Finance and Economics Discussion Series: 2014-58 Screen Reader version ♣

# A Tale of Two Option Markets:

# Pricing Kernels and Volatility Risk *

Zhaogang Song*
Federal Reserve Board
Dacheng Xiu*
University of Chicago
This Version: January, 2014

Abstract:

Using prices of both S&P 500 options and recently introduced VIX options, we study asset pricing implications of volatility risk. While pointing out the joint pricing kernel is not identified nonparametrically, we propose model-free estimates of marginal pricing kernels of the market return and volatility conditional on the VIX. We find that the pricing kernel of market return exhibits a decreasing pattern given either a high or low VIX level, whereas the unconditional estimates present a U-shape. Hence, stochastic volatility is the key state variable responsible for the U-shape puzzle documented in the literature. Finally, our estimates of the volatility pricing kernel feature a U-shape, implying that investors have high marginal utility in both high and low volatility states.

Keywords: Pricing kernel, volatility risk, VIX option, state-price density
JEL Classification: G12, G13

# 1 Introduction

In addition to the uncertainty of market returns, volatility risk has been well documented as an essential component of time-varying investment opportunities. Together with the preferences of economic agents, a priced volatility factor leads to a pricing kernel (or stochastic discount factor) which depends on both market returns and volatility. Nevertheless, because volatility is neither tradable nor observable, existing studies on pricing kernels either impose strong parametric restrictions, or ignore the unobservable volatility factor in nonparametric analysis. The pricing kernel estimates produced by these studies exhibit a puzzling U-shape as a function of market return, in conflict with a standard expected utility theory.

The lack of tradable and observable volatility has changed substantially since the introduction of the Volatility Index (VIX) in 1993 by the Chicago Board of Options Exchange (CBOE),1 and the introduction of VIX derivatives such as futures and options in 2004 and 2006, respectively. The VIX, derived from S&P 500 options as the square root of the expected average variance over the next 30 calendar days, provides investors with a direct measure of volatility; and VIX derivatives offer investors convenient instruments for trading on the volatility of S&P 500 index.2 As a result, the VIX is constantly exposed in the media spotlight, and VIX options have achieved huge liquidity and become the third most active contracts at CBOE as of October 2011.

Taking advantage of the S&P 500 and VIX option markets, we nonparametrically identify and estimate the marginal pricing kernel of market returns and volatility, which equals the ratio of state-price density (or risk-neutral density) to physical density. We show that information in the two option prices is fully captured by the two marginal state-price densities of market returns and volatility separately, whereas the joint state-price density and hence joint pricing kernel cannot be identified nonparametrically as a result of incomplete markets.3 We then provide nonparametric estimates of pricing kernels with respect to return and volatility respectively. Our estimates not only shed light on the puzzling U-shaped pricing kernel, but also provide new empirical stylized facts on the pricing kernel of volatility. In particular, we make several important findings regarding asset pricing implications of volatility.

First, our estimates of pricing kernels with respect to the market return show that stochastic volatility is the key state variable responsible for the "pricing kernel puzzle." More specifically, we find that a pricing kernel of market return conditional on either a high or low VIX level presents a decreasing pattern, whereas an unconditional pricing kernel (i.e. the one that ignores volatility) may become U-shaped. In fact, marginal utility (the pricing kernel up to a scaling factor) conditional on high volatility is above that conditional on low volatility, as low volatility signals a good investment opportunity and hence is preferred by investors. As a mixture of pricing kernel estimates conditional on different volatility levels, unconditional estimates can exhibit an increasing pattern over the high return region (right tail), where high volatility is prevalent. Our finding echoes the conclusions of parametric models in Chabi-Yo et al. (2008) and Christoffersen et al. (2010), which show that missing state variables in the pricing kernel may result in a U-shape. Without restricting the specification of pricing kernels, however, we show that including volatility as a state variable is the solution to this puzzle.

Second, we provide nonparametric estimates of pricing kernels with respect to volatility, for the first time to the best of our knowledge. Our estimates exhibit a pronounced U-shape conditional on either a high or low VIX, indicating that investors attach high marginal utility to payoffs received in both high and low future volatility states, regardless of today's volatility level. Bakshi et al. (2010) also document a U-shape for the unconditional volatility pricing kernel, but indirectly, by exploring the link between the monotonicity of pricing kernel and returns of VIX option portfolios. In contrast, we provide direct estimates of the conditional volatility pricing kernel by nonparametric methods, which provide further information about the shape and tail behavior of the pricing kernel. In particular, we find that the volatility pricing kernel is asymmetric, and the asymmetry conditional on a current high volatility is much stronger than that conditional on a low volatility. This finding implies that market investors price the volatility risk differently according to different scenarios of the economy, which presents new empirical regularities that need to be incorporated into models of volatility risk.

Finally, we evaluate the performance of our nonparametric estimator for in-sample fitting and out-of-sample forecasts against two alternative methods: the nonparametric approach of Aït-Sahalia & Lo (1998) without a volatility factor and a martingale approach commonly used by practitioners that simply predicts tomorrow's implied volatility by interpolating today's implied volatility surface. We find that our estimator outperforms both alternative methods for density and implied volatility forecasts, which again highlights the importance of conditioning on volatility.

Estimating pricing kernels from option prices is discussed in Aït-Sahalia & Lo (1998), Aït-Sahalia & Duarte (2003), Jackwerth (2000), and Rosenberg & Engle (2002), which ignore the volatility risk and discover a puzzling U-shape.4Hereafter, many studies have proposed different explanations for the U-shaped pricing kernel, including models with missing state variables in Chabi-Yo et al. (2008), Chabi-Yo (2011), and Christoffersen et al. (2010), and models with heterogeneous agents in Bakshi & Madan (2008) and Ziegler (2007). Our empirical study contributes to this literature by showing, without imposing any parametric restrictions, that volatility is the missing state variable responsible for the puzzle.

Our paper is also related to the large literature on models with volatility risk, including both reduced-form option pricing models, e.g. Bakshi et al. (1997), Bates (2000), Pan (2002), Eraker (2004), and Broadie et al. (2007), and equilibrium models, such as Bansal et al. (2012), Bollerslev et al. (2012), and Campbell et al. (2012). Unlike these studies, our framework does not depend on any parametric restrictions on volatility dynamics that may obscure the empirical characteristics of pricing kernels. Several recent studies have constructed model-free measures of risk-neutral volatility from S&P 500 options, e.g. Bakshi & Kapadia (2003), Bollerslev et al. (2009), Carr & Wu (2009), and Todorov (2010), and compared them with measures of realized volatility. Their focuses are on the sign, time variation, and return predictability of variance risk premium, which only relates to the conditional mean of variance distributions under different measures. In contrast, we recover the entire volatility pricing kernel.

Methodologically, our paper is also related to Boes et al. (2007) and Li & Zhao (2009) who estimate pricing kernels of stock market returns and interest rates, respectively, conditional on an ex-post volatility proxy filtered from historical time series. Our strategy differs from their approach by using the VIX, which possesses a monotonic functional relationship with the unobservable volatility for almost all state-of-the-art volatility models. Therefore, our method avoids estimation errors from the filtering stage, while making it possible to study volatility pricing kernels with the help of VIX options.

Furthermore, several recent studies document the importance of multiple volatility factors in capturing dynamics of option prices or the term structure of variance swaps, see e.g. Christoffersen et al. (2008), Egloff et al. (2010), Mencia & Sentana (2012), and Bates (2012). Our nonparametric framework can be extended to nest these models by including additional regressors such as a VIX future contract, or CBOE S&P 500 3-Month Volatility Index (VXV). Such an extension, though being interesting and important itself, is beyond the scope of the current focus, to which term structure of volatility is less relevant.

Section 2 discusses the nonparametric identification of pricing kernels of both the market return and volatility. Section 3 provides our nonparametric estimation framework and Monte Carlo simulations. Empirical estimates of pricing kernels are presented in Section 4. Section 5 concludes the paper.

#
2 Pricing Kernels with a Volatility Factor

The pricing kernel equals the ratio of risk-neutral density, also known as state-price density (SPD), to the density under the physical measure. To study pricing kernels, we first discuss the identification of state price densities, by exploring the underlying connection of S&P 500 options, VIX, and VIX options through the latent volatility factor.

##
2.1 Identification of State-Price Densities

To fix ideas, we denote the log price of the S&P 500 index as , the VIX as , and the unobserved volatility as . The information in the derivative markets is driven by the joint evolution of and , which determines endogenously. As is not observable, there exist no Arrow-Debreu securities traded on directly.

In fact, the payoffs of S&P 500 and VIX options depend on their own underlying indices at maturity . Therefore, we focus on the marginal state-price densities with respect to and separately. We show that the marginal densities together span the two option markets, and provide sufficient and necessary information about the dynamics of the market return and its volatility. The joint dynamics, nevertheless, cannot be identified nonparametrically unless certain options whose payoff depends on both and are traded.

We write the time- price of a S&P 500 call option with maturity and strike as: 5

where denotes the log forward price of the S&P 500 index, is the time-to-maturity, and is the deterministic risk-free rate between and at time . Similarly, the price of a VIX call option with strike is given by:

Observe that the two SPDs and completely determine these option prices. Building upon the insight of Breeden & Litzenberger (1978), they can be estimated as the second order derivative of option prices with respect to different strikes. In particular, we can recover

| (1)

from S&P 500 options and

| (2)

from VIX options. It is apparent that and summarize the entire information about these two option markets, hence the joint density of and cannot be identified from the data without additional parametric assumptions.

Nevertheless, these two densities and are not practically feasible to estimate as is unobservable. Alternatively, with the observed VIX from the market,6 we may rewrite the option prices with as a state variable, i.e., and ,7and take second order derivatives to obtain

|  | (3)

In fact, writing options in terms of and amounts to assuming that can be determined from and , which is rigorous under most models of volatility risk in the literature (see Section 2.3 below for details). With state variables being fully observable, and can be identified from the data.

In summary, state-price densities and encapsulate all the information in the two option markets. They complement each other to reveal an intact picture of the market return, its volatility dynamics and the interactions of the two markets.

##
2.2 From State-Price Densities to Pricing Kernels

We now discuss how to obtain the pricing kernels by combining the risk-neutral and physical densities of and . We denote as the pricing kernel and use for short. Not surprisingly, for the same reason described in Section 2.1, the joint pricing kernel cannot be identified nonparametrically. We therefore study the projections of pricing kernel on 8 and , denoted as and , respectively. They are called the pricing kernel of the market return and the pricing kernel of the VIX in the following.

In fact, the price of a S&P 500 call option can be written as

|  | (4)

and the price of a VIX call option is

|  | (5)

where and are conditional densities of and under the physical measure, respectively. Note that the law of iterated expectation is used in the second equality of both (4) and (5).

Similar to (3), equations (4) and (5) imply that the second order derivatives of the S&P 500 and VIX call prices with respect to their strikes are also equal to and , respectively. This fact, combined with (3), further implies that

That is, by combining the risk-neutral and physical densities of and , we obtain the projections of onto and , respectively. These two pricing kernels contain rich information on how risks, especially those associated with volatility shocks, are priced in financial markets. In the equilibrium setup of Aït-Sahalia & Lo (2000) with a representative agent, these pricing kernels represent--up to a scaled factor--the marginal rate of substitution. While Aït-Sahalia & Lo (2000) and Jackwerth (2000) estimate the pricing kernels of S&P 500 returns, our includes the VIX in the conditional information set so that volatility becomes relevant to the price of risk regarding the expected returns. In addition, we are able to identify the pricing kernel of the VIX.

##
2.3 Nested Models

As discussed in Section 2.1, we employ the information set generated by and to replace the information generated by and , because is directly observable. In fact, the information set of and is coarser than the set generated by and , and equating these two effectively assumes that is an invertible function of and . We now show that this assumption is satisfied in most parametric models proposed in the literature, including both reduced-form option pricing models and equilibrium models with a priced stochastic volatility factor. Unlike Boes et al. (2007) and Li & Zhao (2009) who use an ex-post volatility proxy filtered from historical time series, we use VIX instead, which bears no approximation errors in most cases.

We first consider the class of option pricing models that induce an affine relationship between the unobservable variance and squared VIX. This class of models has the following risk-neutral dynamics:9

|  | (6)

where and may be driven by finite activity compound Poisson processes with correlated jump sizes and . Such models include those discussed in Bakshi et al. (1997), Bates (2000), Pan (2002), Chernov & Ghysels (2000), Eraker (2004), Carr et al. (2003), Eraker et al. (2003), and Broadie et al. (2007). Jumps can be driven by Lévy processes such as the CGMY process in Carr et al. (2003) and Bates (2012). Note that this class also includes non-Gaussian OU processes, as introduced in Barndorff-Nielsen & Shephard (2001); see Shephard (2005) for a collection of similar models.

For models of this class, we have

where and are functions of model parameters (see Carr & Wu (2009) for details). That is, is a linear function of , hence and deliver the same information set.

The second class of models introduces a non-affine structure between the squared VIX and variance, such as the exponential-OU-L models in Shephard (2005). In particular, under the risk-neutral measure, such models specify the volatility process as

The squared VIX, as calculated by Tauchen & Todorov (2011), is

| (7)

where is determined by the characteristic exponent of the Lévy process , and and are constants determined by the quadratic variation of . Observe that the function is invertible, so that the information sets generated by and by are equivalent.

Finally, we consider a stylized general equilibrium model, which is a simplified version of Bollerslev et al. (2009) and Drechsler & Yaron (2011) that builds on the long-run risk framework of Bansal & Yaron (2004).10 Specifically, the representative agent's preference over consumption is recursive (Epstein & Zin (1989)). Therefore, the log pricing kernel at time t+1 is

| (8)

where , is the subjective discount factor, is the risk aversion coefficient, is the intertemporal elasticity of substitution, is the growth rate of log consumption, and is the time t to t+1 return on the aggregate wealth claim.11 The state vector of the economy follows

|  | (9)

where and are independent i.i.d. N(0,1) processes, is a compound Poisson process with intensity and i.i.d. jump size , is a compound Poisson process with intensity and i.i.d. jump size , and both jump processes are independent of each other and of the Gaussian shocks. Note that both the Gaussian process and jump process contribute to volatility shocks.

By the standard log-linearization approach following Campbell & Shiller (1988), we have

| (10)

where the price-wealth ratio is conjectured to be affine in the state vector:

| (11)

with and as functions of the model parameters we suppress for notational brevity. With (10) and (11), we have

Therefore, the volatility factor shows up in , and hence in the pricing kernel given in (8). Following the standard practice to proxy the aggregate wealth (consumption) by the aggregate stock market (see Aït-Sahalia & Lo (2000), Bansal & Yaron (2004), and Campbell et al. (2012)), the return corresponds to , and corresponds to the square root of the risk-neutral expectation of the consumption growth variance . Although the state vector dynamics are specified in discrete time, the model (9) is actually a special case of the affine model in Bollerslev et al. (2012). Therefore, is an invertible function of , which represents the variance of consumption growth rate under this equilibrium model.

In summary, most parametric models with volatility risk proposed in the literature, whether reduced-form or structural, can be nested within our nonparametric framework. As a result, we do not lose any information about the dynamics of and when incorporating into the information set; instead, the implementation becomes feasible with the information set fully observable.

#
3 Estimation Strategy

## 3.1 Multivariate Local Linear Estimators for Densities

Here we introduce our nonparametric estimation strategies for SPDs. To fix ideas, we assume the observed prices, and , are contaminated with observation errors, such that12

We then construct nonparametric estimators of and , and take derivatives to estimate the SPDs. Different from the multivariate kernel regression approach adopted by Aït-Sahalia & Lo (1998), we prefer the local linear estimator (Fan & Gijbels (1996)) for two main reasons. First of all, the bias and variance of local polynomial estimators are of the same order of magnitude in the interior or near the boundary, whereas kernel estimators are notorious for the boundary effects. As our empirical studies focus on the tail of pricing kernels, it is advantageous to adopt more efficient estimators. Second, local polynomial regression provides estimates of derivatives, in addition to option prices, which makes it more convenient for our purpose.

Theoretically, it is better to use a local cubic estimator to obtain second-order derivatives. Since we have more than one state variable, including all cross-terms of cubic polynomials into the regression is cumbersome. We avoid this by applying the local linear estimator, so that estimators for SPDs can be obtained simply by a first-order differentiation with respect to the strike.

We write the option price as a function of , and consider the following minimization problem,

where and are the characteristics and price respectively of the -th option in the sample. is a kernel function scaled by a bandwidth vector :

| (12)

where is, for example, the density a of standard normal distribution. The minimizer has a closed-form representation:

| (13)

where

The nonparametric local linear estimator for the option pricing function is

| (14)

with and the estimator for the SPD of is

| (15)

where . The nonparametric estimator and can be constructed similarly.

It may be worth pointing out that in our local linear regression (13) provides estimates of option Greeks. Specifically, option Theta is given by by , Delta by , and Vega by .

## 3.2 Dimension Reduction

One of the major issues of nonparametric estimation is the curse of dimensionality. The rate of convergence decreases rapidly as the dimension of state variables increases. In the most general forms, the pricing functions and depend not only on time-to-maturity, strike, VIX, and the S&P 500 index, but also on interest rates and dividends. Instead of regressing on additional interest rate and dividend variables, we assume that option prices multiplied by depend on these variables only through forward prices. As mentioned in Aït-Sahalia & Lo (1998), models that violate this assumption seem very remote empirically.

Furthermore, following many existing studies such as Aït-Sahalia & Lo (1998) and Li & Zhao (2009), we assume that the S&P 500 option price is homogeneous of degree one in the forward price level:

| (16)

where represents the moneyness of the option. Consequently, we obtain the estimate of through multiplying the nonparametric estimate of by , and write the SPD of as

As for VIX options, we assume that the information about in is fully incorporated into . In other words, conditional on , is independent of , for any . This assumption further implies that the SPD of , obtained from VIX option prices, depends on only through , i.e., . Thus, the number of state variables for the SPD of VIX is also decreased by one. We conduct robustness checks for these assumptions in Section 4.5, and find supportive evidence. Our dimension reduction strategy is motivated from the economic intuition, which is in sharp contrast to the statistical approach proposed by Yao & Hall (2005), who discuss an alternative method in the context of conditional density estimation.

##
3.3 Estimation of Pricing Kernels

Given the homogeneity assumption in (16), the risk neutral density of the return can be estimated using the following formula:

where . Note that homogeneous of degree one in option prices is equivalent to that the conditional density of the log returns is independent of , see, e.g. Joshi (2007) for more details. This property is satisfied by all parametric models discussed in Section 2.3.

While estimating the risk neutral density from option prices, we estimate the physical density using the time series of the S&P 500 index and VIX based on the local linear method. A similar strategy has been adopted by Aït-Sahalia et al. (2009). We collect time series of , , with fixed. We then construct the local linear estimator of the conditional density of returns by minimizing:

where and are the bandwidths to be selected, and and are kernels. Therefore, our density estimator is,

| (17)

Consequently, our pricing kernel estimator can be constructed as

Similarly, we can construct the estimator for the pricing kernel of VIX.

##
3.4 Asymptotic Theory

To provide theoretical guidance for our approach, we derive the asymptotic distribution of the option price and density for S&P 500 options as an example. Suppose the sample size of the S&P 500 options is . Using the equivalent kernels introduced in Fan & Gijbels (1996) and following the derivation in Aït-Sahalia & Lo (1998), we obtain:13

|  | (18)
| as |
|  | (19)
| as |

where is the conditional variance for the local linear regression of on the state variables, and is the joint density of these variables. The estimator for can be constructed using similar nonparametric regressions of squared fitting errors on these state variables. The same asymptotic distributions apply to estimators for VIX option prices and their SPDs. Similar technique has been adopted in Ruppert & Wand (1994).

In addition to estimating the option price and its first and second order derivatives, we apply a local linear method to estimate the conditional density in (17). Its asymptotic theory is given by (see, e.g. Fan et al. (1996)):

The asymptotic theories provided here are applied to construct confidence bands in our empirical studies.

## 3.5 Bandwidth Selection

Bandwidth selection is important especially for multivariate nonparametric regressions. In theory, the optimal rate of bandwidth for estimating the option price is , whereas to estimate densities, we need to adopt a bandwidth with rate due to the curse of differentiation. These bandwidth choices ensure that the nonparametric pricing function achieves the optimal rate of convergence in the mean-squared sense. Empirically, we can choose a bandwidth ( and for S&P 500 options, or for VIX options) as , where is the unconditional standard deviation of the regressor , is the number of regressors, and and 1 for option prices and SPDs, respectively. The constant is chosen by minimizing the mean-squared error of option prices via cross-validation. The cross-validation objective function for regression (14) is given by the weighted mean squared errors:

where means leaving the ith observation out, and is the weighting function. To further accelerate the cross-validation, we adopt the popular K-fold cross-validation, which is faster compared with this leave-one-out method.

The bandwidths of our nonparametric conditional density estimator (17) under the physical measure are chosen by the cross-validation following Fan & Yim (2004):

where the first integral can be calculated in closed-form from (17). Alternative choices of bandwidths have been discussed in Yao & Tong (1998) and Ruppert et al. (1995).

##
3.6 Monte Carlo Simulations

Here we provide simulation studies of our local linear estimators. The Monte Carlo experiments are designed to match our empirical studies. First, we select the same option characteristics as those traded on CBOE in our sample. Second, we select a sample path generated from the following stochastic volatility models with both jumps in volatility and prices:

where and are standard Brownian motions satisfying , and are random jump sizes, is a pure-jump process with intensity , and . The jump sizes follow:

The parameters are taken from Amengual & Xiu (2012), where , , , , , , , , , and . We then calculate S&P 500 and VIX option prices, according to the closed-form formulae given in Amengual & Xiu (2012). Finally, we pollute the prices with multiplicative measurement errors following log-normal distribution with a 5% standard deviation.

Based on the generated sample, we evaluate our nonparametric estimators of option prices on the grid of time-to-maturity and current index level, with the VIX and strike fixed at their sample median. We also calculate the index densities on the grid of and , with fixed at the sample mean, to evaluate our density estimators. The nonparametric estimators of VIX option prices and densities are evaluated similarly. All of these quantities and their percentage errors are reported in Figure 1, averaged over 1000 replications. We observe that the nonparametric estimates are within 5% and 10% of their theoretical Black-Scholes implied volatilities for S&P 500 and VIX options, respectively. The errors for densities are slightly larger, due to the fact that derivatives are estimated with slower rates of convergence, i.e., the so-called curse of differentiation.

#### Figure 1: Monte Carlo Simulations

Note: This figure plots the nonparametric estimation error in the Monte Carlo simulations. The left panel plots the pricing error measured in terms of difference in implied volatility, whereas the right panel plots the percentage error in density estimates. The number of Monte Carlo samples is 1000.

#
4 Empirical Results

In this section, we estimate nonparametric SPDs and pricing kernels using both S&P 500 and VIX options, and present our empirical findings. Before delving into the details, we introduce the dataset.

## 4.1 Data

We obtain daily bid and offer prices of S&P 500 and VIX options, quoted between 3:59 p.m. and 4:00 p.m. EST from the OptionMetrics. Our sample period is chosen as June 1, 2009-May 31, 2011, during which the liquidity of VIX options is satisfactory. We plot the daily open interests of VIX options in Figure 2, along with those of S&P 500 options for comparison. It is obvious from the figure that the liquidity of VIX options has improved dramatically since introduced in 2006, and their open interests have achieved roughly 1/4 of those of S&P 500 options. As a result, our choice of sample ensures that our empirical results are not subject to liquidity issues.

#### Figure 2: Open Interests of the S&P 500 and VIX Options

Note: This figure plots the monthly time series of the open interests of S&P 500 and VIX options from March 1, 2006 to May 31, 2011.

#### Figure 3: Time Series of the S&P 500 Index and the VIX

Note: This figure plots the time series of the S&P 500 index and VIX from Jun 1, 2009 to May 31, 2011.

Figure 3 plots the joint time series of the S&P 500 index and VIX over the sample period, while Table 1 provides their summary statistics. We observe that the VIX ranges between 14.62 and 45.79, which is large enough to have both relatively low and high volatility levels. Moreover, Table 2 also presents summary statistics of option prices. It is worth pointing out that the differences between adjacent strikes of VIX options range from $0.50 to $5 for smaller strikes and from $1 to $10 for large strikes, which are significantly larger percentage-wise than their counterparts for S&P 500 options. Therefore, the impact of price discreteness on the nonparametric estimation of VIX densities could be more severe than on the density estimation of the S&P 500 index, as discussed in Section 3.6.

#### Table 1: Summary Statistics of the S&P 500 Index and VIX

| Mean | Std | Skew | Kurt | Min | 25% | 75% | Max
Index | 1141.670 | 115.806 | 0.065 | 2.414 | 879.130 | 1070.453 | 1221.178 | 1363.610
Return | 0.001 | 0.011 | -0.330 | 4.822 | -0.040 | -0.004 | 0.006 | 0.043
VIX | 22.307 | 4.900 | 0.891 | 4.173 | 14.620 | 18.000 | 25.153 | 45.790

Note: This table reports the summary statistics of the time series of S&P 500 index, return, and VIX from June 1, 2009 to May 31, 2011.

#### Table 2: Summary Statistics of S&P 500 and VIX Options

Moneyness | SPO ITM | SPO ATM | SPO OTM | VXO ITM | VXO ATM | VXO OTM
# of Records | 72161 | 27144 | 33245 | 5573 | 16823 | 17726
Volume | 102.12 | 82.73 | 35.76 | 2.43 | 37.55 | 32.86
Open Interest | 1665.98 | 616.39 | 513.05 | 44.57 | 356.76 | 471.44
Open Interest min | 41.07 | 0.13 | 0.05 | 2.03 | 0.03 | 0.03
Open Interest 25% | 120.19 | 18.15 | 0.35 | 6.40 | 1.83 | 0.18
Derivative Prices 50% | 186.63 | 33.36 | 1.73 | 8.55 | 3.05 | 0.50
Derivative Prices 75% | 284.04 | 49.00 | 6.40 | 11.70 | 4.55 | 1.03
Derivative Prices max | 1128.50 | 110.67 | 60.00 | 25.90 | 11.70 | 4.75
Derivative Prices min | 100 | 845 | 915 | 10 | 14 | 22.5
Derivative Prices 25% | 825 | 1075 | 1160 | 15 | 21 | 35
Strike Prices 50% | 940 | 1135 | 1225 | 17 | 25 | 42.5
Strike Prices 75% | 1040 | 1250 | 1325 | 20 | 30 | 50
Strike Prices max | 1305 | 1415 | 3000 | 40 | 65 | 100
Strike Prices min | 5 | 5 | 5 | 5 | 5 | 5
Strike Prices 25% | 18 | 17 | 21 | 21 | 27 | 28
Time-to-Maturity 50% | 32 | 32 | 35 | 44 | 50 | 50
Time-to-Maturity 75% | 53 | 51 | 56 | 77 | 82 | 78
Time-to-Maturity max | 136 | 136 | 136 | 128 | 128 | 126
Time-to-Maturity min | 16.05 | 9.39 | 9.48 |  |  |
Time-to-Maturity 25% | 27.56 | 16.67 | 15.31 |  |  |
Implied Volatility 50% | 33.85 | 19.84 | 18.13 |  |  |
Implied Volatility 75% | 43.41 | 23.37 | 20.84 |  |  |
Implied Volatility max | 143.48 | 45.81 | 49.44 |  |  |

Note: This table reports the summary statistics (minimum, quantiles, and maximum) for selected S&P 500 and VIX option quotes from June 1, 2009 to May 31, 2011, including the number of records, trading volume, open interest, option price, strike price, time-to-maturity, and implied volatility. In total, there are 132,550 trading records for S&P 500 options, and 40,122 records for VIX options. All options are call options. The prices of S&P 500 ITM call options are computed from OTM put options using the put-call parity for liquidity concerns. For S&P 500 options, ATM is defined as , whereas for VIX options, it is defined as VIX , with as the forward price and as the generic strike.

We follow the data-cleaning routine commonly used in the literature; see, e.g., Aït-Sahalia & Lo (1998). First, observations with bid or ask prices smaller than $0.025 are eliminated to mitigate the effect of pricing errors. For each option, we take the midquote as the observed option price. Due to liquidity concerns, we eliminate any options with zero open interests or trading volumes as well as options with time-to-maturity of less than 5 days. In addition, we only consider options with maturity of less than 136 days, because only VIX option contracts with maturities shorter than 6 months are offered by the CBOE after 2009. It is well known that in-the-money S&P 500 options are less liquid than out-of-the-money options. Therefore, we delete in-the-money options, and use the put-call parity to construct prices of in-the-money call options from out-of-the-money put options. There is no such pattern for VIX options and hence we only consider VIX call options. The last step is to eliminate option contracts that violate no-arbitrage conditions. The resulting sample covers a broad cross section of options, including 420,711 S&P 500 call options, and 53,530 VIX call options, which account for 50.84% and 54.18% of their total number of records, respectively.

## 4.2 Pricing Kernels of the Market Return

The upper panels of Figure 4 provide nonparametric SPD estimates of the S&P 500 index for both low and high levels of VIX, fixed at 18.00 and 25.15 that correspond to the 25% and 75% quantiles of the VIX time series in our sample, respectively. We observe that index densities strongly depend on the VIX level . Conditional on a low , has pronounced spikes, while the density becomes more dispersed when rises to a high level, suggesting that volatility is a key state variable that should be included in the SPDs.

#### Figure 4: State-Price Densities of the S&P 500 Index

Note: The top panels provide our nonparametric estimates of SPDs of the S&P 500 index at various time-to-maturities, with volatility levels at 18.00 (left) and 25.15 (right) that correspond to the 25% and 75% quantiles of the VIX time series in our sample, respectively. The bottom panels compare our estimates (SX) of index SPDs (black, solid) with those using the Aït-Sahalia & Lo (1998) (AL) method (blue, dashed) for the maturity of 42 days, and two current VIX levels at 18.00 and 25.15. Dotted lines around each SPD estimate are the 95% confidence intervals constructed by the asymptotic distribution theory in (19). The interest rate and dividend are fixed at their averages, 2.15% and 2.06%, respectively.

To further demonstrate the importance of volatility in studying the SPDs of market return, the bottom panels of Figure 4 compare the nonparametric SPD estimates proposed by Aït-Sahalia & Lo (1998) (AL) who neglect the volatility variable, with conditional on the two different VIX levels of 18.00 and 25.15. We choose the time-to-maturity as 42 days, and compute the 95% confidence intervals by the asymptotic theory given in (19). Observe that our SPDs differ from the AL densities substantially, with the former more compact and showing higher spikes for a low , which confirms the importance of incorporating volatility into the SPDs.

Given the importance of volatility in SPDs of the S&P 500 index documented above, we now study whether and how volatility affects the shape of the pricing kernel . According to Section 3.3, we further estimate the physical densities of the S&P 500 return conditional on VIX and obtain the pricing kernel estimates. The top two panels of Figure 5 report the pricing kernel estimates with equal to 18.00 (left) and 25.15 (right) and a maturity of 42 days. We observe that the pricing kernels conditional on either a low or high VIX level exhibit a decreasing shape, consistent with a standard expected utility theory, which prescribes that the pricing kernel decrease when expected returns are increasing. In contrast, the bottom left panel of Figure 5 shows that the unconditional estimator of the pricing kernel shows a pronounced U-shape, consistent with what have been found in the literature (Jackwerth (2000) and Bakshi et al. (2010)). Therefore, it is the volatility factor, missing in the unconditional estimates, that may lead to the puzzling U-shape.

#### Figure 5: Pricing Kernels of the S&P 500

Note: The top panels plot the nonparametric estimates of pricing kernels of the S&P 500 index return (black, solid) for the maturity of 42 days, with two current VIX levels at 18.00 and 25.15 that correspond to the 25% and 75% quantiles of the VIX time series in our sample, respectively. Dotted lines are the 95% confidence intervals. The bottom left figure compares the unconditional pricing kernel (red, solid) with the previous two conditional pricing kernels. The bottom right panel presents the scatter plot of S&P 500 returns against the current VIX level .

Specifically, high volatility signals bad future investment opportunities, and investors should have high marginal utility in such a state. Hence, the pricing kernel of market return conditional on a high volatility, which equals the marginal utility up to a re-scaling, is higher than that conditional on a low volatility, as shown in the bottom left panel of Figure 5. The unconditional pricing kernel, however, is a mixture of pricing kernels conditional on different volatility levels, and could exhibit a U-shape when volatility switches from low to high levels. Our finding echoes the conclusions of parametric models in Jackwerth & Brown (2001), Chabi-Yo et al. (2008), Chabi-Yo (2011), and Christoffersen et al. (2010), that missing state variables in the pricing kernel may result in the U-shape. Without restricting the specification of pricing kernels, however, our result shows that stochastic volatility is the key but missing state variable of pricing kernels estimated in the literature.

The pricing kernels conditional on low and high values of have different supporting regions on the left and right tail. For instance, over the interval (0.08, 0.15), we only have the pricing kernel estimates conditional on a high . The reason is that the realized return never exceeds 8% given , as can be seen from the scatter plot of on the bottom right panel of Figure 5. In fact, this observation implies that the unconditional pricing kernel estimates around high levels of market return are dominated by high level volatility, which shifts the unconditional estimates upwards, and explains why they present a U-shape. In other words, large market returns are accompanied by high current volatility , because of which investors have a high marginal utility that leads to the increasing portion of the unconditional pricing kernel on the right tail.

Overall, our nonparametric state-price density estimates differ significantly from those without conditioning on volatility, and confirms that volatility is a key state variable that should be included in the pricing kernel. More importantly, without imposing any restrictions on the dynamics of the market return and volatility, our pricing kernel estimates conditional on VIX show that stochastic volatility is the key variable responsible for the "pricing kernel puzzle."

##
4.3 Pricing Kernels of the VIX

We now present nonparametric estimates of SPDs and pricing kernels of the VIX and investigate their implications for the pricing of volatility risk. The top panels of Figure 6 present the VIX SPDs at various maturities conditional on two different levels of equal to 18.00 and 25.15. We find first that the VIX SPDs are all positively skewed, with the probability of achieving higher VIX levels decreasing given a low time- VIX level. Second, the SPD of VIX conditional on a high (right panel) has a spike around median volatility levels, consistent with the conventional wisdom that volatility reverts to its long-run mean.

#### Figure 6: State-Price Densities and Pricing Kernels of the VIX

Note: The top panels provide the nonparametric estimates of SPDs of the VIX at various time-to-maturities, with volatility level at 18.00 (left) and 25.15 (right) that correspond to the 25% and 75% quantiles of the VIX time series in our sample, respectively. The bottom panels plot the nonparametric estimates of VIX pricing kernels (black, solid) for the maturity of 42 days, and two current VIX levels at 18.00 and 25.15. Dotted lines are the the 95% confidence intervals. The interest rate and dividend are fixed at their averages, 2.15% and 2.06%, respectively.

Furthermore, we estimate the pricing kernel by combining estimates of both risk-neutral and physical densities of the VIX. The bottom panels of Figure 6 provide nonparametric estimates of for a maturity of 42 days and two different levels of at 18.00 and 25.15. We observe that the pricing kernel exhibits a pronounced U-shape as a function of future VIX levels. Therefore, volatility risk is priced, and the price of volatility risk increases when volatility deviates from its median level. In other words, investors attach high marginal utility to payoffs received when the future volatility is either extremely high or low. Bakshi et al. (2010) document the U-shape for the volatility pricing kernel indirectly, by exploring the link between the monotonicity of the pricing kernel and returns of VIX option portfolios. They further provide a model with heterogeneity in beliefs to account for the U-shape, in which the volatility market is dominated by investors with zero market risk. In contrast, we provide direct estimates of the volatility pricing kernel by nonparametric methods, which provide more robust information about the shape. In particular, we find that the volatility pricing kernel is asymmetric, and the asymmetry conditional on a high time- volatility is much stronger than that conditional on a low volatility. This finding implies that investors price the volatility risk differently according to different scenarios of the economy, which presents new empirical regularities that need to be incorporated into models of volatility risk.

In summary, our SPD estimates of VIX document empirical features of risk-neutral dynamics of volatility such as positive skewness and mean reversion. Although the volatility process under the physical measure is well documented as displaying a mean-reverting pattern using historical time series, its risk-neutral behavior is not crystal clear. Our findings uncover the risk-neutral dynamics of volatility without any parametric restrictions. More importantly, our estimates of the volatility pricing kernel show that investors have high marginal utility even in low volatility states, which supports the model with heterogeneity in beliefs.

## 4.4 In-Sample Fitting and Out-of-Sample Forecasts

We evaluate the performance of our nonparametric estimator (SX) by comparing in-sample fitting and out-of-sample forecasts with two alternative methods discussed in Aït-Sahalia & Lo (1998): the nonparametric approach without volatility factor (AL) in terms of both density and option implied volatility forecasts, and the martingale approach (MKT) for option implied volatility forecasts only. As it is widely used by practitioners, the MKT approach simply forecasts tomorrow's implied volatility by interpolation using today's implied volatility surface.

Intuitively, a potential advantage of our estimator over the MKT method lies in the inclusion of historical options with similar characteristics. As opposed to the MKT approach that relies exclusively on the cross section of options on the previous day, the SX estimator is able to capture a more stable pricing function over time, and hence is expected to outperform the MKT approach in out-of-sample forecasting, although not surprisingly, the SX estimator may fit the cross section of option prices worse on certain days, but better on other days. With historical option prices incorporated, the AL estimator is also capable of capturing certain stability in the historical data, which helps make predictions. However, it misses an important volatility factor that is incorporated into the SX approach.

Panel A of Table 3 reports the forecasting performance of the SX, AL, and MKT methods for option prices (quoted in implied volatility). For each date , we adopt a preceding 16-month window, within which the SX, AL, and MKT estimators for the target options are obtained. The selected target options have a maturity of 42 days with moneyness ranging between -0.15 and 0.15 . We forecast such options on day , for (in-sample), and =7, 14, 21, 28, 35, 42, 63 and 84 days (out-of-sample) progressively. We repeat the procedure for each day in the last 8 months of our sample period and average across days to obtain the root-mean-squared percentage difference between the predictions and the realized option prices.

#### Table 3: In-Sample Fitting and Out-of-Sample Forecasts

Panel A: Implied Volatility Forecast Error (%)

| 0 | 7 | 14 | 21 | 28 | 35 | 42 | 63 | 84
SX | 17.32 | 14.21 | 16.68 | 15.82 | 17.07 | 16.92 | 18.88 | 17.56 | 17.56
AL | 28.25 | 28.93 | 32.10 | 32.14 | 33.27 | 32.52 | 34.15 | 35.55 | 36.61
MKT | 13.74 | 15.94 | 16.18 | 15.77 | 17.26 | 18.22 | 18.80 | 20.63 | 22.30

Panel B: Density Forecast Error (%)

| 0 | 7 | 14 | 21 | 28 | 35 | 42 | 63 | 84
SX | 0.00 | 0.28 | 0.52 | 0.73 | 0.96 | 1.15 | 1.34 | 1.76 | 2.20
AL | 5.00 | 5.05 | 5.22 | 5.47 | 5.69 | 6.07 | 6.25 | 6.75 | 7.35

Note: Panel A reports average forecast errors of implied volatility produced by the Aït-Sahalia & Lo (1998) estimator (AL), our estimator conditional on VIX (SX), and a martingale interpolation (MKT) method, while Panel B reports those of risk-neutral densities using the AL and SX methods. The nonparametric option implied volatility and their corresponding SPDs are estimated with a rolling-window of 16 months, and out-of-sample forecasts are generated for various forecast horizons on a daily rolling basis from June 1 2009 to May 31, 2011. The time-to-maturity for both SPDs and option prices is chosen as 42 days.

We observe first that the MKT approach outperforms the AL approach uniformly in forecasting option prices for the sample period we consider, which is in contrast with findings of Aït-Sahalia & Lo (1998). However, this is not surprising as the AL estimator does not include volatility as a conditioning variable which changed substantially over the sample period we consider, i.e., June 1, 2009 - May 31, 2011. In contrast, the SX estimator outperforms both the AL and MKT methods especially for longer horizons. The superior performance of the SX estimator highlights the benefit of predicting by capturing certain stable price patterns in the historical data and incorporating the volatility factor. Not surprisingly, the MKT approach has a better in-sample performance given its implementation.

Panel B of Table 3 reports the forecasting performance of the SX and AL estimators for state-price densities. The empirical design is similar to the forecasting exercise of option implied volatility, with a 16-month window, a target maturity of 42 days, and horizons of 7, 14, 21, 28, 35, 42, 63, and 84 days for the out-of-sample performance. We compute the average forecast error (root-mean-squared percentage difference) as a percentage of the mode value of the realized density over the last 8 months of our sample period. The realized density is computed by the SX approach using a 16-month window including the target day. Results in Panel B show that the SX estimator outperforms the AL density substantially over all horizons, due to the missing volatility factor in AL densities. For example, the forecast error for = 84 is 2.2% and 7.4% for the SX and AL density estimators, respectively.

##
4.5 Robustness Checks

As robustness checks, we verify the two dimension-reduction assumptions employed in our nonparametric procedure: the homogeneity of degree one for S&P 500 options, and the conditional independence of state-price densities of the VIX with respect to .

Figure 7 plots nonparametric estimators of the implied volatility surface of S&P 500 options across both log-moneyness and time-to-maturities: one with the assumption of homogeneity of degree one (left panel) and the other without using it (right panel). We observe that the shape of the two surfaces match each other well in general, although there are slight differences around the boundaries where nonparametric estimators usually incur relatively large biases. Moreover, the estimator without dimension reduction is noiseier as its convergence rate is lower due to the "curse of dimensionality."

#### Figure 7: Robustness Check I

Note: This figure plots the nonparametric estimates for the implied volatility surface of S&P 500 option prices. The left panel plots the estimates based on dimension reduction techniques, whereas the right panel plots the estimates without such techniques.

#### Figure 8: Robustness Check II

Note: This figure plots the nonparametric estimates of VIX state-price densities with both and as conditioning variables. The time-to-maturity is , and is fixed at 18.00 and 25.15 respectively.

Figure 8 plots estimates of VIX SPDs against the S&P 500 index and VIX for , and for and 25.15 respectively. We observe that conditional densities do not vary much with conditional on either the low or high level of , especially for the part away from the boundary. Overall, the dimension-reduction assumption for VIX options, i.e., the dependence of VIX SPD on mainly through , seems valid for the sample period we consider.

#
5 Conclusion

Volatility has been well documented as a priced risk factor, and hence an essential component of pricing kernels. Taking advantage of the rapidly developed volatility derivative markets, we provide nonparametric estimates of both SPDs and pricing kernels with volatility. We show that volatility is the key but missing state variable in the unconditional pricing kernel estimates that exhibit the puzzling U-shape. Moreover, we document a U-shaped pricing kernel of volatility, which cannot be captured by standard models with volatility risk, such as Bollerslev et al. (2009) and Drechsler & Yaron (2011). Therefore, it remains important to develop extensions of these models that are in compliance with our empirical findings.

In addition, our framework extends the nonparametric option pricing method to allow for stochastic volatility, by exploring additional information from the VIX. Existing parametric stochastic volatility models face an unfortunate compromise between model flexibility and tractability. In contrast, our method enjoys several advantages, such as being model-free, robust to model misspecification and pricing measures, and computationally efficient. Hence, our nonparametric option pricing approach with VIX alleviates the compromise to a great extent.

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#### Footnotes

* We benefited from discussions with Yacine Aït-Sahalia, Andrea Buraschi, Bjorn Eraker, Peter Carr, Peter Christoffersen, Fousseni Chabi-Yo, George Constantinides, Jianqing Fan, René Garcia, Kris Jacobs, Jakub Jurek, Ilze Kalnina, Ralph Koijen, Nicholas Polson, Eric Renault, Jeffrey Russell, Neil Shephard, George Tauchen (discussant), Viktor Todorov, Grigory Vilkov (discussant), Hao Zhou, as well as seminar and conference participants at the University of Chicago, Northwestern, Princeton, Toulouse School of Economics, Liverpool School of Management, the 2012 CICF, the 5th Annual SoFiE Conference, the Measuring Risk conference 2012, the 2012 Financial Engineering and Risk Management International Symposium, and the 2012 International Symposium on Risk Management and Derivatives. Xiu acknowledges research support by the Fama-Miller Center for Research in Finance at Chicago Booth. The views expressed herein do not reflect those of the Board of Governors of the Federal Reserve System. Return to Text
* Board of Governors of the Federal Reserve System, Mail Stop 165, 20th Street and Constitution Avenue, Washington, DC, 20551. E-mail: Zhaogang.Song@frb.gov. Return to Text
* University of Chicago Booth School of Business, 5807 S. Woodlawn Avenue, Chicago, IL 60637. Email: dacheng.xiu@chicagobooth.edu. Return to Text
1. The VIX, from its inception, was calculated from S&P 500 index options by inverting the Black-Scholes formula. In 2003, the CBOE amended this approach and adopted a model-free method to calculate the VIX. Return to Text
2. Previously, investors have to take positions in option portfolios, such as straddles or strangles, in order to trade volatility. Return to Text
3. We emphasize that the joint pricing kernel, though not identifiable nonparametrically using the S&P 500 and VIX options, can be estimated with certain parametric correlation restriction on the two marginal pricing kernels. We do not explore this approach because our focus is to recover the pricing kernels without any parametric restrictions. A follow-up paper of our study, Jackwerth & Vilkov (2013), implemented such an exercise using the parametric Frank copula for the two marginal distributions. Return to Text
4. A related study, Fan & Mancini (2009), proposes nonparametric methods for pricing derivatives based on state price distributions. Return to Text
5. In our setting, the time- information set contains stock prices, instantaneous volatility, interest rates and dividends, which can be summarized by the log forward price and the volatility . Return to Text
6. The CBOE constructs form a portfolio of options weighted by strikes according to the formula:

where denotes the quadratic variation of the log return process from to , and are put and call options with time-to-maturity and strike , and is the log price of forward contracts, see e.g. Britten-Jones & Neuberger (2000) and Carr & Wu (2009). Return to Text
7. Strictly speaking, the function here is a composite function, which is different from the previous call option pricing function. We recycle it to simplify our notations. Return to Text
8. The projection of on is defined as . Return to Text
9. The discontinuous part of the quadratic variation of is assumed to be linear in . Return to Text
10. Other equilibrium models that satisfy the invertibility between and include Bansal et al. (2012), Bollerslev et al. (2009), and Campbell et al. (2012). We choose to present the framework of Bollerslev et al. (2009) and Drechsler & Yaron (2011) for simplicity of illustration. Return to Text
11. The literature usually assumes that and , which implies . This assumption ensures that the representative agent has a preference for early resolution of uncertainty, which is the key for the price of volatility risk. Return to Text
12. Hereafter, we multiply all option prices by the corresponding , so that we can omit in and , and reduce one state variable in the following regressions. Again, we recycle the notations and without ambiguity. Return to Text
13. We sketch a proof here for the asymptotic theory as part of it is non-standard. Notice from (13) that

Using the properties of Gaussian kernel, we have

Therefore, we can write the estimators in their equivalent kernel forms:

Using the standard kernel asymptotic results, we can obtain the above asymptotic theory. 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