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Realized GARCH, CBOE VIX, and the Volatility Risk Premium
Keywords:{ Realized GARCH; High Frequency Data; Volatility Risk Premium; Realized Variance; VIX.}
JEL Classification:{ C10; C22; C80}
{}
The variance risk premium or volatility risk premium (VRP) has been the focus of much research since the seminal papers by CovalShumway2001 and BK2003. The VRP is the difference between the expected return variance under the risk neutral measure and the expected return variance under the physical measure,\footnote{This definition of the VRP follows that in BollerslevTauchenZhou2009. Other definitions are used in part of the literature.} where the former can be inferred from option prices. The leading example is the VIX, which is a model-free measure of the expected variance over the next 30 days under the risk neutral measure. Expectations under the physical measure can be based on a suitable volatility model.\footnote{For instance a GARCH model or a reduced-form model for the realized volatility. For alternative methods for computing the expected variance under both the physical and risk neutral measures, see BollerslevTauchenZhou2009, BollerMZ2011, Wu_JOE2011 and Conrad_Loch2015.} The VRP is a measure of volatility risk compensation and it is typically positive. This is to be expected because large increases in volatility tend to coincide with large negative returns. This relationship is observed for a broad range of financial assets, see e.g. CovalShumway2001, BK2003 and CW2009. The VRP is also recognized as a distinct risk factor that predicts both aggregate stock returns and the cross-section of stock returns, see e.g. BollerslevTauchenZhou2009, Bekaert2014, and Cremers_HW2015.
Moreover, the VRP plays an important role in option pricing, see e.g. Byun_JMY2015 and Song_Xiu2016. Given its importance, it is interesting to develop an econometric model that can explain the time variation in the VRP while being coherent with other features of the data.
Conventional GARCH models specify expectations under the physical measure, $\mathbb{P}$, and additional structure is needed before expectations can be computed under the risk neutral measure, $\mathbb{Q}$. Duan1995 pioneered the use of GARCH models for option pricing by introducing a locally risk-neutral valuation relationship (LRNVR). The LRNVR defines a link between expected volatility under the $\mathbb{P}$ and expected volatility under $\mathbb{Q}$. With this additional structure in place, GARCH models can be used to price options and the corresponding VRP can be inferred. Unfortunately, standard GARCH models combined with LRNVR cannot adequately explain the VRP, as shown by HaoZhang2013. They found that the VIX implied by GARCH models is substantially below the observed VIX. HaoZhang2013 explored if this shortcoming could be amended by modifying the objective function to also target the VIX. Unfortunately, this leads to parameter values (in the GARCH model) that contradict the empirical properties under $\mathbb{P}$. In our empirical application, we also reach the conclusion that GARCH models in conjunction with LRNVR cannot explain the dynamic properties under both probability measures. A strong argument for looking beyond standard GARCH models is provided by their diffusion limits. These reveal that GARCH models are unable to fully compensate for volatility risk, because GARCH models lack a separate volatility shock variable. Stochastic volatility (SV) models, such as those by Taylor1986 and Kim1998, are better suited for this situation because they have a dual-shock structure with distinct shocks to returns and volatilities. This property facilitates a distinct compensation for volatility risk, see e.g. BollerMZ2011. The main drawback of SV models is that they are more involved to estimate than observation-driven models, such as GARCH models.\footnote{The same complication arises with Jump-GARCH models that typically rely on particle filters for estimation, see e.g. Ornthanalai2014. }
It is evident that the GARCH framework must be generalized in order to become a coherent model of $\mathbb{P}$ and $\mathbb{Q}$. This requires either a more flexible volatility model or a more sophisticated risk neutralization method. In this paper, we pursue both extensions by combining the Realized GARCH model with an exponentially affine stochastic discount factor. This framework includes compensation for both equity risk and volatility risk and it yields closed-form expressions for both the VIX and the VRP. The Realized GARCH model is an observation-driven model that conveniently has a dual shock structure that is similar to that of SV models. This model is simple to estimate and easy to combine with an\textcolor{blue}{ }exponentially affine stochastic discount factor. The parameter estimation can be adapted to include VIX pricing errors in the objective function, similar to the estimation method used in bardg2019, see also AFT2019 who links the realized volatility to volatility in an SV model. The estimated model has several interesting properties. It delivers a higher level of volatility, a higher volatility-of-volatility, and a stronger (more negative) leverage correlation under $\mathbb{Q}$ than under $\mathbb{P}$. The estimated model also generates higher levels of skewness and kurtosis in cumulative returns under $\mathbb{Q}$ than under $\mathbb{P}$, which are key determinants of the VRP, see Bakshi-Madan2006 and Chabi-Yo2012. The difference between log-volatility under $\mathbb{P}$ and $\mathbb{Q}$ can conveniently be decomposed into two terms, where the first term is compensation for equity risk through the leverage effect and the second term is compensation for volatility-of-volatility.
In an empirical analysis of daily S&P 500 returns, realized volatilities, and the VIX over 15 years, we compare the proposed model with a range of alternative specifications. These include the EGARCH model by Nelson91, the GARCH model by bollerslev:86, Heston-Nandi GARCH by HN2000. These models are combined with either the the LRNVR by Duan1995 or the variance dependent SDF by VDPK2013.\footnote{VDPK2013, introduced a variance-dependent SDF to improve the option pricing performance of the Heston-Nandi GARCH model. The idea was also used in Byun_JMY2015 in the context of Jump-GARCH models.} We find that the new model has the best in-sample and out-of-sample VIX pricing performance, and the proposed model does particularly well during the turmoil period with the global financial crisis. We find that the Realized GARCH model provides the best empirical fit and, importantly, provides superior out-of-sample forecast of all variables of interest.
The improved empirical results are driven by the inclusion of realized volatility in the modeling. The help in two ways. First, the inclusion improves volatility forecasts and this greatly improve the log-likelihood of returns under $\mathbb{P}$. Second, the inclusion of a realized volatility enables us to define a volatility shock that serves as a second state variables in the SDF. This state variable characterizes the compensation for volatility risk, which is important for explaining key differences between $\mathbb{P}$ and $\mathbb{Q}$ and time-variation therein.\footnote{The need for a model to simultaneously explain the variation under $\mathbb{P}$ and $\mathbb{Q}$ was pointed out in Bates1996, and has since received much attention in the option pricing literature, see, e.g., PAN20023, Eraker2004, and SantaYan2010. }
The remainder of this paper is organized as follows. We present the Realized GARCH model, the risk-neutralization, and the model implied VIX/VRP formula in Section 2 and discuss the distinct model dynamics under $\mathbb{P}$ and $\mathbb{Q}$ in Section 3. In Section 4, we present the set of competing models and we present our empirical analysis in Section 5. We conclude in Section 6, present all proofs in Appendix A, and present a range of empirical robustness checks in Appendix B, which are based on different definitions of the VRP, different error specification, and different choice of realized volatility measure in the modeling.
The Realized GARCH framework is a join model of returns and realized volatility measures. Returns are modeled with a GARCH model, which is augmented to include a realized measure of volatility, and the Realized GARCH framework is characterized by a measurement equation that ties the realized measure to the conditional variance. Realized measures of volatility are computed from high frequency data where the realized variance (RV) and the realized kernel (RK) by BHLS2008 are prime examples. Realized GARCH models are generally found to outperform conventional GARCH models in terms of modeling returns as well as forecasting volatility. The reason is simply that the realized measures provide more accurate measurements of volatility than daily returns, and conventional GARCH models rely on the latter for “updating” the time-varying volatility. The Realized GARCH framework was introduced by HansenHuangShek:2012 and later refined in RealizedEGARCH2016 to have a more flexible leverage function and to allow for the inclusion of multiple realized measures.
We adapt the model in RealizedEGARCH2016 to the present context, by adding an appropriate compensation for equity risk. Under the physical measure the model is characterized by:
where $r_{t}$ is the logarithmic return, $\lambda$ is the price of equity risk, $h_{t}=\mathrm{var}_{t-1}(r_{t})$ is the conditional variance, $r$ is the risk-free interest rate, $z_{t}=(r_{t}-\mathbb{E}_{t-1}r_{t})/\sqrt{h_{t}}$ is the standardized return, and $x_{t}$ is the realized measure of volatility. Our addition to this model framework, is a compensation for equity risk that adds the term, $\lambda\sqrt{h_{t}}-\tfrac{1}{2}h_{t}$, to the return equation ((ref)). The two random innovations, $z_{t}$ and $u_{t}$, that are assumed to be independent and iid standard Gaussian, $N(0,1)$. The quadratic functions $\tau(z)=\tau z+\tau_{2}(z^{2}-1)$ and $\delta(z)=\delta_{1}z+\delta_{2}(z^{2}-1)$ are leverage functions that capture dependence between return shocks and volatility shocks. The parameter $\sigma$ can be interpreted as the volatility-of-volatility shock. The model simplifies to a variant of the classical EGARCH\textcolor{black}{ model of Nelson91} when $\gamma=0$.\footnote{We use $z_{t}^{2}$ in place of $|z_{t}|$ that was used the original EGARCH model, which has some advantages, see HansenHuangShek:2012 and RealizedEGARCH2016. For completeness, we have also estimated a Realized GARCH model with $\tau_{1}z_{t}+\tau_{2}(|z_{t}|-\sqrt{2/\pi})$, which led to very similar qualitative and quantitative results.}
A key property of the model is that two shocks, $z_{t}$ and $u_{t}$, are included in the GARCH equation, ((ref)). This is contrast to conventional GARCH models, where the conditional volatility is solely driven by lagged daily returns.\footnote{For derivative pricing, this “single shock” structure is a serious limitation because the equity risk premium parameter, $\lambda$, must be increased to unreasonable high levels in order to explain the variance risk premium, see HaoZhang2013 and our empirical results in Table 2.} The dual-shock structure is important for describing the dynamic properties under both $\mathbb{P}$ and $\mathbb{Q}$ simultaneously, such that the dynamic properties of returns, the VIX, and the VRP, can be explained within a unified coherent framework.
Before we can price the VIX, we need to state how the physical measure, $\mathbb{P}$, relates to the risk neutral counterpart, $\mathbb{Q}$. In the literature on option pricing with GARCH models, the most commonly used risk neutralization methods are the locally risk-neutral valuation relationship (LRNVR) by Duan1995 and the variance-dependent SDF by VDPK2013. These methods are applicable to a single-shock GARCH models and do not apply to the dual-shock structure in our framework. We will instead adopt an exponentially affine stochastic discount factor, which has previously been used for risk neutralization with multiple shocks in Corsi2013JFE.
The stochastic discount factor, $M_{t+1}$, must satisfy $\mathbb{E}_{t}^{\mathbb{Q}}[X_{t+1}]=\mathbb{E}_{t}^{\mathbb{P}}[M_{t+1}X_{t+1}]$ for all asset prices, $X_{t+1}$. In the Realized GARCH framework it is natural to use $z_{t+1}$ and $u_{t+1}$ as state variables, and we will adopt the SDF define by:
Empirically, one would expect $\lambda$ to be positive and $\xi$ to be negative, which correspond to a positive equity premium and a negative variance risk premium, respectively. It should be noted that the parameter, $\lambda$, that appears in ((ref)) is identical to the $\lambda$ in the return equation, ((ref)). This is not by assumption but an implication of a no arbitrage condition. If we, as a starting point, permitted the $\lambda$ in ((ref)), to be a free and, possibly, time-varying parameter, then it can be shown that this coefficient must be constant and equal to $\lambda$ in ((ref)). This is a consequence of a no-arbitrage condition, see Lemma (ref) in the Appendix.
Under the risk neutral measure, $\mathbb{Q}$, it follows that the moment generating function (MGF) is given by
This MGF is identical to $\mathbb{E}_{t}^{\mathbb{P}}[\exp(s_{1}z_{t+1}^{\ast}+s_{2}u_{t+1}^{\ast})]$, where $z_{t+1}^{\ast}=z_{t+1}+\lambda$ and $u_{t+1}^{*}=u_{t+1}+\xi$, and it implies the following dynamic model under the risk neutral measure:{
}where $(z_{t}^{\ast},u_{t}^{\ast})$ has bivariate Gaussian distribution, $N(0,I)$, under $\mathbb{Q}$. The relationships between parameters (under $\mathbb{P}$ and $\mathbb{Q}$) are: $\tilde{\omega}=\omega-\tau_{1}\lambda+\tau_{2}\lambda^{2}-\gamma\sigma\xi$, $\tilde{\tau}_{1}=\tau_{1}-2\tau_{2}\lambda$, $\tilde{\kappa}=\kappa-\delta_{1}\lambda+\delta_{2}\lambda^{2}-\sigma\xi$, and $\tilde{\delta}_{1}=\delta_{1}-2\delta_{2}\lambda$, see Lemma (ref) for details.
The mapping $(z_{t},u_{t})\mapsto(z_{t}^{\ast},u_{t}^{\ast})$ can be viewed as a generalization of LRNVR to the bivariate case. In the present context, this bivariate structure rely on the inclusion of the realized measure in the modeling, which facilitates a more complex dynamic structure than can be achieved with conventional GARCH models. The conventional GARCH model emerges as a special case when $\gamma=0$ and $\xi=0$. In this situation, ((ref)) and ((ref)) simplify to an EGARCH model with the change of measure, $z_{t+1}^{*}=z_{t+1}+\lambda$. This reveals a close relation between the GARCH models with exponentially affine SDF and the simple change of measure that was proposed by Duan1995. This connection appears to have been overlooked in the exiting literature.
The Chicago Board Options Exchange's (CBOE) VIX index is defined as the square-root of the annualized expected variance over the next 30 calendar days, where the expectation is computed under the risk neutral measure. We will model returns and realized measure using daily data and we will use 22 trading days and 252 trading days to represent a month and a year, respectively. The model-based VIX formula is therefore given by \[ \mathrm{VIX}_{t}^{\mathrm{model}}=\sqrt{\tfrac{252}{22}\sum_{k=1}^{22}\mathbb{E}_{t}^{\mathbb{Q}}(h_{t+k})}\times100, \] where the expression for $\mathbb{E}_{t}^{\mathbb{Q}}(h_{t+k})$ is model specific. The combination of the Realized GARCH model and the exponentially affine SDF leads to the following expression:
The expression ((ref)) facilitates an easy comparison of the model implied VIX with the actual VIX index, and it is analogous to the expressions obtained for a range of conventional GARCH models in HaoZhang2013. The proof of Theorem (ref) is given in Appendix (ref).
The literature has proposed several definitions of the VRP, see BollerslevTauchenZhou2009.\footnote{See BollerslevTauchenZhou2009 for detailed discussion of VRP measures, including ex-post and ex-ante measures, and measures in units of variances and in units of volatilities.} We adopt the following definition,
\[ \mathrm{VRP}_{t}^{\mathtt{modelfree}}=\mathrm{VIX}_{t}-\sqrt{\tfrac{252}{22}\mathbb{E}_{t}^{\mathbb{P}}\left(\sum_{i=1}^{22}\mathrm{RVcc}_{t+i}\right)}\times100, \] where $\mathrm{RVcc}_{t}=\mathrm{RV}_{t}+r_{co,t}^{2}$, $\mathrm{RV}_{t}$ is the realized variance estimator for the hours with active trading on day $t$, and $r_{co,t}^{2}$ is the squared overnight return, which is computed from the closing price on day $t-1$ and the opening price of day $t$. This difference between the observed VIX and the expected realized measure of volatility (for the corresponding 22 trading days) is a model-free measure of the VRP. This is a theoretical quantity, because the expectation operator depends on $\mathbb{P}$ that is unknown in practice. The following empirical VRP
\[ \mathrm{VRP}_{t}^{\mathtt{market}}=\mathrm{VIX}_{t}-\sqrt{\tfrac{252}{22}\sum_{i=1}^{22}\mathrm{RVcc}_{t-i+1}}\times100, \] was proposed in BollerslevTauchenZhou2009. This quantity relies on the assumption that realized volatility follows a martingale process, such that the expected monthly realized volatility is given by the observed realized volatility over the most recent month. As a robustness check, we also consider a second, alternative, empirical measure, which is based on the heterogeneous autoregressive (HAR) model by Corsi2009, see Appendix (ref).
Our model-implied VRP is simply \[ \mathrm{VRP}_{t}^{\mathtt{model}}=\left(\sqrt{\tfrac{252}{22}\sum_{k=1}^{22}\mathbb{E}_{t}^{\mathbb{Q}}(h_{t+k})}-\sqrt{\tfrac{252}{22}\sum_{k=1}^{22}\mathbb{E}_{t}^{\mathbb{P}}(h_{t+k})}\right)\times100, \] which is the annualized, one-month ahead, expected volatility using the risk neutral measure less the corresponding quantity under the physical probability measure.
In this section, we analyze the Realized GARCH model with the exponentially affine SDF, under both $\mathbb{P}$ and $\mathbb{Q}$, and we derive key properties of volatility, leverage, and returns under both measures. These results provide theoretical insight about importance of various model parameters and their interpretations. We derive the properties under the assumption that the parameters in the Realized GARCH model and the SDF satisfy: \[ |\beta|<1,\quad\lambda,\gamma,\sigma,\tau_{2},\delta_{2}>0,\quad\xi,\tau_{1},\delta_{1}<0. \] These inequalities guarantee the following properties: (a) the volatility process is stationary, (b) the equity premium is positive, (c) the volatility premium is negative, and (d) the model has a leverage effect. The stated parameter restrictions are consistent with our empirical results in Section (ref).
For the average log-volatility (unconditional mean of $\log h_{t}$) we have that \[ \mathbb{E}^{\mathbb{P}}(\log h)=\frac{\omega}{1-\beta}\qquad\mbox{and}\qquad\mathbb{E}^{\mathbb{Q}}(\log h)=\frac{\omega-\tau_{1}\lambda+\tau_{2}\lambda^{2}-\gamma\sigma\xi}{1-\beta}. \] Given our assumptions above, it follows that the average log-volatility is higher under the $\mathbb{Q}$-measure than under the $\mathbb{P}$-measure. Thus, the logarithmic variant of the VRP,
is positive. The logarithmic variant of the VRP was used in CW2009 and AMMANN2013.
From ((ref)) we see that the logarithmic VRP can be decomposed into two terms. One that is driven by the equity risk premium and the leverage effect and a second term that is driven by the compensation for volatility risk and volatility-of-volatility due to volatility shocks. Their relative contributions to the log VRP are given by $\frac{-\tau_{1}\lambda+\tau_{2}\lambda^{2}}{-\tau_{1}\lambda+\tau_{2}\lambda^{2}-\gamma\sigma\xi}$ and $\frac{-\gamma\sigma\xi}{-\tau_{1}\lambda+\tau_{2}\lambda^{2}-\gamma\sigma\xi}$ , respectively, which, conveniently, do not depend on $\beta$.
It is worth noting that a positive VRP does not require the equity risk premium to be positive if $\xi$ is sufficiently negative. The literature typically finds $\lambda>0$, e.g. French1987, but negative values have also been reported, see e.g. JLA2001.
The volatility-of-volatility (in log-$h$) can be derived similarly and is, under $\mathbb{P}$ and $\mathbb{Q}$, given by \[ \mathrm{var}^{\mathbb{P}}(\log h)=\frac{\tau_{1}^{2}+2\tau_{2}^{2}+\gamma^{2}\sigma^{2}}{1-\beta}\qquad\mbox{and}\qquad\mathrm{var}^{\mathbb{Q}}(\log h)=\frac{(\tau_{1}-2\tau_{2}\lambda)^{2}+2\tau_{2}^{2}+\gamma^{2}\sigma^{2}}{1-\beta^{2}}, \] respectively. Since $\tau_{1}<0$ it follows that their difference, $\mathrm{var}^{\mathbb{Q}}(\log h)-\mathrm{var}^{\mathbb{P}}(\log h)=4\lambda(\lambda\tau_{2}^{2}-\tau_{1}\tau_{2})/(1-\beta^{2})$ is positive, such that the unconditional variance of volatility is higher under $\mathbb{Q}$ than under $\mathbb{P}$.
The dependence between returns and volatility is another important aspect of asset pricing. Here we follow Christoffersen2014 and quantify this dependence using the conditional correlation between $\log h_{t+1}$ and $r_{t}$, (leverage correlations) under $\mathbb{P}$ and $\mathbb{Q}$. Under $\mathbb{P}$ we have \[ \mathrm{cov}_{t}^{\mathbb{P}}\left(\log h_{t+1},r_{t}\right)=\mathbb{E}_{t}^{\mathbb{P}}[(\tau_{1}z_{t}+\tau_{2}z_{t}^{2}+\gamma\sigma u_{t})z_{t}\sqrt{h_{t}}]=\tau_{1}\sqrt{h_{t}}, \] such that the conditional correlation is \[ \rho_{\mathbb{P}}=\mathrm{corr}_{t}^{\mathbb{P}}\left(\log h_{t+1},r_{t}\right)=\frac{\tau_{1}}{\sqrt{\tau_{1}^{2}+2\tau_{2}^{2}+\gamma^{2}\sigma^{2}}}. \] Under the risk neutral measure, $\mathbb{Q}$, we find that $\mathrm{cov}_{t}^{\mathbb{Q}}\left(\log h_{t+1},r_{t}\right)=(\tau_{1}-2\tau_{2}\lambda)\sqrt{h_{t}}$ such that \[ \rho_{\mathbb{Q}}=\mathrm{corr}_{t}^{\mathbb{Q}}\left(\log h_{t+1},r_{t}\right)=\frac{\tau_{1}-2\lambda\tau_{2}}{\sqrt{(\tau_{1}-2\tau_{2}\lambda)^{2}+2\tau_{2}^{2}+\gamma^{2}\sigma^{2}}}. \] These correlations are, as expected, both negative, and it can be shown that $\rho_{\mathbb{Q}}^{2}-\rho_{\mathbb{P}}^{2}>0$, such that the leverage effect is more pronounce under $\mathbb{Q}$ than under $\mathbb{P}$.
While VIX pricing only requires the expectations of future volatility, many other problems, such as option pricing, require an accurate description of the distribution of cumulative returns.\footnote{For example, DGS1999 provided a method to price options with the skewness and kurtosis of cumulative returns.} Figure (ref) presents the skewness and kurtosis of cumulative returns for the Realized GARCH model, for cumulative returns spanning a period from 1 to 250 days (approximately one year). For comparison, we also include the corresponding results based on the EGARCH model. The simulation designs for the two models are the estimates we obtained in our empirical analysis, see Table 2. Because closed-form expressions for skewness and kurtosis of cumulative returns are not readily available, these results are based on simulation methods with 1,000,000 replications. The first 750 days were discarded in each simulation in order to minimize the influence of initial values.
The results in Figure (ref) show that cumulative returns are more left-skewed (have a more negative skewness) under $\mathbb{Q}$ than under $\mathbb{P}$, and the tails are also thicker (larger kurtosis) under $\mathbb{Q}$ than under $\mathbb{P}$. This is true for both the Realized GARCH model and the EGARCH model. However, the magnitude of skewness and kurtosis is much larger for the Realized GARCH model especially under the risk neutral measures. These features of the Realized GARCH model are potentially important because theoretical results in Bakshi-Madan2006 and Chabi-Yo2012 demonstrate that the skewness and the kurtosis of the market index are key determinants of the variance risk premium.
In Figure (ref), we present the simulated densities for standardized cumulative returns over one month (left panels) and six months (right panels). The densities under $\mathbb{P}$ are in the upper panels and those under $\mathbb{Q}$ are presented in the lower panels, where the solid red lines are for the Realized GARCH model and the dashed blue line are for the EGARCH model based on the parameter estimates we obtained in our empirical analysis. A left skew can be seen for both models and it is more pronounced at longer horizons (six months), especially for the Realized GARCH model. The skewness is also more pronounced under the risk neutral measure, $\mathbb{Q}$, which is consistent with the results in Figure (ref).
In this section, we introduce four alternative models, which we will use to benchmark the Realized GARCH model against. We compare the models ability to explain and predict the three variables: return volatility, the market VIX, and the volatility risk premium. The four alternative models are: the GARCH model by bollerslev:86, the EGARCH model by Nelson91, the Heston-Nandi GARCH model by HN2000, which are combined with Duan's LRNVR, and the Heston-Nandi GARCH combined with the variance dependent SDF, as proposed by VDPK2013.\footnote{HaoZhang2013 examined GARCH, EGARCH, TGARCH, AGARCH and CGARCH models. To conserve space, we focus on the GARCH and EGARCH models because the EGARCH had the best performance in the study by HaoZhang2013, and the original GARCH model is a natural benchmark.}
\textcolor{black}{GARCH and EGARCH model are commonly used as benchmarks in comparisons of volatility models. The original GARCH model tends to perform well with exchange rate data, but it is typically outperformed by models that can accommodate a leverage effect when applied to equity returns, see HansenLundeBeatGarch. The volatility dynamics for the GARCH(1,1) is given by} \[ h_{t+1}=\omega+\beta h_{t}+\alpha h_{t}z_{t}^{2}, \] and that of the\textcolor{black}{ EGARCH(1,1) is given by}
From HaoZhang2013 it follows that these models in conjunction with the exponentially affine SDF yield the following model-based pricing formulae for the VIX:{ \[ \mathrm{VIX}_{t}^{\mathrm{G}}=100\times\sqrt{252\sigma_{h}^{2}+\tfrac{252}{22}\frac{1-\beta^{22}}{1-\beta}(h_{t+1}-\sigma_{h}^{2})},\qquad\mbox{with}\quad\sigma_{h}^{2}=\omega/(1-\beta), \] }and
{ \[ \mathrm{VIX}_{t}^{\mathrm{EG}}=100\times\sqrt{\tfrac{252}{22}\left[h_{t+1}+\sum_{k=2}^{22}\left(\prod_{i=0}^{k-2}F_{i}\right)h_{t+1}^{\beta^{k-1}}\right]}, \] }respectively, where{
}
The Heston-Nandi GARCH model is a popular discrete-time model for option pricing. The equity premium is assumed to be proportional to the conditional variance and a specific leverage term is adopted in the GARCH equation,
This structure conveniently yields a closed-form option pricing formula. Under LRNVR risk neutralization the corresponding dynamics under $\mathbb{Q}$ is:
where $z_{t}^{*}=z_{t}+\lambda\sqrt{h_{t}}$. If $|\tilde{\beta}|<1$, then the unconditional mean of $h_{t}$ under $\mathbb{Q}$ is $\sigma_{h}^{2}=\tilde{\omega}/(1-\tilde{\beta})$, and the $k$-step ahead expected conditional variance is
\[ \mathbb{E}_{t}^{\mathbb{Q}}(h_{t+k})=\sigma_{h}^{2}+\tilde{\beta}^{k-1}(h_{t+1}-\sigma_{h}^{2}). \] The model implied VIX pricing formula is therefore given by:
{ \[ \mathrm{VIX}_{t}^{\mathrm{HN}}=\sqrt{\tfrac{252}{22}\sum_{k=1}^{22}\mathbb{E}_{t}^{\mathbb{Q}}(h_{t+k})}\times100=\sqrt{252\sigma_{h}^{2}+\tfrac{252}{22}\tfrac{1-\tilde{\beta}^{22}}{1-\tilde{\beta}}(h_{t+1}-\sigma_{h}^{2})}\times100. \] }
An alternative to LRNVR is the variance dependent SDF by VDPK2013. As suggested by its name, this SDF depends on $h_{t}$, and this dependence has been shown to improve the option pricing performance of the Heston-Nandi GARCH model.
VDPK2013, show that the dynamic properties under $\mathbb{Q}$ are given by
where $z_{t}^{*}$ has a standard normal distribution and
Here $\xi$ is the variance risk aversion parameter, see VDPK2013. The resulting model-implied VIX pricing formula is given by
{ \[ \mathrm{VIX}_{t}^{\mathrm{HN}_{\mathtt{vd}}}=\sqrt{252\sigma_{h}^{*2}+\tfrac{252}{22}\tfrac{1-\beta^{\ast}{}^{22}}{1-\beta^{\ast}}(h_{t+1}^{*}-\sigma_{h}^{*2})}, \] }where $\sigma_{h}^{*2}=(\omega^{\ast}+\alpha^{\ast})/(1-\beta^{\ast})$ with $\beta^{\ast}=\beta+\alpha^{\ast}\delta^{\ast2}$. Unlike LRNVR, the variance dependent SDF will induce a transformation of one-step-ahead conditional variance after the change of measure.\footnote{VDPK2013 suggested to reparametrize the model with $\tilde{\xi}=1/(1+2\alpha\xi)$ in place of $\xi$, when estimating the model.}
For the estimation it is convenient to use a different expression of the GARCH equation,
which is obtained by substituting ((ref)) into ((ref)). This formulation highlights the observation-driven structure of the model, as it shows how the conditional volatility depends on the observable realized measure and (a function of) the lagged standardized return. This makes evaluation and maximization of the log-likelihood straight forward.\footnote{For the related stochastic volatility models, direct maximization of the likelihood for is typically impractical, and other estimation methods, such as GMM and simulation based methods, are often employed for this type of models.}
We estimate the unknown parameters (of the model and the SDF) by maximizing a joint log-likelihood function that is composed of the log-likelihood function of the (Realized) GARCH model and the log-likelihood for VIX pricing errors.
The log-likelihood function for the Realized GARCH model specifies the dynamics for $(r_{t},x_{t})$ while the GARCH models (GARCH, EGARCH, Heston-Nandi) specifies the dynamics for $r_{t}$. These likelihood terms are combined with a log-likelihood for the VIX pricing errors, where the latter is influenced by both the choice of volatility model and the choice of SDF. Following HaoZhang2013 we adopt at Gaussian specification for the pricing error, where $\mathrm{VIX}_{t}^{\mathrm{Model}}-\mathrm{VIX}_{t}\sim iidN(0,\sigma_{\mathtt{vix}}^{2})$, and as a robustness check we also estimate the parameters using a second (multiplicative) specification: $\mathrm{VIX}_{t}=\mathrm{VIX}_{t}^{\mathrm{Model}}\eta_{t}$, where it is assumed that $\log\eta_{t}\sim iidN(-\sigma_{\mathtt{vix}}^{2}/2,\sigma_{\mathtt{vix}}^{2})$, such that $\mathbb{E}(\eta_{t})=1$. The two specifications produce very similar estimates and similar pricing errors, see Appendix (ref).
For the Realized GARCH model, the total (quasi) log-likelihood is given by \[ \ell_{r}+\ell_{x}+\ell_{\mathtt{vix}}, \] where
with $h_{t}^{\mathtt{RG}}$ given from the GARCH equation ((ref)) and $\mu_{t}^{\mathtt{RG}}=r+\lambda\sqrt{h_{t}^{\mathtt{RG}}}-\tfrac{1}{2}h_{t}^{\mathtt{RG}}$.
The likelihood of the other models are define similarly with model-specific definitions of $\mu_{t}$, $h_{t}$, and the model-implied VIX. The conventional GARCH models do not have the second term of the log-likelihood, because they do include the realized measure, $x_{t}$, in the modeling.
The idea of combining the likelihood of a time-series model with a second likelihood for option pricing errors is now standard in this literature. Some papers including pricing errors for the a range of options, see e.g. VDPK2013 and Christoffersen2014, or pricing errors for volatility derivatives, see e.g. Wang2017, bardg2019, or VIX pricing errors as in HaoZhang2013. This is in contrast to an earlier literature that implicitly assumed pricing errors to be zero and adopted the VIX as the volatility variables, see, e.g., Duan2010JEDC who estimated a stochastic volatility model with jumps by exploiting the theoretical link between the VIX and the latent volatility.
Our empirical analysis is based on daily data for S&P 500 stock index and CBOE VIX. We obtain the daily VIX index and the daily returns from Yahoo Finance while the realized measure are downloaded from the Realized Library at Oxford-Man Institute. The primary realized measure is the realized variance from the hours with active trading with the squared overnight return added, see HansenLundeWholeDay. As another robustness check of our main results, we have also used different choices of realized measures, see Appendix (ref).
Our full sample spans 15 years, from January 2004 to December 2018. We will present empirical results based on the full sample period as well as out-of-sample results where the model is estimated recursively using a rolling window sample with 750 days. The out-of-sample performance is evaluated over the years 2007 to 2018. We also present separate out-of-sample results for two subsamples: the years 2007-2012, which include the global financial crisis period, and the years 2013-2018, which span the post-crisis period.
The VIX index is a model-free measure of volatility. Prior to being annualized, it is computed as \[ \mathrm{VIX}_{t}=\sqrt{\frac{2}{T}\sum_{i}\frac{\Delta K_{i}}{K_{i}^{2}}\exp\left(rT\right)Q\left(K_{i}\right)-\frac{1}{T}\left(\frac{F}{K_{0}}-1\right)^{2}}, \] where $T$ is the time to maturity, $F$ is the forward index level, $K_{0}$ is the first strike below $F$, $K_{i}$ is the strike price of the $i$-th out-of-the-money option, $\Delta K_{i}$ is the interval between strike prices, $r$ is the risk-free rate associated with time to maturity, $Q(K_{i})$ is the midpoint of the bid-ask spread for options with strike $K_{i}$. See JiangTian2005 for a detail discussion on the VIX formula, and VIXreview2015 for a review of model-free measures.\footnote{The VIX formula is described in the CBOE white paper, http://www.cboe.com/micro/vix/vixwhite.pdf, and is based on earlier results in, CarrMadan1998, DDKZ1999, and BN2000, who applied similar methods to approximate the expected volatility under $\mathbb{Q}$.}
We present summary statistics for the full sample period in Table (ref). The data consists of daily returns, the daily realized variances (measured in units of annualized standard deviation), the CBOE VIX, and the VRP. A number of interesting observation can be made from Table (ref). First, the distribution of VIX is skewed to the right with one (or more) extremely large values, and the same is seen for the realized volatility. Second, both time series of volatility are highly persistent with large and slowly decaying autocorrelations. Third, the VRP also has a large first-order autocorrelation but its higher-order autocorrelations decay much faster than is the case for the VIX and the realized variance. This suggest that the two variables that the VRP is composed of, have a common stochastic trend that cancels out in the difference between the two variables (a type of “cointegration”). On average, the VIX is larger than the realized volatility, with the average VRP being around 4.8%.
In the following, we use the following abbreviations for the models: $\mathrm{RG}$ for the Realized GARCH model, $\mathrm{EG}$ for EGARCH, $\mathrm{G}$ for GARCH, $\mathrm{HN}$ for Heston-Nandi GARCH, and $\mathrm{HN}_{\mathtt{vd}}$ for Heston-Nandi GARCH with the variance dependent SDF.
We present the parameter estimates for the full sample period for each of the five models in Table (ref) along with robust standard errors in parentheses and some additional statistics.
An interesting observation can be made about the market price of equity risk, $\lambda$. This parameter is similar for the first three models, however the estimated of $\lambda$ in the EGARCH and GARCH models are 10-20 times larger than the estimate for the Realized GARCH model. The estimates of $\lambda$ in the EGARCH and GARCH models are in line with those reported in HaoZhang2013. If the model are estimated solely from return data, then the estimates of $\lambda$ are much smaller, see HaoZhang2013. The reason is that the $\mathrm{EG}$ and $\mathrm{G}$ models lack a separate volatility risk parameters, and the models inflate the value of $\lambda$ in order to compensate for the volatility risk that is embedded in the VIX. The $\lambda$ for the Heston-Nandi model is not directly comparable to those of the other models, because this coefficient is associated with $h$ in the return equation, rather than $\sqrt{h}$ (for the other models).
The persistence parameters under the $\mathbb{P}$-measure is denoted $\pi^{\mathbb{P}}$ and is defined by $\beta$ for RG and EG, by $\alpha+\beta$ for G, and by $\beta+\alpha\delta^{2}$ for $\mathrm{HN}$ and $\mathrm{HN}_{\mathtt{vd}}$. The persistence is quite similar across models and close to unity in all cases. The estimates of $\tau_{1}$ and $\delta_{1}$ are negative for both RG model and the EG model, which reflect a negative correlation between return and volatility shocks. This is the so-called leverage effect and these findings are consistent with the existing literature.
The estimate of the volatility risk parameter in the RG model, $\xi$, is negative and significant. From the decomposition of the (log) VRP we can compute the relative contributions of the two terms in ((ref)) using the estimated RG model. The first term is compensation for the equity risk premium and its contribution ($\propto-\tau_{1}\lambda+\tau_{2}\lambda^{2}$) is estimated to be 2.2%. The second term is the separate compensation for volatility risk and its contribution ($\propto-\gamma\sigma\xi$) is estimated to be 97.8%. This suggests that the majority of VRP is due to compensation for the volatility shock, $u_{t}$, and only a small of fraction of the VRP can be attributed to the leverage effect and the equity premium.\footnote{This finding is specific to the RG model structure, that only includes a short-term leverage effect. So it is possible that that models with a more sophisticated leverage effect and/or long memory feature, would result in different weights on the two terms.} This empirical finding supports the view in HaoZhang2013 who argued that equity risk cannot justify the observed market VRP.
The value of the maximized log-likelihood function is a measure of the model's ability to fit the empirical distribution of the observed data. The Realized GARCH model with the affine exponential SDF clearly has the best fit for all terms of the log-likelihood that are directly comparable. Both $\ell_{r}$ and $\ell_{\mathrm{vix}}$ and their sum $\ell_{r,\mathrm{vix}}$, are much larger for the Realized GARCH model than any of the other models. This is despite the fact that the other models seek to maximize $\ell_{r,\mathrm{vix}}$ while the objective of the Realized GARCH model entails a tradeoff between this term and the log-likelihood for the realized measures, $\ell_{x}$. Following the Realized GARCH model, the $\mathrm{HN}_{\mathtt{vd}}$ has the second best performance in terms of describing returns, $\ell_{r}$, whereas the EGARCH takes the second spot in terms of explaining the variation in the VIX, $\ell_{\mathrm{vix}}$. Below, we evaluate the model's ability to describe the VIX in greater details.
In this section, we focus on the models' ability to explain the variation in the VIX, VRP, and the volatility of cumulative returns. First, we report summary statistics for the full sample, then we report results for various subsamples -- in-sample results as well as out-of-sample results. Most of the existing literature has focused on a single variable. For instance, the focus in HaoZhang2013, Christoffersen2014, and Corsi2015JOE was VIX and derivative pricing, whereas Wang2017 focused on volatility under the physical measure.
In this section, we evaluate the model's ability to describe the VIX in greater details beyond the log-likelihood term, $\ell_{\mathrm{vix}}$, listed above. Table (ref) reports a range of summary statistics based on the full sample, where we compare the model-based measures of VIX with the observed VIX.
In this comparison, the Realized GARCH model is also consistently the best model. It has the smallest bias, the smallest mean squared error, and the smallest mean absolute error. The models: EGARCH, GARCH, and Heston-Nandi with LRNVR tend to underestimate the VIX, whereas the Heston-Nandi GARCH with the variance dependent SDF tends to overestimate the VIX. The Realized GARCH model also has the highest correlation between the model-implied VIX and the market-based VIX. With the Realized GARCH model, the resulting statistical properties of the model-based VIX are closer to those of the market-based VIX, that those of other models.
Table (ref) provides the in-sample pricing performance for the variance risk premium and its two components: the volatility index (VIX) and the annualized model-based volatility, and we report the bias for each of the models.\footnote{The annualized volatility is calculated based on the martingale process assumption made by BollerslevTauchenZhou2009 and our results are robust when annualized volatility is calculated based on the forecast value using HAR model (method used in Bekaert2014 etc.). See section (ref) for details.}
In terms of the volatility risk premium, the Realized GARCH model provides the smallest bias, the smallest root mean square error (RMSE), and the smallest mean absolute error (MAE). The reduction in pricing errors relative to other models ranges from 15.0% to 30.6% in terms of RMSE and 27.1% to 44.7% in terms of MAE. Among the competing models, the Heston-Nandi GARCH model with variance dependent SDF appears to be the best alternative. In contrast, the the Heston-Nandi model with LRNVR, which is arguably a very popular option pricing model, does not fair well in terms of explaining the volatility risk premium. The EGARCH model performs significantly better than other GARCH models using LRNVR, especially in terms of the RMSE.
We observe very similar patterns across models in terms of their ability to price the VIX. The Realized GARCH delivers the best performance while the Heston-Nandi GARCH takes last place. In fact, the non-affine models (RG, EG and G) perform substantially better that the two affine models ($\mathrm{HN}$ and $\mathrm{HN}_{\mathtt{vd}}$), which is consistent with the existing literature on option pricing with GARCH models. The main advantage of the affine models is their analytical expressions for the moment generating function. Fortunately, these are not needed for VIX pricing, so the non-affine model are clearly preferred for this problem.\footnote{It is very difficult, if not impossible, to obtain an analytical moment generation function for cumulative returns or the $k$-step ahead conditional volatility for these non-affine models. Both are needed for quasi-analytical pricing formula for derivatives using a Fourier inverse transformation. For this reason, computationally intensive simulation methods and analytical expansions are commonly used for pricing derivatives with non-affine models, see RGOP2017 for the use of an Edgeworth expansion to price options with the Realized GARCH model.} The performance gain for the Realized GARCH model ranges from 13.6% to 29.8% in terms of RMSE and between 15.1% and 23.5% in terms of MAE.
All models tend to over-estimate the expected volatility under the physical measure. However, the bias is much smaller for the Realized GARCH model. This indicates that the other models, in order to price the VIX, inadvertently increase the level of volatility to compensate for their shortcomings in risk neutralization. The RG and $\mathrm{HN}_{\mathtt{vd}}$ both have additional parameter to compensate for volatility risk, which likely explain their smaller bias. In terms of explaining the annualized volatility, the Realized GARCH model reduces the RMSE by 32.8% to 48.1% and the MAE is reduced by 48.6% to 58.6%.
It is worth emphasizing that the parameter estimation does not target the volatility risk premium directly, the superior performance of Realized GARCH highlights the model's ability of reconcile the physical and risk neutral dynamics within a single model framework. This is some accomplishment by the Realized GARCH framework, because this was considered to be a very difficult empirical problem, see Bates1996.
The proposed pricing model, which is based on the Realized GARCH model and the affine exponential SDF, requires a larger number of parameters to be estimated than the methods based on the conventional GARCH models. The larger number of parameters could entail some overfitting of the model, and this might explain some of the observed empirical improvements. It is therefore important to document that the model also provides improvements out-of-sample.\textcolor{black}{ In this section, we compare the models in terms of their out-of-sample pricing errors using a rolling estimation window, based on the past 750 daily observations. The first forecast is made for the first month (22 trading days) of 2007, and this forecasts is based on parameters that were estimated with the previous 750 daily observations (January 6th 2004 to December 29th, 2006). We report out-of-sample pricing errors for 2007-2018 }and two sub-sample periods, 2007-2012 and 2013-2018. Splitting the out-of-sample period into two subsamples is interesting because some results could potentially be specific to the global financial crisis, which had high volatility and high volatility-of-volatility. The global financial crisis is contained in the first subsample.
We report a range of performance statistics for each of the models and each of the sample periods in Table (ref). The significance of relative performance is evaluated with Diebold-Mariano (DM) statistics where we compare each of the alternative models to the Realized GARCH model. For this purpose, we first compute the tracking errors for the $i$-th model,
\[ e_{it}=X_{i,t}^{\mathtt{model}}-X_{t}^{\mathtt{market}}, \] where $X$ represents the volatility risk premium, the volatility index, or the annualized volatility. These errors are translated into losses using either the mean square error, $g(e_{it})=e_{it}^{2}$ or the mean absolute error $g(e_{it})=|e_{it}|$. The loss of model $i$, relative to the Realized GARCH model ($i=0$) is now defined by
\[ d_{i,t}=g(e_{it})-g(e_{0t}), \] and we proceed to tests the hypothesis, $H_{0}:\thinspace\thinspace\mathbb{E}(d_{i,t})=0$ using the Diebold-Mariano (DM) statistic, $\mathrm{DM}_{i}=\sqrt{T}\bar{d}_{i,\cdot}/\hat{\sigma}_{d_{i}}$, where $\bar{d}_{i,\cdot}=\frac{1}{T}\sum_{t}d_{i,t}$ and $\hat{\sigma}_{d_{i}}^{2}$ is an estimate of the long-run variance of $\{d_{i,t}\}$. Our estimates of $\sigma_{d_{i}}^{2}$, are based on the Parzen kernel with bandwidth $H=42$. Under suitable regularity conditions, it can be shown that $\mathrm{DM}_{i}\overset{d}{\rightarrow}N(0,1)$ under the null hypothesis $\mathbb{E}(d_{i,t})=0$, see DieboldMariano95, and the 10% and 5% critical values are therefore given by 1.64 and 1.96, respectively.
Once again the Realized GARCH provides the best out-of-sample pricing performance for the VRP, the VIX, and the annualized volatility, and this is found in all three sample periods. The model also provides the smallest bias in most cases and the improvement in RMSE/MAE ranges from 10% to 40% in most cases, which is significant in most cases. This shows that superior in-sample performance of the Realized GARCH model cannot be attributed to overfitting. The RMSE and MAE are, as expected, larger in the subsample with the financial crisis. The Realized GARCH model really stands out in terms of explaining the volatility under the physical measure (Annualized Volatility), where the reduction in out-of-sample loss is always larger than 20% and as larger as 52.2%.
The HN is always the worst model for tracking the VIX out-of-sample, as was the case in-sample. The picture is similar for the annualized volatility, albeit $\mathrm{HN}_{\mathtt{vd}}$ is the “best of the rest” in terms of tracking of annualized volatility in the post crisis period. This might be explained by the variance dependent SDF being less misspecified when volatility is low, while it cannot generate enough discrepancy between $\mathbb{P}$ and $\mathbb{Q}$ when volatility is high. Interestingly, even though the $\mathrm{HN}_{\mathtt{vd}}$ is clearly inferior to both the GARCH and EGARCH models in terms of forecasting volatility under $\mathbb{P}$ and $\mathbb{Q}$, it is actually better than these two models in terms of forecasting the VRP.
We have developed a Realized GARCH model for the simultaneous modeling of returns, the VIX, and the VRP, using an exponentially affine SDF that takes advantage of the dual shock structure in the Realized GARCH model. This framework has several attractive features. First, its dual-shock structure lead to a distinct compensation for volatility risk, which is empirically important. Second, it takes advantage of the information contained in realized measures of volatility. Third, it has a flexible leverage function that captures the empirically important return-volatility dependence in a parsimonious manner. Fourth, the model combined with the exponentially affine SDF, conveniently, yields analytical formulae for the VIX and the volatility risk premium. Fifth, the model is an observation-driven model, which makes estimation straight forward. Sixth, its dynamic properties under the physical and risk-neutral measures offer intuitive and theory-consistent explanations for the excellent empirical performance offered by the Realized GARCH model.
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