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Option Valuation with Volatility Components, Fat Tails, and Nonmonotonic Pricing Kernels Option Valuation with Volatility Components, Fat Tails, and Nonmonotonic Pricing Kernels
Kadir, Babaoğlu,;Peter, Christoffersen,;Steven, Heston,;Kris, Jacobs,
2018-12-01 00:00:00
Abstract We nest multiple volatility components, fat tails, and a U-shaped pricing kernel in a single option model and compare their contribution in describing returns and option data. All three features lead to statistically significant model improvements. A U-shaped pricing kernel is economically most important and improves option fit by 17%, on average, and more so for two-factor models. A second volatility component improves the option fit by 9%, on average. Fat tails improve option fit by just over 4%, on average, but more so when a U-shaped pricing kernel is applied. Overall, these three model features are complements rather than substitutes: the importance of one feature increases in conjunction with the others. Received date September 27, 2016; Accepted date July 09, 2017 By Editor Raman Uppal By accounting for heteroscedasticity and volatility clustering, empirical studies on option valuation substantially improve on the Black-Scholes (1973) model prices through the parametric modeling of stochastic volatility (SV) (see, e.g., Heston 1993; Bakshi, Cao, and Chen 1997). The literature has focused on two improvements to capture the stylized facts in the data. First, by accounting for more than one volatility component, the model becomes more flexible and its modeling of the term structure of volatility improves. This approach is advocated by Duffie, Pan, and Singleton (2000) and implemented on option prices by, among others, Bates (2000), Christoffersen, Heston, and Jacobs (2009), and Xu and Taylor (1994).1Christoffersen, Jacobs, Ornthanalai, and Wang (2008) propose a discrete-time GARCH option valuation model with two volatility components that has more structure, by modeling total volatility as evolving around a stochastic long-run mean. The second modeling improvement that reliably improves model fit is to augment stochastic volatility with jumps in returns and/or volatility. A large number of studies have implemented this approach.2 Intuitively, the advantage offered by jump processes is that they allow for conditional nonnormality and therefore for instantaneous skewness and kurtosis. In discrete-time modeling, an equivalent approach uses innovations that are conditionally non-Gaussian. Examples of this approach are Christoffersen, Heston, and Jacobs (2006), who use inverse Gaussian innovations, and Barone-Adesi, Engle, and Mancini (2008), who take a nonparametric approach. The studies cited above convincingly demonstrate that these two modeling approaches improve model fit for both the option prices and the underlying returns. However, the most important challenge faced by these models is the simultaneous modeling of the underlying returns and the options. This position is forcefully articulated by, for example, Bates (1996b, 2003). Andersen, Fusari, and Todorov (2015a) address this by fitting realized (physical) volatility together with option prices, but this still leaves open the question of a pricing kernel that links the observed “physical” measure and the risk-neutral measure inherent in option prices. In particular, deficiencies in a model’s ability to simultaneously describe returns and option prices may not be exclusively due to the specification of the driving process, but could also be caused by a misspecified price of risk or, equivalently, the pricing kernel. The literature focuses on pricing kernels that depend on wealth, originating in the seminal work of Brennan (1979) and Rubinstein (1976). Liu, Pan, and Wang (2004) discuss the specification of the price of risk when SV models are augmented with Poisson jumps. Several papers, including Ait-Sahalia and Lo (1998), Jackwerth (2000), Rosenberg and Engle (2002), Bakshi, Madan, and Panayotov (2010), Brown and Jackwerth (2012), and Chabi-Yo (2012), have documented deviations from and explored extensions to the traditional log-linear pricing kernel. In recent work, Christoffersen, Heston, and Jacobs (2013) specify a more general pricing kernel that depends on volatility as well as wealth. The kernel is nonmonotonic after projecting onto wealth, a result consistent with recent evidence by Cuesdeanu and Jackwerth (2015).3Christoffersen, Heston, and Jacobs (2013) show that the more general pricing kernel provides a superior fit to option prices and returns. The literature suggests at least three important improvements on the benchmark SV option pricing model. First, multiple volatility components; second, conditional nonnormality or jumps; and third, nonmonotonic pricing kernels. It is important to nest these features within a common framework in order to have a “horse race” comparison of their importance. In addition, examining these features jointly shows how they interact in describing returns and options. Ideally, these different model features ought to be complements rather than substitutes. The second volatility factor should improve the modeling of the volatility term structure, and therefore the valuation of options of different maturities, and long-maturity options in particular. Non-Gaussian innovations should prove most useful in capturing the moneyness dimension for short-maturity options, usually referred to as a smirk. Andersen, Fusari, and Todorov (2017) provide recent empirical evidence of this using the increasingly popular one-week maturity options on the S&P500 index. The nonmonotonic pricing kernel has an entirely different purpose, because its relevance lies in the joint modeling of index returns and options, rather than the modeling of options alone. However, the existing literature does not contain any evidence on whether these model features are indeed complements when confronted with the data. The literature does also not address the question of which model feature is statistically and economically most significant. This paper is the first to address this issue by comparing the three features within a nested model. We conduct an extensive empirical evaluation of the three model features using returns data, using options data, and finally using a sequential estimation exercise. We find that all three model features lead to statistically significant model improvements. A U-shaped pricing kernel is economically most important and improves option fit by 17%, on average, and more so for two-factor models. A second volatility factor improves the option fit by 9%, on average. Fat tails improve option fit by just over 4%, on average, but more so when a U-shaped pricing kernel is applied. Our results suggest that the three features are complements rather than substitutes. We develop a class of models in which the dynamic variance of the equity index appears in the pricing kernel. It is natural to ask what type of underlying economies can generate this type of pricing kernel. One approach is to allow for dynamics in the variance of aggregate production in a constant returns to scale production technology, like in Cox, Ingersoll, and Ross (1985). Alternatively, the variance-dependent pricing kernel can also result from the model of Benzoni, Collin-Dufresne, and Goldstein (2011). In their model, uncertainty directly affects Epstein-Zin-style preferences. Bollerslev, Tauchen, and Zhou (2009) generate a variance risk premium in equilibrium by allowing for stochastic volatility of aggregate consumption and this approach can be taken in our framework as well. Determining which equilibrium models are consistent with our pricing kernel is an interesting question that we leave for future work. Calibration or estimation, like in Constantinides and Ghosh (2011), could be used to distinguish between the different structural models consistent with our framework. 1. A Class of GARCH Dynamics for Option Valuation This section introduces a general class of GARCH dynamics for index option valuation. The most general model we will consider is a two-component fat-tailed GARCH model. We will subsequently show how the IG-GARCH(1,1), the Gaussian GARCH(1,1), and other component models can be viewed as special cases of our most general model. One can use observable state variables to value options in any dynamic model. For example, one might use implied volatilities extracted from option prices. Alternatively, one might use a filtering technique such as the particle filter, or rely on realized volatility computed from intraday returns like in Andersen, Fusari, and Todorov (2015a). We choose a GARCH model because we want to assess whether option prices are consistent with observed returns. The straightforward filtering used in this framework facilitates investigating the relationship between option prices and return dynamics. The GARCH approach allows us to impose economic restrictions based on observed returns, without an auxiliary filter that is separate from the assumptions of the option model. The limitation of a GARCH approach is that it does not allow one-step-ahead volatility to evolve independently of returns. This is not a significant problem in practice, because the model allows innovations in variance to be imperfectly correlated with daily (or higher frequency) returns. 1.1 The IG-GARCH component model Our most general model combines two generalizations of the Heston-Nandi (2000) model, Inverse Gaussian innovations and the component structure. Consider first the IG-GARCH(2,2) process given by ln(S(t+Δ))=ln(S(t))+r+μh(t+Δ)+ηy(t+Δ), (1a) h(t+Δ)=w+b1h(t)+b2h(t−Δ)+c1y(t)+c2y(t−Δ)+a1h(t)2/y(t)+a2h(t−Δ)2/y(t−Δ), (1b) where y(t+Δ) has an inverse Gaussian distribution with degrees of freedom h(t+Δ)/η2 . Note that whereas y(t+Δ) is a positive random variable, returns are shifted by μh(t+Δ) and can have both negative and positive values. The advantage of the IG-GARCH is that the innovation is nonnormal, thus allowing for conditional skewness and kurtosis. The dynamic (1a)-(1b) can be written in terms of zero-mean innovations as follows ln(S(t+Δ))=ln(S(t))+r+μ˜h(t+Δ)+h(t+Δ)z(t+Δ), (2a) h(t+Δ)=w˜+b˜1h(t)+b˜2h(t−Δ)+υ1(t)+υ2(t−Δ), (2b) where μ˜=μ+η−1, (3a) w˜=w+a1η4+a2η4, (3b) b˜i=bi+ci/η2+aiη2, (3c) z(t)=ηy(t)−h(t)/ηh(t), (3d) υi(t)=ciy(t)+aih(t)2/y(t)−cih(t)/η2−aiη2h(t)−aiη4. (3e) The conditional means of return and variance are given by Et[ln(S(t+Δ)/S(t))]=r+μ˜h(t+Δ), (4a) Et[h(t+2Δ)]=w˜+b˜1h(t+Δ)+c2y(t)+a2h(t)2/y(t). (4b) GARCH(2,2) models are not typically used in empirical work on option valuation. However, building on Engle and Lee (1999), by imposing some parameter restrictions, we can transform the IG-GARCH(2,2) into a component model that nests Christoffersen et al. (2008). Define the long-run component q(t) of the variance process (2b) as q(t)=−ρ1w˜(1−ρ1)(ρ2−ρ1)+ρ2ρ2−ρ1h(t)+b˜2ρ2−ρ1h(t−Δ)+1ρ2−ρ1υ2(t−Δ), (5) where υ2(t) is given by (3e), and where ρ1 and ρ2 are the respective smaller and larger roots of the quadratic equation ρ2−b˜1ρ−b˜2=0, which are the eigenvalues of the transition equation (1b). The short-run component is the deviation of variance from its long-run mean, h(t) – q(t). Substituting these into the IG-GARCH(2,2) dynamics (1a)-(1b) yields the IG-GARCH component model that we denote IG-GARCH(C): ln(S(t+Δ))=ln(S(t))+r+μh(t+Δ)+ηy(t+Δ), (6a) h(t+Δ)=q(t+Δ)+ρ1(h(t)−q(t))+υh(t), (6b) q(t+Δ)=wq+ρ2q(t)+υq(t), (6c) or, equivalently, q(t+Δ)=σ2+ρ2(q(t)−σ2)+υq(t), where σ2 is the unconditional variance, and σ2=w˜(1−ρ1)(1−ρ2)wq=w˜1−ρ1ah=−ρ1ρ2−ρ1a1−1ρ2−ρ1a2aq=ρ2ρ2−ρ1a1+1ρ2−ρ1a2ch=−ρ1ρ2−ρ1c1−1ρ2−ρ1c2cq=ρ2ρ2−ρ1c1+1ρ2−ρ1c2υi(t)=ciy(t)+aih(t)2/y(t)−cih(t)/η2−aiη2h(t)−aiη4. The unit root condition, ρ2 = 1, corresponds to the restriction b˜2=1−b˜1 . The expression for σ2 shows that total variance persistence in the component model is simply 1−(1−ρ1)(1−ρ2)=ρ2+ρ1(1−ρ2). The component parameters also can be inverted to obtain the IG-GARCH(2,2) parameters: a1=ah+aqa2=−ρ2ah−ρ1aqb˜1=ρ1+ρ2b˜2=−ρ1ρ2c1=ch+cqc2=−ρ2ch−ρ1cq. This proves that the IG-GARCH(2,2) model is equivalent to the component model (6a)–(6c). The component structure helps with interpreting the model. The coefficients of the lagged variables (in the long- and short-run components) are the roots of the process’ characteristic equation. These parameters are more informative about the process than the parameters in the IG-GARCH(2,2) model. This facilitates estimation, including identifying appropriate parameter starting values. 1.2 The IG-GARCH(1,1) model Christoffersen, Heston, and Jacobs (2006) studies the IG-GARCH(1,1) special case in which ln(S(t+Δ))=ln(S(t))+r+μh(t+Δ)+ηy(t+Δ), (7a) h(t+Δ)=w+b1h(t)+c1y(t)+a1h(t)2/y(t). (7b) Note that this can be viewed as effectively removing the long-run component in (6c) from the return dynamics. 1.3 The Gaussian limit of the IG model We now show formally how Gaussian models are nested by the inverse Gaussian models. Consider the normalization of the innovation to the return process in (1a): z(t)=ηy(t)−h(t)/ηh(t). (8) This normalized inverse Gaussian innovation converges to a Gaussian distribution as the degrees of freedom, h(t) /η2, approach infinity. If we fix z(t) and h(t) and take the limit as η approaches zero, then the IG-GARCH(2,2) process (2a)-(2b) weakly converges to the Heston-Nandi (2000) GARCH(2,2) process: ln(S(t+Δ))=ln(S(t))+r+μ˜h(t+Δ)+h(t+Δ)z(t+Δ),h(t+Δ)=ω+β1h(t)+β2h(t−Δ)+α1(z(t)−γ1h(t))2+α2(z(t−Δ)−γ2h(t−Δ))2, (9a) where the limit is taken as follows: w˜=ω−α1−α2,ai=αi/η4,bi=βi+αiγi2+2αiγi/η−2αi/η2,ci=αi(1−2ηγi). 1.4 The Gaussian component model Once again, we can write the GARCH(2,2) model in (9a) as a component model. By imposing some parameter restrictions we obtain the component model of Christoffersen et al. (2008) h(t+Δ)=q(t+Δ)+ρ1(h(t)−q(t))+νh(t), (11a) q(t+Δ)=ωq+ρ2q(t)+νq(t), (11b) where νi(t)=αi[(z(t)−γih(t))2−1−γi2h(t)] i=h,q,γh=−ρ1α1γ1+α2γ2(ρ2−ρ1)αhγq=ρ2α1γ1+α2γ2(ρ2−ρ1)αqαh=−ρ1ρ2−ρ1α1−1ρ2−ρ1α2αq=ρ2ρ2−ρ1α1+1ρ2−ρ1α2, and ρ1 and ρ2 are the respective smaller and larger roots of the quadratic equation: ρ2−(β1+α1γ12)ρ−β2−α2γ22=0. The dynamic for the long-run component can equivalently be expressed as q(t+Δ)=σ2+ρ2(q(t)−σ2)+νq(t), where σ2 is the unconditional variance. The component parameters also can be inverted to recover the GARCH(2,2) parameters: α1=αh+αqα2=−ρ2αh−ρ1αqγ1=αhγh+αqγqα1γ2=−ρ2αhγh−ρ1αqγqα2β1=ρ1+ρ2−α1γ12β2=−ρ1ρ2−α2γ22. Our Inverse Gaussian Component model in (6a)-(6c) therefore corresponds in the limit to the component model of Christoffersen et al. (2008). 1.5 The Gaussian GARCH(1,1) model The final model we consider is the Heston-Nandi (2000) Gaussian GARCH(1,1) process ln(S(t+Δ))=ln(S(t))+r+μ˜h(t+Δ)+h(t+Δ)z(t+Δ), (12a) h(t+Δ)=ω+β1h(t)+α1(z(t)−γ1h(t))2, (12b) where z(t) has a standard normal distribution. This benchmark model has been estimated and tested in several empirical applications.4 Christoffersen, Heston, and Jacobs (2006) show that the inverse Gaussian GARCH(1,1) model in (7a)-(7b) nests the Heston-Nandi (2000) Gaussian GARCH(1,1) model in (12a)-(12b). 2. The Risk-Neutral Model and Option Valuation To value options, we introduce the pricing kernel and the resulting risk-neutral dynamics. We then elaborate on the relationships between the risk-neutral and physical parameters. We first discuss the risk neutralization for the most general IG-GARCH(2,2) process. We subsequently discuss special cases nested by the most general specification. 2.1 Risk neutralization For the purpose of option valuation we need to derive the risk-neutral dynamics from the physical dynamics and pricing kernel. Risk neutralization is more complicated for the inverse Gaussian distribution than for the Gaussian distribution. We implement a volatility-dependent pricing kernel following Christoffersen, Heston, and Jacobs (2013), where M(t+Δ)=M(t)(S(t+Δ)S(t))φ exp (δ0+δ1h(t+Δ)+ξh(t+2Δ)). (13) Recent evidence by Cuesdeanu and Jackwerth (2015) suggests that the pricing kernel may be a nonmonotonic function of returns. Accordingly, Christoffersen, Heston, and Jacobs (2013) show that in a GARCH framework, the log-kernel is a nonlinear and nonmonotonic function of the path of spot returns. Henceforth we refer to it as the nonmonotonic pricing kernel. If ξ > 0, the pricing kernel is U shaped in returns. In Appendix A, we show that the risk-free and the risky assets both satisfy the martingale restriction under the pricing kernel in Equation (13).5 In Appendix B, we show that the scaled return innovation syy(t) is distributed inverse Gaussian under the risk-neutral measure with variance shh(t), where sy=1−2c1ξ−2ηφ,sh=1−2a1ξη4sy−3/2. (14) Inserting these definitions into the IG-GARCH(2,2) dynamics in (1) yields the following risk-neutral process: ln(S(t+Δ))=ln(S(t))+r+μ∗h∗(t+Δ)+η∗y∗(t+Δ),h∗(t+Δ)=w∗+b1h∗(t)+b2h∗(t−Δ)+c1∗y(t)+c2∗y(t−Δ)+a1∗h∗(t)2/y∗(t)+a2∗h∗(t−Δ)2/y∗(t−Δ), (15) where h∗(t)=shh(t), y∗(t)=syy(t), (16a) μ∗=μ/sh, η∗=η/sy, w∗=shw, (16b) ai∗=syai/sh, ci*=shci/sy. (16c) The risk-neutral return process is IG-GARCH because the innovation y*(t+Δ) has an inverse Gaussian distribution under the risk-neutral measure. Notice that b1 and b2 are identical in the physical and risk-neutral processes. The risk-neutral process also can be written as a component model, and Appendix C contains the details. 2.2 Preference parameters and risk-neutral parameters Note that the risk neutralization is specified for convenience in terms of the two reduced-form preference parameters sh and sy. It is worth emphasizing that in fact only one extra parameter is required to convert physical to risk-neutral parameters. The martingale restriction for the risk-neutral dynamics is given by μ∗=1−2η∗−1η∗2. (17) This imposes the following restriction between the physical parameters μ and the preference parameters φ and ξ μ=sh1−2η/sy−1η2/sy2=1−2a1ξη41−2c1ξ−2ηφ−2η−1−2c1ξ−2ηφη2. (18) Given the physical parameters and the value of ξ (or sy), we can thus recover the value of the risk aversion parameter φ (or sh). In other words, it takes only one additional parameter to convert between physical and risk-neutral parameters. To see this, alternatively rewrite these restrictions as sy=(12μ2η4+(1−2a1ξη4)η)2(1−2a1ξη4)μ2η4, (19) sh=μη2sy2(1−2η/sy−1). (20) Because sh is now a function only of sy and physical parameters, this demonstrates that we can write (16a)-(16c) as a function of the physical parameters and one additional parameter, either the reduced form parameter sy or the preference parameter ξ. 2.3 Nested option models The full risk-neutral valuation model has two components with inverse Gaussian innovations. This model contains a number of simpler models as special cases. First consider the Gaussian limit of the risk-neutral dynamic. In the limit, as η approaches zero, μ˜∗=μ∗+η∗−1 approaches −12 . Also in this limit, sh=sy−1 as seen from Equation (14). The risk-neutral process therefore converges to ln(S(t+Δ))=ln(S(t))+r−12h∗(t+Δ)+h∗(t+Δ)z∗(t+Δ),h∗(t+Δ)=ω∗+β1h∗(t)+β2h∗(t−Δ)+α1∗(z(t)−γ1∗h∗(t))2+α2∗(z∗(t−Δ)−γ2∗h∗(t−Δ))2, where z∗(t+Δ)=z(t+Δ)sh+(μ˜sh+sh2)h(t+Δ),ω∗=shω, αi*=sh2αi, γi*=γi+μ˜sh+12. This is the GARCH(2,2) generalization of the risk-neutral version of the Gaussian GARCH(1,1) model studied in Christoffersen, Heston, and Jacobs (2013). Following our previous analysis in Equation (5), one may alternatively express this as the risk-neutral Gaussian component model. Further setting ξ = 0, or equivalently sh = 1, we retrieve the GARCH(2,2) version of the Heston-Nandi (2000) model. Finally, the risk-neutral versions of the GARCH(1,1) models are straightforwardly obtained by setting the appropriate parameters to zero, similar to the restrictions for the physical dynamics discussed in Section 1. 2.4 Option valuation Option valuation with this model is straightforward. Following Heston and Nandi (2000), the value of a European call option at time t with strike price X maturing at T is equal to Call(S(t),h(t+Δ),X,T)=S(t)(12+ exp−r(T−t)π∫0∞Re[X−iϕgt∗(iϕ+1,T)iϕS(t)]dϕ)−X exp −r(T−t)(12+1π∫0∞Re[X−iϕgt∗(iϕ,T)iϕ]dϕ). (23) where gt∗(ϕ,T) is the conditional generating function for the risk-neutral process in (15). The conditional generating function gt(ϕ,T) under the physical measure is given by gt(φ,T)=Et[S(T)ϕ]=S(t)φ exp (A(t)+B(t)h(t+Δ)+C(t)q(t+Δ)), (24) where A(T)=B(T)=C(T)=0, (25) A(t)=A(t+Δ)+φr+(wq−ahη4−aqη4)B(t+Δ)+(wq−aqη4)C(t+Δ)−12ln(1−2(ah+aq)η4B(t+Δ)−2aqη4C(t+Δ)), (26) B(t)=φμ+(ρ1−(ch+cq)η−2−(ah+aq)η2)B(t+Δ)−(cqη−2+aqη2)C(t+Δ)+η−2−(1−2(aq+ah)η4B(t+Δ)−2aqη4C(t+Δ))(1−2ηφ−2(cq+ch)B(t+Δ)−2cqC(t+Δ))η2, (27) C(t)=(ρ2−ρ1)B(t+Δ)+ρ2C(t+Δ). (28) This recursive definition requires computing Equations (26-28) period-by-period with the terminal condition in (25) and then integrating gt(φ,T) like in (23). Note that in Equations (26)–(28) risk-neutral parameters should be used when valuing options. Note also that we have supplied the conditional generating function for the IG-GARCH(C) model. The corresponding functions for the nested models can be obtained as special cases of this function using the results above. Put options can be valued using put-call parity. Armed with the formulas for computing option values, we are now ready to embark on an empirical investigation of our model. 3. Data and Estimation 3.1 Data Our empirical analysis uses out-of-the-money S&P500 call and put options with a maturity between 14 and 365 days for the period January 10, 1996 through December 26, 2012 (in sample), and the period January 2, 2013 through December 30, 2015 (out of sample). We apply the filters proposed by Bakshi, Cao, and Chen (1997) as well as other consistency checks. Rather than using a short time series of daily option data, we use an extended time period, but we select option contracts for one day per week only. This choice is motivated by two constraints. On the one hand, it is important to use as long a time period as possible, in order to be able to identify key aspects of the model including volatility persistence. See, for instance, Broadie, Chernov, and Johannes (2007) for a discussion. On the other hand, despite the numerical efficiency of our model, the optimization problems we conduct are very time intensive, because we use very large panels of option contracts. Selecting one day per week over a long time period is therefore a useful compromise. We use Wednesday data, because it is the day of the week least likely to be a holiday. It is also less likely than other days such as Monday and Friday to be affected by day-of-the-week effects. Moreover, following the work of Dumas, Fleming and Whaley (1998) and Heston and Nandi (2000), several studies have used a long time series of Wednesday contracts. The first Wednesday available in the OptionMetrics database is January 10, 1996. Panel A in Table 1 presents descriptive statistics for three return samples. The return samples are constructed from the S&P500 index returns. The first return sample dates from January 1, 1990 through December 31, 2012. The standard deviation of returns, at 18.61%, is substantially smaller than the average option-implied volatility, at 22.47%. The higher moments of the return sample are consistent with return data in most historical time periods, with a small negative skewness and substantial excess kurtosis. Table 1 also presents descriptive statistics for the return sample from January 10, 1996 through December 26, 2012, that matches the option sample. In comparison to the 1990-2012 sample, the standard deviation is somewhat higher, and average returns are somewhat lower. Average skewness and kurtosis in 1996-2012 are quite similar to the 1990-2012 sample. Finally, panel A in Table 1 also presents descriptive statistics for the 2013-2015 out-of-sample period. This sample has a higher average return, lower volatility, and lower excess kurtosis than the in-sample period 1996-2012. Table 1 Returns and options data A. Return characteristics 1990-2012 1996-2012 2013-2015 Mean (annual) 6.06% 4.99% 12.00% SD (annual) 18.61% 20.57% 12.82% Skewness −0.228 −0.217 −0.354 Excess kurtosis 8.461 7.235 5.217 No. of returns 5,797 4,280 756 No. of options 29,022 9,557 B. Option data by moneyness, in sample F/X ≤ .80 .80 ≤ F/X ≤ .90 .90 ≤ F/X ≤ 1.00 1.00 ≤ F/X ≤ 1.10 1.10 ≤ F/X ≤ 1.20 F/X≥1.20 All Number of contracts 720 3,819 8,413 8,033 5,778 2,259 29,022 Average IV 23.11% 19.65% 18.79% 22.09% 27.03% 30.52% 22.47% Average price 62.94 40.71 43.62 47.93 33.18 28.14 41.63 Average spread 1.30 1.42 1.89 2.06 1.58 1.41 1.76 C. Option data by maturity, in sample DTM ≤ 30 30 ≤ DTM ≤ 60 60 ≤ DTM ≤ 90 90 ≤ DTM ≤ 120 120 ≤ DTM ≤ 180 DTM≥180 All Number of contracts 2,771 6,127 4,565 2,720 4,019 8,820 29,022 Average IV 24.93% 23.33% 22.45% 22.93% 21.74% 21.32% 22.47% Average price 18.06 26.83 31.88 39.29 48.54 61.92 41.63 Average spread 0.94 1.31 1.59 1.87 1.97 2.29 1.76 D. Option data by moneyness, out of sample F/X ≤ .80 .80 ≤ F/X ≤ .90 .90 ≤ F/X ≤ 1.00 1.00 ≤ F/X ≤ 1.10 1.10 ≤ F/X ≤ 1.20 F/X≥1.20 All Number of contracts 67 853 2,972 2,366 2,350 949 9,557 Average IV 12.32% 11.85% 12.43% 17.33% 23.17% 26.56% 17.63% Average price 1.03 4.90 24.96 36.48 13.53 8.66 21.42 Average spread 0.66 0.95 1.14 1.54 1.15 1.01 1.21 E. Option data by maturity, out of sample DTM ≤ 30 30 ≤ DTM ≤ 60 60 ≤ DTM ≤ 90 90 ≤ DTM ≤ 120 120 ≤ DTM ≤ 180 DTM≥180 All Number of contracts 1,819 2,016 1,490 994 1,127 2,111 9,557 Average IV 20.01% 17.93% 17.27% 17.03% 16.59% 16.41% 17.63% Average price 6.77 11.30 16.57 21.27 27.91 43.75 21.42 Average spread 0.42 0.76 1.02 1.28 1.48 2.26 1.21 A. Return characteristics 1990-2012 1996-2012 2013-2015 Mean (annual) 6.06% 4.99% 12.00% SD (annual) 18.61% 20.57% 12.82% Skewness −0.228 −0.217 −0.354 Excess kurtosis 8.461 7.235 5.217 No. of returns 5,797 4,280 756 No. of options 29,022 9,557 B. Option data by moneyness, in sample F/X ≤ .80 .80 ≤ F/X ≤ .90 .90 ≤ F/X ≤ 1.00 1.00 ≤ F/X ≤ 1.10 1.10 ≤ F/X ≤ 1.20 F/X≥1.20 All Number of contracts 720 3,819 8,413 8,033 5,778 2,259 29,022 Average IV 23.11% 19.65% 18.79% 22.09% 27.03% 30.52% 22.47% Average price 62.94 40.71 43.62 47.93 33.18 28.14 41.63 Average spread 1.30 1.42 1.89 2.06 1.58 1.41 1.76 C. Option data by maturity, in sample DTM ≤ 30 30 ≤ DTM ≤ 60 60 ≤ DTM ≤ 90 90 ≤ DTM ≤ 120 120 ≤ DTM ≤ 180 DTM≥180 All Number of contracts 2,771 6,127 4,565 2,720 4,019 8,820 29,022 Average IV 24.93% 23.33% 22.45% 22.93% 21.74% 21.32% 22.47% Average price 18.06 26.83 31.88 39.29 48.54 61.92 41.63 Average spread 0.94 1.31 1.59 1.87 1.97 2.29 1.76 D. Option data by moneyness, out of sample F/X ≤ .80 .80 ≤ F/X ≤ .90 .90 ≤ F/X ≤ 1.00 1.00 ≤ F/X ≤ 1.10 1.10 ≤ F/X ≤ 1.20 F/X≥1.20 All Number of contracts 67 853 2,972 2,366 2,350 949 9,557 Average IV 12.32% 11.85% 12.43% 17.33% 23.17% 26.56% 17.63% Average price 1.03 4.90 24.96 36.48 13.53 8.66 21.42 Average spread 0.66 0.95 1.14 1.54 1.15 1.01 1.21 E. Option data by maturity, out of sample DTM ≤ 30 30 ≤ DTM ≤ 60 60 ≤ DTM ≤ 90 90 ≤ DTM ≤ 120 120 ≤ DTM ≤ 180 DTM≥180 All Number of contracts 1,819 2,016 1,490 994 1,127 2,111 9,557 Average IV 20.01% 17.93% 17.27% 17.03% 16.59% 16.41% 17.63% Average price 6.77 11.30 16.57 21.27 27.91 43.75 21.42 Average spread 0.42 0.76 1.02 1.28 1.48 2.26 1.21 We present descriptive statistics for daily return data from January 2, 1990 through December 31, 2012, for daily return data from January 10, 1996 through December 26, 2012, and for daily return data from January 2, 2013 to December 31, 2015. We use Wednesday closing options contracts from January 10, 1996 through December 26, 2012 as our in-sample option data set, and January 2, 2013 through December 30, 2015 as our out-of-sample option data set. Table 1 Returns and options data A. Return characteristics 1990-2012 1996-2012 2013-2015 Mean (annual) 6.06% 4.99% 12.00% SD (annual) 18.61% 20.57% 12.82% Skewness −0.228 −0.217 −0.354 Excess kurtosis 8.461 7.235 5.217 No. of returns 5,797 4,280 756 No. of options 29,022 9,557 B. Option data by moneyness, in sample F/X ≤ .80 .80 ≤ F/X ≤ .90 .90 ≤ F/X ≤ 1.00 1.00 ≤ F/X ≤ 1.10 1.10 ≤ F/X ≤ 1.20 F/X≥1.20 All Number of contracts 720 3,819 8,413 8,033 5,778 2,259 29,022 Average IV 23.11% 19.65% 18.79% 22.09% 27.03% 30.52% 22.47% Average price 62.94 40.71 43.62 47.93 33.18 28.14 41.63 Average spread 1.30 1.42 1.89 2.06 1.58 1.41 1.76 C. Option data by maturity, in sample DTM ≤ 30 30 ≤ DTM ≤ 60 60 ≤ DTM ≤ 90 90 ≤ DTM ≤ 120 120 ≤ DTM ≤ 180 DTM≥180 All Number of contracts 2,771 6,127 4,565 2,720 4,019 8,820 29,022 Average IV 24.93% 23.33% 22.45% 22.93% 21.74% 21.32% 22.47% Average price 18.06 26.83 31.88 39.29 48.54 61.92 41.63 Average spread 0.94 1.31 1.59 1.87 1.97 2.29 1.76 D. Option data by moneyness, out of sample F/X ≤ .80 .80 ≤ F/X ≤ .90 .90 ≤ F/X ≤ 1.00 1.00 ≤ F/X ≤ 1.10 1.10 ≤ F/X ≤ 1.20 F/X≥1.20 All Number of contracts 67 853 2,972 2,366 2,350 949 9,557 Average IV 12.32% 11.85% 12.43% 17.33% 23.17% 26.56% 17.63% Average price 1.03 4.90 24.96 36.48 13.53 8.66 21.42 Average spread 0.66 0.95 1.14 1.54 1.15 1.01 1.21 E. Option data by maturity, out of sample DTM ≤ 30 30 ≤ DTM ≤ 60 60 ≤ DTM ≤ 90 90 ≤ DTM ≤ 120 120 ≤ DTM ≤ 180 DTM≥180 All Number of contracts 1,819 2,016 1,490 994 1,127 2,111 9,557 Average IV 20.01% 17.93% 17.27% 17.03% 16.59% 16.41% 17.63% Average price 6.77 11.30 16.57 21.27 27.91 43.75 21.42 Average spread 0.42 0.76 1.02 1.28 1.48 2.26 1.21 A. Return characteristics 1990-2012 1996-2012 2013-2015 Mean (annual) 6.06% 4.99% 12.00% SD (annual) 18.61% 20.57% 12.82% Skewness −0.228 −0.217 −0.354 Excess kurtosis 8.461 7.235 5.217 No. of returns 5,797 4,280 756 No. of options 29,022 9,557 B. Option data by moneyness, in sample F/X ≤ .80 .80 ≤ F/X ≤ .90 .90 ≤ F/X ≤ 1.00 1.00 ≤ F/X ≤ 1.10 1.10 ≤ F/X ≤ 1.20 F/X≥1.20 All Number of contracts 720 3,819 8,413 8,033 5,778 2,259 29,022 Average IV 23.11% 19.65% 18.79% 22.09% 27.03% 30.52% 22.47% Average price 62.94 40.71 43.62 47.93 33.18 28.14 41.63 Average spread 1.30 1.42 1.89 2.06 1.58 1.41 1.76 C. Option data by maturity, in sample DTM ≤ 30 30 ≤ DTM ≤ 60 60 ≤ DTM ≤ 90 90 ≤ DTM ≤ 120 120 ≤ DTM ≤ 180 DTM≥180 All Number of contracts 2,771 6,127 4,565 2,720 4,019 8,820 29,022 Average IV 24.93% 23.33% 22.45% 22.93% 21.74% 21.32% 22.47% Average price 18.06 26.83 31.88 39.29 48.54 61.92 41.63 Average spread 0.94 1.31 1.59 1.87 1.97 2.29 1.76 D. Option data by moneyness, out of sample F/X ≤ .80 .80 ≤ F/X ≤ .90 .90 ≤ F/X ≤ 1.00 1.00 ≤ F/X ≤ 1.10 1.10 ≤ F/X ≤ 1.20 F/X≥1.20 All Number of contracts 67 853 2,972 2,366 2,350 949 9,557 Average IV 12.32% 11.85% 12.43% 17.33% 23.17% 26.56% 17.63% Average price 1.03 4.90 24.96 36.48 13.53 8.66 21.42 Average spread 0.66 0.95 1.14 1.54 1.15 1.01 1.21 E. Option data by maturity, out of sample DTM ≤ 30 30 ≤ DTM ≤ 60 60 ≤ DTM ≤ 90 90 ≤ DTM ≤ 120 120 ≤ DTM ≤ 180 DTM≥180 All Number of contracts 1,819 2,016 1,490 994 1,127 2,111 9,557 Average IV 20.01% 17.93% 17.27% 17.03% 16.59% 16.41% 17.63% Average price 6.77 11.30 16.57 21.27 27.91 43.75 21.42 Average spread 0.42 0.76 1.02 1.28 1.48 2.26 1.21 We present descriptive statistics for daily return data from January 2, 1990 through December 31, 2012, for daily return data from January 10, 1996 through December 26, 2012, and for daily return data from January 2, 2013 to December 31, 2015. We use Wednesday closing options contracts from January 10, 1996 through December 26, 2012 as our in-sample option data set, and January 2, 2013 through December 30, 2015 as our out-of-sample option data set. Panels B and C of Table 1 present descriptive statistics for the in-sample option data by moneyness and maturity. Moneyness is defined as the implied futures price F divided by strike price X. When F / X is smaller than one, the contract is an out-of-the-money (OTM) call, and when F / X is larger than one, the contract is an OTM put. The out-of-the-money put prices are converted into call prices using put-call parity. The sample includes a total of 29, 022 option contracts with an average mid-price of 41.63 and average implied volatility of 22.47% as noted above. The implied volatility is largest for the OTM put options in panel B, reflecting the well-known volatility smirk in index options. The implied volatility term structure in panel C is slightly downward sloping, on average, during the sample period. Panels D and E of Table 1 present the corresponding statistics for the 2013-2015 out-of-sample period. Although the average implied volatility is lower, the strong volatility smirk and slightly downward-sloping average term structure are evident in the out-of-sample period as well. 3.2 Estimation We now present a detailed empirical investigation of the models outlined in Section 1. We can separately evaluate the model’s ability to describe return dynamics and to fit option prices. But the model’s ability to capture the differences between the physical and risk-neutral distributions requires fitting both return and option data using the same, internally consistent, set of parameters. We first use an estimation exercise that fits options and returns separately. We also employ sequential estimation following Broadie, Chernov, and Johannes (2007), who first estimate each model on returns only and subsequently assess the fit of each model to option prices in a second step where only risk-premium parameters are estimated. This procedure is also used by Christoffersen, Heston, and Jacobs (2013) in the context of a Gaussian GARCH(1,1) model with a quadratic pricing kernel. First consider returns, which are defined by R(t)≡lnS(t)−lnS(t−Δ) . In the inverse Gaussian case, the conditional density of the daily return is f(R(t)|h(t))=h(t)|η|−32π[R(t)−r−μ˜h(t)]3η−3× exp (−12(R(t)−r−μ˜h(t)η−h(t)η2ηR(t)−r−μ˜h(t))2). The return log-likelihood is summed over all return dates. lnLR∝∑t=1T{ln(f(R(t)|h(t)))}. (29) We can therefore obtain the physical parameters Θ by estimating ΘReturn=argmaxlnLRΘ. (30) Now consider the options data. Define the Black-Scholes Vega (BSV) weighted option valuation errors as ɛi=(CalliMkt−CalliMod)/BSViMkt, where CalliMkt represents the market price of the ith option, CalliMod represents the model price, and BSViMkt represents the Black-Scholes vega of the option (the derivative with respect to volatility) at the market implied level of volatility. Assume these disturbances are i.i.d. normal so that the option log-likelihood is lnLO∝−12∑i=1N{ln(sɛ2)+ɛi2/sɛ2}, (31) where we can concentrate out sɛ2 using the sample analogue s^ɛ2=1N∑i=1Nɛi2 . We use the term structure of interest rates from OptionMetrics when pricing options. The vega-weighted option errors are very useful because it can be shown that they are an approximation to implied volatility based errors, which have desirable statistical properties. Unlike implied volatility errors, they do not require Black-Scholes inversion of model prices at every step in the optimization, which is very costly in large scale empirical estimation exercises such as ours.6 We obtain the risk-neutral parameters Θ* based on options data by estimating ΘOption∗=argmaxlnLOΘ∗. (32) Note that both estimation exercises mentioned above ignore the specification of the pricing kernel, and are therefore uninformative about the choice between the log-linear and nonmonotonic pricing kernels. We thus conduct a third estimation exercise where we sequentially estimate the nonmonotonic pricing kernel parameter, ξ, on options only, keeping all the physical parameters from (30) fixed. We thus estimate ξSeq=argmaxlnLOξ. (33) Sequential estimation is of course only conducted for the models with nonmonotonic pricing kernels. Our sequential estimation approach follows that in Broadie, Chernov, and Johannes (2007) and Christoffersen, Heston, and Jacobs (2013). 4. Empirical Results Because our specification nests several models, it allows for a comparison of the relative importance of model features. Specifically, we can compare the contribution of a second dynamic volatility factor, fat-tailed innovations, and a nonmonotonic (or variance-dependent) pricing kernel. We can quantify the contribution of these features in separately explaining the time series of returns and the cross-section of option prices, as well as in explaining returns and options together, which we do in a sequential estimation exercise. Although a horse race based on model fit is of interest, it is also necessary to verify whether the different model features are complements rather than substitutes. In theory this should be the case: the second volatility factor should improve the modeling of the term structure of volatility, and therefore the valuation of options of different maturities, especially long-maturity options. In contrast, the fat-tailed IG innovation should prove most useful to capture the moneyness dimension for short-maturity out-of-the-money options, usually referred to as a smirk. The nonmonotonic pricing kernel has an entirely different purpose, because its relevance lies in the joint modeling of index returns and options, rather than the modeling of options alone. Tables 2–8 present the empirical results. Table 2 presents estimation results using returns data. The results include parameter estimates and log-likelihoods, as well as several implications of the parameter estimates such as moments and persistence. Table 3 presents similar results for the estimation based on option data, and Table 4 does the same for the sequential estimation based first on returns and subsequently on options. Table 4 also reports the improvement in fit for the nonmonotonic pricing kernel over the linear pricing kernel in terms of log-likelihood values. Tables 5 and 6 provide more details on the models’ fit across moneyness and maturity categories for the three estimation exercises in Tables 2-4. Finally, Table 7 presents information criterion values for the different Models, and Table 8 contains out-of-sample results. Table 2 Maximum likelihood estimation results from returns A. Parameter estimates Gaussian models μ˜ ω Homoscedastic 0.78 1.373E-04 (1.123) (1.12E-6) μ˜ ω β1 α1 γ1 GARCH(1,1) 1.10 −1.396E-06 0.900 3.761E-06 145.7 (1.127) (1.35E-7) (0.008) (2.30E-7) (10.182) μ˜ ωq ρ1 αh γh ρ2 αq γq GARCH(C) 1.26 1.473E-06 0.705 9.979E-07 840.6 0.987 2.832E-06 118.7 (1.128) (1.53E-7) (0.031) (4.32E-7) (2.70E+6) (0.002) (2.20E-7) (10.617) IG models μ˜ w η Homoscedastic 0.78 1.372E-04 −1.279E-04 (1.233) (1.13E-6) (1.64E-5) μ˜ w b1 a1 c1 η IG-GARCH(1,1) 1.16 −1.469E-06 −21.82 3.190E+07 4.047E-06 −5.729E-04 (1.127) (1.28E-7) (3.885) (1.03E+7) (2.15E-7) (4.78E-5) μ˜ wq ρ1 ah ch ρ2 aq cq η IG-GARCH(C) 1.23 1.393E-06 0.743 2.247E+06 6.987E-07 0.988 7.101E+07 3.153E-06 −4.469E-04 (1.116) (1.33E-7) (0.030) (9.17E+6) (4.81E-7) (0.001) (2.42E+7) (2.04E-7) (4.52E-5) B. Model properties and likelihoods C. p-values from Vuong-Shi tests Return Annualized Volatility Uncond. Uncond. Log Model under the null hypothesis mean volatility persistence skewness kurtosis likelihood GARCH(1,1) IG-GARCH(1,1) GARCH(C) Gaussian models Homoscedastic 5.99% 18.60% 0.000 3.000 17,548.0 GARCH(1,1) 6.48% 17.00% 0.979387 0.015 4.750 18,781.4 GARCH(C) 6.91% 16.91% 0.996170 0.024 5.199 18,831.4 0.0004 0.5000 IG models Homoscedastic 5.99% 18.59% −0.033 3.000 17,551.5 IG-GARCH(1,1) 6.61% 16.92% 0.982695 −0.152 4.775 18,829.5 0.0037 IG-GARCH(C) 6.78% 16.81% 0.996802 −0.099 5.247 18,865.0 0.0276 0.0349 0.0024 A. Parameter estimates Gaussian models μ˜ ω Homoscedastic 0.78 1.373E-04 (1.123) (1.12E-6) μ˜ ω β1 α1 γ1 GARCH(1,1) 1.10 −1.396E-06 0.900 3.761E-06 145.7 (1.127) (1.35E-7) (0.008) (2.30E-7) (10.182) μ˜ ωq ρ1 αh γh ρ2 αq γq GARCH(C) 1.26 1.473E-06 0.705 9.979E-07 840.6 0.987 2.832E-06 118.7 (1.128) (1.53E-7) (0.031) (4.32E-7) (2.70E+6) (0.002) (2.20E-7) (10.617) IG models μ˜ w η Homoscedastic 0.78 1.372E-04 −1.279E-04 (1.233) (1.13E-6) (1.64E-5) μ˜ w b1 a1 c1 η IG-GARCH(1,1) 1.16 −1.469E-06 −21.82 3.190E+07 4.047E-06 −5.729E-04 (1.127) (1.28E-7) (3.885) (1.03E+7) (2.15E-7) (4.78E-5) μ˜ wq ρ1 ah ch ρ2 aq cq η IG-GARCH(C) 1.23 1.393E-06 0.743 2.247E+06 6.987E-07 0.988 7.101E+07 3.153E-06 −4.469E-04 (1.116) (1.33E-7) (0.030) (9.17E+6) (4.81E-7) (0.001) (2.42E+7) (2.04E-7) (4.52E-5) B. Model properties and likelihoods C. p-values from Vuong-Shi tests Return Annualized Volatility Uncond. Uncond. Log Model under the null hypothesis mean volatility persistence skewness kurtosis likelihood GARCH(1,1) IG-GARCH(1,1) GARCH(C) Gaussian models Homoscedastic 5.99% 18.60% 0.000 3.000 17,548.0 GARCH(1,1) 6.48% 17.00% 0.979387 0.015 4.750 18,781.4 GARCH(C) 6.91% 16.91% 0.996170 0.024 5.199 18,831.4 0.0004 0.5000 IG models Homoscedastic 5.99% 18.59% −0.033 3.000 17,551.5 IG-GARCH(1,1) 6.61% 16.92% 0.982695 −0.152 4.775 18,829.5 0.0037 IG-GARCH(C) 6.78% 16.81% 0.996802 −0.099 5.247 18,865.0 0.0276 0.0349 0.0024 Panel A reports parameter estimates obtained by an MLE estimation on returns from January 2, 1990 through December 31, 2012. Table 1 provides a description of the data. Robust standard errors (based on the outer product of gradients) are in parantheses below the parameter estimates. Panel B reports the maximum log-likelihood values and some model properties. Panel C reports p-values from Vuong-Shi tests comparing the alternative model in the row to the model under the null hypothesis in the column. We estimate six models. Each model has constant or time-varying volatility (which is either two components or one), and Normal or IG innovations. Table 2 Maximum likelihood estimation results from returns A. Parameter estimates Gaussian models μ˜ ω Homoscedastic 0.78 1.373E-04 (1.123) (1.12E-6) μ˜ ω β1 α1 γ1 GARCH(1,1) 1.10 −1.396E-06 0.900 3.761E-06 145.7 (1.127) (1.35E-7) (0.008) (2.30E-7) (10.182) μ˜ ωq ρ1 αh γh ρ2 αq γq GARCH(C) 1.26 1.473E-06 0.705 9.979E-07 840.6 0.987 2.832E-06 118.7 (1.128) (1.53E-7) (0.031) (4.32E-7) (2.70E+6) (0.002) (2.20E-7) (10.617) IG models μ˜ w η Homoscedastic 0.78 1.372E-04 −1.279E-04 (1.233) (1.13E-6) (1.64E-5) μ˜ w b1 a1 c1 η IG-GARCH(1,1) 1.16 −1.469E-06 −21.82 3.190E+07 4.047E-06 −5.729E-04 (1.127) (1.28E-7) (3.885) (1.03E+7) (2.15E-7) (4.78E-5) μ˜ wq ρ1 ah ch ρ2 aq cq η IG-GARCH(C) 1.23 1.393E-06 0.743 2.247E+06 6.987E-07 0.988 7.101E+07 3.153E-06 −4.469E-04 (1.116) (1.33E-7) (0.030) (9.17E+6) (4.81E-7) (0.001) (2.42E+7) (2.04E-7) (4.52E-5) B. Model properties and likelihoods C. p-values from Vuong-Shi tests Return Annualized Volatility Uncond. Uncond. Log Model under the null hypothesis mean volatility persistence skewness kurtosis likelihood GARCH(1,1) IG-GARCH(1,1) GARCH(C) Gaussian models Homoscedastic 5.99% 18.60% 0.000 3.000 17,548.0 GARCH(1,1) 6.48% 17.00% 0.979387 0.015 4.750 18,781.4 GARCH(C) 6.91% 16.91% 0.996170 0.024 5.199 18,831.4 0.0004 0.5000 IG models Homoscedastic 5.99% 18.59% −0.033 3.000 17,551.5 IG-GARCH(1,1) 6.61% 16.92% 0.982695 −0.152 4.775 18,829.5 0.0037 IG-GARCH(C) 6.78% 16.81% 0.996802 −0.099 5.247 18,865.0 0.0276 0.0349 0.0024 A. Parameter estimates Gaussian models μ˜ ω Homoscedastic 0.78 1.373E-04 (1.123) (1.12E-6) μ˜ ω β1 α1 γ1 GARCH(1,1) 1.10 −1.396E-06 0.900 3.761E-06 145.7 (1.127) (1.35E-7) (0.008) (2.30E-7) (10.182) μ˜ ωq ρ1 αh γh ρ2 αq γq GARCH(C) 1.26 1.473E-06 0.705 9.979E-07 840.6 0.987 2.832E-06 118.7 (1.128) (1.53E-7) (0.031) (4.32E-7) (2.70E+6) (0.002) (2.20E-7) (10.617) IG models μ˜ w η Homoscedastic 0.78 1.372E-04 −1.279E-04 (1.233) (1.13E-6) (1.64E-5) μ˜ w b1 a1 c1 η IG-GARCH(1,1) 1.16 −1.469E-06 −21.82 3.190E+07 4.047E-06 −5.729E-04 (1.127) (1.28E-7) (3.885) (1.03E+7) (2.15E-7) (4.78E-5) μ˜ wq ρ1 ah ch ρ2 aq cq η IG-GARCH(C) 1.23 1.393E-06 0.743 2.247E+06 6.987E-07 0.988 7.101E+07 3.153E-06 −4.469E-04 (1.116) (1.33E-7) (0.030) (9.17E+6) (4.81E-7) (0.001) (2.42E+7) (2.04E-7) (4.52E-5) B. Model properties and likelihoods C. p-values from Vuong-Shi tests Return Annualized Volatility Uncond. Uncond. Log Model under the null hypothesis mean volatility persistence skewness kurtosis likelihood GARCH(1,1) IG-GARCH(1,1) GARCH(C) Gaussian models Homoscedastic 5.99% 18.60% 0.000 3.000 17,548.0 GARCH(1,1) 6.48% 17.00% 0.979387 0.015 4.750 18,781.4 GARCH(C) 6.91% 16.91% 0.996170 0.024 5.199 18,831.4 0.0004 0.5000 IG models Homoscedastic 5.99% 18.59% −0.033 3.000 17,551.5 IG-GARCH(1,1) 6.61% 16.92% 0.982695 −0.152 4.775 18,829.5 0.0037 IG-GARCH(C) 6.78% 16.81% 0.996802 −0.099 5.247 18,865.0 0.0276 0.0349 0.0024 Panel A reports parameter estimates obtained by an MLE estimation on returns from January 2, 1990 through December 31, 2012. Table 1 provides a description of the data. Robust standard errors (based on the outer product of gradients) are in parantheses below the parameter estimates. Panel B reports the maximum log-likelihood values and some model properties. Panel C reports p-values from Vuong-Shi tests comparing the alternative model in the row to the model under the null hypothesis in the column. We estimate six models. Each model has constant or time-varying volatility (which is either two components or one), and Normal or IG innovations. Table 3 Maximum likelihood estimation results from options A. Parameter estimates Gaussian models μ˜* ω* Homoscedastic −0.50 1.272E-04 (2.34E-7) μ˜* ω* β1* α1* γ1* GARCH(1,1) −0.50 −1.260E-06 0.823 2.931E-06 241.23 (6.99E-9) (1.17E-3) (8.76E-9) (8.78E-1) μ˜* ωq* ρ1* αh* γh* ρ2* αq* γq* GARCH(C) −0.50 2.876E-07 0.981 4.595E-07 2092.58 0.9995 1.582E-06 121.76 (8.89E-9) (1.91E-4) (2.21E-8) (9.97E+1) (3.44E-5) (1.58E-8) (1.94E+0) IG models μ˜* w* η* Homoscedastic −0.48 1.593E-04 −3.241E-02 (2.67E-7) (4.46E-6) μ˜* w* b1* a1* c1* η* IG-GARCH(1,1) −0.50 −1.956E-06 −2.50 4.931E+05 5.841E-06 −1.649E-03 (2.68E-8) (9.41E-3) (5.32E+3) (3.46E-8) (7.59E-6) μ˜* wq* ρ1* ah* ch* ρ2* aq* cq* η* IG-GARCH(C) −0.50 2.753E-07 0.985 3.054E+06 3.123E-06 0.9998 3.426E+06 2.101E-06 −8.630E-04 (2.75E-9) (1.26E-4) (5.36E+4) (2.43E-8) (2.23E-7) (6.15E+4) (1.86E-8) (3.34E-6) B. Model properties and likelihoods C. p-values from Vuong-Shi tests Return Annualized Volatility Uncond. Uncond. Log Model under the null hypothesis mean volatility persistence skewness kurtosis likelihood GARCH(1,1) IG-GARCH(1,1) GARCH(C) Gaussian models Homoscedastic 1.69% 17.90% 0.000 3.000 32,632.1 GARCH(1,1) 0.21% 24.85% 0.993182 −0.017 5.113 53,971.2 GARCH(C) −4.10% 38.45% 0.999990 −0.018 5.485 59,124.7 0.0000 0.0000 IG models Homoscedastic 1.35% 20.04% −0.101 3.699 36,945.6 IG-GARCH(1,1) 0.12% 25.24% 0.993322 −0.592 5.379 55,824.0 0.0130 IG-GARCH(C) −13.15% 57.38% 0.999997 −0.147 5.732 60,715.2 0.0000 0.0000 0.0000 A. Parameter estimates Gaussian models μ˜* ω* Homoscedastic −0.50 1.272E-04 (2.34E-7) μ˜* ω* β1* α1* γ1* GARCH(1,1) −0.50 −1.260E-06 0.823 2.931E-06 241.23 (6.99E-9) (1.17E-3) (8.76E-9) (8.78E-1) μ˜* ωq* ρ1* αh* γh* ρ2* αq* γq* GARCH(C) −0.50 2.876E-07 0.981 4.595E-07 2092.58 0.9995 1.582E-06 121.76 (8.89E-9) (1.91E-4) (2.21E-8) (9.97E+1) (3.44E-5) (1.58E-8) (1.94E+0) IG models μ˜* w* η* Homoscedastic −0.48 1.593E-04 −3.241E-02 (2.67E-7) (4.46E-6) μ˜* w* b1* a1* c1* η* IG-GARCH(1,1) −0.50 −1.956E-06 −2.50 4.931E+05 5.841E-06 −1.649E-03 (2.68E-8) (9.41E-3) (5.32E+3) (3.46E-8) (7.59E-6) μ˜* wq* ρ1* ah* ch* ρ2* aq* cq* η* IG-GARCH(C) −0.50 2.753E-07 0.985 3.054E+06 3.123E-06 0.9998 3.426E+06 2.101E-06 −8.630E-04 (2.75E-9) (1.26E-4) (5.36E+4) (2.43E-8) (2.23E-7) (6.15E+4) (1.86E-8) (3.34E-6) B. Model properties and likelihoods C. p-values from Vuong-Shi tests Return Annualized Volatility Uncond. Uncond. Log Model under the null hypothesis mean volatility persistence skewness kurtosis likelihood GARCH(1,1) IG-GARCH(1,1) GARCH(C) Gaussian models Homoscedastic 1.69% 17.90% 0.000 3.000 32,632.1 GARCH(1,1) 0.21% 24.85% 0.993182 −0.017 5.113 53,971.2 GARCH(C) −4.10% 38.45% 0.999990 −0.018 5.485 59,124.7 0.0000 0.0000 IG models Homoscedastic 1.35% 20.04% −0.101 3.699 36,945.6 IG-GARCH(1,1) 0.12% 25.24% 0.993322 −0.592 5.379 55,824.0 0.0130 IG-GARCH(C) −13.15% 57.38% 0.999997 −0.147 5.732 60,715.2 0.0000 0.0000 0.0000 Panel A reports risk-neutral parameter estimates obtained by MLE estimation on options from January 10, 1996 through December 26, 2012. Table 1 provides a description of the data. Robust standard errors (based on the outer product of gradients) are in parantheses below the parameter estimates. Panel B reports maximum log-likelihood values and some model properties. Panel C reports p-values from Vuong-Shi tests comparing the alternative model in the row to the model under the null hypothesis in the column. We estimate six models using only options data. Each model has constant or time-varying volatility (which is either two components or one), and Normal or IG innovations. Table 3 Maximum likelihood estimation results from options A. Parameter estimates Gaussian models μ˜* ω* Homoscedastic −0.50 1.272E-04 (2.34E-7) μ˜* ω* β1* α1* γ1* GARCH(1,1) −0.50 −1.260E-06 0.823 2.931E-06 241.23 (6.99E-9) (1.17E-3) (8.76E-9) (8.78E-1) μ˜* ωq* ρ1* αh* γh* ρ2* αq* γq* GARCH(C) −0.50 2.876E-07 0.981 4.595E-07 2092.58 0.9995 1.582E-06 121.76 (8.89E-9) (1.91E-4) (2.21E-8) (9.97E+1) (3.44E-5) (1.58E-8) (1.94E+0) IG models μ˜* w* η* Homoscedastic −0.48 1.593E-04 −3.241E-02 (2.67E-7) (4.46E-6) μ˜* w* b1* a1* c1* η* IG-GARCH(1,1) −0.50 −1.956E-06 −2.50 4.931E+05 5.841E-06 −1.649E-03 (2.68E-8) (9.41E-3) (5.32E+3) (3.46E-8) (7.59E-6) μ˜* wq* ρ1* ah* ch* ρ2* aq* cq* η* IG-GARCH(C) −0.50 2.753E-07 0.985 3.054E+06 3.123E-06 0.9998 3.426E+06 2.101E-06 −8.630E-04 (2.75E-9) (1.26E-4) (5.36E+4) (2.43E-8) (2.23E-7) (6.15E+4) (1.86E-8) (3.34E-6) B. Model properties and likelihoods C. p-values from Vuong-Shi tests Return Annualized Volatility Uncond. Uncond. Log Model under the null hypothesis mean volatility persistence skewness kurtosis likelihood GARCH(1,1) IG-GARCH(1,1) GARCH(C) Gaussian models Homoscedastic 1.69% 17.90% 0.000 3.000 32,632.1 GARCH(1,1) 0.21% 24.85% 0.993182 −0.017 5.113 53,971.2 GARCH(C) −4.10% 38.45% 0.999990 −0.018 5.485 59,124.7 0.0000 0.0000 IG models Homoscedastic 1.35% 20.04% −0.101 3.699 36,945.6 IG-GARCH(1,1) 0.12% 25.24% 0.993322 −0.592 5.379 55,824.0 0.0130 IG-GARCH(C) −13.15% 57.38% 0.999997 −0.147 5.732 60,715.2 0.0000 0.0000 0.0000 A. Parameter estimates Gaussian models μ˜* ω* Homoscedastic −0.50 1.272E-04 (2.34E-7) μ˜* ω* β1* α1* γ1* GARCH(1,1) −0.50 −1.260E-06 0.823 2.931E-06 241.23 (6.99E-9) (1.17E-3) (8.76E-9) (8.78E-1) μ˜* ωq* ρ1* αh* γh* ρ2* αq* γq* GARCH(C) −0.50 2.876E-07 0.981 4.595E-07 2092.58 0.9995 1.582E-06 121.76 (8.89E-9) (1.91E-4) (2.21E-8) (9.97E+1) (3.44E-5) (1.58E-8) (1.94E+0) IG models μ˜* w* η* Homoscedastic −0.48 1.593E-04 −3.241E-02 (2.67E-7) (4.46E-6) μ˜* w* b1* a1* c1* η* IG-GARCH(1,1) −0.50 −1.956E-06 −2.50 4.931E+05 5.841E-06 −1.649E-03 (2.68E-8) (9.41E-3) (5.32E+3) (3.46E-8) (7.59E-6) μ˜* wq* ρ1* ah* ch* ρ2* aq* cq* η* IG-GARCH(C) −0.50 2.753E-07 0.985 3.054E+06 3.123E-06 0.9998 3.426E+06 2.101E-06 −8.630E-04 (2.75E-9) (1.26E-4) (5.36E+4) (2.43E-8) (2.23E-7) (6.15E+4) (1.86E-8) (3.34E-6) B. Model properties and likelihoods C. p-values from Vuong-Shi tests Return Annualized Volatility Uncond. Uncond. Log Model under the null hypothesis mean volatility persistence skewness kurtosis likelihood GARCH(1,1) IG-GARCH(1,1) GARCH(C) Gaussian models Homoscedastic 1.69% 17.90% 0.000 3.000 32,632.1 GARCH(1,1) 0.21% 24.85% 0.993182 −0.017 5.113 53,971.2 GARCH(C) −4.10% 38.45% 0.999990 −0.018 5.485 59,124.7 0.0000 0.0000 IG models Homoscedastic 1.35% 20.04% −0.101 3.699 36,945.6 IG-GARCH(1,1) 0.12% 25.24% 0.993322 −0.592 5.379 55,824.0 0.0130 IG-GARCH(C) −13.15% 57.38% 0.999997 −0.147 5.732 60,715.2 0.0000 0.0000 0.0000 Panel A reports risk-neutral parameter estimates obtained by MLE estimation on options from January 10, 1996 through December 26, 2012. Table 1 provides a description of the data. Robust standard errors (based on the outer product of gradients) are in parantheses below the parameter estimates. Panel B reports maximum log-likelihood values and some model properties. Panel C reports p-values from Vuong-Shi tests comparing the alternative model in the row to the model under the null hypothesis in the column. We estimate six models using only options data. Each model has constant or time-varying volatility (which is either two components or one), and Normal or IG innovations. Table 4 Sequential maximum likelihood estimation A. Risk-neutral parameters based onTable 2and ξ estimates Gaussian models μ˜* ω* β1* α1* γ1* ξ sh GARCH(1,1) −0.50 −1.641E-06 0.900 5.193E-06 125.89 19791.7 1.1749 (8.13E+1) μ˜* ωq* ρ1* αh* γh* ρ2* αq* γq* ξ sh GARCH(C) −0.50 2.445E-06 0.708 1.369E-06 725.92 0.9883 4.081E-06 102.02 21122.1 1.1930 (8.17E+1) IG models μ˜* w* b1* a1* c1* η* ξ sh sy IG-GARCH(1,1) −0.50 −1.768E-06 −21.82 2.205E+07 5.854E-06 −6.886E-04 24225.6 1.2033 0.8319 (9.05E+1) μ˜* wq* ρ1* ah* ch* ρ2* aq* cq* η* ξ sh sy IG-GARCH(C) −0.50 2.415E-06 0.745 1.033E+06 9.682E-07 0.989 4.911E+07 4.660E-06 −5.399E-04 29269.1 1.2092 0.8276 (9.55E+1) B. Model properties and likelihoods C. p-values from Vuong-Shi tests Return Annualized Volatility Uncond. Uncond. Log LL increase Model under the null hypothesis mean volatility persistence skewness kurtosis likelihood NK over MK GARCH(1,1) IG-GARCH(1,1) GARCH(C) Gaussian models GARCH(1,1) 0.84% 22.18% 0.981805 −0.013 4.745 48,065.9 6,643.5 GARCH(C) 0.67% 22.94% 0.996582 −0.017 5.190 51,034.7 9,336.0 0.0000 0.0001 IG models IG-GARCH(1,1) 0.69% 22.84% 0.984584 −0.202 4.799 50,295.6 9,548.3 0.0002 IG-GARCH(C) 0.55% 23.44% 0.997176 −0.154 5.238 52,280.0 11,180.2 0.0000 0.0000 0.0000 A. Risk-neutral parameters based onTable 2and ξ estimates Gaussian models μ˜* ω* β1* α1* γ1* ξ sh GARCH(1,1) −0.50 −1.641E-06 0.900 5.193E-06 125.89 19791.7 1.1749 (8.13E+1) μ˜* ωq* ρ1* αh* γh* ρ2* αq* γq* ξ sh GARCH(C) −0.50 2.445E-06 0.708 1.369E-06 725.92 0.9883 4.081E-06 102.02 21122.1 1.1930 (8.17E+1) IG models μ˜* w* b1* a1* c1* η* ξ sh sy IG-GARCH(1,1) −0.50 −1.768E-06 −21.82 2.205E+07 5.854E-06 −6.886E-04 24225.6 1.2033 0.8319 (9.05E+1) μ˜* wq* ρ1* ah* ch* ρ2* aq* cq* η* ξ sh sy IG-GARCH(C) −0.50 2.415E-06 0.745 1.033E+06 9.682E-07 0.989 4.911E+07 4.660E-06 −5.399E-04 29269.1 1.2092 0.8276 (9.55E+1) B. Model properties and likelihoods C. p-values from Vuong-Shi tests Return Annualized Volatility Uncond. Uncond. Log LL increase Model under the null hypothesis mean volatility persistence skewness kurtosis likelihood NK over MK GARCH(1,1) IG-GARCH(1,1) GARCH(C) Gaussian models GARCH(1,1) 0.84% 22.18% 0.981805 −0.013 4.745 48,065.9 6,643.5 GARCH(C) 0.67% 22.94% 0.996582 −0.017 5.190 51,034.7 9,336.0 0.0000 0.0001 IG models IG-GARCH(1,1) 0.69% 22.84% 0.984584 −0.202 4.799 50,295.6 9,548.3 0.0002 IG-GARCH(C) 0.55% 23.44% 0.997176 −0.154 5.238 52,280.0 11,180.2 0.0000 0.0000 0.0000 Panel A reports parameter estimates obtained by sequential MLE estimation on options from January 10, 1990 through December 26, 2012. Table 1 provides a description of the data. We estimate the preference parameter ξ of four models using the returns-based parameters reported in Table 2 by applying the transformation (from physical to risk-neutral measure) mentioned in the appendix. Robust standard errors (based on the outer product of gradients) are in parantheses below the parameter estimates. Panel B reports the maximum log-likelihood values and some model properties. The “LL Increase” column in Panel B reports the increase in log-likelihood (LL) going from a monotonic (ξ = 0) to non-monotonic pricing kernel (NK over MK). Each model has constant or time-varying volatility (which is either two components or one), and Normal or IG innovations. Panel C reports P-values from the Vuong-Shi tests, comparing the alternative model in each row to the model under the null hypothesis in each column. Table 4 Sequential maximum likelihood estimation A. Risk-neutral parameters based onTable 2and ξ estimates Gaussian models μ˜* ω* β1* α1* γ1* ξ sh GARCH(1,1) −0.50 −1.641E-06 0.900 5.193E-06 125.89 19791.7 1.1749 (8.13E+1) μ˜* ωq* ρ1* αh* γh* ρ2* αq* γq* ξ sh GARCH(C) −0.50 2.445E-06 0.708 1.369E-06 725.92 0.9883 4.081E-06 102.02 21122.1 1.1930 (8.17E+1) IG models μ˜* w* b1* a1* c1* η* ξ sh sy IG-GARCH(1,1) −0.50 −1.768E-06 −21.82 2.205E+07 5.854E-06 −6.886E-04 24225.6 1.2033 0.8319 (9.05E+1) μ˜* wq* ρ1* ah* ch* ρ2* aq* cq* η* ξ sh sy IG-GARCH(C) −0.50 2.415E-06 0.745 1.033E+06 9.682E-07 0.989 4.911E+07 4.660E-06 −5.399E-04 29269.1 1.2092 0.8276 (9.55E+1) B. Model properties and likelihoods C. p-values from Vuong-Shi tests Return Annualized Volatility Uncond. Uncond. Log LL increase Model under the null hypothesis mean volatility persistence skewness kurtosis likelihood NK over MK GARCH(1,1) IG-GARCH(1,1) GARCH(C) Gaussian models GARCH(1,1) 0.84% 22.18% 0.981805 −0.013 4.745 48,065.9 6,643.5 GARCH(C) 0.67% 22.94% 0.996582 −0.017 5.190 51,034.7 9,336.0 0.0000 0.0001 IG models IG-GARCH(1,1) 0.69% 22.84% 0.984584 −0.202 4.799 50,295.6 9,548.3 0.0002 IG-GARCH(C) 0.55% 23.44% 0.997176 −0.154 5.238 52,280.0 11,180.2 0.0000 0.0000 0.0000 A. Risk-neutral parameters based onTable 2and ξ estimates Gaussian models μ˜* ω* β1* α1* γ1* ξ sh GARCH(1,1) −0.50 −1.641E-06 0.900 5.193E-06 125.89 19791.7 1.1749 (8.13E+1) μ˜* ωq* ρ1* αh* γh* ρ2* αq* γq* ξ sh GARCH(C) −0.50 2.445E-06 0.708 1.369E-06 725.92 0.9883 4.081E-06 102.02 21122.1 1.1930 (8.17E+1) IG models μ˜* w* b1* a1* c1* η* ξ sh sy IG-GARCH(1,1) −0.50 −1.768E-06 −21.82 2.205E+07 5.854E-06 −6.886E-04 24225.6 1.2033 0.8319 (9.05E+1) μ˜* wq* ρ1* ah* ch* ρ2* aq* cq* η* ξ sh sy IG-GARCH(C) −0.50 2.415E-06 0.745 1.033E+06 9.682E-07 0.989 4.911E+07 4.660E-06 −5.399E-04 29269.1 1.2092 0.8276 (9.55E+1) B. Model properties and likelihoods C. p-values from Vuong-Shi tests Return Annualized Volatility Uncond. Uncond. Log LL increase Model under the null hypothesis mean volatility persistence skewness kurtosis likelihood NK over MK GARCH(1,1) IG-GARCH(1,1) GARCH(C) Gaussian models GARCH(1,1) 0.84% 22.18% 0.981805 −0.013 4.745 48,065.9 6,643.5 GARCH(C) 0.67% 22.94% 0.996582 −0.017 5.190 51,034.7 9,336.0 0.0000 0.0001 IG models IG-GARCH(1,1) 0.69% 22.84% 0.984584 −0.202 4.799 50,295.6 9,548.3 0.0002 IG-GARCH(C) 0.55% 23.44% 0.997176 −0.154 5.238 52,280.0 11,180.2 0.0000 0.0000 0.0000 Panel A reports parameter estimates obtained by sequential MLE estimation on options from January 10, 1990 through December 26, 2012. Table 1 provides a description of the data. We estimate the preference parameter ξ of four models using the returns-based parameters reported in Table 2 by applying the transformation (from physical to risk-neutral measure) mentioned in the appendix. Robust standard errors (based on the outer product of gradients) are in parantheses below the parameter estimates. Panel B reports the maximum log-likelihood values and some model properties. The “LL Increase” column in Panel B reports the increase in log-likelihood (LL) going from a monotonic (ξ = 0) to non-monotonic pricing kernel (NK over MK). Each model has constant or time-varying volatility (which is either two components or one), and Normal or IG innovations. Panel C reports P-values from the Vuong-Shi tests, comparing the alternative model in each row to the model under the null hypothesis in each column. Table 5 Implied volatility RMSE and bias by moneyness A. IV RMSE (bias) by moneyness for models fitted to returns only Model F/X ≤ .80 .80 ≤ F/X ≤ .90 .90 ≤ F/X ≤ 1.00 1.00 ≤ F/X ≤ 1.10 1.10 ≤ F/X ≤ 1.20 F/X≥1.20 All GARCH(1,1) 4.2651 (3.4014) 4.0436 (2.3690) 4.4883 (2.1799) 5.4407 (3.4356) 6.3429 (5.0015) 6.9426 (5.8150) 5.3289 (3.4274) GARCH(C) 3.8684 (3.0676) 3.5813 (2.2484) 4.0457 (2.2724) 5.1913 (3.5101) 6.2316 (5.0933) 6.8989 (5.9136) 5.0694 (3.4766) IG-GARCH(1,1) 4.2076 (3.4509) 3.8985 (2.6056) 4.3027 (2.4838) 5.3553 (3.5878) 6.2616 (5.0382) 6.8554 (5.8009) 5.2161 (3.5962) IG-GARCH(C) 3.8812 (3.1466) 3.5629 (2.4280) 4.0048 (2.4566) 5.1453 (3.5727) 6.1570 (5.0590) 6.8110 (5.8486) 5.0179 (3.5610) B. IV RMSE (bias) by moneyness for models fitted to options only Model F/X ≤ .80 .80 ≤ F/X ≤ .90 .90 ≤ F/X ≤ 1.00 1.00 ≤ F/X ≤ 1.10 1.10 ≤ F/X ≤ 1.20 F/X≥1.20 All GARCH(1,1) 3.8553 (1.3958) 3.4887 (0.3345) 3.4791 (-0.1077) 3.5988 (0.4029) 4.1294 (1.9483) 4.7093 (2.9076) 3.7663 (0.7732) GARCH(C) 3.4674 (2.2275) 2.9646 (1.3057) 2.7904 (0.1487) 3.1088 (-0.1795) 3.4522 (0.7840) 3.9121 (1.6311) 3.1545 (0.5035) IG-GARCH(1,1) 3.8871 (1.0528) 3.3177 (0.1329) 3.0903 (-0.1521) 3.3615 (0.3737) 3.9667 (1.8961) 4.5821 (2.8858) 3.5336 (0.7051) IG-GARCH(C) 3.4030 (1.3913) 2.8663 (0.6174) 2.5614 (-0.0095) 2.8312 (0.0758) 3.3569 (1.2359) 3.9144 (2.1325) 2.9875 (0.5460) C. IV RMSE (bias) by moneyness for models fitted to options sequentially Model F/X ≤ .80 .80 ≤ F/X ≤ .90 .90 ≤ F/X ≤ 1.00 1.00 ≤ F/X ≤ 1.10 1.10 ≤ F/X ≤ 1.20 F/X≥1.20 All GARCH(1,1) 3.7330 (1.7271) 4.3861 (-0.1290) 4.3634 (-0.4802) 4.3408 (0.8915) 5.0734 (2.8571) 5.7419 (4.0494) 4.6155 (1.0174) GARCH(C) 3.5061 (0.9489) 3.9614 (-0.5154) 3.6967 (-0.5480) 3.9011 (0.7551) 4.7363 (2.7687) 5.5014 (4.0094) 4.1672 (0.8692) IG-GARCH(1,1) 3.5286 (1.6067) 4.0133 (-0.1417) 3.8638 (-0.4725) 4.0715 (0.7085) 4.8059 (2.5988) 5.4717 (3.7719) 4.2747 (0.8913) IG-GARCH(C) 3.4103 (0.9890) 3.7561 (-0.4212) 3.4667 (-0.4919) 3.7688 (0.6511) 4.5792 (2.5756) 5.3209 (3.7905) 3.9924 (0.8146) A. IV RMSE (bias) by moneyness for models fitted to returns only Model F/X ≤ .80 .80 ≤ F/X ≤ .90 .90 ≤ F/X ≤ 1.00 1.00 ≤ F/X ≤ 1.10 1.10 ≤ F/X ≤ 1.20 F/X≥1.20 All GARCH(1,1) 4.2651 (3.4014) 4.0436 (2.3690) 4.4883 (2.1799) 5.4407 (3.4356) 6.3429 (5.0015) 6.9426 (5.8150) 5.3289 (3.4274) GARCH(C) 3.8684 (3.0676) 3.5813 (2.2484) 4.0457 (2.2724) 5.1913 (3.5101) 6.2316 (5.0933) 6.8989 (5.9136) 5.0694 (3.4766) IG-GARCH(1,1) 4.2076 (3.4509) 3.8985 (2.6056) 4.3027 (2.4838) 5.3553 (3.5878) 6.2616 (5.0382) 6.8554 (5.8009) 5.2161 (3.5962) IG-GARCH(C) 3.8812 (3.1466) 3.5629 (2.4280) 4.0048 (2.4566) 5.1453 (3.5727) 6.1570 (5.0590) 6.8110 (5.8486) 5.0179 (3.5610) B. IV RMSE (bias) by moneyness for models fitted to options only Model F/X ≤ .80 .80 ≤ F/X ≤ .90 .90 ≤ F/X ≤ 1.00 1.00 ≤ F/X ≤ 1.10 1.10 ≤ F/X ≤ 1.20 F/X≥1.20 All GARCH(1,1) 3.8553 (1.3958) 3.4887 (0.3345) 3.4791 (-0.1077) 3.5988 (0.4029) 4.1294 (1.9483) 4.7093 (2.9076) 3.7663 (0.7732) GARCH(C) 3.4674 (2.2275) 2.9646 (1.3057) 2.7904 (0.1487) 3.1088 (-0.1795) 3.4522 (0.7840) 3.9121 (1.6311) 3.1545 (0.5035) IG-GARCH(1,1) 3.8871 (1.0528) 3.3177 (0.1329) 3.0903 (-0.1521) 3.3615 (0.3737) 3.9667 (1.8961) 4.5821 (2.8858) 3.5336 (0.7051) IG-GARCH(C) 3.4030 (1.3913) 2.8663 (0.6174) 2.5614 (-0.0095) 2.8312 (0.0758) 3.3569 (1.2359) 3.9144 (2.1325) 2.9875 (0.5460) C. IV RMSE (bias) by moneyness for models fitted to options sequentially Model F/X ≤ .80 .80 ≤ F/X ≤ .90 .90 ≤ F/X ≤ 1.00 1.00 ≤ F/X ≤ 1.10 1.10 ≤ F/X ≤ 1.20 F/X≥1.20 All GARCH(1,1) 3.7330 (1.7271) 4.3861 (-0.1290) 4.3634 (-0.4802) 4.3408 (0.8915) 5.0734 (2.8571) 5.7419 (4.0494) 4.6155 (1.0174) GARCH(C) 3.5061 (0.9489) 3.9614 (-0.5154) 3.6967 (-0.5480) 3.9011 (0.7551) 4.7363 (2.7687) 5.5014 (4.0094) 4.1672 (0.8692) IG-GARCH(1,1) 3.5286 (1.6067) 4.0133 (-0.1417) 3.8638 (-0.4725) 4.0715 (0.7085) 4.8059 (2.5988) 5.4717 (3.7719) 4.2747 (0.8913) IG-GARCH(C) 3.4103 (0.9890) 3.7561 (-0.4212) 3.4667 (-0.4919) 3.7688 (0.6511) 4.5792 (2.5756) 5.3209 (3.7905) 3.9924 (0.8146) We report implied volatility (IV) RMSE (values before parentheses) and bias (values inside parentheses) in percentage by moneyness. Bias is defined as market IV less model IV. Panel A uses the parameter estimates from the return-based estimation in Table 2; panel B uses the options-based estimates in Table 3; and panel C uses the sequential estimates in Table 4. Table 5 Implied volatility RMSE and bias by moneyness A. IV RMSE (bias) by moneyness for models fitted to returns only Model F/X ≤ .80 .80 ≤ F/X ≤ .90 .90 ≤ F/X ≤ 1.00 1.00 ≤ F/X ≤ 1.10 1.10 ≤ F/X ≤ 1.20 F/X≥1.20 All GARCH(1,1) 4.2651 (3.4014) 4.0436 (2.3690) 4.4883 (2.1799) 5.4407 (3.4356) 6.3429 (5.0015) 6.9426 (5.8150) 5.3289 (3.4274) GARCH(C) 3.8684 (3.0676) 3.5813 (2.2484) 4.0457 (2.2724) 5.1913 (3.5101) 6.2316 (5.0933) 6.8989 (5.9136) 5.0694 (3.4766) IG-GARCH(1,1) 4.2076 (3.4509) 3.8985 (2.6056) 4.3027 (2.4838) 5.3553 (3.5878) 6.2616 (5.0382) 6.8554 (5.8009) 5.2161 (3.5962) IG-GARCH(C) 3.8812 (3.1466) 3.5629 (2.4280) 4.0048 (2.4566) 5.1453 (3.5727) 6.1570 (5.0590) 6.8110 (5.8486) 5.0179 (3.5610) B. IV RMSE (bias) by moneyness for models fitted to options only Model F/X ≤ .80 .80 ≤ F/X ≤ .90 .90 ≤ F/X ≤ 1.00 1.00 ≤ F/X ≤ 1.10 1.10 ≤ F/X ≤ 1.20 F/X≥1.20 All GARCH(1,1) 3.8553 (1.3958) 3.4887 (0.3345) 3.4791 (-0.1077) 3.5988 (0.4029) 4.1294 (1.9483) 4.7093 (2.9076) 3.7663 (0.7732) GARCH(C) 3.4674 (2.2275) 2.9646 (1.3057) 2.7904 (0.1487) 3.1088 (-0.1795) 3.4522 (0.7840) 3.9121 (1.6311) 3.1545 (0.5035) IG-GARCH(1,1) 3.8871 (1.0528) 3.3177 (0.1329) 3.0903 (-0.1521) 3.3615 (0.3737) 3.9667 (1.8961) 4.5821 (2.8858) 3.5336 (0.7051) IG-GARCH(C) 3.4030 (1.3913) 2.8663 (0.6174) 2.5614 (-0.0095) 2.8312 (0.0758) 3.3569 (1.2359) 3.9144 (2.1325) 2.9875 (0.5460) C. IV RMSE (bias) by moneyness for models fitted to options sequentially Model F/X ≤ .80 .80 ≤ F/X ≤ .90 .90 ≤ F/X ≤ 1.00 1.00 ≤ F/X ≤ 1.10 1.10 ≤ F/X ≤ 1.20 F/X≥1.20 All GARCH(1,1) 3.7330 (1.7271) 4.3861 (-0.1290) 4.3634 (-0.4802) 4.3408 (0.8915) 5.0734 (2.8571) 5.7419 (4.0494) 4.6155 (1.0174) GARCH(C) 3.5061 (0.9489) 3.9614 (-0.5154) 3.6967 (-0.5480) 3.9011 (0.7551) 4.7363 (2.7687) 5.5014 (4.0094) 4.1672 (0.8692) IG-GARCH(1,1) 3.5286 (1.6067) 4.0133 (-0.1417) 3.8638 (-0.4725) 4.0715 (0.7085) 4.8059 (2.5988) 5.4717 (3.7719) 4.2747 (0.8913) IG-GARCH(C) 3.4103 (0.9890) 3.7561 (-0.4212) 3.4667 (-0.4919) 3.7688 (0.6511) 4.5792 (2.5756) 5.3209 (3.7905) 3.9924 (0.8146) A. IV RMSE (bias) by moneyness for models fitted to returns only Model F/X ≤ .80 .80 ≤ F/X ≤ .90 .90 ≤ F/X ≤ 1.00 1.00 ≤ F/X ≤ 1.10 1.10 ≤ F/X ≤ 1.20 F/X≥1.20 All GARCH(1,1) 4.2651 (3.4014) 4.0436 (2.3690) 4.4883 (2.1799) 5.4407 (3.4356) 6.3429 (5.0015) 6.9426 (5.8150) 5.3289 (3.4274) GARCH(C) 3.8684 (3.0676) 3.5813 (2.2484) 4.0457 (2.2724) 5.1913 (3.5101) 6.2316 (5.0933) 6.8989 (5.9136) 5.0694 (3.4766) IG-GARCH(1,1) 4.2076 (3.4509) 3.8985 (2.6056) 4.3027 (2.4838) 5.3553 (3.5878) 6.2616 (5.0382) 6.8554 (5.8009) 5.2161 (3.5962) IG-GARCH(C) 3.8812 (3.1466) 3.5629 (2.4280) 4.0048 (2.4566) 5.1453 (3.5727) 6.1570 (5.0590) 6.8110 (5.8486) 5.0179 (3.5610) B. IV RMSE (bias) by moneyness for models fitted to options only Model F/X ≤ .80 .80 ≤ F/X ≤ .90 .90 ≤ F/X ≤ 1.00 1.00 ≤ F/X ≤ 1.10 1.10 ≤ F/X ≤ 1.20 F/X≥1.20 All GARCH(1,1) 3.8553 (1.3958) 3.4887 (0.3345) 3.4791 (-0.1077) 3.5988 (0.4029) 4.1294 (1.9483) 4.7093 (2.9076) 3.7663 (0.7732) GARCH(C) 3.4674 (2.2275) 2.9646 (1.3057) 2.7904 (0.1487) 3.1088 (-0.1795) 3.4522 (0.7840) 3.9121 (1.6311) 3.1545 (0.5035) IG-GARCH(1,1) 3.8871 (1.0528) 3.3177 (0.1329) 3.0903 (-0.1521) 3.3615 (0.3737) 3.9667 (1.8961) 4.5821 (2.8858) 3.5336 (0.7051) IG-GARCH(C) 3.4030 (1.3913) 2.8663 (0.6174) 2.5614 (-0.0095) 2.8312 (0.0758) 3.3569 (1.2359) 3.9144 (2.1325) 2.9875 (0.5460) C. IV RMSE (bias) by moneyness for models fitted to options sequentially Model F/X ≤ .80 .80 ≤ F/X ≤ .90 .90 ≤ F/X ≤ 1.00 1.00 ≤ F/X ≤ 1.10 1.10 ≤ F/X ≤ 1.20 F/X≥1.20 All GARCH(1,1) 3.7330 (1.7271) 4.3861 (-0.1290) 4.3634 (-0.4802) 4.3408 (0.8915) 5.0734 (2.8571) 5.7419 (4.0494) 4.6155 (1.0174) GARCH(C) 3.5061 (0.9489) 3.9614 (-0.5154) 3.6967 (-0.5480) 3.9011 (0.7551) 4.7363 (2.7687) 5.5014 (4.0094) 4.1672 (0.8692) IG-GARCH(1,1) 3.5286 (1.6067) 4.0133 (-0.1417) 3.8638 (-0.4725) 4.0715 (0.7085) 4.8059 (2.5988) 5.4717 (3.7719) 4.2747 (0.8913) IG-GARCH(C) 3.4103 (0.9890) 3.7561 (-0.4212) 3.4667 (-0.4919) 3.7688 (0.6511) 4.5792 (2.5756) 5.3209 (3.7905) 3.9924 (0.8146) We report implied volatility (IV) RMSE (values before parentheses) and bias (values inside parentheses) in percentage by moneyness. Bias is defined as market IV less model IV. Panel A uses the parameter estimates from the return-based estimation in Table 2; panel B uses the options-based estimates in Table 3; and panel C uses the sequential estimates in Table 4. Table 6 Implied volatility RMSE and bias by maturity A. IV RMSE (bias) by maturity for models fitted to returns only Model DTM ≤ 30 30 ≤ DTM ≤ 60 60 ≤ DTM ≤ 90 90 ≤ DTM ≤ 120 120 ≤ DTM ≤ 180 DTM≥180 All GARCH(1,1) 4.6495 (2.9547) 5.2143 (3.3967) 5.2047 (3.3524) 5.6930 (3.9329) 5.3252 (3.5045) 5.5520 (3.4450) 5.3289 (3.4274) GARCH(C) 4.4964 (3.0103) 5.0258 (3.4418) 4.9616 (3.3926) 5.3320 (3.8838) 5.0543 (3.5510) 5.2453 (3.5312) 5.0694 (3.4766) IG-GARCH(1,1) 4.5423 (3.0684) 5.0795 (3.5273) 5.0601 (3.5077) 5.4981 (4.0699) 5.2431 (3.6863) 5.4791 (3.6685) 5.2161 (3.5962) IG-GARCH(C) 4.4689 (3.0707) 4.9698 (3.5111) 4.8928 (3.4777) 5.2497 (3.9643) 5.0139 (3.6417) 5.2034 (3.6316) 5.0179 (3.5610) B. IV RMSE (bias) by maturity for models fitted to options only Model DTM ≤ 30 30 ≤ DTM ≤ 60 60 ≤ DTM ≤ 90 90 ≤ DTM ≤ 120 120 ≤ DTM ≤ 180 DTM≥180 All GARCH(1,1) 4.3171 (1.3157) 4.2394 (1.3154) 3.7668 (0.8594) 3.6197 (1.0708) 3.4035 (0.6858) 3.4165 (0.1294) 3.7663 (0.7732) GARCH(C) 4.0301 (0.8330) 3.7421 (0.9144) 3.2046 (0.5466) 2.9483 (0.6318) 2.7796 (0.4902) 2.5290 (0.0589) 3.1545 (0.5035) IG-GARCH(1,1) 4.0868 (1.3671) 3.9773 (1.3381) 3.4912 (0.8563) 3.2858 (1.0133) 3.2160 (0.5547) 3.2401 (-0.0475) 3.5336 (0.7051) IG-GARCH(C) 3.8771 (1.1875) 3.6164 (1.2356) 3.0171 (0.7659) 2.7624 (0.6886) 2.5303 (0.4240) 2.3618 (-0.2367) 2.9875 (0.5460) C. IV RMSE (bias) by maturity for models fitted to options sequentially Model DTM ≤ 30 30 ≤ DTM ≤ 60 60 ≤ DTM ≤ 90 90 ≤ DTM ≤ 120 120 ≤ DTM ≤ 180 DTM≥180 All GARCH(1,1) 4.3879 (1.7672) 4.6955 (1.7208) 4.5165 (1.1667) 4.7743 (1.5080) 4.4305 (0.7540) 4.7115 (0.1848) 4.6155 (1.0174) GARCH(C) 4.0062 (1.7594) 4.2872 (1.6855) 4.0786 (1.0979) 4.1949 (1.3008) 3.9768 (0.5875) 4.2529 (-0.1007) 4.1672 (0.8692) IG-GARCH(1,1) 4.0903 (1.7646) 4.3269 (1.6860) 4.1321 (1.0911) 4.2528 (1.3719) 4.1266 (0.6011) 4.4375 (-0.0544) 4.2747 (0.8913) IG-GARCH(C) 3.9210 (1.7758) 4.1112 (1.6930) 3.8617 (1.0860) 3.9197 (1.2588) 3.8005 (0.5184) 4.1033 (-0.2402) 3.9924 (0.8146) A. IV RMSE (bias) by maturity for models fitted to returns only Model DTM ≤ 30 30 ≤ DTM ≤ 60 60 ≤ DTM ≤ 90 90 ≤ DTM ≤ 120 120 ≤ DTM ≤ 180 DTM≥180 All GARCH(1,1) 4.6495 (2.9547) 5.2143 (3.3967) 5.2047 (3.3524) 5.6930 (3.9329) 5.3252 (3.5045) 5.5520 (3.4450) 5.3289 (3.4274) GARCH(C) 4.4964 (3.0103) 5.0258 (3.4418) 4.9616 (3.3926) 5.3320 (3.8838) 5.0543 (3.5510) 5.2453 (3.5312) 5.0694 (3.4766) IG-GARCH(1,1) 4.5423 (3.0684) 5.0795 (3.5273) 5.0601 (3.5077) 5.4981 (4.0699) 5.2431 (3.6863) 5.4791 (3.6685) 5.2161 (3.5962) IG-GARCH(C) 4.4689 (3.0707) 4.9698 (3.5111) 4.8928 (3.4777) 5.2497 (3.9643) 5.0139 (3.6417) 5.2034 (3.6316) 5.0179 (3.5610) B. IV RMSE (bias) by maturity for models fitted to options only Model DTM ≤ 30 30 ≤ DTM ≤ 60 60 ≤ DTM ≤ 90 90 ≤ DTM ≤ 120 120 ≤ DTM ≤ 180 DTM≥180 All GARCH(1,1) 4.3171 (1.3157) 4.2394 (1.3154) 3.7668 (0.8594) 3.6197 (1.0708) 3.4035 (0.6858) 3.4165 (0.1294) 3.7663 (0.7732) GARCH(C) 4.0301 (0.8330) 3.7421 (0.9144) 3.2046 (0.5466) 2.9483 (0.6318) 2.7796 (0.4902) 2.5290 (0.0589) 3.1545 (0.5035) IG-GARCH(1,1) 4.0868 (1.3671) 3.9773 (1.3381) 3.4912 (0.8563) 3.2858 (1.0133) 3.2160 (0.5547) 3.2401 (-0.0475) 3.5336 (0.7051) IG-GARCH(C) 3.8771 (1.1875) 3.6164 (1.2356) 3.0171 (0.7659) 2.7624 (0.6886) 2.5303 (0.4240) 2.3618 (-0.2367) 2.9875 (0.5460) C. IV RMSE (bias) by maturity for models fitted to options sequentially Model DTM ≤ 30 30 ≤ DTM ≤ 60 60 ≤ DTM ≤ 90 90 ≤ DTM ≤ 120 120 ≤ DTM ≤ 180 DTM≥180 All GARCH(1,1) 4.3879 (1.7672) 4.6955 (1.7208) 4.5165 (1.1667) 4.7743 (1.5080) 4.4305 (0.7540) 4.7115 (0.1848) 4.6155 (1.0174) GARCH(C) 4.0062 (1.7594) 4.2872 (1.6855) 4.0786 (1.0979) 4.1949 (1.3008) 3.9768 (0.5875) 4.2529 (-0.1007) 4.1672 (0.8692) IG-GARCH(1,1) 4.0903 (1.7646) 4.3269 (1.6860) 4.1321 (1.0911) 4.2528 (1.3719) 4.1266 (0.6011) 4.4375 (-0.0544) 4.2747 (0.8913) IG-GARCH(C) 3.9210 (1.7758) 4.1112 (1.6930) 3.8617 (1.0860) 3.9197 (1.2588) 3.8005 (0.5184) 4.1033 (-0.2402) 3.9924 (0.8146) We report implied volatility (IV) RMSE (values before parentheses) and bias (values inside parentheses) in percentage by maturity. Bias is defined as market IV less model IV. Panel A uses the parameter estimates from the return-based estimation in Table 2; panel B uses the options-based estimates in Table 3; and panel C uses the sequential estimates in Table 4. Table 6 Implied volatility RMSE and bias by maturity A. IV RMSE (bias) by maturity for models fitted to returns only Model DTM ≤ 30 30 ≤ DTM ≤ 60 60 ≤ DTM ≤ 90 90 ≤ DTM ≤ 120 120 ≤ DTM ≤ 180 DTM≥180 All GARCH(1,1) 4.6495 (2.9547) 5.2143 (3.3967) 5.2047 (3.3524) 5.6930 (3.9329) 5.3252 (3.5045) 5.5520 (3.4450) 5.3289 (3.4274) GARCH(C) 4.4964 (3.0103) 5.0258 (3.4418) 4.9616 (3.3926) 5.3320 (3.8838) 5.0543 (3.5510) 5.2453 (3.5312) 5.0694 (3.4766) IG-GARCH(1,1) 4.5423 (3.0684) 5.0795 (3.5273) 5.0601 (3.5077) 5.4981 (4.0699) 5.2431 (3.6863) 5.4791 (3.6685) 5.2161 (3.5962) IG-GARCH(C) 4.4689 (3.0707) 4.9698 (3.5111) 4.8928 (3.4777) 5.2497 (3.9643) 5.0139 (3.6417) 5.2034 (3.6316) 5.0179 (3.5610) B. IV RMSE (bias) by maturity for models fitted to options only Model DTM ≤ 30 30 ≤ DTM ≤ 60 60 ≤ DTM ≤ 90 90 ≤ DTM ≤ 120 120 ≤ DTM ≤ 180 DTM≥180 All GARCH(1,1) 4.3171 (1.3157) 4.2394 (1.3154) 3.7668 (0.8594) 3.6197 (1.0708) 3.4035 (0.6858) 3.4165 (0.1294) 3.7663 (0.7732) GARCH(C) 4.0301 (0.8330) 3.7421 (0.9144) 3.2046 (0.5466) 2.9483 (0.6318) 2.7796 (0.4902) 2.5290 (0.0589) 3.1545 (0.5035) IG-GARCH(1,1) 4.0868 (1.3671) 3.9773 (1.3381) 3.4912 (0.8563) 3.2858 (1.0133) 3.2160 (0.5547) 3.2401 (-0.0475) 3.5336 (0.7051) IG-GARCH(C) 3.8771 (1.1875) 3.6164 (1.2356) 3.0171 (0.7659) 2.7624 (0.6886) 2.5303 (0.4240) 2.3618 (-0.2367) 2.9875 (0.5460) C. IV RMSE (bias) by maturity for models fitted to options sequentially Model DTM ≤ 30 30 ≤ DTM ≤ 60 60 ≤ DTM ≤ 90 90 ≤ DTM ≤ 120 120 ≤ DTM ≤ 180 DTM≥180 All GARCH(1,1) 4.3879 (1.7672) 4.6955 (1.7208) 4.5165 (1.1667) 4.7743 (1.5080) 4.4305 (0.7540) 4.7115 (0.1848) 4.6155 (1.0174) GARCH(C) 4.0062 (1.7594) 4.2872 (1.6855) 4.0786 (1.0979) 4.1949 (1.3008) 3.9768 (0.5875) 4.2529 (-0.1007) 4.1672 (0.8692) IG-GARCH(1,1) 4.0903 (1.7646) 4.3269 (1.6860) 4.1321 (1.0911) 4.2528 (1.3719) 4.1266 (0.6011) 4.4375 (-0.0544) 4.2747 (0.8913) IG-GARCH(C) 3.9210 (1.7758) 4.1112 (1.6930) 3.8617 (1.0860) 3.9197 (1.2588) 3.8005 (0.5184) 4.1033 (-0.2402) 3.9924 (0.8146) A. IV RMSE (bias) by maturity for models fitted to returns only Model DTM ≤ 30 30 ≤ DTM ≤ 60 60 ≤ DTM ≤ 90 90 ≤ DTM ≤ 120 120 ≤ DTM ≤ 180 DTM≥180 All GARCH(1,1) 4.6495 (2.9547) 5.2143 (3.3967) 5.2047 (3.3524) 5.6930 (3.9329) 5.3252 (3.5045) 5.5520 (3.4450) 5.3289 (3.4274) GARCH(C) 4.4964 (3.0103) 5.0258 (3.4418) 4.9616 (3.3926) 5.3320 (3.8838) 5.0543 (3.5510) 5.2453 (3.5312) 5.0694 (3.4766) IG-GARCH(1,1) 4.5423 (3.0684) 5.0795 (3.5273) 5.0601 (3.5077) 5.4981 (4.0699) 5.2431 (3.6863) 5.4791 (3.6685) 5.2161 (3.5962) IG-GARCH(C) 4.4689 (3.0707) 4.9698 (3.5111) 4.8928 (3.4777) 5.2497 (3.9643) 5.0139 (3.6417) 5.2034 (3.6316) 5.0179 (3.5610) B. IV RMSE (bias) by maturity for models fitted to options only Model DTM ≤ 30 30 ≤ DTM ≤ 60 60 ≤ DTM ≤ 90 90 ≤ DTM ≤ 120 120 ≤ DTM ≤ 180 DTM≥180 All GARCH(1,1) 4.3171 (1.3157) 4.2394 (1.3154) 3.7668 (0.8594) 3.6197 (1.0708) 3.4035 (0.6858) 3.4165 (0.1294) 3.7663 (0.7732) GARCH(C) 4.0301 (0.8330) 3.7421 (0.9144) 3.2046 (0.5466) 2.9483 (0.6318) 2.7796 (0.4902) 2.5290 (0.0589) 3.1545 (0.5035) IG-GARCH(1,1) 4.0868 (1.3671) 3.9773 (1.3381) 3.4912 (0.8563) 3.2858 (1.0133) 3.2160 (0.5547) 3.2401 (-0.0475) 3.5336 (0.7051) IG-GARCH(C) 3.8771 (1.1875) 3.6164 (1.2356) 3.0171 (0.7659) 2.7624 (0.6886) 2.5303 (0.4240) 2.3618 (-0.2367) 2.9875 (0.5460) C. IV RMSE (bias) by maturity for models fitted to options sequentially Model DTM ≤ 30 30 ≤ DTM ≤ 60 60 ≤ DTM ≤ 90 90 ≤ DTM ≤ 120 120 ≤ DTM ≤ 180 DTM≥180 All GARCH(1,1) 4.3879 (1.7672) 4.6955 (1.7208) 4.5165 (1.1667) 4.7743 (1.5080) 4.4305 (0.7540) 4.7115 (0.1848) 4.6155 (1.0174) GARCH(C) 4.0062 (1.7594) 4.2872 (1.6855) 4.0786 (1.0979) 4.1949 (1.3008) 3.9768 (0.5875) 4.2529 (-0.1007) 4.1672 (0.8692) IG-GARCH(1,1) 4.0903 (1.7646) 4.3269 (1.6860) 4.1321 (1.0911) 4.2528 (1.3719) 4.1266 (0.6011) 4.4375 (-0.0544) 4.2747 (0.8913) IG-GARCH(C) 3.9210 (1.7758) 4.1112 (1.6930) 3.8617 (1.0860) 3.9197 (1.2588) 3.8005 (0.5184) 4.1033 (-0.2402) 3.9924 (0.8146) We report implied volatility (IV) RMSE (values before parentheses) and bias (values inside parentheses) in percentage by maturity. Bias is defined as market IV less model IV. Panel A uses the parameter estimates from the return-based estimation in Table 2; panel B uses the options-based estimates in Table 3; and panel C uses the sequential estimates in Table 4. Table 7 Information criteria Return-based estimation Option-based estimation Sequential estimation AIC BIC HQIC AIC BIC HQIC AIC BIC HQIC Gaussian models Homoscedastic −35,092 −35,079 −35,087 −65,262 −65,254 −65,259 GARCH(1,1) −37,553 −37,520 −37,541 −107,934 −107,901 −107,924 −96,130 −96,122 −96,127 GARCH(C) −37,647 −37,593 −37,628 −118,235 −118,177 −118,217 −102,067 −102,059 −102,065 IG models Homoscedastic −35,097 −35,077 −35,090 −73,887 −73,871 −73,882 IG-GARCH(1,1) −37,647 −37,607 −37,633 −111,638 −111,597 −111,625 −100,589 −100,581 −100,587 IG-GARCH(C) −37,712 −37,652 −37,691 −121,414 −121,348 −121,393 −104,558 −104,550 −104,555 Return-based estimation Option-based estimation Sequential estimation AIC BIC HQIC AIC BIC HQIC AIC BIC HQIC Gaussian models Homoscedastic −35,092 −35,079 −35,087 −65,262 −65,254 −65,259 GARCH(1,1) −37,553 −37,520 −37,541 −107,934 −107,901 −107,924 −96,130 −96,122 −96,127 GARCH(C) −37,647 −37,593 −37,628 −118,235 −118,177 −118,217 −102,067 −102,059 −102,065 IG models Homoscedastic −35,097 −35,077 −35,090 −73,887 −73,871 −73,882 IG-GARCH(1,1) −37,647 −37,607 −37,633 −111,638 −111,597 −111,625 −100,589 −100,581 −100,587 IG-GARCH(C) −37,712 −37,652 −37,691 −121,414 −121,348 −121,393 −104,558 −104,550 −104,555 We report the AIC, BIC, and Hannan-Quinn information criteria (HQIC) using the likelihood values from Tables 2-4. Table 7 Information criteria Return-based estimation Option-based estimation Sequential estimation AIC BIC HQIC AIC BIC HQIC AIC BIC HQIC Gaussian models Homoscedastic −35,092 −35,079 −35,087 −65,262 −65,254 −65,259 GARCH(1,1) −37,553 −37,520 −37,541 −107,934 −107,901 −107,924 −96,130 −96,122 −96,127 GARCH(C) −37,647 −37,593 −37,628 −118,235 −118,177 −118,217 −102,067 −102,059 −102,065 IG models Homoscedastic −35,097 −35,077 −35,090 −73,887 −73,871 −73,882 IG-GARCH(1,1) −37,647 −37,607 −37,633 −111,638 −111,597 −111,625 −100,589 −100,581 −100,587 IG-GARCH(C) −37,712 −37,652 −37,691 −121,414 −121,348 −121,393 −104,558 −104,550 −104,555 Return-based estimation Option-based estimation Sequential estimation AIC BIC HQIC AIC BIC HQIC AIC BIC HQIC Gaussian models Homoscedastic −35,092 −35,079 −35,087 −65,262 −65,254 −65,259 GARCH(1,1) −37,553 −37,520 −37,541 −107,934 −107,901 −107,924 −96,130 −96,122 −96,127 GARCH(C) −37,647 −37,593 −37,628 −118,235 −118,177 −118,217 −102,067 −102,059 −102,065 IG models Homoscedastic −35,097 −35,077 −35,090 −73,887 −73,871 −73,882 IG-GARCH(1,1) −37,647 −37,607 −37,633 −111,638 −111,597 −111,625 −100,589 −100,581 −100,587 IG-GARCH(C) −37,712 −37,652 −37,691 −121,414 −121,348 −121,393 −104,558 −104,550 −104,555 We report the AIC, BIC, and Hannan-Quinn information criteria (HQIC) using the likelihood values from Tables 2-4. Table 8 Out-of-sample fit A. Out-of-sample log-likelihood B. Out-of-sample IV RMSE Return-based Option-based Sequential Option-based Sequential Gaussian models estimation estimation estimation estimation estimation Homoscedastic 2,487 8,377 10.010 GARCH(1,1) 2,637 19,585 14,951 3.117 4.947 GARCH(C) 2,650 23,682 15,967 2.026 4.496 IG models Homoscedastic 2,488 8,448 9.926 IG-GARCH(1,1) 2,647 19,830 16,230 3.038 4.354 IG-GARCH(C) 2,660 21,423 17,453 2.572 3.866 A. Out-of-sample log-likelihood B. Out-of-sample IV RMSE Return-based Option-based Sequential Option-based Sequential Gaussian models estimation estimation estimation estimation estimation Homoscedastic 2,487 8,377 10.010 GARCH(1,1) 2,637 19,585 14,951 3.117 4.947 GARCH(C) 2,650 23,682 15,967 2.026 4.496 IG models Homoscedastic 2,488 8,448 9.926 IG-GARCH(1,1) 2,647 19,830 16,230 3.038 4.354 IG-GARCH(C) 2,660 21,423 17,453 2.572 3.866 Panel A reports out-of-sample log likelihood values for the 2013-2015 period using the estimates in Tables 2–4. Panel B reports implied volatility (IV) RMSE in percentage for the out-of-sample period using estimates in Tables 3 and 4. Table 8 Out-of-sample fit A. Out-of-sample log-likelihood B. Out-of-sample IV RMSE Return-based Option-based Sequential Option-based Sequential Gaussian models estimation estimation estimation estimation estimation Homoscedastic 2,487 8,377 10.010 GARCH(1,1) 2,637 19,585 14,951 3.117 4.947 GARCH(C) 2,650 23,682 15,967 2.026 4.496 IG models Homoscedastic 2,488 8,448 9.926 IG-GARCH(1,1) 2,647 19,830 16,230 3.038 4.354 IG-GARCH(C) 2,660 21,423 17,453 2.572 3.866 A. Out-of-sample log-likelihood B. Out-of-sample IV RMSE Return-based Option-based Sequential Option-based Sequential Gaussian models estimation estimation estimation estimation estimation Homoscedastic 2,487 8,377 10.010 GARCH(1,1) 2,637 19,585 14,951 3.117 4.947 GARCH(C) 2,650 23,682 15,967 2.026 4.496 IG models Homoscedastic 2,488 8,448 9.926 IG-GARCH(1,1) 2,647 19,830 16,230 3.038 4.354 IG-GARCH(C) 2,660 21,423 17,453 2.572 3.866 Panel A reports out-of-sample log likelihood values for the 2013-2015 period using the estimates in Tables 2–4. Panel B reports implied volatility (IV) RMSE in percentage for the out-of-sample period using estimates in Tables 3 and 4. 4.1 Fitting returns and fitting options We will organize our initial discussion around the measures of fit (i.e. log-likelihood values) for the different models contained in Table 2 (return fitting) and Table 3 (option fitting). We have results for the fit of six models in these tables. Of these six models, three have Gaussian innovations and three are characterized by fat-tailed inverse Gaussian innovations. Two models have two variance factors, and two have one factor. For comparison, we also estimate two models that have no variance dynamics; we refer to them as homoscedastic models. The most highly parameterized two-factor model with fat tails fits the returns and options data best, as can be seen in Tables 2 and 3, whereas the most restrictive single-factor Gaussian model fits worst. This is not surprising in an in-sample exercise. All the two-factor models have substantially higher likelihood values than all the one-factor models. The two-factor models have three more parameters than the corresponding one-factor models, and two times the difference in the log-likelihoods is asymptotically distributed chi-square with three degrees of freedom. The 99.9% p-level for this test is 16.3. In the case of the option-based estimation in Table 3, the improvement provided by the second factor is very large, with a likelihood improvement of approximately 5,000. This suggests that the most important feature in accurately modelling option prices is the correct specification of the volatility dynamics. This finding confirms the results in Andersen, Fusari, and Todorov (2015a), where the biggest improvement in fit also stems from a second factor. Bates (2000), Duffie, Pan, and Singleton (2000), Christoffersen, Heston, and Jacobs (2009), and Christoffersen et al. (2008) also emphasize the importance of a second volatility factor. Andersen, Fusari, and Todorov (2015b) additionally find support for a latent factor that drives the risk-neutral left tail of the distribution. The inclusion of a second factor also significantly improves the return fit in Table 2. For example, for the Gaussian case, twice the difference in the log-likelihood between the two-factor and one-factor models is 100, and for the fat-tailed case the corresponding number is 71. These test statistics are highly significant. We conclude that a second factor is important in describing the underlying returns as well as option prices. When comparing IG versus Gaussian models, Tables 2 and 3 show that adding the single parameter η in the IG models increases the return and option likelihoods substantially. In Table 3 the likelihood improvements are again in the thousands. The improvements in the return likelihoods in Table 2 are less dramatic but still statistically significant at conventional confidence levels. These conclusions confirm existing findings that incorporating jumps leads to improved option valuation (see, for instance, Bakshi, Cao, and Chen (1997), Bates (2000), Pan (2002), and Eraker, Johannes, and Polson (2003). Our horse race approach incorporates several model features that allow us to conclude that the magnitude of improvement contributed by the fat-tailed IG feature depends on the other model features. For the option-based estimation in Table 3, the improvement in option log-likelihood is 4,314 for the homoscedastic case, 1,853 for the one-factor GARCH, and 1,590 for the two-factor GARCH. For the return-based estimation in Table 2, the improvements in log-likelihood are 3.5 for the homoscedastic case, 48.1 for the one-factor GARCH, and 33.7 for the two-factor GARCH. The IG feature improves the fit more in the case of the simpler one-factor model than in the case of the two-factor model. Nonnormal innovations and a second volatility component are therefore to some extent substitutes in model specification. In order to assess the significance of the differences in likelihoods we rely on Shi’s (2015) modification of the classic Vuong (1989) likelihood ratio test. Vuong (1989) develops tests of the hypothesis that two potentially nonnested parametric models are equally distant (in the Kullback–Leibler sense) from the distribution of the data. Shi (2015) provides various modifications that improve the size and power of Vuong’s original tests. The p-values in the first column of panel C in Table 2 show that the null of the GARCH(1,1) model is rejected in favor of the GARCH(C), IG-GARCH(1,1) and IG-GARCH(C) models at the 5% level. The second column shows that the null of the IG-GARCH(1,1) model in turn is rejected in favor of the IG-GARCH(C) model but the test fails to reject the IG-GARCH(1,1) when compared with the GARCH(C) model. Finally, the null of the GARCH(C) model is rejected in favor of the IG-GARCH(C) model. Panel C of Table 3 shows that the model under the null is rejected in all cases when the models are estimated using options. Altogether, we find strong evidence in favor of our new IG-GARCH(C) model. 4.2 Sequential estimation of the nonmonotonic pricing kernel parameter Table 2 contains return-based estimates of the physical distributions. Table 3 contains option-based estimates of the risk-neutral distribution. Neither table is informative about the pricing kernel. In Table 4 we therefore use the physical parameter estimates from Table 2 and estimate only the nonmonotonic pricing kernel parameter ξ by fitting options.7Table 4 reports risk-neutral values of all parameters, but only ξ is estimated from options. The penultimate column in panel B of Table 4 reports the option likelihoods for the four dynamic models with nonmonotone pricing kernel. The last column in panel B shows the difference between the option likelihood for optimal ξ and that for ξ = 0, where the options are valued using the risk-neutralized parameters from Table 2. The increase in option log-likelihood when allowing for a nonmonotonic pricing kernel and adding just a single parameter is again in the thousands. Table 4 shows that the log-likelihood increase due to the more general pricing kernel is 6,644 in the single-factor Gaussian model, and 9,548 in the corresponding inverse Gaussian model. In case of the two-factor models, the improvements are even higher: the nonmonotonic kernel improves the two-factor likelihoods by 9,336 in the Gaussian model and 11,180 in the inverse Gaussian model. These findings confirm the results in Christoffersen, Heston, and Jacobs (2013). The results in Table 4 additionally indicate that the importance of modeling a more general pricing kernel depends on the models’ ability to capture the tails of the distribution. The richer dynamics of two-factor models allow them to better fit the fat tails, and a nonmonotonic pricing kernel captures this property by allowing the model’s physical parameters to fit the returns and risk-neutral parameters to fit options in the same model. Complex modelling of risk premia complements adequate modelling of return dynamics. Panel C in Table 4 shows that when using the Vuong-Shi test, the more specific model under the null in each column is always rejected in favor of the more general model in the row. This shows that the option data fit improves significantly when allowing for both a volatility component structure and non-Gaussian innovations. Although not reported in Table 4, the Vuong-Shi test also rejects the null of a monotonic pricing kernel (MK) in favor of the nonmonotonic (NK) alternative at the 0.01% significance level. Table 4 is also interesting in that it shows that the two key conclusions from Tables 2 and 3 still obtain: allowing for inverse Gaussian innovations improves the fit, as does allowing for a second variance component. In Table 4, these conclusions are based on option fit but use return-based estimates. This means that these findings are not merely in-sample phenomena. Figure 1 complements Table 4 by plotting the implied volatility root-mean-square error (RMSE) percentages (top panel) and log-likelihood values (bottom panel) for different values of the ξ parameter in the models we consider. Figure 1 shows that the IG-GARCH component model we propose has lower RMSE and higher log-likelihood values for the optimal ξ parameter and indeed for a wide range of values around the optimum. The linear pricing kernel corresponds to the left-most point on the curves, where ξ = 0. Figure 1 View largeDownload slide RMSE and option likelihood values versus ξ This figure plots the RMSE (top panel) and the option log likelihood (bottom panel) as a function of the nonmonotonic pricing kernel parameter, ξ. All other parameter values are fixed at their optimal values from Table 2. Figure 1 View largeDownload slide RMSE and option likelihood values versus ξ This figure plots the RMSE (top panel) and the option log likelihood (bottom panel) as a function of the nonmonotonic pricing kernel parameter, ξ. All other parameter values are fixed at their optimal values from Table 2. 4.3 Capturing dynamics in higher moments Examination of the parameter estimates in Tables 2-4 reveals the main reason for the superior performance of the two-factor models. For the returns-based estimation in Table 2, the persistence of the single-factor estimates is 0.98 at a daily frequency for the Gaussian and the inverse Gaussian model. For the two-factor models, the long-run factor is always very persistent (ρ2 is around 0.99), but the persistence of the short-run factor, ρ1, is 0.71 in the Gaussian model and 0.74 in the inverse Gaussian model. The single-factor models are forced to compromise between slow and fast mean reversion, leading to a deterioration in fit in some parts of the sample. This confirms existing findings by, among others, Bates (2000), Christoffersen, Heston, and Jacobs (2009), and Andersen, Fusari, and Todorov (2015a, 2015b). Figures 2 and 3 provide additional perspective on the differences between the GARCH(1,1) and component models. Figure 2 plots the spot variance for all models using the return-based estimates. Figure 3 also uses the return-based estimates to plot conditional (“leverage”) correlation between returns and variance, Corrt[R(t+Δ),h(t+2Δ)] , which is informative about the third moment dynamics, and conditional standard deviation of variance, Vart[h(t+2Δ)] , which is informative about the fourth moment dynamics. Appendix D provides the formulas used for these conditional moments. Figure 2 View largeDownload slide Spot variance paths using return-based estimates For each model, we plot the spot variance components over time. The parameter values used are the MLE estimates from returns in Table 2. Figure 2 View largeDownload slide Spot variance paths using return-based estimates For each model, we plot the spot variance components over time. The parameter values used are the MLE estimates from returns in Table 2. Figure 3 View largeDownload slide Leverage correlation and volatility of variance using return-based estimates For each model, we plot the conditional correlation and the conditional standard deviation of variance. In the left panels, we plot the conditional correlation between return and variance as implied by the models. In the right panels, we plot the conditional standard deviation of conditional variance. The scales are identical across the rows of panels to facilitate comparison across models. The parameter values used are the MLE estimates from returns in Table 2. Figure 3 View largeDownload slide Leverage correlation and volatility of variance using return-based estimates For each model, we plot the conditional correlation and the conditional standard deviation of variance. In the left panels, we plot the conditional correlation between return and variance as implied by the models. In the right panels, we plot the conditional standard deviation of conditional variance. The scales are identical across the rows of panels to facilitate comparison across models. The parameter values used are the MLE estimates from returns in Table 2. In Figure 2, we can see that component model total variance (i.e. h(t)) is more variable and has the ability to increase faster than the GARCH(1,1), thanks to its short-run component (i.e. h(t) – q(t)). During the recent financial crisis the variances in the component models jump to a higher level than do the GARCH(1,1) variances. Consistent with this finding, the conditional standard deviation of variance (conditional correlation between returns and variance) of the component models in Figure 3, is higher in level (more negative) and more noisy than those of GARCH(1,1) models. Figure 4 graphs the term structure of variance, skewness and kurtosis using the derivatives of the moment generating function. Variance, skewness and kurtosis are defined by Vart(T)=∂2lngt(φ,T)/∂φ2|φ=0, (34) Skewt(T)=∂3lngt(φ,T)/∂φ3|φ=0(∂2lngt(φ,T)/∂φ2|φ=0)3/2, (35) Kurtt(T)=∂4lngt(φ,T)/∂φ4|φ=0(∂2lngt(φ,T)/∂φ2|φ=0)2−3. (36) Appendix E provides the analytical expressions for the derivatives required to compute these moments. Figure 4 View largeDownload slide Term structure of variance, skewness, and kurtosis We plot the term structure of variance, skewness, and excess kurtosis with high (solid) and low (dashed) initial variance for 1 to 250 trading days. Conditional variance is normalized by the unconditional variance, σ2. For the low initial variance, the initial value of q(t+Δ) is set to 0.75σ2 , and the initial value of h(t+Δ) is set to 0.5σ2 . For the high initial variance, the initial value of q(t+Δ) is set to 1.75σ2 , and the initial value of h(t+Δ) is set to 2σ2 . The return-based parameter values from Table 2 are used. Figure 4 View largeDownload slide Term structure of variance, skewness, and kurtosis We plot the term structure of variance, skewness, and excess kurtosis with high (solid) and low (dashed) initial variance for 1 to 250 trading days. Conditional variance is normalized by the unconditional variance, σ2. For the low initial variance, the initial value of q(t+Δ) is set to 0.75σ2 , and the initial value of h(t+Δ) is set to 0.5σ2 . For the high initial variance, the initial value of q(t+Δ) is set to 1.75σ2 , and the initial value of h(t+Δ) is set to 2σ2 . The return-based parameter values from Table 2 are used. The plots in the first column of Figure 4 show variance normalized by unconditional variance of each model, the second column shows skewness and the third column shows kurtosis. Each row corresponds to a different model. The initial variance is set to twice the unconditional model variance in the solid lines and the initial variance is set to one-half the unconditional variance in the dashed lines. For the component models we set the long-run variance component, q(t) equal to three-quarters of total variance, h(t). We use the return-based parameters in Table 2 to plot Figure 4. The left-side panels in Figure 4 highlight the differences between the GARCH(1,1) and component models. The impact of the current conditions on the future variance is much larger for the component models, and this is, of course, due to the persistence of the long-run component. For the GARCH(1,1) model, the conditional variance converges much quicker to the long-run variance. Figure 4 also shows that the term structures of skewness and kurtosis in the models differ between one-factor and component models. The one-factor models generate strongly hump-shaped term structures whereas the component models do so to a much lesser degree. Figure 4 confirms that the Gaussian and inverse Gaussian models do not differ much in the term structure dimension, and also indicates that the effects of shocks last much longer in the component models. Figures 5 and 6 repeat Figures 2 and 3 but use the option-based parameters in Table 3 rather than the physical parameters in Table 2. In Figure 5, the variance paths for the GARCH(1,1) and component models are very different compared to the return-implied paths in Figure 2. Note in particular that the short-run component in the component models strongly differs between Figures 2 and 5. In Figure 6, the time path of the conditional standard deviation of variance in the right-side panels is rather similar to the one from Figure 3, but this is not the case for the conditional correlation in the left-side panels. Figure 5 View largeDownload slide Spot variance paths using option-based estimates We plot the spot variance components over time. The parameter values used are the MLE on options in Table 3. Figure 5 View largeDownload slide Spot variance paths using option-based estimates We plot the spot variance components over time. The parameter values used are the MLE on options in Table 3. Figure 6 View largeDownload slide Leverage correlation and volatility of variance using option-based estimates For each model, we plot the conditional correlation and the conditional standard deviation of variance. In the left panels, we plot the conditional correlation between return and variance as implied by the models. In the right panels, we plot the conditional standard deviation of conditional variance. The scales are identical across the rows of panels to facilitate comparison across models. The parameter values used are the MLE estimates estimated from options in Table 3. Figure 6 View largeDownload slide Leverage correlation and volatility of variance using option-based estimates For each model, we plot the conditional correlation and the conditional standard deviation of variance. In the left panels, we plot the conditional correlation between return and variance as implied by the models. In the right panels, we plot the conditional standard deviation of conditional variance. The scales are identical across the rows of panels to facilitate comparison across models. The parameter values used are the MLE estimates estimated from options in Table 3. Most model implications can be easily understood by inspecting the parameter estimates in Tables 2-4. In the case of the risk-neutral estimates from options in Table 3, a first important conclusion is that the component models are more persistent than the GARCH(1,1) models, but the differences are smaller than in the case of the return-based estimates in Table 2. As a result, the impact of the current conditions on the future variance is larger for the component models, but the differences with the GARCH(1,1) model are larger for the return-based estimates. Second, and unsurprisingly, results are always very similar for the Gaussian and inverse Gaussian models. Third, and most importantly, the risk-neutral dynamics are more persistent than physical dynamics. As a result, the impact of the current conditions on the future variance is much larger for the option-implied risk-neutral estimates, regardless of the model. When estimating the models using returns and options sequentially in Table 4, the persistence of the models, and consequently the impact of the current conditions on the future variance, is close to the physical persistence based on returns in Table 2 since we fix the physical parameters in this estimation to the optimized returns-based parameter estimates. 4.4 Capturing smiles and smirks In Tables 5 and 6 we further investigate the model option fit across the moneyness and maturity categories defined in Table 1. Tables 5 and 6 report implied volatility RMSE and bias (in percentage) by moneyness, and maturity, respectively. Table 5 shows that the IG-GARCH(C) model we propose fits the data best in almost all moneyness categories. Not surprisingly, all models have most difficulty fitting the deep in-the-money calls (corresponding to deep out-of-the-money puts) that are very expensive. It is also not surprising that the fit in panel B is almost always better than in panel C, which, in turn, is better than in panel A. In panel B, the option fit drives all the parameter estimates, in panel C only ξ is estimated on options, whereas in panel A no parameters are fitted to option prices. Again, the most important conclusion from Table 5 is that the IG-GARCH(C) model performs well regardless of implementation and moneyness category. Panel A of Table 5 also shows that the large RMSEs are largely driven by bias, which is defined as market IV less model IV. Positive numbers thus indicate that the model underprices options, on average. Panel A shows that only the models with linear pricing kernel estimated on returns have large positive biases in every moneyness category. In panel B, where all parameters are estimated on options, the bias is much closer to zero. In panel C, the bias is much smaller than in panel A, but it is still fairly large for deep in-the-money calls. Table 6 reports the implied volatility RMSE and bias by maturity. The IG-GARCH component model now performs the best in all categories. Table 6 also shows that all models tend to underprice options (i.e. positive bias) at most maturities except for the very long-dated options. Tables 5 and 6 indicate that the fat-tailed inverse Gaussian distribution is also helpful in fitting the data. As emphasized by Eraker, Johannes, and Polson (2003), for instance, fat-tailed innovations increase the values of short-term out-of-the-money options. Two-factor dynamics increase the tails and values of long-term out-of-the-money options. Tables 5 and 6 demonstrate that these model features are to some extent complementary. The increases in likelihood due to fat-tailed innovations are much smaller than those due to the second volatility factor. This observation is consistent across estimation exercises and is confirmed by inspecting stylized facts. It also confirms the existing literature that emphasizes the importance of the second volatility factor (Bates 2000; Christoffersen, Heston, and Jacobs 2009; Andersen, Fusari, and Todorov 2015a, 2015b). Figure 5 indicates that the variance paths are very similar for the models with Gaussian and inverse Gaussian innovations for the option-based estimation results. However, this is unsurprising and not necessarily very relevant for the purpose of option valuation. Models with very similar variance paths can greatly differ with respect to their (conditional) third and fourth moments, and these model properties are of critical importance for option valuation, and for capturing smiles and smirks in particular. Therefore, we again look at conditional correlation and standard deviation of variance paths for the options-based estimations in Figure 6, which indicates substantial differences between the conditional correlation and standard deviation of variance paths for the Gaussian and inverse Gaussian models. However, perhaps somewhat surprisingly, Figures 5 and 6 clearly indicate that the differences between the GARCH(1,1) and component models are actually larger than the differences between the Gaussian and Inverse Gaussian models in this dimension. This is surprising because a priori we expect the second factor to be more important for term structure modeling, as confirmed by Figure 4. The conditional moments in Figures 5 and 6 are more important for the modeling of smiles and smirks, and a priori we expect the modeling of the conditional innovation to be more important in this dimension. However, it seems that the second volatility factor is also of first-order importance in this dimension. Figure 7 further illustrates the component model’s flexibility. We plot model-based implied volatility smiles using our proposed IG component model and the parameter values from Table 4. The total spot volatility, h(t) , is fixed at 25% per year in all panels. In the top panel, the long run volatility factor, q(t) is set to 20%, in the middle panel it is set to 25%, and in the bottom the top panel it is set to 30%. We also show the IG-GARCH(1,1) model for reference. It is of course the same across the three panels. Figure 7 shows that the second volatility factor gives the model a great deal of flexibility in modeling the implied volatility smile. Figure 7 View largeDownload slide Model-based implied volatility smiles in the IG-GARCH component model We plot model-based implied volatility smiles for 30 days to maturity from the IG-GARCH(1,1) and IG-GARCH(C) models. Long-run volatility, q(t) , is set to 20% (top) panel, 25% (middle panel), and 30% (bottom panel). Total volatility, h(t) , is set to 25% in all panels. The parameter estimates from Table 4 are used to generate the model prices. Model-implied volatilities are calculated by inverting the Black-Scholes formula based on model prices. Figure 7 View largeDownload slide Model-based implied volatility smiles in the IG-GARCH component model We plot model-based implied volatility smiles for 30 days to maturity from the IG-GARCH(1,1) and IG-GARCH(C) models. Long-run volatility, q(t) , is set to 20% (top) panel, 25% (middle panel), and 30% (bottom panel). Total volatility, h(t) , is set to 25% in all panels. The parameter estimates from Table 4 are used to generate the model prices. Model-implied volatilities are calculated by inverting the Black-Scholes formula based on model prices. 4.5 The relative importance of model features for option RMSE We now perform an assessment of the relative importance of the three model features for option fitting. To this end, consider the “All” RMSE in the last column of Table 5 because it contains the implied volatility RMSEs across all options. Panel A uses the return-based estimates from Table 2; panel B uses the option-based estimates from Table 3; and panel C uses the sequential estimates from Table 4. The last column in Table 5 enables us to make six pairwise comparisons of GARCH(1,1) and component GARCH(C) models. The improvement from adding a second volatility factor ranges from 4.87% (1 – 5.0694/5.3289) and 3.8% in panel A, to 16.24% and 15.45% in panel B, and finally 9.71% and 6.6% in panel C. On average, the improvement from adding a second volatility factor is 9.45%. The improvement from adding a second volatility factor is largest in panels B and C where the nonmonotonic pricing kernel affects the results. The second volatility component and the U-shaped pricing kernel thus appear to be complements. The last column in Table 5 also enables us to compute six pairwise comparisons of GARCH versus IG-GARCH models. The IV-RMSE improvement from adding fat tails ranges from 2.12% and 1.02% in panel A, to 6.18% and 5.29% in panel B, and 7.38% and 4.20% in panel C. The overall improvement from adding fat tails is 4.4% and thus considerably lower than from adding a second volatility factor. The improvement from adding fat tails is again largest in panels B and C where the nonmonotonic pricing kernel affects the results. Fat tails and a U-shaped pricing kernel thus also appear to be complements rather than substitutes. Comparing panels A and C in Table 5 allows us assess the importance of a U-shaped versus a linear pricing kernel. The improvement from allowing for a U-shaped kernel is 13.39% (1 – 4.6155/5.3289) for the GARCH(1,1) model, 18.05% for the IG-GARCH(1,1) model, 17.80% for the GARCH(C) model, and 20.44% for the IG-GARCH(C) model. On average, the improvement is 17.42%. The improvement from allowing for a U-shaped kernel is larger for IG than for Gaussian GARCH models, and it is larger for two-factor than for single-factor models. This, again, suggests that the three features we investigate are complements rather than substitutes. In Table 7, we compare the models using the Akaike (AIC), the Bayesian (BIC), and the Hannan-Quinn (HQIC) information criteria. The AIC weights together the number of parameters and the negative of the in-sample model log likelihood using AIC=2k−2 lnL,BIC uses BIC=k lnN−2 lnL, whereas HQIC is based on HQIC=2k ln(lnN)−2 lnL, where k is the number of parameters, N is the number of observations, and lnL is the log-likelihood. Table 7 shows that the more general model is always preferred, typically by quite a large margin. The differences in information criteria are particularly large when using option-based and sequential parameter estimation. 4.6 Out-of-sample fit Although our results indicate that the relative importance of the three model features differs, the more general models are always supported over the more parsimonious models. This is perhaps not surprising because we are implementing in-sample tests that tend to favor less parsimonious models. One way to address this is to construct out-of-sample tests. Panel A of Table 8 reports the log-likelihood values computed for the 2013-2015 out-of-sample period, using the in-sample parameter estimates from Tables 2-4. Panel A shows that the model rankings out-of-sample are virtually everywhere consistent with the in-sample results in Tables 2-4. The out-of-sample IV RMSEs in panel B of Table 8 yield the same conclusion. These findings suggest that the evidence in favor of the more elaborate models is not simply due to overfitting the data in-sample. One important difference between the out-of-sample results in Table 8 and the in-sample results is the option-based estimates of the non-Gaussian IG-GARCH(C) model, which are dominated by the corresponding Gaussian GARCH(C). Christoffersen, Heston, and Jacobs (2006) find that a Gaussian GARCH(1,1) model dominates an IG-GARCH(1,1) model out-of-sample. These findings may indicate that modeling an IG innovation does not produce sufficient improvements over a Gaussian innovation in practically relevant out-of-sample applications. However, this finding needs to be interpreted with some caution. An alternative potential explanation for this finding is that the 2013–2015 out-of-sample period that we rely on is relatively tranquil. Note also that the results for the sequential estimation in Table 8 favor the IG-GARCH(C) model. 4.7 The pricing kernel and model-implied relative-risk aversion Figure 8 shows the log pricing kernel for the one-month maturity by reporting the risk-neutral to physical log density ratio for four dynamic models with nonmonotonic pricing kernel (solid line) as well as the special Rubinstein (1976) and Brennan (1979) case with a standard log-linear pricing kernel (dashed line). We set the conditional variance to its unconditional level, the risk-free rate to 5% per year, and use the empirical estimates from Table 4. Figure 8 shows that the log pricing kernel is U shaped, not only at the one-day maturity at which it is defined in the model in equation (13), but also at the empirically more relevant one-month maturity. Figure 8 View largeDownload slide Monotonic and nonmonotonic one-month pricing kernels For the four dynamic models, we plot the natural logarithm of the ratio of the risk-neutral and physical conditional one-month densities implied by the nonmonotonic pricing kernel using solid lines and the special case of a monotonic pricing kernel using dashed lines. We use the parameters from Table 4. The conditional variance is set to its unconditional level. The risk-free rate is set to 5% per year. Figure 8 View largeDownload slide Monotonic and nonmonotonic one-month pricing kernels For the four dynamic models, we plot the natural logarithm of the ratio of the risk-neutral and physical conditional one-month densities implied by the nonmonotonic pricing kernel using solid lines and the special case of a monotonic pricing kernel using dashed lines. We use the parameters from Table 4. The conditional variance is set to its unconditional level. The risk-free rate is set to 5% per year. When using the standard log-linear pricing kernel, the coefficient of relative-risk aversion is simply (the negative of) ϕ. For the nonmonotonic pricing kernel the computation of risk-aversion is slightly more involved and we therefore provide some discussion here. Assume a representative agent with utility function U(S(t)), then the one-period coefficient of relative-risk aversion can be written RRA(t)≡−S(t)U′′(S(t))U′(S(t))=−S(t)M′(t)M(t)=−S(t)∂ln(M(t))∂S(t), (37) where we have used the insight of Jackwerth (2000) to link risk aversion to the pricing kernel. From (13), we have that ∂ln(M(t))∂S(t)=ϕS(t)+ξ∂h(t+Δ)∂S(t). (38) In the Gaussian model, we have ∂h(t+Δ)∂S(t)=∂h(t+Δ)∂z(t)∂z(t)∂S(t)=2α1(z(t)−γ1h(t))h(t)S(t). (39) Combining (38) and (39) we obtain a relative-risk aversion of RRA(t)=−ϕ−2α1ξ(z(t)−γ1h(t))h(t). Note as indicated above that the parameter ϕ does not in itself capture relative-risk aversion unless ξ = 0, which corresponds to the linear pricing kernel. Using the law of iterated expectations we can now compute the expected RRA as E[RRA(t)]=−ϕ+2α1ξγ1. Using the GARCH(1,1) parameter estimates in Tables 2 and 4, and the results in Appendix B of Christoffersen, Heston, and Jacobs (2013), we obtain ϕ=−(μ˜+γ1)(1−2α1ξ)+γ1−12≈20.26, so that we can write E[RRA(t)]≈−20.26+2α1ξγ1≈1.44. This result shows that the nonmonotonic pricing kernel delivers reasonable coefficients of relative-risk aversion, and furthermore that it is important not to rely on (the negative of) ϕ as a measure of RRA when using the nonmonotonic pricing kernel. 5. Conclusion We have found that multiple volatility factors, fat-tailed return innovations, and a variance-dependent pricing kernel all provide economically and statistically significant improvements in describing S&P500 returns and option prices. A U-shaped pricing kernel is economically most important and improves option fit by 17%, on average, and more so for two-factor models. A second volatility factor improves the option fit by 9%, on average. Fat tails improve option fit by just over 4%, on average, and more so when a U-shaped pricing kernel is applied. Overall, our results show that these three features are complements rather than substitutes. This indicates that although proper specification of volatility dynamics is quantitatively most important in option models, the interdependent explanatory power of different features makes it essential to evaluate them in a properly specified model that nests all of these features. Appendix A. Martingale Restrictions A.1 Restrictions Implied by the Risk-free Asset We first impose on the pricing kernel that the risk-free bond price is a martingale under the risk-neutral measure. We need Et[M(t+Δ)M(t)Bτ(t+Δ)]=Bτ(t), (40) where Bτ(t) is a bond with maturity τ at time t and M(t+Δ)/M(t)=(S(t+Δ)/S(t))ϕ exp (δ(t+Δ)+ξh(t+2Δ)) where δ(t+Δ)≡δ0+δ1h(t+Δ) . WLOG we assume that the risk-free rate is constant so that Bτ(t+Δ)/Bτ(t)≡ exp (r) . We can now write 1=Et[exp (ϕR(t+Δ)+δ0+δ1h(t)+ξh(t+2Δ)+r)]. The martingale restriction in Equation (40) implies that we need to impose the following parameter restrictions on the pricing kernel, δ0=−(1+ϕ)r−ξw+12ln(1−2ξa1η4), (41) δ1=−ϕμ−ξb1−η−2(1−(1−2ξa1η4)(1−2(ϕη+ξc1))), (42) where we have used the following property of the IG distribution Et[exp (αy(t+Δ)+β/y(t+Δ))]=11−2βh(t+Δ)−2η4× exp [h(t+Δ)/η2(1−(1−2βh(t+Δ)−2η4)(1−2α))]. (43) A.2 Restrictions Implied by the Risky Asset Next, we impose on the pricing kernel that the risky stock is a martingale under the risk-neutral measure. We now need Et[M(t+Δ)M(t)S(t+Δ)]=S(t). We can thus write 1=Et[exp (ϕR(t+Δ)+δ0+δ1h(t+Δ)+ξh(t+2Δ)+R(t+Δ))]. Taking logs in the equation above implies the following restriction on μ, μ=η−2(1−2ξa1η4)[1−2(η+ϕη+ξc1)−1−2(ϕη+ξc1)], where we have used Equation (43). Appendix B. The Risk-Neutral Distribution Consider the physical probability density function of the IG stock price: ft−Δ(S(t))=ft−Δ(y(t))|∂y(t)∂S(t)|=h(t)/|η3|2πy(t)3S(t)exp (−12[y(t)−h(t)/η2y(t)]2). (44) To find the risk-neutral dynamic, we use the price kernel as follows: ft−Δ∗(S(t))=ft−Δ(S(t)) exp (r)M(t)/M(t−Δ), (45) where M(t−Δ) is (t−Δ)-measurable. Using the pricing kernel definition in (13) and the IG-GARCH(1,1) return dynamic, we can write ft−Δ∗(S(t))=ft−Δ(S(t)) exp [r+δ0+δ1h(t)+ϕln(S(t)/S(t−Δ))+ξh(t+2Δ)]=h(t)/|η3|1−2ξaη42πy(t)3S(t)exp [−12((1−2ϕη−2ξc)y(t)−h(t)/η2y(t)1−2ξaη4)2]. Substituting the physical distribution from Equation (44) and rearranging terms yields ft−Δ∗(S(t))=h(t)(1−2ξaη4)(1−2ϕη−2ξc)−3(1−2ϕη−2ξc)3/|η3|2πy(t)3(1−2ϕη−2ξc)3S(t)× exp [−12(y(t)(1−2ϕη−2ξc)−h(t)(1−2ξaη4)(1−2ϕη−2ξc)−3(1−2ϕη−2ξc)2/η2y(t)(1−2ϕη−2ξc))2]. This enables us to define the risk-neutral counterparts to y(t), h(t) , and η by y∗(t)=y(t)(1−2ϕη−2ξc)=y(t)sy,h∗(t)=h(t)(1−2ξaη4)(1−2ϕη−2ξc)−3=h(t)sh,η∗=η/(1−2ϕη−2ξc)=η/sy, where we have used the definitions sy=1−2ϕη−2ξc,sh=1−2ξaη4sy−3/2, like in the text. Using these mappings yields the risk-neutral density: ft−Δ∗(S(t))=h∗(t)/|η∗|32π(y∗(t))3S(t)exp [−12(y∗(t)−h∗(t)/(η∗)2y∗(t))2]. So that ft−Δ∗(y∗(t))=ft−Δ∗(S(t))|∂S(t)∂y∗(t)|=ft−Δ∗(S(t))|S(t)×(−η∗)|=h∗(t)/(η∗)22π(y∗(t))3exp [−12(y∗(t)−h∗(t)/(η∗)2y∗(t))2]. Therefore y∗(t) is distributed inverse Gaussian, and we can write y∗(t)∼IG(h∗(t)(η∗)2). Using the physical return process and the above mappings we can write the risk-neutral return process as ln(S(t+Δ))=ln(S(t))+r+μh∗(t+Δ)/sh+ηy∗(t+Δ)/syh∗(t+Δ)=wsh+bh∗(t)+cy∗(t)sh/sy+asyshh∗(t)2/y∗(t), or, equivalently, ln(S(t+Δ))=ln(S(t))+r+μ∗h∗(t+Δ)+η∗y∗(t+Δ)h∗(t+Δ)=w∗+bh∗(t)+c∗y∗(t)+a∗h∗(t)2/y∗(t), where we have used the parameter mapping in Equation (16b) and (16c). Appendix C. The Risk-Neutral Component Model The component representation of the risk-neutral process (15) is given by ln(S(t+Δ))=ln(S(t))+r+μ˜∗h(t+Δ)+(η∗y∗(t+Δ)−h∗(t+Δ)/η∗),h∗(t+Δ)=q∗(t+Δ)+ρ1∗(h∗(t)−q∗(t))+νh∗(t),q∗(t+Δ)=σ∗2+ρ2∗(q∗(t)−σ∗2)+νq∗(t), where q∗(t)=−ρ1∗w˜∗(1−ρ1∗)(ρ2∗−ρ1∗)+ρ2∗ρ2∗−ρ1∗h∗(t)+b˜2∗ρ2∗−ρ1∗h∗(t−Δ)+1ρ2∗−ρ1∗υ2∗(t−Δ),μ˜∗=μ∗+η∗−1=μ/sh+syη−1,σ∗2=(ρ2∗1−ρ2∗−ρ1∗1−ρ1∗)w˜∗ρ2∗−ρ1∗,w˜∗=w∗+a1∗η∗4+a2∗η∗4=shw+a1η4+a2η4shsy3, υh∗(t)=ch∗y∗(t)+ah∗h∗(t)2/y∗(t)−ch∗h∗(t)/η∗2−ah∗η∗2h∗(t)−ah∗η∗4,υq∗(t)=cq∗y∗(t)+aq∗h∗(t)2/y∗(t)−cq∗h∗(t)/η∗2−aq∗η∗2h∗(t)−aq∗η∗4, ah∗=−ρ1∗ρ2∗−ρ1∗a1∗−1ρ2∗−ρ1∗a2∗,ch∗=−ρ1∗ρ2∗−ρ1∗c1∗−1ρ2∗−ρ1∗c2∗,aq∗=ρ2∗ρ2∗−ρ1∗a1∗+1ρ2∗−ρ1∗a2∗,cq∗=ρ2∗ρ2∗−ρ1∗c1∗+1ρ2∗−ρ1∗c2∗,b˜i∗=bi+shsyci/η2+aiη2shsy, and where ρ1∗ and ρ2∗ are the smaller and larger respective roots of the equation ρ∗2−b˜1∗ρ−b˜2∗=0 . Appendix D. Conditional Moments Consider the following basic definitions: Vart[h(t+2Δ)]≡Et[(h(t+2Δ)−Et[h(t+2Δ)])2],Covt[R(t+Δ),h(t+2Δ)]≡Et[(R(t+Δ)−Et[R(t+Δ)])(h(t+2Δ)−Et[h(t+2Δ)])], where R(t+Δ)≡lnS(t+Δ)−lnS(t) . In this section, we only focus on the derivation of conditional correlation, and conditional standard deviation of variance for IG-GARCH(C) model, since derivations for other models are similar. Recall that the standardized conditional moments of an inverse Gaussian random variable y(t+1) are given by Et[y(t+Δ)]=ψ(t+Δ),Vart[y(t+Δ)]=ψ(t+Δ),Et[1/y(t+Δ)]=1/ψ(t+Δ)+1/ψ(t+Δ)2,Vart[1/(t+Δ)]=1/ψ(t+Δ)3+2/ψ(t+Δ)4,Covt[y(t+Δ),1/y(t+Δ)]=−1/ψ(t+Δ), where the degree of freedom is defined by ψ(t+Δ)=h(t+Δ)/η2. The variance process is defined as h(t+Δ)=q(t+Δ)+ρ1[h(t)−q(t)]+vh(t),q(t+Δ)=wq+ρ2q(t)+vq(t),vh(t)=ch[y(t)−ψ(t)]+ahh(t)2[1/y(t)−1/ψ(t)−1/ψ(t)2],vq(t)=cq[y(t)−ψ(t)]+aqh(t)2[1/y(t)−1/ψ(t)−1/ψ(t)2]. Conditional variance of variance is given by Vart[h(t+2Δ)]=(ch+cq)2h(t+Δ)/η2−2(ah+aq)(ch+cq)η2h(t)+(ah+aq)2η6h(t)+2(ah+aq)2η8. We thus can write Stdt[h(t+2Δ)]=2(ah+aq)2η8+[(ch+cq)/η−(ah+aq)η3]2h(t+Δ). Consider now the innovation to returns: R(t+Δ)−Et[R(t+Δ)]=η(y(t+Δ)−ψ(t+Δ)). We can then derive covariance and correlation as Covt[R(t+Δ),h(t+2Δ)]=(ch+cq)ηVart[y(t+Δ)]+(ah+aq)ηh(t+Δ)2Covt[y(t+Δ),1/y(t+Δ)]=(ch+cq)/ηh(t+Δ)−(ah+aq)η3h(t+Δ),Corrt[R(t+Δ),h(t+2Δ)]≡Covt[R(t+Δ),h(t+2Δ)]Vart[R(t+Δ)]Vart[h(t+2Δ)]=[(ch+cq)/η−(ah+aq)η3]h(t+Δ)2(ah+aq)2η8+[(ch+cq)/η−(ah+aq)η3]2h(t+Δ). Appendix E. Conditional Cumulants In this appendix, we derive analytical expressions for the cumulants so that the term-structure of higher moments can be computed without relying on numerical derivatives. The cumulants are the derivatives of the logarithm of the generating function with respect to ϕ , evaluated at ϕ=0 . The conditional generating function gt(φ,T) is given by gt(φ,T)=Et[S(T)ϕ]=S(t)ϕ exp (A(t)+B(t)h(t+Δ)+C(t)q(t+Δ)), (50) where A(T)=B(T)=C(T)=0, (51a) A(t)=A(t+Δ)+ϕr+(wq−ahη4−aqη4)B(t+Δ)+(wq−aqη4)C(t+Δ)−12ln(D1(t)), (51b) B(t)=ϕμ+(ρ1−(ch+cq)η−2−(ah+aq)η2)B(t+Δ)− (51c) (cqη−2+aqη2)C(t+Δ)+η−2−D(t)η2, (51d) C(t)=(ρ2−ρ1)B(t+Δ)+ρ2C(t+Δ), (51e) D(t)=D1(t)D2(t), (51f) D1(t)=1−2(aq+ah)η4B(t+Δ)−2aqη4C(t+Δ), (51g) D2(t)=1−2ηϕ−2(cq+ch)B(t+Δ)−2cqC(t+Δ). (51h) Note that the functions D1(t) and D2(t) are used to simplify the notation. The ith cumulant is simply A(i)(t)+B(i)(t)h(t+Δ)+C(i)(t)q(t+Δ)|ϕ=0 . We can write the terminal conditions as A(i)(T)=B(i)(T)=C(i)(T)=0. (52a) The recursive definition of A(i)(.) is A′(t)=A′(t+Δ)+r+(wq−ahη4−aqη4)B′(t+Δ)+(wq−aqη4)C′(t+Δ)−12D1′(t)D1(t), (53a) A′′(t)=A′′(t+Δ)+(wq−ahη4−aqη4)B′′(t+Δ)+(wq−aqη4)C′′(t+Δ)−12(D1′′(t)D1(t)−[D1′(t)D1(t)]2), (53b) A′′′(t)=A′′′(t+Δ)+(wq−ahη4−aqη4)B′′′(t+Δ)+(wq−aqη4)C′′′(t+Δ)−12(D1′′′(t)D1(t)−3D1′′(t)D1′(t)(D1(t))2+2[D1′(t)D1(t)]3), (53c) A(iv)(t)=A(iv)(t+Δ)+(wq−ahη4−aqη4)B(iv)(t+Δ)+(wq−aqη4)C(iv)(t+Δ)−12(D1(iv)(t)D1(t)−4D1′′′(t)D1′(t)(D1(t))2+12D1′′(t)(D1′(t))2(D1(t))3−3[D1′′(t)D1(t)]2−6[D1′(t)D1(t)]4), (53d) where we have used primes to denote derivatives. The recursive definition of B(i)(.) is B′(t)=μ+(ρ1−(ch+cq)η−2−(ah+aq)η2)B′(t+Δ)−(cqη−2+aqη2)C′(t+Δ)−D′(t+Δ)D(t+Δ)−1/22η2, (54a) B′′(t)=(ρ1−(ch+cq)η−2−(ah+aq)η2)B′′(t+Δ)−(cqη−2+aqη2)C′′(t+Δ)−D′′(t)D(t)−1/22η2+D′(t)2D(t)−3/24η2, (54b) B′′′(t)=(ρ1−(ch+cq)η−2−(ah+aq)η2)B′′′(t+Δ)−(cqη−2+aqη2)C′′′(t+Δ)−D′′′(t)D(t)−1/22η2+3D′′(t)D′(t)D(t)−3/24η2−3D′(t)3D(t)−5/28η2, (54c) B(iv)(t)=(ρ1−(ch+cq)η−2−(ah+aq)η2)B(iv)(t+Δ)−(cqη−2+aqη2)C(iv)(t+Δ)−D(iv)(t)D(t)−1/22η2+D′′′(t)D′(t)D(t)−3/2η2+3D′′(t)2D(t)−3/24η2−9D′′(t)D′(t)2D(t)−5/24η2+15D′(t)4D(t)−7/216η2. (54d) The recursive definition of C(i)(.) is C(i)(t)=(ρ2−ρ1)B(i)(t+Δ)+ρ2C(i)(t+Δ). Finally, the definition for D(i)(.), D1(i)(.) , and D2(i)(.) are D1(i)(t)=−2(aq+ah)η4B(i)(t+Δ)−2aqη4C(i)(t+Δ), (56a) D2′(t)=−2η−2(cq+ch)B′(t+Δ)−2cqC′(t+Δ), (56b) D2(i)(t)=−2(cq+ch)B(i)(t+Δ)−2cqC(i)(t+Δ), for i>1, (56c) D(i)(t)=∑j=0i(ij)D1(i−j)(t)D2(j)(t), where D1(0)(t)=D1(t) and D2(0)(t)=D1(t). (56d) Footnotes 1 See, for instance, Chernov, Gallant, Ghysels, and Tauchen (2003) for a study of multiple volatility components in the underlying return series. 2 See, for instance, Andersen, Benzoni, and Lund (2002), Bakshi, Cao, and Chen (1997), Bates (1996a, 2000), Broadie, Chernov, and Johannes (2007), Chernov and Ghysels (2000), Eraker (2004), Jones (2003), and Pan (2002) for studies that estimate SV models with jumps using options and/or return data. 3 Linn, Shive, and Shumway (2014) argue that the finding of a nonmonotone pricing kernel could be an artifact of the econometric method used. Cuesdeanu and Jackwerth (2015) show that the finding of a nonmonotone kernel is robust across a range of econometric techniques. 4 See, for example, Hsieh and Ritchken (2005), Barone-Adesi, Engle and Mancini (2008), and Christoffersen et al. 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