Volatility Model Specification: Evidence from the Pricing of VIX Derivatives. Chien-Ling Lo, Pai-Ta Shih, Yaw-Huei Wang, Min-Teh Yu *

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1 Volatility Model Specification: Evidence from the Pricing of VIX Derivatives Chien-Ling Lo, Pai-Ta Shih, Yaw-Huei Wang, Min-Teh Yu * Abstract This study examines whether a jump component or an additional factor better supports volatility modeling by investigating the pricing of VIX derivatives. To reduce the computational burdens for the empirical estimation significantly, we propose an efficient and easily implemented numerical approximation for the pricing of VIX derivatives. In terms of the term structure of VIX futures, we show that the additional volatility factor can replicate the common empirical patterns and explain the changes in the term structure, but the jump component cannot. In terms of the pricing of VIX options, we find that the two-factor volatility models significantly outperform the jump volatility models and that adding jumps in volatility only provides a minor improvement. Therefore, our general findings support the merit of the two-factor volatility specification. Key Words: VIX; VIX option; VIX futures; Two-factor Model; Jump in Volatility; Affine Model; Closed-form approximation; Term structure. JEL classification: G13, C32 First version: November 16, 2012 This version: July 22, 2013 * Lo, Shih, and Wang are at the Department of Finance, National Taiwan University, No. 1, Section 4, Roosevelt Road, Taipei 10617, Taiwan. Yu is at the Institute of Finance, National Chiao Tung University, Hsinchu 30010, Taiwan. Tel.: (Shih), (Wang), (Yu). d @ntu.edu.tw (Lo), ptshih@management.ntu.edu.tw (Shih), wangyh@ ntu.edu.tw (Wang), mtyu@nctu.edu.tw (Yu). The authors thank San-Lin Chung, Ming-Hsien Lin, Wei-Che Tsai, Pei-Shih Weng, and the seminar participants at National Chengchi University for constructive comments. The authors express their sincere gratitude to the National Science Council of Taiwan for the financial support. 1

2 1 Introduction Specifying the volatility of equity returns as a time-varying process is without doubt the most crucial step for volatility modeling and derivatives pricing. Among various alternatives of time-varying volatility models, the stochastic volatility (SV) model proposed by Heston (1993) is the most well-cited model for derivatives pricing due to its advantage on the plausibility of obtaining a closed-form solution for European options. 1 Following Heston s seminal work, several studies theoretically extended the SV model by either introducing jump components for returns (and volatility) or specifying an additional volatility factor (see, e.g., Bates 1996, 2000; Duffie, Pan, and Singleton, 2000). 2 Numerous studies have investigated the empirical performance of these models using historical returns and/or option prices of equity assets. 3 Most empirical studies support the valuable role of jump components, particularly for the jumps in returns, but the evidence is mixed on the role of the second volatility factor. Bates (2000) and Eraker, Johannes, and Polson (2003) find little improvement for an additional volatility factor, because it cannot capture the tails of the return distribution. Conversely, Alizadeh, Brandt, and Diebold (2002) and Christoffersen, Heston, and Jacobs (2009) find strong evidence for the presence of the additional volatility factor using currency futures and the S&P 500 index options, respectively. Although the models with jump components in returns (SVJ) substantially improve the SV model in the pricing of equity options, additionally including jumps in volatility (SVCJ) only slightly improves the SVJ model. In terms of the root mean square errors of Black Scholes implied volatility, Broadie, Chernov, and Johannes (2007) show that the SVJ model improves the SV model by 51.5%, while the SVCJ model improves the SVJ model by 4.9% only. 4 Therefore, whether specifying the volatility dynamic with a jump component is promising is debatable. So far, a comprehensive comparison between the models with a jump component and those with an additional factor for volatility modeling has not been well explored because almost all previous studies extract information from equity derivatives. However, volatility-related derivatives essentially contain more and purer volatility information than equity options and thus provide a direct channel to investigate the volatility model specification. The most prominent volatility-related derivatives to date are the volatility index (VIX) futures and options launched by the Chicago Board of Option Exchange 1 Hull and White (1987), Scott (1987), and Wiggins (1987), among others, also propose the stochastic volatility models for options pricing but do not derive a closed-form solution. See also Duan (1995) for the discrete-time type generalized autoregressive conditional heteroskedasticity option pricing model. 2 Bates (1996) introduces jumps in returns to explain large price movements, and Duffie, Pan, and Singleton (2000) further incorporate correlated jumps in volatility to capture the volatility clustering. Alternatively, Bates (2000) extends Bates (1996) by adding an independent volatility factor to allow flexibility of the term structure. 3 Bakshi, Cao, and Chen (1997) and Christoffersen, Heston, and Jacobs (2009) use the S&P 500 option data, Bates (2000) and Broadie, Chernov, and Johannes (2007) use the S&P 500 futures options data, Eraker, Johannes, and Polson (2003) use the S&P 500 and Nasdaq 100 index returns data, and Eraker (2004) uses the S&P 500 options and historical returns data simultaneously. 4 We only include the models without constraints on parameters in this comparison. 2

3 (CBOE) in 2004 and 2006, respectively. 5 The open interests and trading volumes of both derivatives have grown rapidly. In addition, several financial applications prove the information implied in the prices and trading activities of these products to be superior. For example, Jiang and Tian (2005) support the outstanding information content of the VIX for the future realized volatility. Given this discussion, we investigate the appropriateness of the volatility model specification by examining the empirical performance on the pricing of VIX derivatives with a focus on the comparison between the models including a jump component in volatility and those specifying an additional volatility factor. According to the definition of the CBOE, VIX is calculated from the market prices of the S&P 500 index options, and, intuitively, the dynamic of VIX should be in line with that of the S&P 500 index. Therefore, we adopt the framework that the S&P 500 index follows a generalized dynamic process with both the jump and two-factor volatility, and thus derivatives on the S&P 500 and VIX are consistently priced. However, under the consistent framework, the VIX is a nonlinear function of the instantaneous variance, which increases the difficulty of the valuation of its derivatives. 6 To tackle the empirical works, we propose an efficient and easily implemented approach to value the VIX derivatives as long as the state variables are affine. 7 The basic idea is similar to the well-known linear approximation, but instead we use exponential curves to fit the nonlinear payoff function. An exponential curve fits the target payoff function more effectively than the linear function does. More important, the closed-form solution of those option prices with exponential payoffs can be more easily derived by the Fourier transform as shown in Duffie, Pan, and Singleton (2000). 8 The proposed approximation allows us to conduct the empirical analysis efficiently. According to our empirical analysis for VIX futures, we first demonstrate that 53% of trading days during our sample period exhibit humped patterns in the term structure and these patterns can be replicated by the two-factor models. We also show that the one-factor jump models cannot generate a hump-shaped term structure of VIX futures and therefore cause serious mispricing. 9 In addition, we observe the term structure for two consecutive trading days during the period from In 1993, the CBOE first published the volatility index VIX calculated from a series of at-the-money S&P 100 index options. In 2003, it updated the definition of VIX by a model-free method using the market prices of S&P 500 index options to estimate the expected volatility of the S&P 500 index during the following 30-calendar-day period. 6 Duan and Yeh (2010), Egloff, Leippold, and Wu (2010), Lin and Chang (2010), Lian and Zhu (2011), Amengual and Xiu (2012), Branger and Völkert (2012), and Zhu and Lian (2012) also adopt this framework. Alternatively, Grünbichler and Longstaff (1996), Cont and Kokholm (2013), and Mencía and Sentana (2013) directly assume the dynamic of the VIX level. 7 Although Lian and Zhu (2011) derive the closed-form solution for VIX options under the SVCJ model of Duffie, Pan, and Singleton (2000), they do not consider the two-factor models, and our proposed approximations are much more efficient as much as hundredfold faster than their formulas within a tolerable error. Amengual and Xiu (2012) and Branger and Völkert (2012) derive similar closed-form solutions to Lian and Zhu (2011) for more complicated models, but their formulas require some further numerical methods such as the Runge-Kutta method to solve ordinary differential equations (ODEs). 8 As shown in Duffie, Pan, and Singleton (2000), the expectation of a linear function conditioned on requires the solution of one more ODE than an exponential function does. In addition, to derive the price of VIX derivatives, the additional ODEs cannot be analytically solved if there are jumps in volatility. 9 Christoffersen, Jacobs, Ornthanalai, and Wang (2008), Christoffersen, Heston, and Jacobs (2009), Egloff, Leippold, and Wu (2010), Aït-Sahalia, Amengual, and Manresa (2011), Aït-Sahalia, Karaman, and Mancini (2012), Branger and Völkert (2012), and Mencía and Sentana (2013) also indicate that the two-factor models can generate a more flexible term structure for volatility. 3

4 to 2010 and find that 36% of observations reveal opposite daily changes in short- and long-run expectations, which can be captured by the two-factor but not by the jump models. This finding is in line with Adrian and Rosenberg (2008), who report that the market volatility consists of short- and long-run components. As pointed out by Bates (2000), the jump models allow more flexibility for the skewness and kurtosis of the underlying distribution whereas the two-factor models allow more flexible for the term structure of volatility. Our exploration on VIX futures indicates that allowing flexibility for the volatility term structure is more crucial for volatility modeling, which is also supported by our investigation on the pricing of VIX options. We find that the models with the second volatility factor significantly outperform those with jump components and that the inclusion of the jump component in the volatility dynamic only provides a minor improvement of the pricing of VIX options. These results are robust across frequencies of estimation, trading dates, and categories of moneyness and time to maturity. In general, our empirical findings are consistent with Branger and Völkert (2012), who show that rendering sufficient flexibility for the volatility of volatility is particularly crucial for volatility modeling. Although both an additional factor and a jump component in volatility allow flexibility for the volatility of volatility, the mean-reverting property significantly mitigates the impact of jumps on the volatility of volatility because the price of a European VIX derivative depends only on the terminal VIX level. In contrast, as the level of the S&P 500 index, like stock prices, does not exhibit mean reversion, jumps in volatility directly impact the change in the index level. Therefore, including the jump components can substantially improve the specification of the price dynamic along with the price distribution and consequently the pricing of S&P 500 index options. To price VIX derivatives, ideally the volatility dynamic should be specified as a two-factor model to capture the short- and long-term changes simultaneously, which allows for flexibility in the term structure of volatility. Following from our empirical results, we conjecture that the superior performance of the two-factor models on the pricing of VIX derivatives is likely associated with the well-documented property of mean reversion. In other words, the information about how volatility persists and mean-reverts is useful for pricing VIX derivatives. The study proceeds as follows. In Section 2, we introduce a generalized consistent derivative pricing model that includes several frequently adopted option valuation models as special cases. In Section 3, we propose an efficient and easily implemented approximation for the prices of VIX derivatives and present the numerical analysis. In Section 4, we describe the data of VIX derivatives and the methodology used for our empirical analysis. Sections 5 and 6 present the empirical results from VIX futures and options, respectively. Section 7 concludes. 2 The Model We begin from a general framework that includes several popular option valuation models frequently adopted in the literature. Let be the level of the S&P 500 index (SPX) at time t. We assume that 4

5 follows the following the risk-neutral dynamic:,,,,, 1,,,,, 2,,,,, 3 Cov,,,, 1, 2, 4 Cov,,, Cov,,, 0, 5 where is the risk-free rate; is the continuous dividend yield;, and,, respectively, capture the short- and long-term instantaneous variance at time t; and, respectively, denote the mean-reverting speed and the long-run mean level of, ; depicts the volatility of variance ;, and, are two Wiener processes with the correlation structure specified above; is a Poisson process with constant intensity ; the random jump sizes in SPX and its variance are and, respectively, and e 1 is the random percentage jump in SPX with mean. Regarding the correlated jump sizes, we follow the literature and assume that is exponentially distributed with positive mean, and that is normally distributed with mean and variance. 10 The general model can nest most existing models. For example, we obtain (i) the SV model of Heston (1993) by setting, 0; (ii) the SVJ model of Bates (1996) by setting, 0; (iii) the SVCJ model of Duffie, Pan, and Singleton (2000) by setting, 0; (iv) the 2-SV model of Christoffersen, Heston, and Jacobs (2009) by setting 0; and (v) the 2-SVJ model of Bates (2000) by setting However, this study does not attempt to create a new valuation model. The VIX squared index introduced by the CBOE in 2003 is defined as a portfolio of all OTM S&P 500 index options with the time to maturity of 30 days: 100 2,,,,, 6 where, and,, and,, are the values of put and call options traded at time with strike price, time to maturity, and discount rate, respectively. Applying the 10 It follows that Q e 1 Q Q e 1 / Our model can also reduce to the SVVJ model of Sepp (2008) by setting, 0 and the SVIJ model of Duffie, Pan, and Singleton (2000) by setting, 0. However, for the one-factor jump models, this study focuses on the SVJ and SVCJ models for simplicity. 5

6 spanning formula of Bakshi and Madan (2000), the VIX squared under (1) (5) can be derived as: Q ln, 7 where Q denotes the risk-neutral expectation under the martingale measure Q,,, defines a weighted variance,, and,1,2, 1, 1, 2, Note that this formula can be reduced to Duan and Yeh (2010) by setting, 0 and to Cheng, Ibraimi, Leippold, and Zhang (2012) by setting, According to the risk-neutral valuation, the value of a VIX call option can be expressed as 8,, Q Q. 9 Similarly, the value of a VIX put option can be expressed as,, Q or derived from the VIX put call parity:,,,,, 10 where 0,, 0 denotes the level of VIX futures at time with time to maturity and can be regarded as a special case of the VIX call option by setting the strike price and discount rate equal to zero. 3 Valuation of VIX Derivatives In this section, we first introduce how to fit the nonlinear payoff of VIX derivatives in (9) by using the exponential curves and present the closed-form approximation consequently. Next, we examine the accuracy and the efficiency of our approximations for pricing VIX options under the SVCJ and the 2-SVJ models. 12 An alternative definition of the VIX squared is the quadratic variation of the log return process Q See, for example, Amengual and Xiu (2012), Branger and Völkert (2012), Britten-Jones and Neuberger (2000), Carr and Wu (2009), and Wu (2011). As Jiang and Tian (2005) point out, the definition of the quadratic variation is an approximation under jump models by using the second-order Taylor expansion 2 ln1. Based on this definition, the VIX squared under our model can be rewritten as Although the approximation error is small, our untabulated results show that the option pricing error caused by the second-order Taylor expansion may underestimate the VIX option prices by 12.95% in terms of relative root mean-squared error for the short-term and at-the-money options under the SVCJ model with the parameters estimated by Duffie, Pan, and Singleton (2000). The details are available on request from the authors. When conducting the empirical analysis in this study, we adopt the definition detailed in (7) to be consistent with the CBOE definition. 6.

7 3.1 Closed-Form Approximation Following the standard representation in the option theory, the pricing formula of VIX calls consists of two parts:,, ΠK Q, 11 where ΠK Q, denotes the indicator function, and Q denotes the risk-neutral probability measure. Using the methodology described in Duffie, Pan, and Singleton (2000), we define an instrumental function, Q and provide its closed-form solution in Appendix A. Hence, the last term in 11 can be easily solved by, and presents the probability of the VIX option expiring in the money. The first term on the right-hand side of 11, however, contains a squared root payoff function that is difficult to solve. Consequently, we attempt to propose an approximation that connects to the function,. The idea is straightforward but feasible. We use N exponential curves to approximate the squared root payoff in the interval,, which covers standard deviations around the mean of the state variable. 13 For each exponential curve, ;,,, 1,2,,, we fit the two endpoints and the midpoint of each subinterval, to solve the corresponding parameters,, as /, /, 12, where denotes the midpoint of,. 14 As a result, the terminal payoff of ΠK is approximated by. 13 In other words, the payoff in the range of, is approximated by the corresponding 13 This trick makes sense of how much information is incorporated. In detail, max, and max,, where and denote the mean and the standard deviation of, respectively, which are derived in the Lemma in Appendix B. Note that we ignore the interval 0,, which follows zero payoff, to enhance the fitting performance. 14 Choosing the midpoint is the simplest but not the only way to solve the parameters. Nonetheless, the linear approximation only has two parameters that can be uniquely determined by the endpoints and. 7

8 exponential function, 1,2,,. For the remainder range of, the payoff is approximated by the terminal function, and this trick can reduce the fitting error especially when the tail probabilities are not trivial. To link the payoff to the instrumental function,, we rewrite (13) as, 14 and therefore obtain an approximation for Π Q as,,,,,. 15 Regarding the choice of fitting points,,,, the equally divided partition is the simplest candidate. However, to fit the target more effectively, we follow Liu (2010) to determine the partition according to their curvature levels as, 16 where 1, 2,, Therefore, all fitting points,,,,,,, are uniquely determined for given N and k. 16 In Figure 1 we demonstrate a concrete example under the SVCJ model with the parameters based on Duffie, Pan, and Singleton (2000): =3.5, =0.01, =0.15, 0.5, 0.1, =0.05, =0.0001, 0.4,, =0.008, 0.03, =1, and the moneyness << INSERT FIGURE 1 HERE >> 15 Liu (2010) shows that the linear approximation with partition points satisfying the nonlinear equations can fit the target payoff more effectively than that with the equally divided partition, where denotes the target function. Applying mathematical induction and setting, we can derive (16). Although Liu (2010) considers the linear curve fitting, our untabulated results indicate that the performance of exponential curve fitting using the curvature partition also dominates that using the equally divided partition. 16 If we choose the according to this curvature rule, the corresponding parameters,, are difficult to solved. Therefore, we still choose by the midpoint of and. 17 For simplicity, we slightly adjust the parameters of Duffie, Pan, and Singleton (2000). The parameters estimated by Duffie, Pan, and Singleton are 3.46, 0.008, 0.14, 0.47, 0.1, 0.05, , 0.38, and

9 The function, includes one integral that can be efficiently computed, while the approximation for VIX derivatives requires 3 integrals. 18 As a result, we rearrange the formula as one with an integral only, which yields a more computationally efficient formula. PROPOSITION 1. (CLOSED-FORM APPROXIMATION) Using the exponential curve fitting method, the VIX call option can be approximated by,, Φ Φ Φ, 17 where Φ Φ,, where,, and defined in Appendix A. are defined as (12) and (16), respectively, and is an analytic term PROOF. See Appendix A. The pricing errors of the proposed approximation are caused by the fitting error multiplied by the corresponding probability. Due to the continuity of the squared root function, the fitting error within, converges to zero for large N. However, the probability beyond, is controllable because for any tolerable error 0, we can choose the such that Q Q,,. 18 Therefore, for large N and k, our approximation can converge to the true value of VIX derivatives. Basically, our idea is similar to the well-known linear approximation, but exponential curves fit the target payoff more effectively. More important, to apply the methodology of Duffie, Pan, and Singleton (2000), the ordinary differential equations (ODEs) for exponential curves can be analytically solved, but those for linear curves cannot when jumps in volatility occur Numerical Analysis To examine the accuracy of our approximation, we first construct the benchmark values by a double-integral method which is similar to that of Cheng, Ibraimi, Leippold, and Zhang (2012). By definition, the VIX option can be expressed as an integral that contains the conditional probability density function of the instantaneous variance as 18 In particular, the computation of Π requires 3 1 integrals, and the last term in (11) requires one integral. 19 The ODEs for the exponential and the linear curves are referred to the equations (2.5) (2.6) and (2.15) (2.16), respectively, in Duffie, Pan, and Singleton (2000). The technical details are available on request from the authors. 9

10 ,,, 19 where includes an integral; see Appendix A. 20 Table 1 provides the VIX option prices under the SVCJ model with the parameters based on Duffie, Pan, and Singleton (2000): =3.5, =0.01, {0.10, 0.15, 0.20}, {0.4, 0.5, 0.6}, 0.1, =0.05, =0.0001, 0.4,, =0.008, and Table 2 shows that under the 2-SVJ model with the parameters based on Bates (2000): =1, =0.01, {0.50, 0.55, 0.60}, =2, =0.02, {0.30, 0.35, 0.40}, =1, 0.05, =0.1,, =0.010, = Other parameters are set as 0.03, =1, and moneyness {0.85, 1.00, 1.15}. The RRMSE is the relative root mean-squared error defined as RRMSE /, 20 where is the number of the observations, denotes the approximated values, and denotes the benchmark values. Both Tables 1 and 2 also provide the total computing time obtained by the Matlab package on Microsoft Windows 7 and based on Intel Core i5-2410m 2.30GHz and 8.00GB RAM. << INSERT TABLE 1 HERE >> << INSERT TABLE 2 HERE >> In Table 1, the results show that the RRMSEs for our approximations with (k, N) = (3,1), (3,2), (3,4), and (3,8) are 4.42%, 0.97%, 0.39%, and 0.29%, respectively, and the RRMSEs for those with (k, N) = (6,1), (6,2), (6,4), and (6,8) are 8.67%, 1.76%, 0.19%, and 0.02%, respectively. In Table 2, the RRMSEs for our approximations with (k, N) = (3,1), (3,2), (3,4), and (3,8) are 3.38%, 0.98%, 0.56%, and 0.44%, respectively, and the RRMSEs for those with (k, N) = (6,1), (6,2), (6,4), and (6,8) are 6.59%, 1.11%, 0.15%, and 0.06%, respectively. For given k, the role of N is a trade-off between accuracy and efficiency. As the components in the VIX option formula (17) are proportional to the number of exponential curves N, the growth of computing time is with order. However, the pricing error decays rapidly. Regarding the fitting error, our untabulated results indicate that the exterior probabilities calculated by (18) are about 3% and 0.5% for the cases k = 3 and k = 6, respectively, in both Tables 1 and 2. Choosing larger k covers some extreme events that may accompany the devastating impacts; however, it also generates a higher interior fitting error for given N. It explains why the cases for k = 6 perform worse than the cases for k = 3 when N = 1 and To calculate the improper integral in (19), we take the upper limit of the integral as 20, which is sufficiently large, and adopt the Simpson method with 20,000 subintervals of equal length. To calculate the improper integrals in and equation (17), we choose the upper limit of the integral as For simplicity, we slightly adjust the parameters of Bates (2000). The parameters estimated by Bates (2000) are =0.91, = , =0.582, =1.76, = , =0.346, =0.9035, 0.057, =0.102,, = , and, =

11 In terms of the efficiency, the computing time depends on the number of exponential curves N and the fitting area k. The VIX option prices under two-factor models require more computing time due to the more complicated integrands. For example, the approximation with k = 6 and N = 2 and 4 for the SVCJ model take 0.18 and 0.41 seconds, respectively, and those for the 2-SVJ model take 0.54 and 1.47 seconds, respectively, to calculate 27 VIX option prices (see Tables 1 and 2). Notwithstanding, the proposed method provides an efficient way to value VIX derivatives Data Description and Empirical Methodology 4.1 Data Description Our primary dataset is obtained from the CBOE and includes the end-of-day bid and ask quotes of VIX options for the period from 2007 to VIX options are European-style options traded on the CBOE, and we use mid-quotes to represent option prices. We obtain the daily closing levels of VIX along with the option data. In addition, to investigate the performance of various models on pricing VIX futures, we also obtain the daily settlement prices of VIX futures from the CBOE. We calculate the risk-free rate for each derivative contract via the interpolation of zero curve surfaces obtained from the OptionMetrics database to fit its maturity. We adopt some commonly adopted filtering rules in the literature for the VIX options. First, we omit from the sample options with fewer than 7 days and more than 365 days to maturity due to the liquidity concern. Second, we eliminate all observations for which the trading volumes are equal to zero to avoid the non-synchrony effect. Next, we exclude all observations for which the bid prices are lower than the minimum tick size US$0.1 or the ask prices are lower than the bid prices. Finally, we eliminate observations that violate the arbitrage conditions. 24 As a result, our sample consists of a total of 86,149 observations, which includes 55,965 calls and 30,184 puts. Following Bakshi, Cao, and Chen (1997), we classify the VIX options into 15 categories by moneyness () and time to maturity (). In particular, the moneyness is defined as the strike price divided by the VIX futures price. A VIX call option is classified as deep-in-the-money (DITM) if 0.9, in-the-money (ITM) if 0.9,0.97, at-the-money (ATM) if 0.97,1.03, out-of-the-money (OTM) if 1.03,1.1, and deep-out-of-the-money (DOTM) if In 22 Our approximation is much more efficient than the formula of Lian and Zhu (2011). To implement their formula, we take the upper limits of integral as 10, which is identical to ours, and adopt the Matlab function mfun( erf,-) to calculate the complex error function in their formula. As a result, to calculate the same 27 VIX option prices under the SVCJ model, the option values are very close to ours, but their formula requires seconds when we choose 1. Note that is a positive number in their formula defined as their equation (A7). 23 Although VIX options were launched on February 24, 2006, we choose the sample period from 2007 to avoid the liquidity issue at the early stage of this newly introduced product. 24 First, we eliminate prices violate the arbitrage bounds max0,,, and max0,,,. Next, for each option maturity, if the prices are not monotonic in the strike prices, we eliminate the observations with lower volumes. Note that the observations that are not monotonic in time to maturity should not be eliminated due to the mean-reverting property of volatility. 25 Similarly, a VIX put option is classified as DOTM if 0.9, OTM if 0.9,0.97, ATM if 0.97,1.03, 11

12 terms of time to maturity, we classify VIX options into short-term, medium-term, and long-term groups. For each category, we report the average mid-quote, the number of observations, and the percentage of the total number in Table 3. Panels A and B report the results for call and put options, respectively. We also report the average trading volume and open interest for each moneyness and time to maturity, respectively. < INSERT TABLE 3 HERE > The average prices range from $0.74 (short-term, DOTM) to $8.35 (short-term, DITM) for VIX call options and from $0.63 (short-term, DOTM) to $12.47 (mid-term, DITM) for VIX put options. In general, a longer maturity does not imply a higher option price. The main reason is that long-term contracts are clustered in 2007, a period with relatively low VIX levels, and the alternative reason comes from the mean-reverting nature of volatility. We also find that ITM VIX options are less liquid than OTM options in terms of both the trading volume and open interest. Therefore, we only adopt the OTM VIX option prices and generate ITM VIX call options from their corresponding OTM VIX put options by applying the VIX put call parity (10). In addition, both the trading volume and open interest of short-term VIX options are much higher than those of mid-term and long-term options. 4.2 Empirical Methodology To estimate the model parameters from VIX options, we minimize the sum of squared pricing errors Θ argmin, (21) where Θ,,,,,,,,,, denotes the structure and jump- related parameters, denotes the total number of days, and denotes the sum of squared error at time, defined as Θ,,,,, (22) where denotes the observations at time, and Θ,,,, and denote the theoretical and market prices of a call option, respectively. Note that the instantaneous variance is time-varying, while the parameters in Θ are assumed to be constant over time. A standard approach in the VIX-related literature is to employ the VIX time series and the constraint VIX formula (7), and, consequently, the instantaneous variance can be uniquely determined by Θ for the one-factor models. 26 However, for the multi-factor models, the variance swap rate cannot uniquely identify the two components of instantaneous variance in one constraint. A straightforward approach implemented by ITM if 1.03,1.1, and DITM if See, for example, Duan and Yeh (2010), Branger and Vo lkert (2012), and Song and Xiu (2012). 12

13 many prior option-related empirical studies such as Bakshi, Cao, and Chen (1997) is to assume that Θ is updated with the same frequency as the instantaneous volatility and thus all parameters are estimated simultaneously. In other words, all parameters are estimated day by day when we use daily data. Alternatively, we can adopt the iterative two-stage procedure used by Bates (2000), Huang and Wu (2004), and Christoffersen, Heston, and Jacobs (2009), which is detailed as follows: Step 1. Given the set of model parameters Θ, we minimize (22) to obtain the sequences of,,, at each date. Step 2. Given the set of spot variances,,, obtained by Step 1, we solve (21) to obtain the model parameters Θ. This procedure iterates between Step 1 and Step 2 until the aggregate converges. 27 Although the iteration is time-consuming, our proposed approximation offers an effective alternative to implement this approach. When the strike price is set as zero, a call option reduces to a futures contract. Therefore, the same estimation methodology can be applied to VIX futures with the theoretical and market prices of call options being replaced by those of futures. 5 Empirical Results from VIX Futures The term structure of VIX futures presents the market expectation of the VIX level across maturities. When the current VIX level is relatively low (high), the term structure of VIX futures is in general upward sloping (downward sloping). However, due to the mean-reverting stylized fact for volatility, the term structure of VIX futures often exhibits a U-shaped or hump-shaped pattern. This section offers a deep look into its empirical patterns. For each trading date during the period from 2007 to 2010, we identify the term structure patterns by examining the market prices of VIX futures. Among the total of 1,006 observed days, we find that the term structure exhibits a monotonic pattern for only 382 days. In particular, the term structure is strictly increasing and decreasing for 266 and 116 days, respectively. However, it exhibits at least one smile for 389 days, and at least one hump for 533 days. 28 These nonmonotonic patterns may be explained by multiple volatility components or may simply come from investors mean-reverting expectation. For an alternative analysis, we observe the changes in the term structure of VIX futures for two consecutive days. Because the observed time to maturities are not constant across days, we adopt the cubic spline to interpolate a smooth term structure function. 29 Consequently, we compare the synthetic term structure for two consecutive trading days for their common range of time to maturity and find that 364 out of 1,005 pairs exhibit crosses, which indicate opposite changes for the shortand long-run expectations. Figure 2 exhibits two pairs of the term structures of VIX futures observed 27 Both steps employ the VIX constraint (7). 28 Suppose the observed prices for each trading date t are,,,,,,. If there exists one,, 2,3,, 1, simultaneously satisfying,, and,,, we conclude that the term structure contains at least one hump. Similar rules can be used to examine the other patterns. 29 To avoid unnecessary errors, we do not extrapolate the term structure beyond the observed ranges. 13

14 on August 10 and 11, 2009 (left figure) and March 14 and 15, 2007 (right figure). The left (right) figure presents the hump-shaped (U-shaped) pattern and reveals monotonic (opposite) changes in short- and long-run expectations. Consequently, these empirical observations on the patterns and the changes of the term structure of VIX futures motivate us to investigate whether any of considered models can capture these patterns and changes. < INSERT FIGURE 2 HERE > As shown in the empirical data, 36% of trading days during the period 2007 to 2010 indicate that the changes in the term structure of VIX futures for two consecutive days are not monotonic. This result can be explained by two-factor models when one of the volatilities is lower but another one is larger than its long-run mean level. 30 However, as proved in Proposition 2 of Appendix B, one-factor jump models cannot capture this common situation. In other words, the one-factor jump models can only generate the monotonic changes in the term structure of VIX futures for two consecutive days among all time to maturities. We also investigate the appropriateness of the models from their performance on fitting the term structure of VIX futures with the estimated parameters. Pondering the trade-off between computational accuracy and efficiency, we choose (k, N) = (6, 4) for the following empirical analysis when implementing the pricing formula of VIX futures. In Figure 3 we demonstrate the fitting performance of the SV, SVCJ, and 2-SV models in terms of four typical patterns of the empirical term structure of VIX futures: upward-sloping (December 11, 2009), downward-sloping (October 24, 2008), U-shaped (March 6, 2007), and hump-shaped (April 7, 2009). 31 < INSERT FIGURE 3 HERE > Interestingly, Figure 3 indicates that while the SV and the SVCJ models perform as well as the 2-SV model does, the 2-SV model outperforms in the hump-shaped pattern. To offer a simple intuition, we follow Bates (2006) using the Taylor expansion and examine the theoretical patterns for the term structure of VIX futures generated by one factor jump models, Q Q, 1 Q, Q,. By the Lemma in Appendix B, the expectation of, is either strictly increasing or strictly decreasing in when, is less than or greater than, respectively, while the variance of,, which can be simplified as, 2, is surely increasing in the The technical details are available on request from the authors. 31 Because the number of available futures contracts is usually smaller than that of the parameters of the models investigated in this study, we use synthetic observations of futures prices generated by fitting all available market prices of futures with the cubic spline to estimate the model parameters. 14

15 short run and thus forces the VIX futures to decrease. However, the variance effect disappears in the long-run futures because of the rapidly growing denominator. This explanation clarifies why the one-factor jump models cannot capture the hump-shaped pattern, even if the SVCJ model uses more parameters than the 2-SV model. 32 By contrast, two-factor models can easily generate the hump-shaped pattern when the short-run volatility is lower but the long-run volatility is larger than their corresponding long-run mean levels. 6 Empirical Results from VIX Options As shown in the numerical analysis, our formulas can provide a precise approximation rapidly. Consequently, this approach is ideal for this type of empirical analysis, which involves a large number of data and highly complicated functions. Using the effective approximation formulas proposed in this study, we investigate the empirical performance for various models frequently adopted in the literature and discuss some properties of VIX options that are different from those of S&P 500 index options. Following the empirical analysis for VIX futures, we also choose (k, N) = (6, 4) for the following analysis for VIX options, in which computing an option price takes about 0.02 and 0.05 seconds for the one- and two-factor models, respectively, with the pricing error being controlled less than 0.2%. 6.1 Parameter Estimation and In-Sample Performance In addition to following most of the prior option-related studies to estimate all parameters day by day, we also estimate the dynamic parameters with an alternative updating frequency (monthly). 33 The summary statistics of the parameter estimates from the daily and monthly updating frequencies are reported in Table 4. Panels A and B present the results for one-factor jump and two-factor models along with the SV model, respectively, with the RMSE representing the root of mean squared error defined as RMSE /, (24) where 50,345 is the total number of options. For the two-factor models, we follow Alizadeh, Brandt, and Diebold (2002) and Christoffersen, Heston, and Jacobs (2009) to identify the two volatility factors by the speed of mean-reversion, in which the factor with a faster (slower) reverting speed captures the short-run (long-run) volatility. < INSERT TABLE 4 HERE > 32 The one-factor jump models can generate the U-shaped pattern when, is smaller than but close to. In this case, the variance effect dominates the mean effect in the short run and is dominated by the mean effect in the long run. 33 The estimation for a two-factor model with an updating frequency other than daily is implemented by the two-stage approach. 15

16 We inevitably obtain nonidentical estimates for different updating frequencies because the degree of freedom and cross-contract weighting systems depend on the updating frequency. As shown in Table 4, the signs of all parameter estimates are consistent across updating frequencies, although the differences of the magnitudes of some parameter estimates may not be trivial. The daily updated speed of mean-reversion is much higher than the monthly updated, which implies different half-life periods of variance shocks. For the two-factor models, the slower is about 1.8 and 0.3 for daily and monthly results, respectively, while the faster is more than 9.9 for both frequencies. These results mean that the two-factor models indeed capture different volatility components in terms of the speed of mean reversion. 34 Regarding the jump parameters, the estimated jump size in SPX and the correlation between two jumps are both negative, which is consistent with the option literature. The estimated jump sizes in volatility are and 0.061, respectively, for the daily and the monthly results, and our untabulated results show zero values of in 6.9% days and 29.2% months, respectively. 35 These findings mean that the significance of jumps in volatility is much lower for monthly estimates. In terms of the RMSE, the model with more parameters always outperforms the model with fewer parameters, because the later can be regarded as a constrained model. However, the advantage becomes smaller when we switch from daily to monthly updating frequencies for one-factor jump models because the lower degree of freedom for the monthly updating frequency partially offsets the advantage from additional parameters. In particular, the RMSE of the monthly SVCJ model is lower than that of the monthly SVJ by only whereas the daily difference is Our untabulated results also show that the advantages for both SVJ and SVCJ models are close to zero when we consider lower updating frequencies such as the quarterly or the annually estimates. Conversely, the improvements for the two-factor models remain significant despite the frequency of estimation. These results may not be surprising because, according to Proposition 2 in Appendix B, models without the second volatility factor cannot capture the various changes in the term structure of volatility, and this frailty does not appear in the daily estimates. To examine whether the difference between the pricing errors from any pair of models is statistically significant, we adopt the Wilcoxon signed-rank test to investigate the null hypothesis that the median of the difference between the absolute errors of Models x and y is zero. The test statistic is: 34 The daily updated long-run mean level of volatility is typically lower and the volatility of variance is higher than their corresponding monthly updated parameters. For the two-factor models, one of the is as high as about 2.4 and 1.5 for daily and monthly results, respectively, which is in line with Bates (2000) and Christoffersen, Heston, and Jacobs (2009). In particular, the larger in Christoffersen, Heston, and Jacobs is estimated by on average and could be as large as 9.43 in If we consider the non-zero jump size in volatility, the estimated values of are and for the daily and the monthly results, respectively. The difference is still economically significant. 36 According to the mean-equality tests on monthly squared errors, the SVCJ and 2-SVJ models do not significantly improve the SVJ and 2-SV models, respectively. 16

17 ,,,, (25) where, is the difference of the absolute pricing errors of Models x and y for the jth VIX option in our sample. The standardized version of this statistic is asymptotically normal. We focus on the discussion on the median test rather than the mean test because the former is less sensitive to extreme values. In addition, our untabulated results show that both tests provide qualitatively similar results. Table 5 presents the median of,, where x stands for the selected model specified in the first column and y stands for the compared model specified in the first row. The standardized test statistics are reported in the parentheses. Panels A and B are the daily and monthly results, respectively. All the statistics, except that for the monthly SVCJ-SVJ pair, are negatively significant at 1% level, which means that the more sophisticated model significantly improves the less sophisticated model, and that models with the second volatility factor significantly improve the one-factor jump models. In addition, the median of the differences between the 2-SVJ and 2-SV in both panels is close to zero. This result indicates that the improvement made by adding the jumps in returns may not be economically considerable for the pricing of VIX options once the second volatility factor is incorporated. < INSERT TABLE 5 HERE > The positive median for the SVCJ-SVJ differences, even though it is very small, may signal that the advantage of the SVCJ model over the SVJ model may not always exist, which is quite different from the findings of most of the prior studies on S&P 500 index options. To look into the contribution of the volatility jump, in Table 6 we further investigate the RMSEs across moneyness and time to maturity. The results show that the models including jumps may not necessarily yield lower pricing errors. Particularly, the monthly results in Panel B show that the fitting errors of the SVCJ model are typically larger than those of the SVJ model for the near-the-money options, although the former outperforms the latter for the DITM VIX options. In other words, the SVCJ model improves the SVJ model by better fitting the tails of the distribution of volatility but may fit the central mass less satisfactorily. < INSERT TABLE 6 HERE > According to Tables 4 and 6, the pricing errors of the two-factor models are substantially smaller than those of the SVCJ model in all categories. To examine whether specifying the volatility dynamic with an additional factor is more useful than that with a jump component for the pricing of VIX options, we further compare the performance across the SVCJ and the 2-SVJ models because they are two alternative ways to improve the SVJ model. 37 Using the SVJ model as the benchmark, 37 Although the SVVJ model of Sepp (2008) and the 2-SV model are two alternative ways to improve the SV model, 17

18 we find that the 2-SVJ model produces remarkably lower RMSEs, while the SVCJ model does not. The Wilcoxon signed-rank test for the pairs of SVJ-SVCJ and SVJ-2SVJ shown in Table 7 statistically support these results. The results suggest that the RMSEs of the 2-SVJ model are significantly lower than those of the SVJ model at the 1% significance level for all cases, while the RMSEs of the SVCJ model are not significantly lower than those of the SVJ model for numerous cases. Although an argue may be made that the better performance may benefit from the number of parameters, we find that the RMSEs of the 2-SV model are also smaller than those of the SVCJ model even though the latter has more parameters than the former. Our untabulated results examining the pairs of SVJ-SVCJ and SVJ-2SV are also consistent with these findings. < INSERT TABLE 7 HERE > Figure 4 depicts the time series of the RMSEs for the SVJ, SVCJ, 2-SV, and 2-SVJ models. Clearly, while the SVJ and SVCJ lines are very close to each other and the 2-SV and 2-SVJ lines are hard to distinguish, the RMSEs of the latter group are consistently lower than those of the former group. Even if we observe only the period of the crash ( ), the second volatility factor obviously plays the most crucial role for the VIX option fits. Thus, Figure 4 shows that our results are robust across trading dates. < INSERT FIGURE 4 HERE > In sum, in terms of pricing VIX options, while the SVCJ model may not necessarily improve the SVJ model and the 2-SVJ model only slightly improves the 2-SV model, in general the 2-SV model outperforms the jump models. In other words, to price VIX options, specifying the volatility dynamic by including one additional factor is a more effective way than including the jump component. As Bates (2000) points out, the jump models allow more flexibility for the skewness and kurtosis of the underlying distribution whereas the two-factor models allow more flexible for the term structure. These findings indicate that allowing flexibility for the term structure is more crucial for VIX option pricing because of the well-documented stylized fact for volatility: mean-reversion. 6.2 Out-of-Sample Results When a model with more parameters fits the in-sample data more appropriately, whether the out-of-sample performance is also superior becomes a question. In this section, we present the out-of-sample results to examine whether the extra parameters cause over-fitting. We detail the procedure used to construct the out-of-sample analysis as follows. For the daily estimation, following Bakshi, Cao, and Chen (1997), we use the in-sample daily estimated parameters and volatilities to compute the next day s VIX option prices for each day except the final day. For the monthly estimation, we employ the parameters and volatility estimated from the previous month to generate a models incorporating the jumps in variance only are unappreciated in the option literature. 18