serve as an approximation. In particular, we use the simple setup of the Heston (1993) stochastic volatility model and we demonstrate that Lin and Chang formula misprices VIX futures and options in general and the error could be substantially large. We further point out that for the simultaneous pricing of index and VIX options, an exact formula has actually been provided by Sepp (2008) under the assumption of a stochastic volatility process with volatility jumps but no jumps in asset return.1 This note is structured as follows. In the next section, we briefly review some general results on affine jump diffusions and their characteristic function. In Section 3, we present the main result of Lin and Chang (2009, 2010). In Section 4, we provide a formal proof showing that the result of Lin and Chang cannot be correct and we also show that their formula cannot serve as an appropriate approximation of the true pricing formula. Section 5 concludes. 2. Affine jump diffusion Let X  R be a closed set with non-empty interior. Throughout this note we assume that for every x 2 X there exists a solution X ¼ X x of the one-dimensional stochastic differential equation dX t ¼ mðX t Þ dt þ sðX t Þ dBt þ dJt ,

Xð0Þ ¼ x,

ð1Þ

where J is a pure-jump process with jump arrival intensity LðSt Þ at time t for some L : R-½0,1Þ. Jump sizes Z 1 ,Z 2 , . . . are iid and independent of the Brownian motion B, which is defined on a filtered probability space ðO,F ,ðF t Þ, PÞ. Definition 1. We call X affine if the F t -conditional characteristic function of XT is exponential affine in Xt for all t rT. That is, there exist C-valued functions fðTt,zÞ and cðTt,zÞ with jointly continuous t-derivatives such that X ¼ X x satisfies

E½ezX T 9F t  ¼ Et ½ezX T  ¼ efðTt,zÞ þ cðTt,zÞX t

ð2Þ

for all z 2 iR, t rT and x 2 X . Before we explain why the calculations of Lin and Chang are wrong, we briefly elaborate on an example in which the problem of determining a characteristic function is reduced to solving a system of ordinary differential equations (ODEs). The same strategy is followed by Lin and Chang (2010) to find a solution for the characteristic function of the logarithm of the VIX squared and therefore deserves some attention. Example 1. We consider the calculation of the following expectation f ðX t ,tÞ ¼ EðeX T 9X t Þ

ð3Þ 2

2

under the assumption of affine dependence of m and s on X, i.e., we assume mðxÞ ¼ a þ bx, sðxÞ ¼ cx and lðxÞ ¼ lo þ l1 x for some coefficients a,b,c,l0 ,l1 2 R. If f has two continuous derivatives, the application of Itˆo’s formula for jump diffusions gives Z t Z t X f ðX t ,tÞ ¼ f ðX 0 ,tÞ þ gðX s ,sÞ ds þ f x ðX s ,sÞ dBs þ ½f ðX s ,sÞf ðX s ,sÞ, ð4Þ 0

0

0osrt

where

gðx,tÞ ¼ f t ðx,tÞ þ f x ðx,tÞmðxÞ þ 12 f xx ðx,tÞsðxÞ2 :

ð5Þ

1 Only recently, the authors became aware of the paper by Lian and Zhu (2011), who also seriously question the correctness of the Lin and Chang formula.

710

J. Cheng et al. / Journal of Economic Dynamics & Control 36 (2012) 708–715

Under some technical regularity conditions we can show that f ðX t ,tÞ is a martingale. Therefore, we get 0 ¼ f t ðx,tÞ þ f x ðx,tÞmðxÞ þ 12 f xx ðx,tÞsðxÞ2 þ E½lðx þ Z i Þf ðx þ Z i ,tÞlðxÞf ðx,tÞ:

ð6Þ

To solve the above partial differential equation (PDE) we conjecture a solution of the form f ðx,tÞ ¼ eaðTtÞ þ bðTtÞx . Substituting this conjectured solution into (6) we obtain eaðTtÞ þ bðTtÞx ða0 ðsÞb ðsÞx þ bðsÞða þ bxÞ þ 12bðsÞ2 c2 x þ l0 ½EðebðsÞZi Þ1 þ l1 ðE½Z i ebðsÞZi  þ E½ðebðsÞZi 1ÞxÞ ¼ 0: 0

Dividing by

eaðTtÞ þ bðTtÞx

ð7Þ

and collecting terms in x, we get

uðsÞx þ vðsÞ ¼ 0,

ð8Þ

where uðsÞ ¼ b ðsÞ þ bðsÞb þ 12bðsÞ2 þl1 ½EðebðsÞZi Þ1Þ, 0

vðsÞ ¼ a0 ðsÞ þ bðsÞa þ l0 ½EðebðsÞZ i Þ1 þl1 E½Z i ebðsÞZ i :

ð9Þ

Because (8) must hold for all x, we have uðsÞ ¼ vðsÞ ¼ 0 for all s 2 R. Therefore, we can reduce the PDE to a set of ODE’s, namely

b0 ðsÞ ¼ bðsÞbþ 12bðsÞ2 þ l1 ½EðebðsÞZ i Þ1Þ,

a0 ðsÞ ¼ bðsÞa þ l0 ½EðebðsÞZi Þ1 þ l1 E½Z i ebðsÞZi :

ð10Þ

Solving this system of ODE’s leads to a solution of the PDE and therefore to a solution for (3), i.e., for the characteristic function of X. Theorem 1. Let X ¼ X x be the solution of the stochastic differential equation defined in (1) with initial condition X 0 ¼ x for all x 2 X for some closed subset X  R. Assume that X is affine as in Definition1. Further, assume that the jump intensity l is affine in X. Then the drift and the variance have affine dependence on the current state Xs. Proof. See Appendix A. Remark 1. To simplify the proof of Theorem 1 and as it is enough for our purpose, we assume an affine jump intensity l. For a more general result, we refer to Duffie et al. (2003, Theorem 2.12). 3. A review of Lin and Chang’s results T

In Lin and Chang’s model, the forward price of the S&P 500 index, denoted as Ft , is modeled as a jump-diffusion process with stochastic instantaneous variance vt. Under the risk-neutral measure Q  P, these processes are defined as pffiffiffiffiffi d ln F Tt ¼ 12vt dt þ vt doS,t þzS dN t klt dt, ð11Þ pffiffiffiffiffi dvt ¼ kv ðyv vt Þ dt þ sv vt dov,t þ zv dN t ,

ð12Þ

where oS,t and ov,t are two Q-Brownian motions with correlation coefficient r. Asset returns and variance jump at the same time according to the poisson process Nt. The variance jump size zv is exponentially distributed with mean mv 40, i.e., its probability density is given by pðzv Þ ¼ ð1=mv Þezv =mv , 0 rzv o þ1. To introduce correlated jump sizes, the asset return jump size zS is conditioned on the realization of zv. In particular, zS is normally distributed with mean mj þ rj zv and variance s2j . The jump intensity is assumed to be lt ¼ l0 þ l1 vt and the relative forward price jump size, J  ezS 1, has a mean given by2 2

2

k  EQ ðezS 1Þ ¼ EQ ½EQ ðezS 9zv Þ1 ¼ EQ ðemj þ rj zv þ ð1=2Þsj Þ1 ¼

emj þ ð1=2Þsj 1: 1rj mv

The variance and covariance of the two jump sizes, zv and zS, are given by Varðzv Þ ¼ EQ ½ðzv mv Þ2  ¼ EQ ðz2v Þm2v ¼ m2v , VarðzS Þ ¼ EQ ½ðzS mj rj mv Þ2  ¼ EQ f½ðzS mj rj zv Þ þ rj ðzv mv Þ2 g ¼ s2j þ r2j m2v , Cov ðzS ,zv Þ ¼ EQ ½ðzS mj rj mv Þðzv mv Þ ¼ EQ ½rj ðzv mv Þ2  ¼ rj m2v ,

2 It seems to us that the notation Jt, frequently used in the literature including Lin and Chang, is not appropriate because J is a random number instead of a process.

J. Cheng et al. / Journal of Economic Dynamics & Control 36 (2012) 708–715

711

hence the correlation coefficient between zS and zv is given by

rj mv Cov ðzS ,zv Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 VarðzS Þ  Varðzv Þ s þ r2 m2v j

j

The variance process can be rewritten as pffiffiffiffiffi n dvt ¼ knv ðyv vt Þ dt þ sv vt dov,t þzv dN t ðl0 þ l1 vt Þmv dt, where

knv ¼ kv l1 mv , ynv ¼

kv yv þ l0 mv kv l1 mv

are the effective mean-reverting speed and long-term mean level under the risk-neutral measure Q. Note that the mean of jump process, EQ ðzv dN t Þ ¼ ðl0 þ l1 vt Þmv dt, affects the parameter values of the mean-reversion process. Based on the CBOE definition, the VIX squared can be derived from3 "Z # " # Z t þ t  T tþt 2 St þ t 2 Q dSt 2 Q dF t T VIX2t  EQ ln E dðln S Þ ¼ E dðln F Þ , F Tt ¼ St erðTtÞ ¼ t t t t t t t St t t t F tt þ t F Tt Z t þ t    Z t þ t  2 1 z z vt þ ðezS 1zS Þðl0 þ l1 vt Þ dt ¼ 1 EQ ¼ EQ vt dt þ z2 ¼ 1 ðat vt þbt Þ þ z2 , ð13Þ t t 2 t t t t t where t ¼ 30=365 and

z1 ¼ 1 þ2l1 ½kðmj þ rj mv Þ, z2 ¼ 2l0 ½kðmj þ rj mv Þ, 1ekv t n

at ¼

knv

n

bt ¼ yv ðtat Þ:

,

Denoting L ¼ ln S, the price of a European call option CðtC ,L,vÞ written on VIX with the strike price K and time-to-maturity tC  Tt satisfies the following integro-partial differential equation (IPDE)4     1 @2 C 1 @C @2 C 1 @2 C @C @C v 2 þ rl0 k l1 k þ v þ rsv v þ s2 v  þ kv ðyv vÞ rC 2 @L 2 @L @L@v 2 v @v2 @v @tC þ EQ t f½l0 þ l1 ðv þzv ÞCðtC ,Lþ zS ,v þ zv Þðl0 þ l1 vÞCðtC ,L,vÞg ¼ 0, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with final condition CðtC ¼ 0,L,vÞ ¼ maxðVIXT K,0Þ, where VIXT ¼ z1 at vT =t þ z1 bt =t þ z2 . Lin and Chang claim that they have obtained a closed-form VIX option pricing formula as follows: r tC P1 KertC P2 , CðtC ,L,vÞ ¼ F VIX t ðTÞe

where

P1 ¼

1 1 þ 2 p

1 1 P2 ¼ þ 2 p

Z

1

" Re

0

Z

1 0

# 2 eif ln K f 2 ðtC ; if þ 1=2Þ df, iff 2 ðtC ; 1=2Þ

# 2 eif ln K f 2 ðtC ; ifÞ df, Re if

ð14Þ

ð15Þ

"

ð16Þ

2

if ln VIXT , the characteristic function of ln VIX2T is given by and f 2 ðtC ; ifÞ ¼ EQ t ½e

f 2 ðtC ; ifÞ ¼ exp½C 2 ðtC Þ þ J2 ðtC Þ þD2 ðtC Þ ln VIX2t ,

ð17Þ

where D2 ðtC Þ, C 2 ðtC Þ and J 2 ðtC Þ are defined in Lin and Chang (2010, Eq. (B.10)). 4. Disproving the correctness of Lin and Chang’s formula 4.1. Formal argument We start by presenting the following result, which is based on a formal argument outlined in Appendix. Proposition 1. Lin and Chang’s formula (14)–(17) with D2 ðtC Þ, C 2 ðtC Þ and J2 ðtC Þ given by their Eq. (B.10) in Lin and Chang (2010) is not an exact solution of their pricing equation (14). 3 The result here is the same as Lin and Chang’s, but the derivation is slightly different from that of Lin and Chang, in which they introduce an approximation on ln ð1 þ JÞ, which seems to be unnecessary at least in our view. 4 In Lin and Chang, both variables t and tC are used as independent variables in the option price function, Cðt, tC Þ. Here we choose to use one of them, tC , as they are related by tC  Tt.

712

J. Cheng et al. / Journal of Economic Dynamics & Control 36 (2012) 708–715

Proof. See Appendix B. We first note that using Eqs. (14)–(16), Lin and Chang describe VIX option price in terms of the characteristic function of ln VIX2T , i.e., f 2 ðtC ; ifÞ. This representation is fine because it is consistent with Bakshi and Madan (2000). The key issue here is the analytical tractability of f 2 ðtC ; ifÞ, without which the representation does not help us much in computing the VIX option prices. When Lin and Chang solve the problem, they conjecture in their Eq. (B.4) that the characteristic function of ln VIX2T has the following form: 2

2

if ln VIXT  ¼ eC 2 ðtC Þ þ J2 ðtC Þ þ D2 ðtC Þ ln VIXt þ G2 ðtC ÞLt : f 2 ðtC ; ifÞ  EQ t ½e

ð18Þ

By imposing such a structure, they implicitly assume that C 2 ðtC Þ, J2 ðtC Þ and D2 ðtC Þ are not functions of VIXt when they derive ODEs for them. However, in the final result of their Eq. (B.10), C 2 ðtC Þ, J2 ðtC Þ, and D2 ðtC Þ are indeed functions of VIXt , which contradicts their original assumption. Therefore, their conjecture (18) cannot be appropriate. We also note that during the process of solving for f 2 ðtC ; ifÞ (Lin and Chang, 2010) introduce an approximation in their Eq. (B.6) for exp½if ln ð1 þðmv =VIX2T ÞÞ by using Taylor’s expansion at VIX2t . However, the error of this approximation is not analyzed.5 What is the reason the method used by Lin and Chang (2010) fails? To give an answer to this question, we observe the following: Proposition 2. The characteristic function of the stochastic process lnðVIX2t Þ cannot be exponentially affine in lnðVIX2t Þ. Proof. See Appendix C. The derivation of Proposition 2 in Appendix makes it obvious why the method used by Lin and Chang (2010) fails, namely because of non-affine dependence of the drift, variance and jump on lnðVIX2t Þ. A potential remedy to obtain at least a closed-form approximation for the characteristic function would be to apply a second-order perturbation of lnðVIX2t Þ around some fixed volatility level. Such an approximation would lead to a characteristic function that is exponential linear-quadratic in VIX2t . However, in such a setting, additional care has to be applied to the specification of the volatility dynamics in a setting with jumps (see, e.g., Cheng and Scaillet, 2007). 4.2. Numerical investigation So far, we have presented a formal argument that Lin and Chang’s formula for VIX option pricing cannot be correct. However, one might argue that their formula may produce reasonable prices and may therefore serve as an approximation of the true option pricing formula. Being an approximate formula for the prices of VIX options and futures, its accuracy is important for users. Unfortunately, with some numerical analysis, we find that in general, Lin and Chang’s formula ((14)–(17)) clearly misprices VIX options and futures. Furthermore, the error could be substantially large. To substantiate our claim, we use a simplified case to analyze the error of Lin and Chang’s formula. In particular, we use the classical Heston model for stochastic volatility (Heston, 1993). Under such a specification, the conditional risk-neutral Q probability density function of VIXT , f ðVIXT 9VIXt Þ has been provided by Zhang and Zhu (2006), which can be used to calculate the prices of VIX futures and options given by VIXFTt ¼ EQ t ½VIXT ,

ð19Þ

CðTt,L,vÞ ¼ erðTtÞ EQ t ½maxðVIXT K,0Þ:

ð20Þ

Using the parameter values estimated from the VIX time series from January 2, 1990 to March 1, 2005 by Zhang and Zhu (2006), ðkv , yv , sv Þ ¼ ð4:9179,0:04874,0:4868Þ, and we assume the current VIX level is at 15% and the riskfree rate is r ¼ 2%. The prices of VIX futures and options with different maturities are presented in Tables 1 and 2. As we can see from the tables, Lin and Chang’s formula ((14)–(17)) misprices VIX options and futures. The error could be substantially large.6 Lin and Chang formula overprices two-month VIX options by about 40%. The overpricing could be even higher than 100% for one-year VIX options. The overpricing for VIX futures is also large even though it is smaller than that for VIX options. In Eq. (B.10) in Lin and Chang (2010), the variable B appears in eBtC , therefore BtC has to be dimensionless. However, from the formula for B, we can tell that it is not dimensionless due to the last term 1=ln VIX2t . This indicates that the formula for B has some problems. Indeed, note that for VIX futures and options with a very long maturity, i.e., Tt-þ 1, we have lim

n

vT ¼ yv

Tt- þ 1

5 6

Also, the variable M in equations below their Eq. (B.6) is never defined in Lin and Chang (2010). Note, the VIX options with a maturity of one to two months are the most liquid ones.

J. Cheng et al. / Journal of Economic Dynamics & Control 36 (2012) 708–715

713

Table 1 The prices of VIX futures with different maturities. The parameter values of the Heston (1993) model are taken to be ðkv , yv , sv Þ ¼ ð4:9179,0:04874,0:4868Þ that are estimated from the VIX time series from January 2, 1990 to March 1, 2005 by Zhang and Zhu (2006). The current VIX level is VIX0 ¼ 15. LC is obtained by using Lin and Chang’s (2010) formula. ZZ is obtained by using Zhang and Zhu (2006) formula. RE is the relative error between LC and ZZ, computed as LC=ZZ1. Maturity (year)

LC

ZZ

RE (%)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

15.00 18.13 20.87 23.20 25.16 26.79 28.13 29.23 30.13 30.87 31.47 31.95 32.35

15.00 17.60 19.09 19.95 20.46 20.77 20.96 21.07 21.14 21.18 21.20 21.22 21.23

0.0 3.0 9.3 16.3 23.0 29.0 34.2 38.7 42.5 45.7 48.4 50.5 52.3

Table 2 The prices of VIX call options with different maturities. The parameter values of Heston’s (1993) model are taken to be ðkv , yv , sv Þ ¼ ð4:9179, 0:04874,0:4868Þ that are estimated from the VIX time series from January 2, 1990 to March 1, 2005 by Zhang and Zhu (2006). The current VIX level is VIX0 ¼ 15 and riskfree rate is r ¼ 2%. LC is obtained by using Lin and Chang’s (2010) formula. ZZ is obtained by using Zhang and Zhu (2006) approach. RE is the relative error between LC and ZZ, computed as LC=ZZ1. Maturity (year)

LC

ZZ

RE (%)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

0.00 4.03 6.69 8.90 10.73 12.23 13.47 14.47 15.28 15.94 16.46 16.88 17.21

0.00 3.17 4.51 5.29 5.75 6.02 6.18 6.27 6.32 6.35 6.36 6.36 6.36

0.0 27.1 48.5 68.4 86.7 103.2 117.9 130.7 141.7 151.0 158.8 165.3 170.0

and lim

VIXT ¼

Tt- þ 1

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z1 ynv þ z2 :

Then the VIX futures price has the same limit qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n lim VIXFTt ¼ z1 yv þ z2 ,

ð21Þ

and the forward VIX call option price has the limit as follows: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n lim erðTtÞ CðTt,L,vÞ ¼ maxð z1 yv þ z2 K,0Þ:

ð22Þ

Tt- þ 1

Tt- þ 1

The asymptotic behavior of Lin and Chang’s formula, depending on the sign of the value of B, does not follow the property above in general.

5. Conclusion In this note, we prove that Lin and Chang’s (2009, 2010) formula is not an exact solution of their pricing equation. Using as a reduced specification the simple case of the Heston (1993) model, we demonstrate that Lin and Chang’s formula misprices VIX futures and options in general and the error could be substantially large. We further point out that an exact formula has actually been provided by Sepp (2008).

714

J. Cheng et al. / Journal of Economic Dynamics & Control 36 (2012) 708–715

The empirical features on VIX options market provided by Lin and Chang (2010) are based on their in-accurate formula. They need to be reexamined immediately by using the correct VIX option pricing formula. Other research that uses Lin and Chang’s formula such as, e.g., Wang and Daigler (2011) and Chung et al. (2011) and also needs to be reexamined.

Acknowledgments We thank Carl Chiarella and an anonymous referee for helpful comments. Markus Leippold and Meriton Ibraimi gratefully acknowledge financial support from the Swiss Finance Institute (SFI) and Bank Vontobel. Jun Cheng gratefully acknowledges financial support from National Natural Science Foundation of China (51002195, 70932003) and China Postdoctoral Science Foundation (20110490127). Jin E. Zhang’s work was supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. HKU 7549/09H). Appendix A. Proof of Theorem 1 Define the function MðX s ,sÞ ¼ efðts,zÞ þ cðts,zÞX s :

ð23Þ

Using Itˆo’s formula as in (6) we obtain the equation 0 ¼ MðX s ,sÞð@t fðts,zÞ@t cðts,zÞX s þ cðts,zÞmðX s Þ þ 12cðts,zÞ2 sðX s Þ2 þ E½lðX s þZ i ÞMðZ i ,sÞlðX s ÞÞ

ð24Þ

for all s r t. Letting s-0 and dividing by Mðx,0Þ, we thus obtain @t fðt,zÞ þ @t cðt,zÞx ¼ cðt,zÞmðxÞ þ 12cðt,zÞ2 sðxÞ2 þ l0 E½MðZ i Þ1 þ l1 E½Z i MðZ i ,0Þ þ l1 E½MðZ i ,0Þ1x

ð25Þ

for all x 2 X and t Z 0, where we have written lðxÞ : ¼ l0 þ l1 x. Now since cð0,zÞ ¼ z we see that m and s2 have to be affine in x. Appendix B. Proof of Proposition 1 Consider the special case of no-jump, i.e., zS ¼ zv ¼ 0, l0 ¼ l1 ¼ 0, hence k  EðezS 1Þ ¼ 0, knv ¼ kv and yv ¼ yv . Then, the VIX formula simplifies to n

VIX2t ¼

1

t

ðat vt þ bt Þ,

where at ¼ ð1ekv t Þ=kv , bt ¼ yv ðtat Þ. Note that z1 ¼ 1 and z2 ¼ 0. The VIX option pricing problem becomes   1 @2 C 1 @C @2 C 1 @2 C @C @C v 2 þ r v þ rsv v þ s2 v  þ kv ðyv vÞ rC ¼ 0, 2 @L 2 @L @L@v 2 v @v2 @v @tC

ð26Þ

CðtC ¼ 0,L,vÞ ¼ maxðVIXT K,0Þ: Lin and Chang’s formula becomes rtC P1 KertC P2 , CðtC ,L,vÞ ¼ F VIX t ðTÞe

ð27Þ

where 1 1 P1 ¼ þ 2 p 1 1 P2 ¼ þ 2 p

Z

1 0

Z

1 0

"

# 2 eif ln K f 2 ðtC ; if þ1=2Þ Re df, iff 2 ðtC ; 1=2Þ # 2 eif ln K f 2 ðtC ; ifÞ Re df, if

ð28Þ

"

ð29Þ

and f 2 ðtC ; ifÞ ¼ exp½C 2 ðtC Þ þ D2 ðtC Þ ln VIX2t , C 2 ðtC Þ ¼

B k kv tC  v A A

(

" # ( ) "   #) A B 1 A BtC B 1 þ if þ  e þln if þ , B A B A

BtC ln

( )1 " #  B A B 1 A BtC þ if þ  e , D2 ðtC Þ ¼  þ A B A B

ð30Þ

J. Cheng et al. / Journal of Economic Dynamics & Control 36 (2012) 708–715

1 tVIX2t bt A ¼ s2v  2 at at

!

" 1 at B ¼ kv yv  s2v 2 tVIX2t

!2

at

tVIX2t tVIX2t at

1

!

ln VIX2t 

715

,

! # ! ! bt bt at 1 þ kv : at at tVIX2t ln VIX2t

2

if ln VIXT  is the characteristic function of ln VIX2T , it must be a solution of pricing PDE (26). Because f 2 ðtC ; ifÞ  EQ t ½e However, by substituting Eq. (30) into Eq. (26), we can show that it is not a solution of (26). Therefore, Lin and Chang’s formula ((14)–(17)) with D2 ðtC Þ, C 2 ðtC Þ and J2 ðtC Þ given by their Eq. (B.10) in Lin and Chang (2010) is not an exact solution of their pricing equation (14).

Appendix C. Proof of Proposition 2 Recall the equation VIX2t ¼ a  nt þb

ð31Þ

by (8) of Lin and Chang (2010), where a,b 2 R are defined as in Lin and Chang. Equivalently, we can write lnðVIX2t Þ ¼ lnða  nt þ bÞ:

ð32Þ

Using Itˆo’s formula, Eq. (32) transforms to d

lnðVIX2t Þ ¼

a VIX2t

kn ðyn a

1

1 VIX2t þ bÞ

a2 yn

2 ðVIX2t Þ2 þ ðlnðVIX2t þazn þ bÞlnðVIX2t ÞÞ dN t :

Eq. (33) shows that the drift, the variance and the Remark 1, the characteristic function of lnðVIX2t Þ

! ða

1

VIX2t bÞ

dt þ

a VIX2t

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

sn ð a1 VIX2t bÞ don,t

jump intensity are not affine in lnðVIX2t Þ and cannot be exponential affine in lnðVIX2t Þ.

ð33Þ therefore, by Theorem 1 and

References Albanese, C., Lo, H., Mijatovic´, A., 2009. Spectral methods for volatility derivatives. Quantitative Finance 9 (6), 663–692. Bakshi, G., Madan, D., 2000. Spanning and derivative-security valuation. Journal of Financial Economics 55 (3), 205–238. Carr, P., Lee, R., 2009. Volatility derivatives. Annual Review of Financial Economics 1 (1), 319–339. Chen, H., Chung, S., Ho, K., 2011. The diversification effects of volatility-related assets. Journal of Banking Finance 35 (5), 1179–1189 /http://www. sciencedirect.com/science/journal/03784266/35/5S. Cheng, P., Scaillet, O., 2007. Linear-quadratic jump-diffusion modeling. Mathematical Finance 17 (4), 575–598. Chung, S.-L., Tsai, W.-C., Wang, Y.-H., Weng, P.-S., 2011. The information content of the S&P 500 index and VIX options on the dynamics of the S&P 500 index. Journal of Futures Markets 31 (12), 1170–1201 /http://onlinelibrary.wiley.com/doi/10.1002/fut.v31.12/issuetocS. Cont, R., Kokholm, T., 2011. A consistent pricing model for index options and volatility derivatives. Mathematical Finance doi:10.1111/j.14679965.2011.00492.x. Dotsis, G., Psychoyios, D., Skiadopoulos, G., 2007. An empirical comparison of continuous-time models of implied volatility indices. Journal of Banking & Finance 31 (12), 3584–3603. Duffie, D., Filipovic, D., Schachermayer, W., 2003. Affine processes and applications in finance. Annals of Applied Probabilities 13 (3), 984–1053. Dupoyet, B., Daigler, R.T., Chen, Z., 2011. A simplified pricing model for volatility futures. Journal of Futures Markets 31 (4), 307–339. Heston, S., 1993. A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies 6 (2), 327–343. Hilal, S., Poon, S.-H., Tawn, J., 2011. Hedging the black swan: conditional heteroskedasticity and tail dependence in S&P500 and VIX. Journal of Banking & Finance 35 (9), 2374–2387. Konstantinidi, E., Skiadopoulos, G., 2011. Are VIX futures prices predictable? An empirical investigation. International Journal of Forecasting 27 (2), 543–560. Lian, G.-H., Zhu, S.-P., 2011. Pricing VIX options with stochastic volatility and random jumps. Decisions in Economics and Finance, 1–18. Lin, Y.-N., 2007. Pricing VIX futures: evidence from integrated physical and risk-neutral probability measures. Journal of Futures Markets 27 (12), 1175–1217. Lin, Y.-N., Chang, C.-H., 2009. VIX option pricing. Journal of Futures Markets 29 (6), 523–543. Lin, Y.-N., Chang, C.-H., 2010. Consistent modeling of S&P 500 and VIX derivatives. Journal of Economic Dynamics and Control 34 (11), 2302–2319. Lu, Z., Zhu, Y., 2010. Volatility components: the term structure dynamics of VIX futures. Journal of Futures Markets 30 (3), 230–256. Sepp, A., 2008. VIX option pricing in jump-diffusion model. Risk, 84–89. Shu, J., Zhang, J.E., 2012. Causality in the VIX futures market. Journal of Futures Markets 32 (1), 24–46 /http://onlinelibrary.wiley.com/doi/10.1002/fut. v32.1/issuetocS. Wang, Z., Daigler, R.T., 2011. The performance of VIX option pricing models: empirical evidence beyond simulation. Journal of Futures Markets 31 (3), 251–281. Zhang, J.E., Huang, Y., 2010. The CBOE S&P 500 three-month variance futures. Journal of Futures Markets 30 (1), 48–70. Zhang, J.E., Shu, J., Brenner, M., 2010. The new market for volatility trading. Journal of Futures Markets 30 (9), 809–833. Zhang, J.E., Zhu, Y., 2006. VIX futures. Journal of Futures Markets 5 (26), 521–531. Zhu, S.-P., Lian, G.-H., 2012. An analytical formula for VIX futures and its applications. Journal of Futures Markets 32 (2), 166–190 /http://onlinelibrary. wiley.com/doi/10.1002/fut.v32.2/issuetocS. Zhu, Y., Zhang, J.E., 2007. Variance term structure and VIX futures pricing. International Journal of Theoretical and Applied Finance (IJTAF) 10 (1), 111–127.