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Finance and Economics Discussion Series: 2015-071 Screen Reader version ^♣

The Effects of Asymmetric Volatility and Jumps on
the Pricing of VIX Derivatives

Yang-Ho Park*

Federal Reserve Board

September 11, 2015

Abstract:

This paper proposes a new collection of affine jump-diffusion models for the valuation of VIX derivatives. The models have two distinctive features. First, we allow for a positive correlation between changes in the VIX and in its stochastic volatility to accommodate asymmetric volatility. Second, upward and downward jumps in the VIX are separately modeled to accommodate the possibility that investors react differently to good and bad surprises. Using the VIX futures and options data from July 2006 through January 2013, we find conclusive evidence for the benefits of including both asymmetric volatility and upward jumps in models of VIX derivatives pricing. We do not, however, find evidence supporting downward jumps.

Keywords: VIX options; VIX futures; stochastic volatility; volatility smile; jump-diffusion

JEL Classification: G12; G13

1 Introduction

Over the past few years, there has been a great deal of financial innovation in volatility trading markets. A new collection of volatility derivatives, such as VIX futures, options, and exchange-traded products, have been introduced, making volatility trading more accessible to a broad range of investors.¹ The increasing volume of trading in those products is largely due to the fact that they can be used as a so-called tail risk hedging strategy against stock market downturns. That is, because changes in market volatility are negatively correlated with stock market returns, investors can limit the loss of an equity portfolio by taking a long position in VIX futures or call options. Given the explosive growth in volatility trading, the objective of this paper is to understand the effects of asymmetric volatility of and jumps in the VIX on the pricing of VIX futures and options.

We use the term "asymmetric volatility" to refer to the fact that the volatility of the VIX is not merely stochastic but also varies asymmetrically in response to changes in the VIX. Such asymmetry can be seen, for example, by looking at the dynamic relation between the VIX and the VVIX of the CBOE (Chicago Board Options Exchange). The VIX is a risk-neutral, forward-looking measure of market volatility implied by a cross section of S&P 500 index options, while the VVIX is a risk-neutral, forward-looking measure of market volatility of volatility implied by a cross section of VIX options.² The scatter plot in Figure 1 shows that there is a positive relation between changes in the VVIX and in the VIX. That is, the volatility of the VIX, as measured by the VVIX, tends to increase (decrease) as the VIX increases (decreases).

An important implication of this empirical regularity is that the stochastic volatility factor implicit in the VIX options comprises two components--one that can be spanned by VIX futures and another that cannot. In other words, the VIX options market is nonredundant with the VIX futures market. To gauge the extent to which the options market can be spanned by the futures market, we run a regression of VVIX returns onto VIX returns, and find that the VIX changes can explain only 51 percent of the variation in the VVIX based on the adjusted R² of the regression.

Contrary to the empirical fact just discussed, the existing VIX option papers generally assume that the volatility of the VIX is either completely spanned or completely unspanned. For example, some authors, such as Grünbichler & Longstaff (1996), Detemple & Osakwe (2000), and Goard & Mazur (2013), apply fully spanned volatility models in which VIX options are completely hedgeable by VIX futures, whereas others, such as Mencía & Sentana (2012), introduce a completely unspanned volatility model in which innovations in the VIX index are uncorrelated with those in the volatility of the VIX index. It is evident that all of those models are unable to accommodate the important feature of the data that we have found-- namely, asymmetric volatility. Hence, the chief goal of this paper is to contribute to the literature by studying the effects of asymmetric volatility on the pricing of VIX derivatives.

Intuitively, a model that allows for asymmetry in volatility may have great potential to improve the pricing of VIX options because it is capable of explaining the positive skewness implicit in the options.³ The VIX options market shows a persistent deviation from a geometric Brownian motion regardless of time to maturity. In particular, out-of-the-money (OTM) VIX calls tend to have higher Black-Scholes implied volatility than OTM VIX puts, a tendency that is sometimes referred to as a volatility smile, implying that the option-implied VIX return distributions are positively skewed. This positive skewness may be in part attributable to asymmetric volatility.

On top of asymmetric volatility, we also look at the impact of jumps in the VIX on the valuation of VIX futures and options because they can be another channel for explaining the positive skewness implied by VIX options. Good and bad surprises may arrive with different rates and sizes and investors may react differently to them.⁴ Hence, our models assume that upward and downward jumps occur independently with different frequencies and magnitudes. In this respect, our jump treatment is close in spirit to the currency option pricing model of Carr & Wu (2007), which separately models upward and downward jumps using time-changed Levy processes.

Building on the affine jump-diffusion framework of Duffie et al. (2000), we introduce a new family of dynamic models for the VIX with and without the features of asymmetric volatility and jumps and derive quasi-analytic solutions to the prices of futures and options. The models are characterized by three dynamic factors: one observed VIX index and two latent factors to capture time variations in the stochastic volatility and central tendency of the VIX. To reflect asymmetric volatility, we allow for a nonzero correlation between innovations in the VIX and in its stochastic volatility. Upward and downward jumps are both assumed to follow independent compound Poisson processes with each having its own jump intensity and exponential jump-size distribution. The models are then tested on the VIX futures and options data covering July 2006 through January 2013, via an unscented Kalman filter.

Turning to the empirical results, we first evaluate the effects of asymmetric volatility on the pricing of VIX derivatives. By comparing the model with symmetric volatility and no jumps (SVV) with the model with asymmetric volatility and no jumps (AVV), we find that the latter is strongly preferred to the former in both in-sample and out-of-sample tests. The performance difference is statistically significant at the 1 percent level based on the Diebold & Mariano (2002) test. That is, allowing for asymmetry in volatility can make large improvements in fitting the prices of VIX futures and options.

We next look at the effects of including upward jumps on top of asymmetric volatility by comparing the AVV model with the AVV-UJ model, the model with asymmetric volatility and upward jumps. The comparison shows decisive evidence in favor of the AVV-UJ model over the AVV model, except for in-sample futures pricing. The results are statistically significant at the 1 percent level according to the Diebold & Mariano (2002) test. That is, including upward jumps in the AVV model can make considerable improvements in VIX derivatives pricing.

Lastly, we investigate whether downward jumps can have an incremental effect on the pricing of VIX derivatives by comparing the AVV-UJ model with the model with asymmetric volatility and asymmetric upward and downward jumps (AVV-AJ). The comparison does not result in any significant ranking of one model over the other. Including downward jumps makes little, if any, improvement in the pricing of both futures and options as long as asymmetric volatility and upward jumps are accounted for. However, because we consider finite-activity jumps in this paper, we do not interpret our finding as implying that downward jumps are of no use in explaining VIX derivatives prices; rather, we argue that upward jumps play a far more important role in the pricing of VIX derivatives than downward jumps do.

Aside from the VIX option pricing literature, this paper also adds to the literature on unspanned stochastic volatility. Unspanned volatility has been found to be important in many other asset classes. For example, unspanned volatility in the interest rate market has been reported by Collin-Dufresne & Goldstein (2002), Li & Zhao (2006), and Bikbov & Chernov (2009), among others. Trolle & Schwartz (2009b) emphasize the importance of including unspanned volatility in pricing commodity options.

This paper is also related to the literature on the structure of jumps in the VIX. For example, Todorov & Tauchen (2011) and Wu (2011) both find that jumps in the VIX are more likely to have infinite activity, although they rely on different sources of information and estimation. More recently, Todorov et al. (2014) examine asymmetry in upward and downward jumps in VIX dynamics and find that upward and downward jumps are approximately symmetric, although the former are slightly more active than the latter. In more loosely related papers, Eraker (2005) and Broadie et al. (2007) study the impact of volatility jumps in joint analyses of stock index returns and options data. Our paper is distinct from all of these papers in that our application takes a direct look at VIX options and futures, which may offer a better environment in which to scrutinize the characteristic of volatility jumps.

The rest of the paper is organized as follows. Section 2 develops the models and methods used in this paper. The VIX options and futures data are described in Section 3. Section 4 shows the parameter estimates and some preliminary analysis, while Section 5 provides the main empirical results on pricing performance across the different model specifications. Section 6 concludes.

2 Models and Methods

The affine jump-diffusion framework of Duffie et al. (2000) has been widely used in the term structure modeling of interest rates and the valuation of derivatives because it allows for analytic tractability. Dai & Singleton (2000); Heston (1993); Bakshi et al. (1997); and Bates (2000) are well-known examples. In this section, we introduce a new family of affine jump-diffusion models for the dynamics of the logarithm of the VIX and derive semi-closed-form solutions to the prices of futures and options. In addition, we provide a description of our estimation method and introduce a competing model from Mencía & Sentana (2012), which will be used as a benchmark.

2.1 VIX dynamics

We assume that the logarithm of the VIX follows an affine jump-diffusion process. This assumption is motivated by a well-known empirical finding that a logarithmic model does a better job of describing the volatility dynamics of a stock index than the square root model of Heston (1993) (see, for example, Aït-Sahalia & Kimmel (2007), Jones (2003), and Durham (2013)). More importantly, Mencía & Sentana (2012) compare logarithmic and square root models with respect to the pricing of VIX futures and options and find that the former is preferred to the latter. Furthermore, an affine modeling of the logarithm of the VIX is important because it permits negative jumps as well as positive ones, whereas an affine modeling of the VIX itself leaves no room for modeling negative jumps.

As stated earlier, the objective of this paper is to understand the effects of asymmetric volatility of and jumps in the VIX on the pricing of VIX derivatives. Toward this end, it is desirable to start with a fairly realistic baseline model upon which new model characteristics are built. Our baseline model contains two latent factors in addition to the logarithmic VIX index, denoted by $$ v_{t}=\log($$VIX$$ _t)$$ . Specifically, a latent factor, u_t, is introduced to reflect time variation in the long-run mean of the VIX, which greatly helps to match VIX future prices along increasing times to maturity.⁵ The volatility of the VIX is also allowed to vary over time, driven by another latent factor, w_t.

Given a risk-neutral probability space $$ (\Omega,\mathcal{F},\mathbb{Q})$$ and information filtration $$ \{\mathcal{F}_{t}\}$$ , we assume that the (logarithmic) VIX dynamics take the following form:

where $$ \kappa_v$$ , $$ \kappa_u$$ , and $$ \kappa_w$$ capture the persistence of v_t, u_t, and w_t, respectively; $$ \sigma_u$$ and $$ \sigma_w$$ control for the volatility of u_t and w_t, respectively; $$ \overline{u}$$ and $$ \overline{w}$$ capture the long-run means of u_t and w_t, respectively; and $$ B_{1t}^{\text{Q}}$$ , $$ B_{2t}^{\text{Q}}$$ , and $$ B_{3t}^{\text{Q}}$$ are standard Brownian motions under the risk-neutral $$ \mathbb{Q}$$ measure.

To capture the asymmetric behavior of volatility, we allow for a nonzero correlation, denoted by $$ \rho$$ , between $$ dB_{1t}^{\text{Q}}$$ and $$ dB_{3t}^{\text{Q}}$$ . Although we expect $$ \rho$$ to be positive, we do not restrict the sign of $$ \rho$$ , letting the data speak as to its direction.

We allow for jumps in v_t. Specifically, as one of our main research goals is to understand the differential pricing implications of upward and downward jumps, we assume that they are driven by independent compound Poisson processes, with each having its own jump intensity and jump-size distribution. $$ N_{1t}^{\text{Q}}$$ and $$ N_{2t}^{\text{Q}}$$ denote risk-neutral Poisson processes driving upward and downward jumps with jump intensities $$ \lambda_+$$ and $$ \lambda_-$$ , respectively. Upward jump magnitudes, $$ J_{1}^{\mathbb{Q}}$$ , are assumed to follow an independent exponential distribution with a positive mean, $$ \delta_+>0$$ , with the probability density function taking $$ \tfrac{1}{\delta_+}\exp(-x/\delta_+)$$ if $$ x>0$$ and 0 otherwise. Similarly, downward jump magnitudes, $$ J_{2}^{\mathbb{Q}}$$ , are assumed to follow an independent exponential distribution with a negative mean, $$ \delta_-<0$$ , with the probability density function taking $$ \tfrac{1}{\vert\delta_-\vert}\exp(-x/\delta_-)$$ if $$ x<0$$ and 0 otherwise.

An affine jump-diffusion model is a Markov process whose drift vector, variance matrix, and jump intensities all have affine dependence on the state vector (see Duffie et al. (2000)). To satisfy this condition, we assume zero correlations between $$ dB_{1t}^{\text{Q}}$$ and $$ dB_{2t}^{\text{Q}}$$ and between $$ dB_{2t}^{\text{Q}}$$ and $$ dB_{3t}^{\text{Q}}$$ .⁶

The general form described in Equation (1) nests a range of model specifications with different features. Among those, we select four different specifications depending on whether there are restrictions on $$ \rho$$ , $$ \lambda_+$$ , or $$ \lambda_-$$ . The model specifications considered in this paper are summarized in Table 1.

2.2 Characteristic function

The virtue of affine models is that various transforms, such as a characteristic function, can be quasi-analytically calculated by solving a system of Riccati ordinary differential equations (ODEs). Specifically, given the model in Equation (1), we consider the characteristic function of v_T at time t under the Q measure, which is denoted by $$ g(v_t,u_t,w_t,t,T;\phi)$$ :

where $$ E_t^{\text{Q}}$$ denotes the conditional expectation using the information up to time t under the Q measure and $$ \mathrm{i}=\sqrt{-1}$$ .

Proposition 1  

Under some regularity conditions, the characteristic function takes an exponentially affine form of $$ (v_t, u_t, w_t)$$ :

where $$ s=T-t$$ and the coefficients, $$ \alpha(s)$$ , $$ \beta_v(s)$$ , $$ \beta_u(s)$$ , and $$ \beta_w(s)$$ , satisfy the system of ODEs:

where the boundary conditions are given as $$ \alpha(0)=0$$ , $$ \beta_v(0)=\mathrm{i}\phi$$ , $$ \beta_u(0)=0$$ , and $$ \beta_w(0)=0$$ .

Proof. See Appendix A.

The system of ODEs in Equation (4) can be solved by numerical solvers such as the Runge-Kutta method. Using this method, however, raises a numerical issue. When ODEs are stiff, a numerical solver requires a smaller step size in order to achieve absolute stability, impairing computational efficiency (see, for example, Huang & Yu (2007)). The stiffness of a system of ODEs is associated with how much the eigenvalues of the system differ in orders of magnitude; the larger the difference, the stiffer the system. The eigenvalues of our system of ODEs are equal to the persistence parameters, $$ \kappa_v$$ , $$ \kappa_u$$ , and $$ \kappa_w$$ . Among the parameters, $$ \kappa_v$$ and $$ \kappa_u$$ have a wide discrepancy in orders of magnitude, as can be seen in Table 3. Specifically, $$ \kappa_v$$ is roughly 20 to 25 times as large as $$ \kappa_u$$ . Because of this stiffness in our system, it is desirable to use an implicit method instead of an explicit method, and we find that the Matlab ode15s solver works well in our application.⁷

2.3 Valuation of VIX futures and options

Once the characteristic function has been calculated, it is straightforward to compute the prices of futures and options. Let $$ F(t,T)$$ denote the time-t price of a futures contract with a maturity of T. $$ F(t,T)$$ represents the risk-neutral expectation of the time-T VIX index given the information at time t.

Proposition 2   $$ F(t,T)$$ is given by

Proof. See Appendix A.

Let $$ C(t,T,K)$$ and $$ P(t,T,K)$$ denote the time-t prices of call and put options, respectively, with a maturity of T and a strike price of K. These prices are determined by

where r_t is the risk-free rate at time t and $$ \tau=T-t$$ is the time to maturity. Note that the value of a futures contract should be equal to the VIX on its expiration date: $$ F(T,T)=$$VIX$$ _T$$ . Because of this relation, the prices of call and put options can be determined by the characteristic function of v_t.

Proposition 3   $$ C(t,T,K)$$ and $$ P(t,T,K)$$ are given by

where the complementary distribution functions, $$ \Pi_1(t,T,K)$$ and $$ \Pi_2(t,T,K)$$ , may be expressed as

where $$ h(v_t,u_t,w_t,t,T;\phi)=g(v_t,u_t,w_t,t,T;\phi-\mathrm{i})/g(v_t,u_t,w_t,t,T;-\mathrm{i})$$ and $$ \Re$$ indicates the real part of a complex number.⁸

Proof. See Appendix A.

$$ \Pi_2(t,T,K)$$ indicates the probability that a call option with a maturity of T and a strike price of K ends up in the money on its expiration date. Among different kinds of numerical integration, we choose to use the Gauss-Laguerre quadrature method with an order of 20 in our application.

2.4 Estimation

The option pricing literature often estimates the models using a quasi maximum likelihood estimation based on a variant of Kalman filtering (see, for example, Carr & Wu (2003,2007), Trolle & Schwartz (2009b,a), and Mencía & Sentana (2012)). Christoffersen et al. (2014) show that unscented Kalman filtering is superior to extended Kalman filtering in the application of interest rate derivatives and comparable to particle filtering, which is computationally far more expensive. Hence, in this paper, we apply a quasi maximum likelihood method via an unscented Kalman filter.⁹ The Kalman filter requires us to recast the models into a state space form that comprises state equations and observation equations. First, it is necessary to introduce state equations for $$ (u_t,w_t)$$ for the physical $$ \mathbb{P}$$ measure. To do so, we assume that u_t and w_t take the following physical dynamics:

where $$ B_{2t}^{\mathbb{P}}$$ and $$ B_{3t}^{\mathbb{P}}$$ are independent standard Brownian motions under the physical $$ \mathbb{P}$$ measure and $$ \eta_u u_t$$ and $$ \eta_w w_t$$ capture the risk premiums for the u_t and w_t processes, respectively. Note that we do not need the physical dynamics for v_t because our observation equations involve only the risk-neutral dynamics. For this reason, we are unable to pin down jump risk premiums.

By applying an Euler approximation to Equation (9), we are able to define discrete-time state equations:

where $$ \Delta$$ is the time interval and $$ \varepsilon_{2,t+\Delta}$$ and $$ \varepsilon_{3,t+\Delta}$$ are independent standard normal random variables.

With respect to observation equations, we assume that VIX futures and options are observed with measurement errors. Specifically, following Trolle & Schwartz (2009b), we assume that measurement errors in log future prices are constant and that measurement errors in options prices are proportional to their market vegas, denoted by $$ \widehat{\nu}(t,T,K)$$ . That is,

where the hat symbol indicates the market price; $$ \sigma_F$$ and $$ \sigma_E$$ capture the sizes of measurement errors for futures and options, respectively; and $$ \xi_{1,t}$$ , $$ \xi_{2,t}$$ , and $$ \xi_{3,t}$$ are independent standard normal random variables. Details on unscented Kalman filtering are described in Appendix C.

2.5 Mencía & Sentana (2012) model

Recently, Mencía & Sentana (2012) studied the impact of stochastic volatility on the valuation of VIX derivatives. Although ours is not the first paper to address the importance of this subject, we take a different approach to modeling stochastic volatility than Mencía & Sentana (2012) do. That is, stochastic volatility is diffusion driven in our models, whereas it is jump driven in the Mencía & Sentana (2012) model. To compare the diffusion-driven model to the jump-driven model, we estimate one of the extended models of Mencía & Sentana (2012), which is hereafter referred to as the MS model. The MS model is specified under the risk-neutral measure as

where z_t is a volatility factor and $$ q_{t}^{\text{Q}}$$ is assumed to follow a compound Poisson process with intensity $$ \lambda_z$$ and an exponential jump distribution with a positive mean of $$ \delta_z$$ . Note that the dynamics of z_t in Equation (12) belong to a class of the non-Gaussian Ornstein-Uhlenbeck process of Barndorff-Nielsen & Shephard (2001).

3 Data

Our application looks at the valuation of VIX futures and options, among other volatility derivatives. The markets for these derivatives have seen explosive growth in trading activity in recent years, as can be seen in Figure 2. The top panel of the figure shows that the number of VIX futures contracts traded increased dramatically from about 1 million in 2007 to about 24 million in 2012, and that most of the growth occurred after 2009, likely provoked by the recent financial crisis. The bottom panel of the figure shows that the dollar trading volume for the VIX options also increased substantially from around $3.3 billion in 2007 to around $16.1 billion in 2012. Note that VIX call options are more actively traded than VIX put options, which may be associated with the fact that the former can be used as a means of hedging a stock market crash unlike the latter.

Our data set comprises the daily prices of VIX futures and options. The futures data come from Thomson Reuters, Datastream, while the options data are from OptionMetrics, Ivy DB. The VIX futures market started on March 26, 2004, and the VIX options market opened about two years later on February 24, 2006. As we decided to include only the dates for which data on both VIX futures and options are available, our sample starts on July 1, 2006, and ends on January 31, 2013, with a total of 1,657 trading days.¹⁰ There are no early exercise premiums because VIX options are European.

In general, VIX futures are available for a wider range of maturities than VIX options. We delete VIX futures for which there are no paired VIX options with the same maturity. This leaves us a total of 9,330 futures contracts.

Option prices are taken from the bid-ask midpoint at each day's close of the options market (3:15 p.m). A filtering scheme is applied to eliminate inaccurate or illiquid options. Specifically, we delete VIX options for which the mid price is less than 0.1, the time to maturity is less than eight days, the Black-Scholes implied volatility is null, or the relative bid-ask spread (defined as the bid-ask spread divided by the mid price) is larger than 0.3. We also discard the VIX options that violate the lower bound constraints:

where the risk-free rates are obtained from OptionMetrics, Ivy DB. Option prices comprise intrinsic and time values, and only the time value portion is subject to model specifications. Moreover, OTM options tend to be more liquid than in-the-money ones. For this reason, we use only OTM call and put options in our empirical tests. Finally, a total of 102,525 options are included in our sample.

Table 2 shows summary statistics such as the number of observations and the average price for VIX futures and options. In this table, the data are divided into two categories depending on time to maturity $$ \tau$$ : short-term contracts with $$ \tau \leq$$   three months and long-term contracts with $$ \tau>$$   three months . For options, we further break down the data into four categories according to their moneyness: deep OTM puts with $$ d<0.85$$ , moderate OTM puts with $$ 0.85<d<1.0$$ , moderate OTM calls with $$ 1.0<d<1.3$$ , and deep OTM calls with $$ d>1.3$$ , where moneyness is defined as $$ d=K/F(t,T)$$ . Note that call options have a larger number of observations and higher levels of Black-Scholes implied volatility than put options.

Figure 3 shows the time series of the VIX, the VVIX, and the option-implied skewness. The VIX and the VVIX are provided by the CBOE, while the one-month option-implied skewness is computed using a modified version of the model-free approach of Bakshi et al. (2003). The detailed procedure is provided in Appendix D. Two features are worth noting. First, the VIX and the VVIX share some pronounced spikes, such as the bankruptcy of Lehman Brothers, which is associated with asymmetric volatility. Second, the implied skewness takes a positive value most of the time, implying that a nonnormal stochastic mechanism is essential for describing the return distribution implied by VIX options.

4 Parameter estimates and preliminary analysis

This section starts by looking at two sets of parameter estimates. The first set is obtained using the entire data set from July 2006 through January 2013, whereas the second set is obtained using a subsample of four years from July 2006 through June 2010. The second set is used to test different model specifications for an out-of-sample period from July 2010 through January 2013. In addition, we will examine whether asymmetric volatility and jumps help to match the observed term structure of the implied volatility and skewness, on average, which will give us a hint as to the pricing performance results discussed in the section that follows.

4.1 Parameter estimates

Table 3 presents in-sample parameter estimates across various model specifications. The models with asymmetry in volatility yield very different volatility dynamics than the model without it. To see this, let us compare the SVV model with the AVV-type models (AVV, AVV-UJ, and AVV-AJ). First, we observe that the volatility process is not very persistent in the SVV model ( $$ \kappa_w=5.506$$ ), whereas it is more persistent in the AVV-type models ( $$ 0.847 \leq \kappa_w \leq 1.553$$ ). Second, the SVV model ( $$ \sigma_w=7.271$$ ) has far higher volatility of volatility than the AVV-type models ( $$ 1.976 \leq \sigma_w\leq 2.610$$ ). Third, we define the long-run mean of volatility under the physical measure as $$ \overline{w}^{\mathbb{P}}=\tfrac{\kappa_w \overline{w}}{\kappa_w -\eta_w}$$ and find that the SVV model ( $$ \overline{w}^{\mathbb{P}}=0.262$$ ) has a lower level of the long-run mean than the AVV-type models ( $$ 0.553 \leq \overline{w}^{\mathbb{P}} \leq 0.677$$ ). Overall, allowing for asymmetric volatility makes a large difference in the estimation of the volatility dynamics of the VIX, which in turn will have a large effect on the pricing of VIX derivatives.

The degree of asymmetry in volatility differs depending on whether upward and downward jumps are included. The asymmetry parameter is highest in the AVV model ( $$ \rho=0.870$$ ) and lowest in the AVV-UJ model ( $$ \rho=0.422$$ ). The AVV-AJ model ( $$ \rho=0.794$$ ) falls in between. Because of this result, a jump process would have an impact on the pricing of VIX derivatives not only per se but also through different degrees of asymmetric volatility.

Interestingly, the variance risk premium parameter, $$ \eta_w$$ , takes a negative value for each of the models considered, implying that the market volatility of volatility is negatively priced. A similar discovery is also made by Song (2012) and Park (2015) in a non-parametric fashion. This negative pricing is not surprising because a market volatility of volatility asset can be used as a hedge against stock market downturns as it has a negative exposure (or beta) to the stock market index.

Now let us look at the frequencies and sizes of upward and downward jumps in the AVV-UJ and AVV-AJ models. The table shows that upward jumps take place about 2.7 times per annum with an average size of about 0.27, regardless of whether downward jumps are included. As expected, in the AVV-AJ model, upward jumps have a higher occurrence rate and a larger size than downward jumps; that is, $$ \lambda_+>\lambda_-$$ and $$ \delta_+>\vert\delta_-\vert$$ .

Table 3 also contains two model comparison metrics: Akaike (AIC) and Schwarz (SIC) information criteria. These are based on some penalty scores, which depend on the number of free parameters, less log likelihood values. The lower the information criteria, the better the model. Both information criteria offer decisive rankings of the models: AVV-AJ > AVV-UJ > AVV > SVV > MS. In short, asymmetric volatility, upward jumps, and downward jumps are all critical modeling features, and the SVV model is preferred to the MS model.

Each panel of Figure 4 shows a time series of the filtered volatility states that are obtained using the in-sample parameter estimates. The filtered volatility states in the SVV model tend to be higher and more volatile than those in the AVV model. In addition, the filtered volatility states are substantially higher in the models with no jumps (SVV and AVV) than in the models with jumps (AVV-UJ and AVV-AJ), although all models have similar evolutions over time.

Table 4 presents out-of-sample parameter estimates across various model specifications. This table also indicates that the decision to allow for asymmetry in volatility has a large effect on the volatility dynamics and the decision to include jumps affects the extent of asymmetry in volatility.

4.2 Fitting the term structure of the implied volatility

Here, we investigate the ability of the models to fit the average term structure of the option-implied volatility. The option-implied volatility is computed by applying the model-free approach of Bakshi et al. (2003) to a cross section of VIX options, with little modification. The detailed procedure is provided in Appendix D.

The top panel of Figure 5 shows the scatter plot and kernel regression (solid line) of the implied volatility against months to maturity; the kernel regression is applied to capture the average term structure of the implied volatility. An interesting fact is that the term structure of the implied volatility is downward sloping, with the short end being nearly twice as high as the long end. This decreasing pattern is associated with the mean-reverting behavior of the VIX; the stronger the mean reversion of the VIX, the steeper the slope.

To see whether the models under consideration can capture the decreasing pattern in the implied volatility, we compute the average term structure of volatility implied by each model. The model-implied volatility, denoted by IV$$ (t,T)$$ , can be computed by differentiating the cumulant generating function for each model:

where the cumulant generating function is defined as $$ \Psi(t,T;\phi)=\log(g(v_t,u_t,w_t,t,T;\phi))$$ . Note that IV$$ (t,T)$$ is annualized. The model-implied volatility is computed on each day and for each time to maturity throughout the sample period using the in-sample parameter estimates in Table 3. We then run a kernel regression of the model-implied volatility onto months to maturity for each model.

The bottom panel of Figure 5 shows that all of the four models--SVV (solid line), AVV (dashed line), AVV-UJ (dotted line), and AVV-AJ (dashed-dotted line)--do a good job of capturing the decreasing pattern in the observed implied volatility, although there are some minor discrepancies across the models. In particular, the SVV model tends to slightly overestimate the term structure of volatility, whereas the AVV-type models tend to slightly underestimate it. Overall, the models considered in this paper are all good at capturing the average term structure of the implied volatility, and the decision whether to allow for asymmetric volatility and jumps does not make a notable difference in this regard.

4.3 Fitting the term structure of the implied skewness

This subsection addresses whether asymmetric volatility and jumps can help explain the average term structure of the implied skewness. As for the option-implied volatility, the option-implied skewness is computed by applying the model-free approach of Bakshi et al. (2003) to a cross section of VIX options, with little modification.

The top panel of Figure 6 shows the scatter plot and kernel regression (solid line) of the implied skewness against months to maturity; the kernel regression is applied to capture the average term structure of the implied skewness. This figure reveals a surprising stylized fact: the average term structure of the implied skewness is nearly flat up to a one-year horizon.¹¹ Put simply, the short-term skewness is approximately as high as the long-term skewness.

To examine whether the models under consideration can produce such a flat pattern, we compute the model-implied skewness. Similar to the model-implied volatility, the model-implied skewness, denoted by SKEW$$ (t,T)$$ , can be computed by differentiating the cumulant generating function:

The bottom panel of Figure 6 shows that the AVV (dashed line) and AVV-UJ (dotted line) models both do a fairly good job of capturing the flat term structure of skewness. The AVV-UJ model tends to generate more of a positive skewness for short horizons than the AVV model, implying that upward jumps have a greater effect for short horizons than for long ones. In contrast, the AVV-AJ model (dashed-dotted line) does a poorer job of fitting the term structure of the implied skewness than the AVV and AVV-UJ models. Finally, the SVV model (solid line) has no ability to produce a positive skewness at all, (although it is capable of generating a positive excess kurtosis).

5 Pricing performance

Our performance analysis focuses on four main comparisons. First, we compare the diffusion-driven SVV model with the jump-driven MS model. Second, we compare the AVV model with the SVV model to test the importance of allowing for asymmetry in volatility. The third comparison is made between the AVV-UJ model and the AVV model in order to examine the effects of including upward jumps on top of asymmetric volatility. Lastly, we compare the AVV-AJ model with the AVV-UJ model to investigate whether the addition of downward jumps can make an incremental improvement once asymmetric volatility and upward jumps are included.

To facilitate our analysis, we compute a performance metric, a root mean squared error (RMSE), which is separately defined for futures and options as

where RMSE$$ _{i}^F$$ and RMSE$$ _{i}^O$$ denote the RMSEs implied by a model i for futures and options, respectively; N^F and N^O indicate the total numbers of futures and options included in the sample, respectively; and $$ \widehat{O}_{i}$$ and O_i denote the market-implied and model-implied option prices, respectively. Hereafter, RMSE$$ _{i}$$ is used to denote either RMSE$$ _{i}^F$$ or RMSE$$ _{i}^O$$ according to the context of analysis. Note that the performance metric is consistent with the assumption that we make about measurement errors in the observation equations.

To measure the extent to which one model is better or worse than another, we compute pricing differences between two models. Let $$ \Delta$$   RMSE$$ _{i\vert j}$$ denote the pricing difference of a model i over a model j. $$ \Delta$$   RMSE$$ _{i\vert j}$$ is defined as

where RMSE$$ _i$$ and RMSE$$ _j$$ denotes the RMSEs implied by models i and j, respectively. A negative (positive) value of $$ \Delta$$   RMSE$$ _{i\vert j}$$ means that model i yields lower (higher) pricing errors than model j, implying that the pricing performance of the former is better (worse) than that of the latter by a percentage of that value.

Tables 5 and 6 show in-sample and out-of-sample pricing performance, respectively, across the different models. From left to right, the groups of columns show the pricing errors as measured by RMSE$$ _{i}$$ ; the pricing improvements relative to the SVV model as measured by $$ \Delta$$   RMSE$$ _{i\vert\text{SVV}}$$ ; the pricing improvements relative to the AVV model as measured by $$ \Delta$$   RMSE$$ _{i\vert\text{AVV}}$$ ; and the pricing improvements relative to the AVV-UJ model as measured by $$ \Delta$$   RMSE$$ _{i\vert\text{AVV-UJ}}$$ , respectively. The main results for all futures and options are shown in the top two panels, and the options are broken down into calls and puts and by moneyness levels in the lower panels.

To assess the statistical significance of a pricing performance difference, we make pairwise model comparisons based on the Diebold & Mariano (2002) test. To do so, we compute a time series of RMSEs for each model, $$ \{$$RMSE$$ _{i,t}\}_{t=1}^{D}$$ , where D is the total number of evaluation days and RMSE$$ _{i,t}$$ is the date-t RMSE implied by a model i. Note that $$ \{$$RMSE$$ _{i,t}\}_{t=1}^{D}$$ is separately obtained for in-sample and out-of-sample tests.

Next, we obtain a series of the differences in RMSEs between every pair of the models, $$ d_{i,j,t}=$$RMSE$$ _{i,t}-$$RMSE$$ _{j,t}$$ , $$ t=1,\ldots,D$$ , and define a test statistic between models i and j as

where $$ \overline{d_{i,j,t}}$$ is a sample mean of the differences and $$ \sigma(d_{i,j,t})$$ is a heteroskedasticity- and autocorrelation-consistent estimate of the sample standard deviation of the differences.¹² Diebold & Mariano (2002) show that the test statistic in Equation (18) follows a standard normal distribution under the null hypothesis that there is no statistically significant difference between models i and j.

Tables 7 and 8 display in-sample and out-of-sample test statistics, respectively. A negative (positive) statistic in a cell $$ (i,j)$$ indicates that model i outperforms (underperforms) model j. Panels A and B correspond to VIX futures and options, respectively. The symbols *, **, and *** represent statistical significance at the 10, 5, and 1 percent levels, respectively.

5.1 Diffusion-driven versus jump-driven volatility

This subsection focuses on investigating whether the SVV model can price VIX futures and options better than the MS model. With respect to futures, the pricing errors and pairwise model comparisons are shown in Panel A of Tables 5-8. The SVV model has an in-sample RMSE of 0.1842 and an out-of-sample RMSE of 0.2252, while the MS model has an in-sample RMSE of 0.1853 and an out-of-sample RMSE of 0.2332. The $$ \Delta$$   RMSE$$ _{i\vert\text{SVV}}$$ metrics indicate that the futures pricing performance of the SVV model is better than that of the MS model by 1.18 percent in the in-sample test and by 7.02 percent in the out-of-sample test. The pricing difference is statistically significant at the 1 percent level only in the out-of-sample test, with a z statistic of $$ \pm 7.58$$ .

With respect to options, the pricing errors and pairwise model comparisons are shown in Panel B of Tables 5-8. The SVV model has an in-sample RMSE of 0.0939 and an out-of-sample RMSE of 0.1199, while the MS model has an in-sample RMSE of 0.1037 and an out-of-sample RMSE of 0.1475. The $$ \Delta$$   RMSE$$ _{i\vert\text{SVV}}$$ metrics indicate that the options pricing performance of the SVV model is better than that of the MS model by 10.00 percent in the in-sample test and by 20.71 percent in the out-of-sample test. The performance difference is statistically significant at the 1 percent level with a z statistic of $$ \pm 6.44$$ in the in-sample test and at the 1 percent level with a z statistic of $$ \pm 12.26$$ in the out-of-sample test.

To summarize, the SVV model outperforms the MS model except for the in-sample futures pricing. The superiority of the SVV model over the MS model implies that the volatility of the VIX is better described by a diffusive process rather than by a finite-activity jump process.¹³ For this reason, the SVV model is preferred over the MS model as a baseline model upon which to add asymmetric volatility and jumps.

5.2 Effects of asymmetric volatility

This subsection focuses on discussing the effects of allowing for asymmetry in volatility on the pricing of futures and options by comparing the AVV model with the SVV model. With respect to futures, the AVV model has an in-sample RMSE of 0.1804 and an out-of-sample RMSE of 0.2201. Based on the $$ \Delta$$   RMSE$$ _{i\vert\text{SVV}}$$ metrics, the futures pricing performance of the AVV model is better than that of the SVV model by 4.22 percent in the in-sample test and by 4.58 percent in the out-of-sample test. The pricing difference is statistically significant at the 5 percent level with a z statistic of $$ \pm 2.52$$ in the in-sample test and at the 1 percent level with a z statistic of $$ \pm 8.17$$ in the out-of-sample test. Furthermore, the futures pricing improvements vary by time to maturity. Specifically, the AVV model generates better futures pricing than the SVV model for short-term horizons both in-sample and out-of-sample but not for long-term horizons.

With respect to options, the AVV model has an in-sample RMSE of 0.0885 and an out-of-sample RMSE of 0.1154. The $$ \Delta$$   RMSE$$ _{i\vert\text{SVV}}$$ metrics suggest that the options pricing performance of the AVV model is better than that of the SVV model by 5.87 percent in the in-sample test and by 3.87 percent in the out-of-sample test. The performance difference is statistically significant at the 1 percent level with a z statistic of $$ \pm 7.08$$ in the in-sample test and at the 1 percent level with a z statistic of $$ \pm 6.49$$ in the out-of-sample test. Furthermore, we find that allowing for asymmetry in volatility reduces option pricing errors for all horizons in both in-sample and out-of-sample tests.

Taking our analysis one step further, we break down option pricing errors into puts and calls. Panels C and D of Table 5 show the in-sample RMSEs for OTM puts and calls, respectively. Compared to the SVV model, the AVV model substantially improves the pricing of put options by 14.60 percent but improves the pricing of call options only by 1.02 percent. Similarly, Panels C and D of Table 6 show the out-of-sample RMSEs for OTM puts and calls, respectively. Again, compared with the SVV model, the AVV model makes a large improvement in the pricing of put options by 11.45 percent, but only a very small improvement in the pricing of call options by 1.86 percent.

To sum up, we find decisive evidence that asymmetry in volatility has an enormous and statistically significant effect on the valuation of VIX futures and options. The AVV model is strongly preferred to the SVV model in both in-sample and out-of-sample tests. Especially, asymmetry in volatility plays a crucial role in fitting the prices of puts and short-term futures. Given the fact that OTM put prices are usually found to be cheaper than OTM call prices with an equivalent moneyness, our result implies that the underpricing of VIX puts is more likely to be driven by asymmetric volatility rather than upward jumps.

5.3 Effects of upward jumps

We now investigate the effects of including upward jumps on top of asymmetric volatility on the pricing of futures and options by comparing the AVV-UJ model with the AVV model. With respect to futures, the AVV-UJ model has an in-sample RMSE of 0.1799 and an out-of-sample RMSE of 0.2163. The $$ \Delta$$   RMSE$$ _{i\vert\text{AVV}}$$ metrics indicate that the futures pricing performance of the AVV-UJ model is better than that of the AVV model by 0.53 percent in the in-sample test and by 3.51 percent in the out-of-sample test. Hence, the statistical significance is obtained only for the out-of-sample test at the 1 percent level with a z statistic of $$ \pm 6.50$$ .

Furthermore, the futures pricing improvement depends on time to maturity in both in-sample and out-of-sample tests. Incorporating upward jumps improves the pricing of long-term futures but worsens the pricing of short-term futures. This finding can be contrasted with the previous finding that allowing for asymmetry in volatility helps to improve the pricing of short-term contracts only. Taken together, asymmetric volatility and upward jumps have opposing yet complementary influences on futures pricing performance.

With respect to options, the AVV-UJ model has an in-sample RMSE of 0.0817 and an out-of-sample RMSE of 0.1094. The $$ \Delta$$   RMSE$$ _{i\vert\text{AVV}}$$ metrics suggest that the options pricing performance of the AVV-UJ model is better than that of the AVV model by 7.96 percent in the in-sample test and by 5.33 percent in the out-of-sample test. The performance difference is statistically significant at the 1 percent level with a z statistic of $$ \pm 8.16$$ in the in-sample test and at the 1 percent level with a z statistic of $$ \pm 11.71$$ in the out-of-sample test. Furthermore, we find that including upward jumps is effective regardless of option maturities in both in-sample and out-of-sample tests, although the improvement is greater for long-term options than for short-term ones.

We now break down option pricing errors into puts and calls. Panels C and D of Tables 5 and 6 show the in-sample and out-of-sample RMSEs for OTM puts and calls, respectively. The AVV-UJ model greatly improves the pricing of cal in both in-sample and out-of-sample tests, but there is no significant pricing difference between the two models with respect to the pricing of put options.

Based on the empirical results above, we conclude that upward jumps have a sizable and statistically significant impact on the valuation of VIX futures and options. The AVV-UJ model is strongly preferred to the AVV model in both in-sample and out-of-sample tests, except for the in-sample futures pricing. Especially, upward jumps play a pivotal role in pricing calls and long-term futures. Given the fact that OTM call prices are usually more expensive than OTM put prices with an equivalent moneyness, our result implies that high levels of call prices are more likely to be driven by upward jumps rather than by asymmetric volatility.

5.4 Effects of downward jumps

Lastly, we evaluate the incremental effect of downward jumps on the pricing of futures and option by comparing the AVV-AJ model with the AVV-UJ model. Based on the $$ \Delta$$   RMSE$$ _{i\vert\text{AVV-UJ}}$$ metrics, including downward jumps makes little improvement in the pricing of futures and options to the extent that asymmetric volatility and upward jumps are already accounted for. The statistical significance is obtained only for in-sample futures pricing but the size of improvement is quite small with $$ \Delta$$   RMSE$$ _{i\vert\text{AVV-UJ}}=-0.86$$ .

To see if the inefficacy of downward jumps relies on our particular choice of jump specifications, we also consider an alternative specification of upward and downward jumps. In particular, we estimate the AVV model with double exponential jumps as in Kou (2002) and Kou & Wang (2004), where upward and downward jumps are driven by a single compound Poisson process (see Appendix E for details on the model specification). Once again, we fail to find evidence that the AVV model with double exponential jumps is better at pricing the VIX derivatives than the AVV-UJ model, although the results are not reported in this paper.

6 Conclusion

Given the growing interest in volatility trading, this paper suggests a class of affine jump-diffusion models for the pricing of VIX derivatives. The models have two innovative features. First, to assess the effects of asymmetric volatility on the pricing of VIX derivatives, we allow for a positive correlation between changes in the VIX and in its stochastic volatility. Second, we separate out the roles of upward and downward jumps to better consider the reality that investors react differently to good and bad shocks, which may arrive independently with different rates and sizes.

Our empirical analysis shows decisive evidence for the benefits of including both asymmetric volatility and upward jumps in pricing models of VIX futures and options. In particular, we find that asymmetric volatility is essential for pricing put options and short-term futures, whereas upward jumps are crucial for pricing call options and long-term futures. That is, low levels of put prices are associated with asymmetric volatility, whereas high levels of call prices are likely to be driven by the possibility of upward jumps. In sum, asymmetric volatility and upward jumps both are required to price VIX derivatives, having opposing yet complementary effects on them.

However, we fail to find consistent evidence for the benefit of including downward jumps in the pricing of the VIX futures and options. Given that all the jump specifications that we consider are of finite activity, we do not interpret our results as implying that downward jumps are of no use in explaining the VIX derivative prices. Indeed, authors such as Todorov & Tauchen (2011) and Wu (2011) show that volatility jumps may be better described by an infinite-activity jump process than a finite-activity one. Therefore, it would be interesting to extend our research to models with infinite-activity jump processes (on top of asymmetric volatility).

Lastly, we report some interesting features of the term structures of the implied volatility and skewness. Specifically, the term structure of the implied volatility has a downward slope, whereas the term structure of the implied skewness is nearly flat. Although all of the models considered do a nice job of capturing the decreasing pattern in the implied volatility, only the AVV and AVV-UJ models successfully match the flat pattern in the implied skewness, consistent with the result that asymmetric volatility and upward jumps are both crucial for pricing the VIX options.

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A. Derivation of the VIX derivative prices

We start by deriving the characteristic function, $$ g(v_t,u_t,w_t,t,T;\phi)=E_t^{\text{Q}}\left[ \exp(\mathrm{i} \phi v_T \right)]$$ , as in Heston (1993). Given the model in Equation (1), Ito's lemma leads to

If a Markov process is affine, its cumulant generating function has affine dependence on the state vector:

Substituting Equation (A.2) into Equation (A.1), we obtain

where

Note that g is a martingale by the law of iterated expectations. Rearranging Equation (A.3) as an affine form of $$ (v,u,w)$$ and using the fact that $$ E_t[dg]=0$$ , we obtain the system of the following ODEs:

where the boundary conditions are given as $$ \alpha(0)=0$$ , $$ \beta_v(0)=\mathrm{i}\phi$$ , $$ \beta_u(0)=0$$ , and $$ \beta_w(0)=0$$ .

Let $$ F(t,T)$$ denote the time-t price of a futures contract with a maturity of T. $$ F(t,T)$$ is equal to the characteristic function evaluated at $$ \phi=-\mathrm{i}$$ :

To derive the pricing formula for $$ C(t,T,K)$$ , let us introduce a change of measure from Q to $$ \tilde{\text{Q}}$$ by a Radon-Nikodym derivative $$ \tfrac{d\tilde{\text{Q}}}{d\text{Q}}=\tfrac{\exp{(v_T)}}{E_t^{\text{Q}}[\exp{(v_T)}]}$$ . Let us denote the characteristic function of v_T under the $$ \tilde{\text{Q}}$$ measure by $$ h(v_t,u_t,w_t,t,T;\phi)$$ . $$ h(v_t,u_t,w_t,t,T;\phi)$$ is given by

$$ C(t,T,K)$$ is calculated using $$ g(v_t,u_t,w_t,t,T;\phi)$$ and $$ h(v_t,u_t,w_t,t,T;\phi)$$ :

where $$ \Pi_1(t,T,K)$$ and $$ \Pi_2(t,T,K)$$ are given by the Levy inversion formula (see Appendix B):

where $$ \Re$$ indicates the real part of a complex number. A similar procedure can be applied to derive the price of a put option.

B. Levy inversion formula

Let $$ F(x)$$ be the distribution function of a random variable. The Levy inversion formula involves computing $$ F(x)$$ given the characteristic function, $$ g(\phi)=\int_{-\infty}^{\infty}\exp(\mathrm{i}\phi x) dF(x)$$ . To begin with, let us bring the following mathematical results:

where $$ \operatorname{sgn}(x)=-1$$ if x<0; 0 if x=0; and 1 if x>0. Let us derive the following relation:

where $$ \Re$$ and $$ \Im$$ denote the real and imaginary parts of a complex number, respectively. The distribution function is thus given by

The option pricing formula calls for the complementary distribution function, $$ \Pi(x)$$ :

C. Unscented Kalman filtering

Let $$ X_t=(v_t,u_t)$$ be the latent state vector. Suppose that the latent state vector is of the form:

where var$$ _t[\epsilon_{t+1}]=Q(X_t)$$ . Kalman filtering comprises two stages: the prediction and the update stages. The prediction stage involves computing one-step-ahead predictive mean and variance of X_t given $$ \hat{X}_t=E_t[X_t]$$ and $$ P_t=$$var$$ _t[X_t]$$ :

The update stage starts by defining a set of sigma points that match the predictive mean and variance of the latent vector. To be specific, a set of $$ 2L+1$$ sigma points and weights are selected based on the scaled unscented transformation of Julier (2002), where L is the dimension of the latent vector:

whose weights are given by

where $$ \xi=\alpha^2(L+\kappa)-L$$ and $$ \left( \sqrt{(L+\xi) P_{t+1\vert t}}\right)_i$$ is the ith column of the matrix square root. In our applications, we set $$ \alpha=0.01$$ , $$ \beta=2$$ , and $$ \kappa=0$$ .

Suppose that the observation equations are of the form:

where $$ H(X_t)$$ represents the pricing formula for futures and options and $$ \Omega$$ denotes the measurement errors. The predictive mean and variance of Y_t are computed using the sigma points and weights:

where $$ \hat{Y}_{t+1\vert t} =E_t[Y_{t+1}]$$ and $$ P_{t+1\vert t}^y =$$var$$ _t[Y_{t+1}]$$ .

Finally, we are able to update the conditional mean and variance of X_t:

where

The log likelihood function is then given by

where N_t is the dimension of the observation equations at time t. The log likelihood function is optimized via the Berndt et al. (1974) algorithm. The standard errors are computed by inverting the Hessian matrix at the optimal solution.

D. Computation of the option-implied volatility and skewness

The model-free, option-implied moments are obtained with the approach of Bakshi et al. (2003) with light modification. The original formula is based on the no-arbitrage relation that the underlying asset is tradable and so grows at a risk-free rate under the risk-neutral measure. That is, $$ E_t^{\text{Q}}(S_T)=S_t\exp(r_t(T-t))$$ , where S_t is the price of an underlying asset at time t. However, this relation no longer holds in the VIX derivatives market because the underlying VIX index is not tradable. Instead, we use the fact that a VIX futures price reflects the risk-neutral expectation of the VIX, $$ F(t,T)=E_t^{\text{Q}}(\text{VIX}_T)$$ , where $$ F(t,T)$$ is the time-t price of a VIX future with a maturity of T. The computation proceeds as follows.

Let $$ H_{1}(t,T),$$ $$ H_{2}(t,T),$$ and $$ H_{3}(t,T)$$ denote the prices of three hypothetical securities that pay quadratic, cubic, and quartic payoffs, respectively. These prices are given by

$$\displaystyle H_{1}(t,T)$$ $$\displaystyle =$$ $$\displaystyle \int_{\text{VIX}_t}^{\infty }\frac{2\left( 1-\ln \left( \frac{K}{\text{VIX}_t}\right) \right) }{K^{2}}\widehat{C}(t,T ,K)dK$$

$$\displaystyle +\int_{0}^{\text{VIX}_t}\frac{2\left( 1+\ln \left( \frac{\text{VIX}_t}{K}\right) \right) }{K^{2}}\widehat{P}(t,T ,K)dK,$$ (D.1)

$$\displaystyle H_{2}(t,T)$$ $$\displaystyle =$$ $$\displaystyle \int_{\text{VIX}_t}^{\infty }\frac{6\ln \left( \frac{K}{\text{VIX}_t}\right) -3\left( \ln \left( \frac{K}{\text{VIX}_t}\right) \right) ^{2}}{K^{2}}\widehat{C}(t,T ,K)dK$$

$$\displaystyle -\int_{0}^{\text{VIX}_t}\frac{6\ln \left( \frac{\text{VIX}_t}{K}\right) +3\left( \ln \left( \frac{\text{VIX}_t}{K}\right) \right) ^{2}}{K^{2}}\widehat{P}(t,T ,K)dK,$$ (D.2)

and

$$\displaystyle H_{3}(t,T )$$ $$\displaystyle =$$ $$\displaystyle \int_{\text{VIX}_t}^{\infty }\frac{12\left( \ln \left( \frac{K}{\text{VIX}_t}\right) \right) ^{2}-4\left( \ln \left( \frac{K}{\text{VIX}_t}\right) \right) ^{3}}{K^{2}}\widehat{C}(t,T ,K)dK$$

$$\displaystyle +\int_{0}^{\text{VIX}_t}\frac{12\left( \ln \left( \frac{\text{VIX}_t}{K}\right) \right) ^{2}+4\left( \ln \left( \frac{\text{VIX}_t}{K}\right) \right) ^{3}}{K^{2}}\widehat{P}(t,T ,K)dK,$$ (D.3)

where $$ \widehat{C}(t,T,K)$$ and $$ \widehat{P}(t,T,K)$$ are the market prices of OTM VIX call and put options with a maturity of T and a strike price of K. The implied volatility and skewness are then computed by

where IV$$ (t,T)$$ is annualized and

$$\displaystyle \mu(t,T )= \frac{F(t,T)}{\text{VIX}_t} -1-\frac{\exp (r_t(T-t))}{2}H_{1}(t,T)-\frac{\exp (r_t(T-t) )}{6}H_{2}(t,T )-\frac{\exp (r_t(T-t))}{24}H_{3}(t,T).$$ (D.5)

In practice, the computation of the implied volatility and skewness is prone to truncation and discretization errors, as reported by Jiang & Tian (2007). To reduce such errors, we interpolate or extrapolate option prices on a fine grid of varying strike prices following the approach of Carr & Wu (2009). First, for a given date and a given maturity, we require at least two observations for both puts and calls. We then generate a grid of strike prices with one-point increments, interpolate Black-Scholes implied volatility at each point, and translate the interpolated implied volatility into the option price. In addition, an extrapolation is used for grid points beyond the last observed strike price.

E. AVV model with double exponential jumps

Kou (2002) and Kou & Wang (2004) develop a double exponential jump diffusion model in which both upward and downward jumps are driven by a single compound Poisson process. To be specific, we introduce the dynamics for v_t as follows:

where $$ N_{t}^{\text{Q}}$$ denotes a risk-neutral Poisson process driving jumps with the intensity $$ \lambda$$ and $$ J^{\text{Q}}$$ has a probability density function that takes $$ \tfrac{p}{\delta_+}\exp(-x/\delta_+)$$ if $$ x>0$$ and $$ \tfrac{1-p}{\vert\delta_-\vert}\exp(-x/\delta_-)$$ if $$ x<0$$ . The coefficients of the cumulant generating function of this model satisfy the following system of ODEs:

where the boundary conditions are given as $$ \alpha(0)=0$$ , $$ \beta_v(0)=\mathrm{i}\phi$$ , $$ \beta_u(0)=0$$ , and $$ \beta_w(0)=0$$ .

Note that the AVV model with double exponential jumps is different from the AVV-AJ model where upward and downward jumps are driven by independent compound Poisson processes with each having its own jump intensity and jump-size distribution. In the AVV-AJ model, upward and downward jumps have distinct jump-size distributions so a combination of upward and downward jumps cannot collapse to a single compound Poisson process as in the double exponential jumps.

Table 1: Summary of model specifications

Model	Description	Constraints
SVV	Symmetric volatility, no jumps	$$ \rho=0$$ , $$ \lambda_+=0$$ , and $$ \lambda_-=0$$
AVV	Asymmetric volatility, no jumps	$$ \lambda_+=0$$ and $$ \lambda_-=0$$
AVV-UJ	Asymmetric volatility, upward jumps only	$$ \lambda_-=0$$
AVV-AJ	Asymmetric volatility, asymmetric jumps	Not applicable

Table 2: Summary statistics for VIX futures and options

The moneyness is defined as $$ d=K/F(t,T)$$ , where K is the strike price and $$ F(t,T)$$ is the futures price with a strike price of K and a maturity of T. Short-term contracts are those with no more than three months to maturity, and long-term contracts are those with more than three months to maturity. BSIV stands for Black-Scholes implied volatility. The futures data come from Thomson Reuters, Datastream, while the options data are from OptionMetrics, Ivy DB.

Months to maturity

	Short	Long	Total
All futures No. of futures	4,578	4,752	9,330
All futures Average price	24.55	25.21	24.89
All OTM options No. of options	49,992	52,533	102,525
All OTM Average price	1.25	1.86	1.56
All OTM Average BSIV	1.05	0.84	0.94
All OTM Average vega	2.58	4.60	3.61
OTM puts No. of options	13,758	13,949	27,707
OTM puts Average price	1.33	1.90	1.62
OTM puts Average BSIV	0.68	0.52	0.60
OTM puts Average vega	2.86	5.01	3.94
OTM calls No. of options	36,403	38,696	75,099
OTM calls Average price	1.23	1.85	1.55
OTM calls Average BSIV	1.20	0.95	1.07
OTM calls Average vega	2.47	4.45	3.49
Deep OTM puts (d < 0.85) No. of options	6,382	7,639	14,021
Deep OTM puts (d < 0.85) Average price	0.75	1.14	0.96
Deep OTM puts (d < 0.85) Average BSIV	0.65	0.48	0.56
Deep OTM puts (d < 0.85) Average vega	2.61	4.38	3.58
Moderate OTM puts (0.85 < d < 1.0) No. of options	7,384	6,312	13,696
Moderate OTM puts (0.85 < d < 1.0) Average price	1.83	2.82	2.29
Moderate OTM puts (0.85 < d < 1.0) Average BSIV	0.70	0.57	0.64
Moderate OTM puts (0.85 < d < 1.0) Average vega	3.08	5.76	4.31
Moderate OTM calls (1.0 < d < 1.3) No. of options	17,424	17,433	34,857
Moderate OTM calls (1.0 < d < 1.3) Average price	1.77	2.68	2.23
Moderate OTM calls (1.0 < d < 1.3) Average BSIV	1.16	0.94	1.05
Moderate OTM calls (1.0 < d < 1.3) Average vega	2.85	5.14	4.00
Deep OTM calls (d > 1.3) No. of options	18,987	21,275	40,262
Deep OTM calls (d > 1.3) Average price	0.73	1.16	0.96
Deep OTM calls (d > 1.3) Average BSIV	1.23	0.96	1.09
Deep OTM calls (d > 1.3) Average vega	2.12	3.89	3.06

Table 3: Parameter estimates: In-sample

This table shows the parameter estimates that are obtained using the full sample period from July 2006 through January 2013. AIC and SIC stand for Akaike and Schwarz information criteria, respectively. Standard errors are in parentheses.

	MS	SVV	AVV	AVV-UJ	AVV-AJ
$$ \kappa_v$$	8.956	6.384	7.393	6.488	6.576
$$ \kappa_v$$ , standard error	(0.008)	(0.006)	(0.006)	(0.005)	(0.006)
$$ \rho$$			0.870	0.422	0.794
$$ \rho$$ , standard error			(0.004)	(0.005)	(0.009)
$$ \kappa_u$$	0.386	0.324	0.319	0.256	0.258
$$ \kappa_u$$ , standard error	(0.001)	(0.002)	(0.001)	(0.001)	(0.001)
$$ \overline{u}$$	2.979	2.877	2.997	3.073	3.106
$$ \overline{u}$$ , standard error	(0.001)	(0.002)	(0.002)	(0.002)	(0.003)
$$ \sigma_u$$	0.088	0.442	0.419	0.282	0.293
$$ \sigma_u$$ , standard error	(0.004)	(0.001)	(0.002)	(0.003)	(0.003)
$$ \eta_u$$	-0.039	-0.003	-0.022	-0.019	-0.024
$$ \eta_u$$ , standard error	(0.013)	(0.013)	(0.027)	(0.016)	(0.017)
$$ \kappa_w$$		5.506	1.553	1.050	0.847
$$ \kappa_w$$ , standard error		(0.015)	(0.008)	(0.008)	(0.008)
$$ \overline{w}$$		1.613	1.629	2.137	1.956
$$ \overline{w}$$ , standard error		(0.003)	(0.012)	(0.019)	(0.027)
$$ \sigma_w$$		7.271	2.610	2.379	1.976
$$ \sigma_w$$ , standard error		(0.011)	(0.009)	(0.013)	(0.012)
$$ \eta_w$$		-28.429	-2.184	-2.926	-2.150
$$ \eta_w$$ , standard error		(1.767)	(0.642)	(1.036)	(0.869)
$$ \lambda_z$$	3.358
$$ \lambda_z$$ , standard error	(0.013)
$$ \delta_z$$	3.391
$$ \delta_z$$ , standard error	(0.007)
$$ \lambda_+$$				2.664	2.682
$$ \lambda_+$$ , standard error				(0.039)	(0.036)
$$ \delta_+$$				0.267	0.266
$$ \delta_+$$ , standard error				(0.001)	(0.001)
$$ \lambda_-$$					2.042
$$ \lambda_-$$ , standard error					(0.152)
$$ \delta_-$$					-0.217
$$ \delta_-$$ , standard error					(0.009)
$$ \sigma_e$$	0.080	0.082	0.078	0.072	0.071
$$ \sigma_e$$ , standard error	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)
$$ \sigma_f$$	0.031	0.031	0.030	0.030	0.029
$$ \sigma_f$$ , standard error	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)
log(L)	8,071	8,600	15,803	24,642	24,826
AIC	-16,124	-17,177	-31,582	-49,255	-49,620
SIC	-16,075	-17,118	-31,517	-49,179	-49,534

Table 4: Parameter estimates: Out-of-sample

This table shows the parameter estimates that are obtained using a subsample of four years from July 2006 through June 2010. This parameter set is used for an out-of-sample test spanning July 2010 through January 2013. AIC and SIC stand for Akaike and Schwarz information criteria, respectively. Standard errors are in parentheses.

	MS	SVV	AVV	AVV-UJ	AVV-AJ
$$ \kappa_v$$	8.956	6.384	7.393	6.488	6.576
$$ \kappa_v$$ , standard error	(0.008)	(0.006)	(0.006)	(0.005)	(0.006)
$$ \rho$$			0.870	0.422	0.794
$$ \rho$$ , standard error			(0.004)	(0.005)	(0.009)
$$ \kappa_u$$	0.386	0.324	0.319	0.256	0.258
$$ \kappa_u$$ , standard error	(0.001)	(0.002)	(0.001)	(0.001)	(0.001)
$$ \overline{u}$$	2.979	2.877	2.997	3.073	3.106
$$ \overline{u}$$ , standard error	(0.001)	(0.002)	(0.002)	(0.002)	(0.003)
$$ \sigma_u$$	0.088	0.442	0.419	0.282	0.293
$$ \sigma_u$$ , standard error	(0.004)	(0.001)	(0.002)	(0.003)	(0.003)
$$ \eta_u$$	-0.039	-0.003	-0.022	-0.019	-0.024
$$ \eta_u$$ , standard error	(0.013)	(0.013)	(0.027)	(0.016)	(0.017)
$$ \kappa_w$$		5.506	1.553	1.050	0.847
$$ \kappa_w$$ , standard error		(0.015)	(0.008)	(0.008)	(0.008)
$$ \overline{w}$$		1.613	1.629	2.137	1.956
$$ \overline{w}$$ , standard error		(0.003)	(0.012)	(0.019)	(0.027)
$$ \sigma_w$$		7.271	2.610	2.379	1.976
$$ \sigma_w$$ , standard error		(0.011)	(0.009)	(0.013)	(0.012)
$$ \eta_w$$		-28.429	-2.184	-2.926	-2.150
$$ \eta_w$$ , standard error		(1.767)	(0.642)	(1.036)	(0.869)
$$ \lambda_z$$	3.358
$$ \lambda_z$$ , standard error	(0.013)
$$ \delta_z$$	3.391
$$ \delta_z$$ , standard error	(0.007)
$$ \lambda_+$$				2.664	2.682
$$ \lambda_+$$ , standard error				(0.039)	(0.036)
$$ \delta_+$$				0.267	0.266
$$ \delta_+$$ , standard error				(0.001)	(0.001)
$$ \lambda_-$$					2.042
$$ \lambda_-$$ , standard error					(0.152)
$$ \delta_-$$					-0.217
$$ \delta_-$$ , standard error					(0.009)
$$ \sigma_e$$	0.080	0.082	0.078	0.072	0.071
$$ \sigma_e$$ , standard error	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)
$$ \sigma_f$$	0.031	0.031	0.030	0.030	0.029
$$ \sigma_f$$ , standard error	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)
log(L)	8,071	8,600	15,803	24,642	24,826
AIC	-16,124	-17,177	-31,582	-49,255	-49,620
SIC	-16,075	-17,118	-31,517	-49,179	-49,534

Table 5: Pricing performance across different models: In-sample

This table shows the in-sample performance metrics across the different models. The metric RMSE$$ _{i}$$ stands for the root mean squared error of a model i as given by Equation (16). The metric $$ \Delta$$   RMSE$$ _{i\vert j}$$ measures the pricing performance difference of a model i relative to a model j, which is defined as $$ \Delta$$   RMSE$$ _{i\vert j}=100 \times \log ($$   RMSE$$ _{i}/$$RMSE$$ _{j})$$ . A negative (positive) value of $$ \Delta$$   RMSE$$ _{i\vert j}$$ means that the pricing performance of the model i is better (worse) than that of the model j by a percentage of that value. The moneyness is defined as $$ d=K/F(t,T)$$ , where K is the strike price and $$ F(t,T)$$ is the futures price with a strike price of K and a maturity of T. Short-term contracts are those with no more than three months to maturity, and long-term contracts are those with more than three months to maturity.

Panel A: All futures

	RMSE$$ _{i}$$ Short	RMSE$$ _{i}$$ long	RMSE$$ _{i}$$ Total	$$ \Delta$$   RMSE$$ _{i\vert\text{SVV}}$$ Short	$$ \Delta$$   RMSE$$ _{i\vert\text{SVV}}$$ Long	$$ \Delta$$   RMSE$$ _{i\vert\text{SVV}}$$ Total	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV}}$$ Short	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV}}$$ Long	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV}}$$ Total	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV-UJ}}$$ Short	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV-UJ}}$$ Long	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV-UJ}}$$ Total
MS	0.1867	0.1839	0.1853	-5.22	8.71	1.18	6.02	4.76	5.39	1.49	10.91	5.93
SVV	0.1916	0.1761	0.1842	0.00	0.00	0.00	11.24	-3.95	4.22	6.71	2.21	4.75
AVV	0.1812	0.1796	0.1804	-11.24	3.95	-4.22	0.00	0.00	0.00	-4.53	6.15	0.53
AVV-UJ	0.1853	0.1741	0.1799	-6.71	-2.21	-4.75	4.53	-6.15	-0.53	0.00	0.00	0.00
AVV-AJ	0.1836	0.1744	0.1791	-8.55	-1.88	-5.60	2.70	-5.83	-1.39	-1.83	0.32	-0.86

Panel B: All options

	RMSE$$ _{i}$$ Short	RMSE$$ _{i}$$ long	RMSE$$ _{i}$$ Total	$$ \Delta$$   RMSE$$ _{i\vert\text{SVV}}$$ Short	$$ \Delta$$   RMSE$$ _{i\vert\text{SVV}}$$ Long	$$ \Delta$$   RMSE$$ _{i\vert\text{SVV}}$$ Total	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV}}$$ Short	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV}}$$ Long	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV}}$$ Total	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV-UJ}}$$ Short	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV-UJ}}$$ Long	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV-UJ}}$$ Total
MS	0.1218	0.0829	0.1037	8.25	13.81	10.00	13.43	21.30	15.87	20.83	30.61	23.83
SVV	0.1122	0.0722	0.0939	0.00	0.00	0.00	5.18	7.48	5.87	12.59	16.80	13.83
AVV	0.1065	0.0670	0.0885	-5.18	-7.48	-5.87	0.00	0.00	0.00	7.41	9.32	7.96
AVV-UJ	0.0989	0.0611	0.0817	-12.59	-16.80	-13.83	-7.41	-9.32	-7.96	0.00	0.00	0.00
AVV-AJ	0.0985	0.0609	0.0814	-13.01	-17.10	-14.21	-7.82	-9.62	-8.35	-0.42	-0.30	-0.38

Panel C: OTM puts

	RMSE$$ _{i}$$ Short	RMSE$$ _{i}$$ long	RMSE$$ _{i}$$ Total	$$ \Delta$$   RMSE$$ _{i\vert\text{SVV}}$$ Short	$$ \Delta$$   RMSE$$ _{i\vert\text{SVV}}$$ Long	$$ \Delta$$   RMSE$$ _{i\vert\text{SVV}}$$ Total	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV}}$$ Short	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV}}$$ Long	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV}}$$ Total	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV-UJ}}$$ Short	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV-UJ}}$$ Long	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV-UJ}}$$ Total
MS	0.1331	0.0785	0.1091	-7.91	0.35	-5.88	7.87	11.20	8.72	5.21	12.98	7.12
SVV	0.1440	0.0782	0.1157	0.00	0.00	0.00	15.78	10.85	14.60	13.12	12.63	13.01
AVV	0.1230	0.0702	0.1000	-15.78	-10.85	-14.60	0.00	0.00	0.00	-2.66	1.78	-1.60
AVV-UJ	0.1263	0.0689	0.1016	-13.12	-12.63	-13.01	2.66	-1.78	1.60	0.00	0.00	0.00
AVV-AJ	0.1245	0.0685	0.1003	-14.61	-13.23	-14.29	1.17	-2.37	0.32	-1.49	-0.60	-1.28

Panel D: OTM calls

	RMSE$$ _{i}$$ Short	RMSE$$ _{i}$$ long	RMSE$$ _{i}$$ Total	$$ \Delta$$   RMSE$$ _{i\vert\text{SVV}}$$ Short	$$ \Delta$$   RMSE$$ _{i\vert\text{SVV}}$$ Long	$$ \Delta$$   RMSE$$ _{i\vert\text{SVV}}$$ Total	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV}}$$ Short	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV}}$$ Long	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV}}$$ Total	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV-UJ}}$$ Short	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV-UJ}}$$ Long	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV-UJ}}$$ Total
MS	0.1183	0.0846	0.1023	18.00	18.93	18.33	16.56	24.92	19.35	30.59	37.72	32.98
SVV	0.0988	0.0700	0.0852	0.00	0.00	0.00	-1.44	5.99	1.02	12.59	18.79	14.66
AVV	0.1002	0.0659	0.0843	1.44	-5.99	-1.02	0.00	0.00	0.00	14.03	12.80	13.64
AVV-UJ	0.0871	0.0580	0.0736	-12.59	-18.79	-14.66	-14.03	-12.80	-13.64	0.00	0.00	0.00
AVV-AJ	0.0874	0.0579	0.0737	-12.22	-18.94	-14.45	-13.66	-12.94	-13.43	0.37	-0.15	0.21

Panel E: Deep OTM puts ($$ d<0.85$$ )

	RMSE$$ _{i}$$ Short	RMSE$$ _{i}$$ long	RMSE$$ _{i}$$ Total	$$ \Delta$$   RMSE$$ _{i\vert\text{SVV}}$$ Short	$$ \Delta$$   RMSE$$ _{i\vert\text{SVV}}$$ Long	$$ \Delta$$   RMSE$$ _{i\vert\text{SVV}}$$ Total	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV}}$$ Short	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV}}$$ Long	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV}}$$ Total	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV-UJ}}$$ Short	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV-UJ}}$$ Long	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV-UJ}}$$ Total
MS	0.1152	0.0695	0.0932	-7.38	-4.78	-6.61	19.07	9.68	16.03	18.07	10.69	15.72
SVV	0.1241	0.0729	0.0995	0.00	0.00	0.00	26.45	14.46	22.64	25.46	15.48	22.33
AVV	0.0952	0.0631	0.0794	-26.45	-14.46	-22.64	0.00	0.00	0.00	-0.99	1.01	-0.31
AVV-UJ	0.0962	0.0625	0.0796	-25.46	-15.48	-22.33	0.99	-1.01	0.31	0.00	0.00	0.00
AVV-AJ	0.0923	0.0617	0.0772	-29.54	-16.69	-25.43	-3.09	-2.23	-2.79	-4.09	-1.21	-3.10

Panel F: Moderate OTM puts ( $$ 0.85<d<1.0$$ )

	RMSE$$ _{i}$$ Short	RMSE$$ _{i}$$ long	RMSE$$ _{i}$$ Total	$$ \Delta$$   RMSE$$ _{i\vert\text{SVV}}$$ Short	$$ \Delta$$   RMSE$$ _{i\vert\text{SVV}}$$ Long	$$ \Delta$$   RMSE$$ _{i\vert\text{SVV}}$$ Total	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV}}$$ Short	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV}}$$ Long	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV}}$$ Total	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV-UJ}}$$ Short	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV-UJ}}$$ Long	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV-UJ}}$$ Total
MS	0.1467	0.0881	0.1232	-8.19	4.59	-5.46	2.79	12.38	4.89	-0.50	14.77	2.70
SVV	0.1592	0.0842	0.1301	0.00	0.00	0.00	10.98	7.78	10.35	7.69	10.18	8.16
AVV	0.1427	0.0779	0.1173	-10.98	-7.78	-10.35	0.00	0.00	0.00	-3.29	2.39	-2.19
AVV-UJ	0.1475	0.0760	0.1199	-7.69	-10.18	-8.16	3.29	-2.39	2.19	0.00	0.00	0.00
AVV-AJ	0.1466	0.0760	0.1194	-8.25	-10.28	-8.64	2.72	-2.49	1.71	-0.57	-0.10	-0.48

Panel G: Moderate OTM calls ($$ 1.0<d<1.3$$ )

	RMSE$$ _{i}$$ Short	RMSE$$ _{i}$$ long	RMSE$$ _{i}$$ Total	$$ \Delta$$   RMSE$$ _{i\vert\text{SVV}}$$ Short	$$ \Delta$$   RMSE$$ _{i\vert\text{SVV}}$$ Long	$$ \Delta$$   RMSE$$ _{i\vert\text{SVV}}$$ Total	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV}}$$ Short	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV}}$$ Long	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV}}$$ Total	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV-UJ}}$$ Short	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV-UJ}}$$ Long	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV-UJ}}$$ Total
MS	0.1260	0.1017	0.1145	10.46	24.75	15.62	8.73	30.78	16.32	17.27	45.14	26.50
SVV	0.1135	0.0794	0.0979	0.00	0.00	0.00	-1.73	6.03	0.69	6.81	20.39	10.88
AVV	0.1154	0.0748	0.0972	1.73	-6.03	-0.69	0.00	0.00	0.00	8.54	14.36	10.19
AVV-UJ	0.1060	0.0648	0.0878	-6.81	-20.39	-10.88	-8.54	-14.36	-10.19	0.00	0.00	0.00
AVV-AJ	0.1066	0.0647	0.0882	-6.27	-20.42	-10.50	-8.00	-14.39	-9.81	0.53	-0.03	0.38

Panel H: Deep OTM calls ($$ d>1.3$$ )

	RMSE$$ _{i}$$ Short	RMSE$$ _{i}$$ long	RMSE$$ _{i}$$ Total	$$ \Delta$$   RMSE$$ _{i\vert\text{SVV}}$$ Short	$$ \Delta$$   RMSE$$ _{i\vert\text{SVV}}$$ Long	$$ \Delta$$   RMSE$$ _{i\vert\text{SVV}}$$ Total	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV}}$$ Short	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV}}$$ Long	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV}}$$ Total	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV-UJ}}$$ Short	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV-UJ}}$$ Long	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV-UJ}}$$ Total
MS	0.1107	0.0674	0.0904	28.76	9.60	22.34	27.82	15.54	23.90	53.14	26.26	43.67
SVV	0.0831	0.0612	0.0723	0.00	0.00	0.00	-0.93	5.94	1.56	24.39	16.66	21.32
AVV	0.0838	0.0577	0.0712	0.93	-5.94	-1.56	0.00	0.00	0.00	25.32	10.72	19.77
AVV-UJ	0.0651	0.0518	0.0584	-24.39	-16.66	-21.32	-25.32	-10.72	-19.77	0.00	0.00	0.00
AVV-AJ	0.0651	0.0516	0.0584	-24.42	-16.96	-21.47	-25.36	-11.02	-19.91	-0.03	-0.30	-0.14

Table 6: Pricing performance across different models: Out-of-sample

This table shows the out-of-sample performance metrics across the different models. The metric RMSE$$ _{i}$$ stands for the root mean squared error of a model i as given by Equation (16). The metric $$ \Delta$$   RMSE$$ _{i\vert j}$$ measures the pricing performance difference of a model i relative to a model j, which is defined as $$ \Delta$$   RMSE$$ _{i\vert j}=100 \times \log ($$   RMSE$$ _{i}/$$RMSE$$ _{j})$$ . A negative (positive) value of $$ \Delta$$   RMSE$$ _{i\vert j}$$ means that the pricing performance of the model i is better (worse) than that of the model j by a percentage of that value. The moneyness is defined as $$ d=K/F(t,T)$$ , where K is the strike price and $$ F(t,T)$$ is the futures price with a strike price of K and a maturity of T. Short-term contracts are those with no more than three months to maturity, and long-term contracts are those with more than three months to maturity.

Panel A: All futures

	RMSE$$ _{i}$$ Short	RMSE$$ _{i}$$ long	RMSE$$ _{i}$$ Total	$$ \Delta$$   RMSE$$ _{i\vert\text{SVV}}$$ Short	$$ \Delta$$   RMSE$$ _{i\vert\text{SVV}}$$ Long	$$ \Delta$$   RMSE$$ _{i\vert\text{SVV}}$$ Total	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV}}$$ Short	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV}}$$ Long	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV}}$$ Total	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV-UJ}}$$ Short	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV-UJ}}$$ Long	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV-UJ}}$$ Total
MS	0.2415	0.2242	0.2332	1.12	16.03	7.02	14.62	7.88	11.60	13.26	17.64	15.10
SVV	0.2402	0.2069	0.2252	0.00	0.00	0.00	13.50	-8.15	4.58	12.14	1.61	8.08
AVV	0.2245	0.2155	0.2201	-13.50	8.15	-4.58	0.00	0.00	0.00	-1.36	9.76	3.51
AVV-UJ	0.2261	0.2052	0.2163	-12.14	-1.61	-8.08	1.36	-9.76	-3.51	0.00	0.00	0.00
AVV-AJ	0.2251	0.2073	0.2166	-13.02	0.34	-7.78	0.48	-7.81	-3.21	-0.89	1.95	0.30

Panel B: All options

	RMSE$$ _{i}$$ Short	RMSE$$ _{i}$$ long	RMSE$$ _{i}$$ Total	$$ \Delta$$   RMSE$$ _{i\vert\text{SVV}}$$ Short	$$ \Delta$$   RMSE$$ _{i\vert\text{SVV}}$$ Long	$$ \Delta$$   RMSE$$ _{i\vert\text{SVV}}$$ Total	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV}}$$ Short	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV}}$$ Long	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV}}$$ Total	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV-UJ}}$$ Short	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV-UJ}}$$ Long	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV-UJ}}$$ Total
MS	0.1625	0.1320	0.1475	15.87	28.46	20.71	19.95	31.97	24.58	24.71	38.34	29.91
SVV	0.1386	0.0993	0.1199	0.00	0.00	0.00	4.07	3.51	3.87	8.84	9.88	9.20
AVV	0.1331	0.0959	0.1154	-4.07	-3.51	-3.87	0.00	0.00	0.00	4.76	6.37	5.33
AVV-UJ	0.1269	0.0900	0.1094	-8.84	-9.88	-9.20	-4.76	-6.37	-5.33	0.00	0.00	0.00
AVV-AJ	0.1263	0.0911	0.1095	-9.30	-8.56	-9.04	-5.23	-5.05	-5.16	-0.47	1.32	0.16

Panel C: OTM puts

	RMSE$$ _{i}$$ Short	RMSE$$ _{i}$$ long	RMSE$$ _{i}$$ Total	$$ \Delta$$   RMSE$$ _{i\vert\text{SVV}}$$ Short	$$ \Delta$$   RMSE$$ _{i\vert\text{SVV}}$$ Long	$$ \Delta$$   RMSE$$ _{i\vert\text{SVV}}$$ Total	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV}}$$ Short	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV}}$$ Long	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV}}$$ Total	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV-UJ}}$$ Short	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV-UJ}}$$ Long	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV-UJ}}$$ Total
MS	0.1728	0.0794	0.1342	8.05	-1.02	6.33	20.56	6.49	17.78	21.87	10.24	19.62
SVV	0.1594	0.0802	0.1260	0.00	0.00	0.00	12.51	7.51	11.45	13.82	11.25	13.29
AVV	0.1407	0.0744	0.1123	-12.51	-7.51	-11.45	0.00	0.00	0.00	1.31	3.74	1.84
AVV-UJ	0.1388	0.0716	0.1103	-13.82	-11.25	-13.29	-1.31	-3.74	-1.84	0.00	0.00	0.00
AVV-AJ	0.1361	0.0719	0.1087	-15.78	-10.93	-14.76	-3.27	-3.42	-3.30	-1.96	0.32	-1.47

Panel D: OTM calls

	RMSE$$ _{i}$$ Short	RMSE$$ _{i}$$ long	RMSE$$ _{i}$$ Total	$$ \Delta$$   RMSE$$ _{i\vert\text{SVV}}$$ Short	$$ \Delta$$   RMSE$$ _{i\vert\text{SVV}}$$ Long	$$ \Delta$$   RMSE$$ _{i\vert\text{SVV}}$$ Total	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV}}$$ Short	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV}}$$ Long	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV}}$$ Total	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV-UJ}}$$ Short	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV-UJ}}$$ Long	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV-UJ}}$$ Total
MS	0.1612	0.1440	0.1525	18.53	32.18	24.41	19.77	35.01	26.27	25.69	41.84	32.55
SVV	0.1339	0.1044	0.1194	0.00	0.00	0.00	1.24	2.83	1.86	7.16	9.66	8.14
AVV	0.1323	0.1015	0.1172	-1.24	-2.83	-1.86	0.00	0.00	0.00	5.93	6.83	6.28
AVV-UJ	0.1246	0.0948	0.1101	-7.16	-9.66	-8.14	-5.93	-6.83	-6.28	0.00	0.00	0.00
AVV-AJ	0.1247	0.0962	0.1108	-7.10	-8.17	-7.52	-5.86	-5.35	-5.66	0.06	1.49	0.62

Panel E: Deep OTM puts ($$ d<0.85$$ )

	RMSE$$ _{i}$$ Short	RMSE$$ _{i}$$ long	RMSE$$ _{i}$$ Total	$$ \Delta$$   RMSE$$ _{i\vert\text{SVV}}$$ Short	$$ \Delta$$   RMSE$$ _{i\vert\text{SVV}}$$ Long	$$ \Delta$$   RMSE$$ _{i\vert\text{SVV}}$$ Total	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV}}$$ Short	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV}}$$ Long	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV}}$$ Total	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV-UJ}}$$ Short	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV-UJ}}$$ Long	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV-UJ}}$$ Total
MS	0.0896	0.0628	0.0754	-13.83	-7.55	-11.41	3.23	0.46	2.11	7.15	2.71	5.33
SVV	0.1029	0.0678	0.0845	0.00	0.00	0.00	17.05	8.01	13.52	20.98	10.25	16.74
AVV	0.0867	0.0626	0.0738	-17.05	-8.01	-13.52	0.00	0.00	0.00	3.92	2.24	3.22
AVV-UJ	0.0834	0.0612	0.0715	-20.98	-10.25	-16.74	-3.92	-2.24	-3.22	0.00	0.00	0.00
AVV-AJ	0.0803	0.0613	0.0700	-24.78	-10.12	-18.84	-7.72	-2.10	-5.33	-3.80	0.14	-2.10

Panel F: Moderate OTM puts ( $$ 0.85<d<1.0$$ )

	RMSE$$ _{i}$$ Short	RMSE$$ _{i}$$ long	RMSE$$ _{i}$$ Total	$$ \Delta$$   RMSE$$ _{i\vert\text{SVV}}$$ Short	$$ \Delta$$   RMSE$$ _{i\vert\text{SVV}}$$ Long	$$ \Delta$$   RMSE$$ _{i\vert\text{SVV}}$$ Total	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV}}$$ Short	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV}}$$ Long	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV}}$$ Total	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV-UJ}}$$ Short	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV-UJ}}$$ Long	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV-UJ}}$$ Total
MS	0.2309	0.1079	0.1924	12.72	4.69	11.66	24.05	11.70	22.36	24.74	16.94	23.71
SVV	0.2033	0.1030	0.1712	0.00	0.00	0.00	11.33	7.01	10.69	12.02	12.25	12.05
AVV	0.1815	0.0960	0.1539	-11.33	-7.01	-10.69	0.00	0.00	0.00	0.69	5.24	1.36
AVV-UJ	0.1803	0.0911	0.1518	-12.02	-12.25	-12.05	-0.69	-5.24	-1.36	0.00	0.00	0.00
AVV-AJ	0.1775	0.0916	0.1499	-13.56	-11.74	-13.30	-2.24	-4.73	-2.61	-1.55	0.51	-1.25

Panel G: Moderate OTM calls ($$ 1.0<d<1.3$$ )

	RMSE$$ _{i}$$ Short	RMSE$$ _{i}$$ long	RMSE$$ _{i}$$ Total	$$ \Delta$$   RMSE$$ _{i\vert\text{SVV}}$$ Short	$$ \Delta$$   RMSE$$ _{i\vert\text{SVV}}$$ Long	$$ \Delta$$   RMSE$$ _{i\vert\text{SVV}}$$ Total	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV}}$$ Short	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV}}$$ Long	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV}}$$ Total	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV-UJ}}$$ Short	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV-UJ}}$$ Long	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV-UJ}}$$ Total
MS	0.1817	0.1976	0.1897	-0.07	32.47	14.69	0.10	35.70	15.98	4.78	46.37	22.74
SVV	0.1819	0.1428	0.1638	0.00	0.00	0.00	0.16	3.23	1.29	4.85	13.90	8.05
AVV	0.1816	0.1383	0.1617	-0.16	-3.23	-1.29	0.00	0.00	0.00	4.69	10.67	6.76
AVV-UJ	0.1732	0.1243	0.1511	-4.85	-13.90	-8.05	-4.69	-10.67	-6.76	0.00	0.00	0.00
AVV-AJ	0.1733	0.1266	0.1521	-4.85	-12.07	-7.43	-4.68	-8.83	-6.14	0.01	1.84	0.62

Panel H: Deep OTM calls ($$ d>1.3$$ )

	RMSE$$ _{i}$$ Short	RMSE$$ _{i}$$ long	RMSE$$ _{i}$$ Total	$$ \Delta$$   RMSE$$ _{i\vert\text{SVV}}$$ Short	$$ \Delta$$   RMSE$$ _{i\vert\text{SVV}}$$ Long	$$ \Delta$$   RMSE$$ _{i\vert\text{SVV}}$$ Total	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV}}$$ Short	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV}}$$ Long	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV}}$$ Total	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV-UJ}}$$ Short	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV-UJ}}$$ Long	$$ \Delta$$   RMSE$$ _{i\vert\text{AVV-UJ}}$$ Total
MS	0.1448	0.0996	0.1225	53.73	31.53	44.65	58.67	33.44	48.17	69.36	32.68	53.06
SVV	0.0846	0.0727	0.0784	0.00	0.00	0.00	4.94	1.92	3.52	15.63	1.15	8.40
AVV	0.0805	0.0713	0.0757	-4.94	-1.92	-3.52	0.00	0.00	0.00	10.69	-0.77	4.88
AVV-UJ	0.0724	0.0718	0.0721	-15.63	-1.15	-8.40	-10.69	0.77	-4.88	0.00	0.00	0.00
AVV-AJ	0.0726	0.0724	0.0725	-15.33	-0.28	-7.80	-10.39	1.63	-4.28	0.29	0.86	0.60

Table 7: Pairwise model comparisons: In-sample

This table shows the Diebold & Mariano (2002) test statistics of the in-sample performance differences. A negative (positive) statistic in a cell (i,j) indicates that the model i outperforms (underperforms) the model j. The symbols *, **, and *** represent the statistical significance of the performance difference at the 10, 5, and 1 percent levels, respectively.

Panel A: Futures

	MS	SVV	AVV	AVV-UJ	AVV-AJ
MS	0.00	0.35	3.20***	1.96*	2.56**
SVV	-0.35	0.00	2.52**	2.18**	2.79***
AVV	-3.20***	-2.52**	0.00	-0.50	0.60
AVV-UJ	-1.96*	-2.18**	0.50	0.00	4.93***
AVV-AJ	-2.56**	-2.79***	-0.60	-4.93***	0.00

Panel B: Options

	MS	SVV	AVV	AVV-UJ	AVV-AJ
MS	0.00	6.44***	11.50***	11.82***	11.76***
SVV	-6.44***	0.00	7.08***	11.22***	10.72***
AVV	-11.50***	-7.08***	0.00	8.16***	8.44***
AVV-UJ	-11.82***	-11.22***	-8.16***	0.00	1.45
AVV-AJ	-11.76***	-10.72***	-8.44***	-1.45	0.00

Table 8: Pairwise model comparisons: Out-of-sample

This table shows the Diebold & Mariano (2002) test statistics of the out-of-sample performance differences. A negative (positive) statistic in a cell (i,j) indicates that the model i outperforms (underperforms) the model j. The symbols *, **, and *** represent the statistical significance of the performance difference at the 10, 5, and 1 percent levels, respectively.

Panel A: Futures

	MS	SVV	AVV	AVV-UJ	AVV-AJ
MS	0.00	7.58***	14.25***	17.46***	17.15***
SVV	-7.58***	0.00	8.17***	15.77***	15.51***
AVV	-14.25***	-8.17***	0.00	6.50***	8.27***
AVV-UJ	-17.46***	-15.77***	-6.50***	0.00	-1.16
AVV-AJ	-17.15***	-15.51***	-8.27***	1.16	0.00

Panel B: Options

	MS	SVV	AVV	AVV-UJ	AVV-AJ
MS	0.00	12.26***	14.01***	15.07***	14.96***
SVV	-12.26***	0.00	6.49***	11.87***	10.79***
AVV	-14.01***	-6.49***	0.00	11.71***	12.10***
AVV-UJ	-15.07***	-11.87***	-11.71***	0.00	-1.13
AVV-AJ	-14.96***	-10.79***	-12.10***	1.13	0.00

Figure 1: Scatter plot of VVIX log returns against VIX log returns.

Accessible version

Figure 2: Trading volume of VIX futures and options.

The top panel shows the numbers of futures contracts traded over time, and the bottom panel presents the dollar trading volumes for options contracts over time.

Accessible version

Figure 3: Time series of the VIX, the VVIX, and the option-implied skewness.

The VIX and the VVIX are provided by the CBOE, and the skewness is computed using the model-free approach of Bakshi et al. (2003) with a modification. The options data are from OptionMetrics, Ivy DB.

Accessible version

Figure 4: Filtered volatility states across different models.

Each panel displays a time series of the filtered volatility states that are obtained by applying an extended Kalman filter to the sample data from July 2006 through January 2013.

Accessible version

Figure 5: Term structure of volatility.

The top panel shows the scatter plot and the average term structure of the market-implied volatility, and the bottom panel shows the average term structures of the model-implied volatility.

Accessible version

Figure 6: Term structure of skewness.

The top panel shows the scatter plot and the average term structure of the market-implied skewness, and the bottom panel shows the average term structures of the model-implied skewness.

Accessible version

Footnotes

* We are very grateful for comments and suggestions from Yacine Aït-Sahalia, Michael Gordy, Jaideep Oberoi, and seminar participants at the IFSID Third Conference on Derivatives. This paper benefits from the excellent research assistance of Juliette Lu. Disclaimer: The analysis and conclusions set forth are those of the authors and do not indicate concurrence by the Board of Governors or other members of its research staff. Send correspondence to Yang-Ho Park, Risk Analysis Section, Board of Governors of the Federal Reserve System, 20th & C Streets, NW, Washington, D.C. 20551, Tel: (202) 452-3177; e-mail: yang-ho.park@frb.gov. Return to Text

1. See, for example, Whaley (2013) and Alexander & Korovilas (2012) as references on exchange-traded volatility products linked to the VIX. Return to Text

2. The VVIX is computed by applying the VIX formula to the VIX options market. Return to Text

3. This echoes a well-known finding that the negative skewness implicit in stock index options is partly associated with the leverage or volatility feedback effect, which indicates a negative correlation between stock index returns and volatility changes. Return to Text

4. The market's asymmetric response to shocks or news announcements has been documented by Andersen et al. (2007) and Bakshi et al. (2008), among others. Return to Text

5. See Gallant et al. (1999), Christoffersen et al. (2009), and Aït-Sahalia et al. (2015), among others, for the importance of a long-run volatility factor. Return to Text

6. Otherwise the conditional variance matrix of the model in Equation (1) will lose affine dependence on the state vector. Return to Text

7. Note that the ODEs are state-independent so we do not need to repeat running a numerical ODE solver on every date for a given parameter set. In other words, given a parameter set, we numerically solve the ODEs just once and save them in a storage space. The stored ODE solution can be repeatedly used to compute the characteristic function on each date throughout the sample period. Return to Text

8. One may want to replace the term $$ g(v_t,u_t,w_t,t,T;\phi=-\mathrm{i})$$ with the observed futures prices (see, for example, the footnote 10 of Mencía & Sentana (2012)). However, we do not take this approach for several reasons. First, we find that the observed VIX future prices have often departed from those implied by the put-call parity condition to a large extent, although this departure is not reported in this paper. Second, when the term $$ g(v_t,u_t,w_t,t,T;\phi=-\mathrm{i})$$ is replaced with the observed prices, we face the numerical issue that the correlation parameter is estimated to be virtually one. Lastly, using the observed future prices in lieu of $$ g(v_t,u_t,w_t,t,T;-\mathrm{i})$$ cannot guarantee the positivity of the option pricing formula in Equation (7). Return to Text

9. We also implement an extended Kalman filter and find that the results are robust to the type of a Kalman filter. Return to Text

10. Because the trading of VIX options was inactive in the very first few months, we exclude those months. Return to Text

11. This finding echoes Carr & Wu (2003), who find that the stock index options market has the almost same slope of the implied volatility smile across different times to maturity. To imitate this pattern, they propose a finite moment log stable process. Return to Text

12. To control for autocorrelations, we use an optimal number of lags following Newey & West (1994). Return to Text

13. This paper considers only two specifications of stochastic volatility of the VIX: the MS and SVV models. A further analysis of other specifications, such as infinite-activity jumps or a combination of diffusive and jump processes, would be an interesting topic for future research. Return to Text

^♣ This version is optimized for use by screen readers. Descriptions for all mathematical expressions are provided in LaTex format. A printable pdf version is available. Return to Text