# Realized GARCH, CBOE VIX, and the Volatility Risk Premium††thanks: Zhuo
Huang acknowledges financial support by the National Natural Science
Foundation of China (71671004) and Tianyi Wang acknowledges financial support
by the National Natural Science Foundation of China (71871060).

Peter Reinhard Hansena & Zhuo Huangb & Chen Tongc & Tianyi Wangd

_a University of North Carolina & Copenhagen Business School_
_b Peking University, National School of Development_
c _Department of Finance, School of Economics & Wang Yanan Institute_ _for_
_Studies in Economics (WISE), Xiamen University_
d _University of International Business and Economics School of Banking and
Finance_

(_August 27, 2024_)

###### Abstract

We show that the Realized GARCH model yields close-form expression for both
the Volatility Index (VIX) and the volatility risk premium (VRP). The Realized
GARCH model is driven by two shocks, a return shock and a volatility shock,
and these are natural state variables in the stochastic discount factor (SDF).
The volatility shock endows the exponentially affine SDF with a compensation
for volatility risk. This leads to dissimilar dynamic properties under the
physical and risk-neutral measures that can explain time-variation in the VRP.
In an empirical application with the S&P 500 returns, the VIX, and the VRP, we
find that the Realized GARCH model significantly outperforms conventional
GARCH models.

Keywords: Realized GARCH; High Frequency Data; Volatility Risk Premium;
Realized Variance; VIX.

JEL Classification: C10; C22; C80

##  1 Introduction

The variance risk premium or volatility risk premium (VRP) has been the focus
of much research since the seminal papers by Coval and Shumway (2001) and
Bakshi and Kapadia (2003). The VRP is the difference between the expected
return variance under the risk neutral measure and the expected return
variance under the physical measure,111This definition of the VRP follows that
in Bollerslev et al. (2009). 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.222For 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 Bollerslev et al. (2009), Bollerslev
et al. (2011), Wu (2011) and Conrad and Loch (2015). 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. Coval and Shumway (2001), Bakshi and Kapadia (2003) and Carr
and Wu (2009). 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. Bollerslev et al. (2009), Bekaert and Hoerova (2014), and Cremers et
al. (2015). Moreover, the VRP plays an important role in option pricing, see
e.g. Byun et al. (2015) and Song and Xiu (2016). 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, $\mathbb{Q}$. Duan (1995) 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 Hao and Zhang (2013). They found that the VIX
implied by GARCH models is substantially below the observed VIX. Hao and Zhang
(2013) 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 Taylor (1986) and Kim et al. (1998),
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. Bollerslev et al. (2011).
The main drawback of SV models is that they are more involved to estimate than
observation-driven models, such as GARCH models.333The same complication
arises with Jump-GARCH models that typically rely on particle filters for
estimation, see e.g. Ornthanalai (2014).

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 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 Bardgett et al.
(2019), see also Andersen et al. (2019) 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 and Madan (2006) and Chabi-Yo (2012). 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 Nelson (1991),
the GARCH model by Bollerslev (1986), Heston-Nandi GARCH by Heston and Nandi
(2000). These models are combined with either the the LRNVR by Duan (1995) or
the variance dependent SDF by Christoffersen et al. (2013).444Christoffersen
et al. (2013), introduced a variance-dependent SDF to improve the option
pricing performance of the Heston-Nandi GARCH model. The idea was also used in
Byun et al. (2015) 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.555The need for a
model to simultaneously explain the variation under $\mathbb{P}$ and
$\mathbb{Q}$ was pointed out in Bates (1996), and has since received much
attention in the option pricing literature, see, e.g., Pan (2002), Eraker
(2004), and Santa-Clara and Yan (2010).

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.

##  2 The Realized GARCH Model and VIX Pricing

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 Barndorff-Nielsen et al. (2008) 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 Hansen et al. (2012) and later refined in Hansen
and Huang (2016) to have a more flexible leverage function and to allow for
the inclusion of multiple realized measures.

###  2.1 Model under the Physical Measure

We adapt the model in Hansen and Huang (2016) to the present context, by
adding an appropriate compensation for equity risk. Under the physical measure
the model is characterized by:

$\displaystyle r_{t}$ $\displaystyle=$ $\displaystyle
r+\lambda\sqrt{h_{t}}-\tfrac{1}{2}h_{t}+\sqrt{h_{t}}z_{t},$ (1)
$\displaystyle\log h_{t+1}$ $\displaystyle=$ $\displaystyle\omega+\beta\log
h_{t}+\tau(z_{t})+\gamma\sigma u_{t},$ (2) $\displaystyle\log x_{t}$
$\displaystyle=$ $\displaystyle\kappa+\phi\log h_{t}+\delta(z_{t})+\sigma
u_{t},$ (3)

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 (1). 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 model of Nelson (1991) when $\gamma=0$.666We use $z_{t}^{2}$
in place of $|z_{t}|$ that was used the original EGARCH model, which has some
advantages, see Hansen et al. (2012) and Hansen and Huang (2016). 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, (2). This is contrast to conventional GARCH
models, where the conditional volatility is solely driven by lagged daily
returns.777For 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 Hao and Zhang (2013) 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.

###  2.2 Risk Neutralization and Properties under the Risk Neutral Measure

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 Duan (1995) and the variance-dependent SDF by Christoffersen et al.
(2013). 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 Corsi et al. (2013).

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:

$M_{t+1}=\frac{\exp(-\lambda z_{t+1}-\xi
u_{t+1})}{\mathbb{E}_{t}^{\mathbb{P}}\exp(-\lambda z_{t+1}-\xi
u_{t+1})}=\exp\left\\{-\lambda z_{t+1}-\xi
u_{t+1}-\tfrac{1}{2}(\lambda^{2}+\xi^{2})\right\\}.$ (4)

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 (4) is identical to the $\lambda$ in the return
equation, (1). This is not by assumption but an implication of a no arbitrage
condition. If we, as a starting point, permitted the $\lambda$ in (4), 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 (1). This is a
consequence of a no-arbitrage condition, see Lemma A.1 in the Appendix.

####  2.2.1 Dynamic Properties under the Risk Neutral Measure

Under the risk neutral measure, $\mathbb{Q}$, it follows that the moment
generating function (MGF) is given by

$\displaystyle\Psi(s_{1},s_{2})=\mathbb{E}_{t}^{\mathbb{Q}}[\exp(s_{1}z_{t+1}+s_{2}u{}_{t+1})]$
$\displaystyle=$
$\displaystyle\mathbb{E}_{t}^{\mathbb{P}}[M_{t+1}\exp(s_{1}z_{t+1}+s_{2}u_{t+1})]$
$\displaystyle=$ $\displaystyle\exp[-s_{1}\lambda-
s_{2}\xi+\tfrac{1}{2}(s_{1}^{2}+s_{2}^{2})].$

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:

$\displaystyle r_{t+1}$ $\displaystyle=$ $\displaystyle
r-\tfrac{1}{2}h_{t+1}+\sqrt{h_{t+1}}z_{t+1}^{\ast},$ (5) $\displaystyle\log
h_{t+1}$ $\displaystyle=$ $\displaystyle\tilde{\omega}+\beta\log
h_{t}+\tilde{\tau}_{1}z_{t}^{\ast}+\tau_{2}(z_{t}^{\ast 2}-1)+\gamma\sigma
u_{t}^{\ast},$ (6) $\displaystyle\log x_{t}$ $\displaystyle=$
$\displaystyle\tilde{\kappa}+\phi\log
h_{t}+\tilde{\delta}_{1}z_{t}^{\ast}+\delta_{2}(z_{t}^{\ast 2}-1)+\sigma
u_{t}^{\ast},$ (7)

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 A.2 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, (5) and (6)
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 Duan (1995). This connection appears to have been overlooked in
the exiting literature.

###  2.3 The VIX Pricing Formula

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

$\mathrm{VIX}_{t}^{\mathrm{model}}=\sqrt{\tfrac{252}{22}\sum_{k=1}^{22}\mathbb{E}_{t}^{\mathbb{Q}}(h_{t+k})}\times
100,$

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:

######  Theorem 1.

For the Realized GARCH model (1)-(3) and the SDF (4), the model-implied VIX is
given by:

$\mathrm{VIX}_{t}^{\mathrm{RG}}=100\times\sqrt{\frac{252}{22}\left[h_{t+1}+\sum_{k=2}^{22}\left(\prod_{i=0}^{k-2}F_{i}\right)h_{t+1}^{\beta^{k-1}}\right]},$
(8)

where
$F_{i}=(1-2\beta^{i}\tau_{2})^{-1/2}\exp\left\\{\beta^{i}(\tilde{\omega}-\tau_{2})+\tfrac{1}{2}\beta^{2i}[\tfrac{\tilde{\tau}_{1}^{2}}{1-2\beta^{i}\tau_{2}}+\gamma^{2}\sigma^{2}]\right\\}$,
with
$\tilde{\omega}=\omega-\tau_{1}\lambda+\tau_{2}\lambda^{2}-\delta\sigma\xi$
and $\tilde{\tau}_{1}=\tau_{1}-2\lambda\tau_{2}$.

The expression (8) 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 Hao and Zhang (2013). The proof of
Theorem 1 is given in Appendix A.

###  2.4 Volatility risk premium

The literature has proposed several definitions of the VRP, see Bollerslev et
al. (2009).888See Bollerslev et al. (2009) 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,

$\mathrm{VRP}_{t}^{\mathtt{modelfree}}=\mathrm{VIX}_{t}-\sqrt{\tfrac{252}{22}\mathbb{E}_{t}^{\mathbb{P}}\left(\sum_{i=1}^{22}\mathrm{RVcc}_{t+i}\right)}\times
100,$

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

$\mathrm{VRP}_{t}^{\mathtt{market}}=\mathrm{VIX}_{t}-\sqrt{\tfrac{252}{22}\sum_{i=1}^{22}\mathrm{RVcc}_{t-i+1}}\times
100,$

was proposed in Bollerslev et al. (2009). 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 Corsi (2009), see Appendix B.1.

Our model-implied VRP is simply

$\mathrm{VRP}_{t}^{\mathtt{model}}=\left(\sqrt{\tfrac{252}{22}\sum_{k=1}^{22}\mathbb{E}_{t}^{\mathbb{Q}}(h_{t+k})}-\sqrt{\tfrac{252}{22}\sum_{k=1}^{22}\mathbb{E}_{t}^{\mathbb{P}}(h_{t+k})}\right)\times
100,$

which is the annualized, one-month ahead, expected volatility using the risk
neutral measure less the corresponding quantity under the physical probability
measure.

##  3 Key Model Properties under $\mathbb{P}$ and $\mathbb{Q}$

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 5.

###  3.1 Average Volatility

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,

$\mathbb{E}^{\mathbb{Q}}(\log h)\text{-}\mathbb{E}^{\mathbb{P}}(\log
h)=\frac{-\tau_{1}\lambda+\tau_{2}\lambda^{2}-\gamma\sigma\xi}{1-\beta}=\frac{-\tau_{1}\lambda+\tau_{2}\lambda^{2}}{1-\beta}+\frac{-\gamma\sigma\xi}{1-\beta},$
(9)

is positive. The logarithmic variant of the VRP was used in Carr and Wu (2009)
and Ammann and Buesser (2013).

From (9) 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. French et al. (1987), but negative values
have also been reported, see e.g. Jensen and Lunde (2001).

###  3.2 Volatility-of-Volatility

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 $\mathbb{P}$.

###  3.3 Dependence between Returns and Volatility (Leverage)

The dependence between returns and volatility is another important aspect of
asset pricing. Here we follow Christoffersen et al. (2014) 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 $\mathbb{P}$.

###  3.4 Skewness and Kurtosis of Multi-period Returns

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.999For example, Duan et al. (1999) provided
a method to price options with the skewness and kurtosis of cumulative
returns. Figure 1 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 1 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 and Madan (2006) and Chabi-Yo (2012) demonstrate
that the skewness and the kurtosis of the market index are key determinants of
the variance risk premium.

Figure 1: Skewness and kurtosis of cumulative returns under $\mathbb{P}$ and
$\mathbb{Q}$ for the EGARCH and the Realized GARCH model.

In Figure 2, 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 1.

Figure 2: Density of Cumulative Returns via Simulation

##  4 Some Competing Models and their Properties

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 (1986), the EGARCH model by Nelson (1991), the Heston-
Nandi GARCH model by Heston and Nandi (2000), which are combined with Duan’s
LRNVR, and the Heston-Nandi GARCH combined with the variance dependent SDF, as
proposed by Christoffersen et al. (2013).101010Hao and Zhang (2013) 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 Hao and Zhang (2013), and the original GARCH model is a natural
benchmark.

###  4.1 GARCH and EGARCH Model

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 Hansen and
Lunde (2005a). 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 EGARCH(1,1) is given by

$\displaystyle\log h_{t+1}$ $\displaystyle=$ $\displaystyle\omega+\beta\log
h_{t}+\tau_{1}z_{t}+\tau_{2}(|z_{t}|-\sqrt{2/\pi}).$

From Hao and Zhang (2013, propositions 1-4) 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

$\displaystyle F_{i}$ $\displaystyle=$
$\displaystyle\exp\left[\beta\left(\omega-\tau_{2}\sqrt{\frac{2}{\pi}}\right)\right]\left\\{\exp\left[-\beta^{i}(\tau_{1}-\tau_{2})\lambda+\frac{\beta^{2i}(\tau_{2}-\tau_{1})^{2}}{2}\right]\Phi[\lambda-\beta^{i}(\tau_{1}-\tau_{2})]\right.$
$\displaystyle\left.+\exp\left[-\beta^{i}(\tau_{1}+\tau_{2})\lambda+\frac{\beta^{2i}(\tau_{2}+\tau_{1})^{2}}{2}\right]\Phi[\beta^{i}(\tau_{1}+\tau_{2})-\lambda]\right\\}.$

###  4.2 Heston-Nandi GARCH Model under LRNVR

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,

$\displaystyle r_{t+1}$ $\displaystyle=$ $\displaystyle r+\lambda
h_{t+1}-\tfrac{1}{2}h_{t+1}+\sqrt{h_{t+1}}z_{t+1},$ $\displaystyle h_{t+1}$
$\displaystyle=$ $\displaystyle\omega+\beta
h_{t}+\alpha(z_{t}-\delta\sqrt{h_{t}})^{2}.$

This structure conveniently yields a closed-form option pricing formula. Under
LRNVR risk neutralization the corresponding dynamics under $\mathbb{Q}$ is:

$\displaystyle r_{t+1}$ $\displaystyle=$ $\displaystyle
r-\tfrac{1}{2}h_{t+1}+\sqrt{h_{t+1}}z_{t+1}^{\ast},$ $\displaystyle h_{t+1}$
$\displaystyle=$ $\displaystyle\omega+\beta
h_{t}+\alpha(z_{t}^{\ast}-(\delta+\lambda)\sqrt{h_{t}})^{2},$ $\displaystyle=$
$\displaystyle\underset{=\tilde{\omega}}{\underbrace{\omega+\alpha}}+\underset{=\tilde{\beta}}{\underbrace{\left[\beta+\alpha(\delta+\lambda)^{2}\right]}}h_{t}-2\alpha(\delta+\lambda)\sqrt{h_{t}}z_{t}^{\ast}+\alpha(z_{t}^{\ast
2}-1).$

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:

$\mathrm{VIX}_{t}^{\mathrm{HN}}=\sqrt{\tfrac{252}{22}\sum_{k=1}^{22}\mathbb{E}_{t}^{\mathbb{Q}}(h_{t+k})}\times
100=\sqrt{252\sigma_{h}^{2}+\tfrac{252}{22}\tfrac{1-\tilde{\beta}^{22}}{1-\tilde{\beta}}(h_{t+1}-\sigma_{h}^{2})}\times
100.$

###  4.3 Heston-Nandi GARCH under Variance Dependent SDF

An alternative to LRNVR is the variance dependent SDF by Christoffersen et al.
(2013). 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.

Christoffersen et al. (2013), show that the dynamic properties under
$\mathbb{Q}$ are given by

$\displaystyle r_{t+1}$ $\displaystyle=$ $\displaystyle
r-\tfrac{1}{2}h_{t+1}^{*}+\sqrt{h_{t+1}^{*}}z_{t+1}^{*},$ $\displaystyle
h_{t+1}^{*}$ $\displaystyle=$ $\displaystyle\omega^{*}+\beta
h_{t}^{*}+\alpha^{*}(z_{t}^{*}-\delta^{*}\sqrt{h_{t}^{*}})^{2},$

where $z_{t}^{*}$ has a standard normal distribution and

$\displaystyle h_{t}^{*}$ $\displaystyle=$ $\displaystyle
h_{t}/(1+2\alpha\xi),\qquad\qquad\omega^{*}=\omega/(1+2\alpha\xi),$
$\displaystyle\alpha^{*}$ $\displaystyle=$
$\displaystyle\alpha/(1+2\alpha\xi)^{2},\qquad\quad\delta^{*}=(\lambda+\delta-\tfrac{1}{2})(1+2\alpha\xi)+\tfrac{1}{2}.$

Here $\xi$ is the variance risk aversion parameter, see Christoffersen et al.
(2013). 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
$\beta^{\ast}=\beta+\alpha^{\ast}\delta^{\ast 2}$. Unlike LRNVR, the variance
dependent SDF will induce a transformation of one-step-ahead conditional
variance after the change of measure.111111Christoffersen et al. (2013)
suggested to reparametrize the model with $\tilde{\xi}=1/(1+2\alpha\xi)$ in
place of $\xi$, when estimating the model.

##  5 Empirical Analysis

For the estimation it is convenient to use a different expression of the GARCH
equation,

$\log h_{t+1}=(\omega-\gamma\kappa)+(\beta-\gamma\phi)\log
h_{t}+(\tau(z_{t})-\gamma\delta(z_{t}))+\gamma\log x_{t},$ (2’)

which is obtained by substituting (3) into (2). 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.121212For 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.

###  5.1 Model Estimation

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 Hao and Zhang (2013) 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 B.2.

For the Realized GARCH model, the total (quasi) log-likelihood is given by

$\ell_{r}+\ell_{x}+\ell_{\mathtt{vix}},$

where

$\displaystyle\ell_{r}$ $\displaystyle=$
$\displaystyle-\tfrac{1}{2}\sum_{t=1}^{T}\\{\log 2\pi+\log
h_{t}^{\mathtt{RG}}+(r_{t}-\mu_{t}^{\mathtt{RG}})^{2}/h_{t}^{\mathtt{RG}}\\},$
$\displaystyle\ell_{x}$ $\displaystyle=$
$\displaystyle-\tfrac{1}{2}\sum_{t=1}^{T}\\{\log 2\pi+\log\sigma^{2}+[\log
x_{t}-\omega-\beta\log h_{t}^{\mathtt{RG}}-\delta(z_{t})]^{2}/\sigma^{2}\\},$
$\displaystyle\ell_{\mathtt{vix}}$ $\displaystyle=$
$\displaystyle-\tfrac{1}{2}\sum_{t=1}^{T}\\{\log
2\pi+\log\sigma_{\mathtt{vix}}^{2}+(\mathrm{VIX}_{t}^{\mathtt{RG}}-\mathrm{VIX}_{t})^{2}/\sigma_{\mathtt{vix}}^{2}\\},$

with $h_{t}^{\mathtt{RG}}$ given from the GARCH equation (2) 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.
Christoffersen et al. (2013) and Christoffersen et al. (2014), or pricing
errors for volatility derivatives, see e.g. Wang et al. (2017), Bardgett et
al. (2019), or VIX pricing errors as in Hao and Zhang (2013). 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., Duan and Yeh
(2010) who estimated a stochastic volatility model with jumps by exploiting
the theoretical link between the VIX and the latent volatility.

###  5.2 Data

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 Hansen
and Lunde (2005b). As another robustness check of our main results, we have
also used different choices of realized measures, see Appendix B.3.

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.

####  5.2.1 CBOE VIX calculation

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 Jiang and
Tian (2005) for a detail discussion on the VIX formula, and Gonzalez-Perez
(2015) for a review of model-free measures.131313The VIX formula is described
in the CBOE white paper, http://www.cboe.com/micro/vix/vixwhite.pdf, and is
based on earlier results in, Carr and Madan (1998), Demeterfi et al. (1999),
and Britten-Jones and Neuberger (2000), who applied similar methods to
approximate the expected volatility under $\mathbb{Q}$.

Table 1: Summary of Statistics

| Mean | Median | Min | Q1 | Q3 | Max
---|---|---|---|---|---|---
$VIX$ | 18.410 | 15.690 | 9.140 | 12.940 | 20.948 | 80.860
$\sqrt{AnnRV}$ | 13.573 | 10.726 | 4.211 | 8.517 | 15.157 | 73.553
$Ret(\%)$ | 0.017 | 0.058 | -9.127 | -0.378 | 0.485 | 10.246
$VRP$ | 4.837 | 4.737 | -25.284 | 3.147 | 6.510 | 28.316
| Std | Skew | Kurt | AR1 | AR10 | AR22
$VIX$ | 8.812 | 2.657 | 12.662 | 0.980 | 0.898 | 0.810
$\sqrt{AnnRV}$ | 8.757 | 3.082 | 16.113 | 0.998 | 0.924 | 0.771
$Ret(\%)$ | 1.094 | -0.410 | 15.224 | -0.091 | 0.030 | 0.039
$VRP$ | 3.300 | -0.214 | 10.088 | 0.860 | 0.286 | 0.030

Note: Variables are measure in percent of annualized volatility. For instance,
$\sqrt{AnnRV}=100\sqrt{\tfrac{252}{22}\sum_{i=1}^{22}\mathrm{RVcc}_{t-i+1}}$.

We present summary statistics for the full sample period in Table 1. 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 1. 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%.

Table 2: Parameter estimates (full sample)

Model |  $\mathrm{RG}$ |  $\mathrm{EG}$ |  $\mathrm{G}$ |  $\mathrm{HN}$ |  $\mathrm{HN}_{\mathtt{vd}}$
---|---|---|---|---|---
$\lambda$ |  0.015 |  0.153 |  0.305 |  4.518 |  9.128
|  (0.010) |  (0.004) |  (0.018) |  (0.259) |  (1.635)
$\omega$ |  -0.088 |  -0.086 |  1.60E-06 |  -1.44E-06 |  -1.39E-06
|  (0.014) |  (0.002) |  (1.09E-07) |  (9.27E-08) |  (1.13E-07)
$\beta$ |  0.991 |  0.990 |  0.940 |  0.870 |  0.895
|  (0.001) |  2.22E-04 |  (0.004) |  (0.010) |  (0.009)
$\alpha$ |  |  |  0.054 |  3.10E-06 |  2.24E-06
|  |  |  (0.004) |  (1.50E-07) |  (2.16E-07)
$\delta$ |  |  |  |  197.183 |  202.167
|  |  |  |  (12.852) |  (18.782)
$\tau_{1}$ |  -0.073 |  -0.062 |  |  |
|  (0.005) |  (0.002) |  |  |
$\tau_{2}$ |  0.012 |  0.096 |  |  |
|  (0.002) |  (0.001) |  |  |
$\gamma$ |  0.080 |  |  |  |
|  (0.009) |  |  |  |
$\kappa$ |  0.427 |  |  |  |
|  (0.278) |  |  |  |
$\phi$ |  1.078 |  |  |  |
|  (0.029) |  |  |  |
$\delta_{1}$ |  -0.083 |  |  |  |
|  (0.010) |  |  |  |
$\delta_{2}$ |  0.129 |  |  |  |
|  (0.010) |  |  |  |
$\sigma^{2}$ |  0.325 |  |  |  |
|  (0.010) |  |  |  |
$\xi$ |  -1.07 |  |  |  |
|  (0.130) |  |  |  |
$\eta$ |  |  |  |  |  1.143
|  |  |  |  |  (0.061)
$\pi^{\mathbb{P}}$ |  0.991 |  0.990 |  0.993 |  0.990 |  0.986
$\ell_{r}$ |  12863.96 |  12661.65 |  12481.99 |  12610.18 |  12693.12
$\ell_{x}$ |  -3229.12 |  |  |  |
$\ell_{\mathtt{vix}}$ |  -8811.76 |  -9362.61 |  -9509.19 |  -10144.56 |  -10079.12
$\ell_{r,x}$ |  9634.84 |  |  |  |
$\ell_{r,\mathtt{vix}}$ |  4052.202 |  3299.045 |  2972.800 |  2465.615 |  2613.997
$\ell_{r,x,\mathtt{vix}}$ |  823.083 |  |  |  |

Note: Parameter estimates are reported with robust standard errors in
parenthesis. The persistence parameter $\pi^{\mathbb{P}}=\beta$ is in the
$\mathrm{RG}$ and $\mathrm{EG}$ models, $\pi^{\mathbb{P}}=\alpha+\beta$ in the
model, and $\pi^{\mathbb{P}}=\beta+\alpha\delta^{2}$ in the two Heston-Nandi
models. For the $\mathrm{HN}_{\mathtt{vd}}$ model we report ,
$\eta=(1+2\alpha\xi)^{-1}$ (which is the the variance risk ratio
$h_{t}^{*}/h_{t}$) instead of $\xi$. (The implied value for $\xi$ is here
-55,768.95).

###  5.3 Model Estimates (Full Sample)

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 2 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 Hao and Zhang
(2013). If the model are estimated solely from return data, then the estimates
of $\lambda$ are much smaller, see Hao and Zhang (2013). 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 (9) 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.141414This 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 Hao and Zhang (2013) 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.

###  5.4 Model Performance’s for VIX, VRP, and Volatility

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 Hao and Zhang (2013), Christoffersen et al. (2014), and Majewski et al.
(2015) was VIX and derivative pricing, whereas Wang et al. (2017) focused on
volatility under the physical measure.

####  5.4.1 Comparison of market and model-based VIX

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 3 reports a range of summary statistics based on the full sample,
where we compare the model-based measures of VIX with the observed VIX.

Table 3: VIX pricing performance (full sample)

Model | $\mathrm{RG}$ | $\mathrm{EG}$ | $\mathrm{G}$ | $\mathrm{HN}$ | $\mathrm{HN_{vd}}$ | VIX
---|---|---|---|---|---|---
Bias | -0.048 | -0.203 | -0.318 | -0.203 | 0.164 |
MAE | 1.866 | 2.222 | 2.199 | 2.439 | 2.419 |
RMSE | 2.504 | 2.898 | 3.012 | 3.565 | 3.504 |
Corr | 0.959 | 0.945 | 0.941 | 0.916 | 0.919 |
AR1 | 0.994 | 0.994 | 0.996 | 0.993 | 0.992 | 0.980
AR10 | 0.901 | 0.935 | 0.943 | 0.933 | 0.926 | 0.898
AR22 | 0.775 | 0.836 | 0.841 | 0.854 | 0.836 | 0.810
Mean | 18.362 | 18.207 | 18.093 | 18.207 | 18.574 | 18.410
Var | 73.173 | 71.775 | 71.594 | 59.951 | 58.190 | 77.652
Skew | 2.560 | 2.560 | 3.556 | 1.714 | 1.779 | 2.657
Kurt | 12.301 | 12.282 | 18.909 | 6.580 | 6.862 | 12.662

Note: Summary statistics for the VIX errors,
$e_{t}=\mathrm{VIX}_{t}^{\mathrm{model}}-\mathrm{VIX}_{t}^{\mathrm{market}}$.
We report the the sample average of $e_{t}$ (Bias), the mean absolute errors
(MAE), the root of mean squared errors (RMSE), the sample correlation between
$\mathrm{VIX}_{t}^{\mathrm{model}}$ and $\mathrm{VIX}_{t}^{\mathrm{market}}$
(Corr), and the sample autocorrelations of $e_{t}$ for lags 1, 10, and 22,
that are denoted AR1, AR10, and AR22, respectively. For
$\mathrm{VIX}_{t}^{\mathrm{model}}$ we report its sample average (Mean), its
sample variance (Var), its sample skewness, (Skew), and its sample excess
kurtosis (Kurt).

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.

####  5.4.2 In-Sample Comparison of the VRP and and Its Components

Table 4 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.151515The annualized volatility is calculated based on the martingale
process assumption made by Bollerslev et al. (2009) and our results are robust
when annualized volatility is calculated based on the forecast value using HAR
model (method used in Bekaert and Hoerova (2014) etc.). See section B.1 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.161616It 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 Huang et al.
(2017) 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 Bates (1996).

Table 4: In-sample Statistics of Model Performance

Model | $\mathrm{RG}$ | $\mathrm{EG}$ | $\mathrm{G}$ | $\mathrm{HN}$ | $\mathrm{HN}_{\mathtt{vd}}$
---|---|---|---|---|---
Volatility Risk Premium |  |  |
Bias | -0.497 | -3.827 | -4.397 | -4.350 | -3.053
RMSE | 3.825 | 5.078 | 5.509 | 5.456 | 4.499
$\triangle\%$ |  | 24.7% | 30.6% | 29.9% | 15.0%
MAE | 2.635 | 4.316 | 4.766 | 4.720 | 3.612
$\triangle\%$ |  | 39.0% | 44.7% | 44.2% | 27.1%
Volatility Index (VIX) |  |  |
Bias | -0.048 | -0.203 | -0.318 | -0.203 | 0.164
RMSE | 2.504 | 2.898 | 3.012 | 3.565 | 3.504
$\triangle\%$ |  | 13.6% | 16.9% | 29.8% | 28.5%
MAE | 1.866 | 2.222 | 2.199 | 2.439 | 2.419
$\triangle\%$ |  | 16.0% | 15.1% | 23.5% | 22.9%
Annualized Volatility |  |  |
Bias | 0.449 | 3.624 | 4.080 | 4.146 | 3.217
RMSE | 2.943 | 4.380 | 4.658 | 5.666 | 4.920
$\triangle\%$ |  | 32.8% | 36.8% | 48.1% | 40.2%
MAE | 2.002 | 3.898 | 4.285 | 4.839 | 3.979
$\triangle\%$ |  | 48.6% | 53.3% | 58.6% | 49.7%

Note: Bias denotes the difference between the model-implied quantity and the
market-based quantity. The rows indicated with “$\triangle\%$” present the
increase in RMSE and MAE for all models relative to the $\mathrm{RG}$. The
market-based VRP in this table is defined by
$\mathrm{VRP}_{t}^{\mathrm{market}}=\mathrm{VIX}_{t}-\sqrt{\tfrac{252}{22}\sum_{i=1}^{22}\mathrm{RVcc}_{t-i+1}}\times
100$.

####  5.4.3 Out-of-sample pricing performance

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. 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 5.4.3. 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 Diebold and Mariano (1995), 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. Out-of-sample Statistics of Model Performance Volatility
Risk Premium Volatility Index (VIX) Annualized Volatility Full: 20070103 -
20181230 Model $\mathrm{RG}$ $\mathrm{EG}$ $\mathrm{G}$ $\mathrm{HN}$
$\mathrm{HN}_{\mathtt{vd}}$ $\mathrm{RG}$ $\mathrm{EG}$ $\mathrm{G}$
$\mathrm{HN}$ $\mathrm{HN}_{\mathtt{vd}}$ $\mathrm{RG}$ $\mathrm{EG}$
$\mathrm{G}$ $\mathrm{HN}$ $\mathrm{HN}_{\mathtt{vd}}$ Bias -0.688 -3.824
-3.762 -3.814 -2.753 0.155 -0.212 0.063 -0.533 0.132 0.844 3.612 3.825 3.282
2.885 RMSE 3.975 5.420 5.378 5.387 4.668 2.870 3.259 3.278 4.298 3.903 3.271
5.154 4.961 5.856 5.360 DM stat. 8.266 8.064 8.386 4.853 3.976 3.579 2.659
1.887 6.609 5.400 5.314 4.066 $\triangle\%$ 26.7% 26.1% 26.2% 14.8% 11.9%
12.4% 33.2% 26.5% 36.5% 34.1% 44.2% 39.0% MAE 2.921 4.565 4.512 4.543 3.676
2.113 2.402 2.387 2.842 2.615 2.466 4.420 4.422 4.750 4.203 DM stat. 10.179
10.170 10.621 6.250 3.995 3.333 4.222 3.137 9.034 8.894 11.298 8.624
$\triangle\%$ 36.0% 35.3% 35.7% 20.5% 12.0% 11.5% 25.6% 19.2% 44.2% 44.2%
48.1% 41.3% Crisis Period: 20070103 - 20121230 Model $\mathrm{RG}$
$\mathrm{EG}$ $\mathrm{G}$ $\mathrm{HN}$ $\mathrm{HN}_{\mathtt{vd}}$
$\mathrm{RG}$ $\mathrm{EG}$ $\mathrm{G}$ $\mathrm{HN}$
$\mathrm{HN}_{\mathtt{vd}}$ $\mathrm{RG}$ $\mathrm{EG}$ $\mathrm{G}$
$\mathrm{HN}$ $\mathrm{HN}_{\mathtt{vd}}$ Bias -1.011 -4.311 -4.439 -4.315
-3.376 -0.294 -0.824 -0.484 -1.427 -0.321 0.717 3.487 3.956 2.888 3.055 RMSE
4.570 6.060 6.187 6.014 5.339 3.383 3.972 3.944 5.497 4.866 3.493 5.520 5.158
6.844 6.471 DM stat. 6.012 6.489 5.940 3.756 4.028 3.277 2.515 1.676 4.307
3.018 4.141 3.540 $\triangle\%$ 24.6% 26.1% 24.0% 14.4% 14.8% 14.2% 38.5%
30.5% 36.7% 32.3% 49.0% 46.0% MAE 3.420 5.164 5.261 5.142 4.301 2.557 3.069
2.946 3.798 3.327 2.514 4.505 4.476 5.261 4.992 DM stat. 7.760 8.200 7.997
5.092 4.254 2.820 3.906 2.541 5.559 5.287 8.462 7.740 $\triangle\%$ 33.8%
35.0% 33.5% 20.5% 16.7% 13.2% 32.7% 23.1% 44.2% 43.8% 52.2% 49.6% Post-crisis
Period: 20130103-20181230 Model $\mathrm{RG}$ $\mathrm{EG}$ $\mathrm{G}$
$\mathrm{HN}$ $\mathrm{HN}_{\mathtt{vd}}$ $\mathrm{RG}$ $\mathrm{EG}$
$\mathrm{G}$ $\mathrm{HN}$ $\mathrm{HN}_{\mathtt{vd}}$ $\mathrm{RG}$
$\mathrm{EG}$ $\mathrm{G}$ $\mathrm{HN}$ $\mathrm{HN}_{\mathtt{vd}}$ Bias
-0.365 -3.336 -3.084 -3.313 -2.129 0.605 0.401 0.611 0.362 0.585 0.970 3.737
3.695 3.675 2.714 RMSE 3.271 4.694 4.423 4.676 3.881 2.242 2.339 2.435 2.591
2.603 3.032 4.758 4.755 4.663 3.946 DM stat. 5.903 5.400 6.254 3.188 1.256
1.704 2.879 2.846 6.162 6.981 6.516 4.014 $\triangle\%$ 30.3% 26.0% 30.0%
15.7% 4.2% 7.9% 13.5% 13.9% 36.3% 36.2% 35.0% 23.2% MAE 2.421 3.966 3.763
3.943 3.050 1.670 1.734 1.828 1.884 1.902 2.418 4.335 4.367 4.238 3.414 DM
stat. 6.644 6.276 7.005 3.767 0.984 1.821 2.679 2.749 7.965 8.344 8.247 5.231
$\triangle\%$ 39.0% 35.7% 38.6% 20.6% 3.7% 8.7% 11.4% 12.2% 44.2% 44.6% 42.9%
29.2%

Note: Let the “model error” be the difference between the model-implied
quantity and the market-based quantity. It sample average is denoted Bias and
the rows indicated with “$\triangle\%$” state who much larger the RMSE or MAE
is for each alternative models, measured relative to $\mathrm{RG}$. Diebold
and Mariano statistics (DM stat. ) are computed for the relative MSE or MAE
losses, where the standard errors are calculated with the Parzen kernel with
$H=42$ as bandwidth. The market VRP in this table is defined as
$\mathrm{VRP}_{t}^{\mathrm{market}}=\mathrm{VIX}_{t}-\sqrt{\tfrac{252}{22}\sum_{i=1}^{22}\mathrm{RVcc}_{t-i+1}}\times
100$.

##  6 Conclusion

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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##  Appendix A Appendix of Proofs

######  Lemma A.1.

Suppose

$M_{t+1}=\frac{\exp(-\lambda_{t}z_{t+1}-\xi_{t}u_{t+1})}{\mathrm{\mathbb{E}}\exp(-\lambda_{t}z_{t+1}-\xi_{t}u_{t+1})}=\exp\left\\{-\lambda_{t}z_{t+1}-\xi_{t}u_{t+1}-\tfrac{1}{2}(\lambda_{t}^{2}+\xi_{t}^{2})\right\\},$

then by non-arbitrage we have $\lambda_{t}=\lambda$.

###### Proof.

The non-arbitrage condition is
$\mathbb{E}_{t}^{\mathbb{Q}}(\exp(r_{t+1}))=\exp(r)$ and the result follows by

$\displaystyle\mathbb{E}_{t}^{\mathbb{Q}}(\exp(r_{t+1}))$ $\displaystyle=$
$\displaystyle\mathbb{E}_{t}(M_{t+1}\exp(r_{t+1}))$ $\displaystyle=$
$\displaystyle\mathbb{E}_{t}\exp\left\\{(\sqrt{h_{t+1}}-\lambda_{t})z_{t+1}-\xi_{t}u_{t+1}-\tfrac{1}{2}(\lambda_{t}^{2}+\xi_{t}^{2})+r+\lambda\sqrt{h_{t+1}}-\tfrac{1}{2}h_{t+1}\right\\}$
$\displaystyle=$
$\displaystyle\mathbb{E}_{t}\exp\left\\{r+(\lambda-\lambda_{t})\sqrt{h_{t+1}}\right\\}.$

In order to have the above equation hold for all $h_{t}$, we need
$\lambda_{t}=\lambda$. ∎

######  Lemma A.2.

The Realized GARCH model defined by (1)-(3) and the affine exponential SDF,
implied the model (5)-(7) under the risk neutral measure, $\mathbb{Q}$.

###### Proof.

Substituting $(z_{t}^{\ast}-\lambda,u_{t}^{\ast}-\xi)$ for $(z_{t},u_{t})$
immediately yields

$\displaystyle\log h_{t+1}$ $\displaystyle=$ $\displaystyle\omega+\beta\log
h_{t}+\tau_{1}(z_{t}^{\ast}-\lambda)+\tau_{2}[(z_{t}^{\ast}-\lambda)^{2}-1]+\gamma\sigma(u_{t}^{\ast}-\xi)$
$\displaystyle=$
$\displaystyle(\omega-\tau_{1}\lambda+\tau_{2}\lambda^{2}-\gamma\sigma\xi)+\beta\log
h_{t}+(\tau_{1}-2\tau_{2}\lambda)z_{t}^{\ast}+\tau_{2}(z_{t}^{\ast
2}-1)+\gamma\sigma u_{t}^{\ast},$ $\displaystyle\log x_{t}$ $\displaystyle=$
$\displaystyle\kappa+\phi\log
h_{t}+\delta_{1}(z_{t}^{\ast}-\lambda)+\delta_{2}[(z_{t}^{\ast}-\lambda)^{2}-1]+\sigma(u_{t}^{\ast}-\xi)$
$\displaystyle=$
$\displaystyle(\kappa-\delta_{1}\lambda+\delta_{2}\lambda^{2}-\sigma\xi)+\phi\log
h_{t}+(\delta_{1}-2\delta_{2}\lambda)z_{t}^{\ast}+\delta_{2}(z_{t}^{\ast
2}-1)+\sigma u_{t}^{\ast}.$

∎

######  Lemma A.3.

Suppose that $X\sim N(0,1)$ then for $b<\tfrac{1}{2}$
$\mathbb{E}\exp\\{aX+bX^{2}\\}=\tfrac{1}{\sqrt{1-2b}}\exp\\{\tfrac{a^{2}/2}{1-2b}\\}$

###### Proof.

We have

$e^{ax+bx^{2}}e^{-\tfrac{x^{2}}{2}}=e^{ax-\tfrac{x^{2}}{2}(1-2b)}=e^{\tfrac{a}{\sqrt{1-2b}}u-\tfrac{u^{2}}{2}},$

where $u=\sqrt{1-2b}x$. So integration by substitution yields

$\displaystyle\mathbb{E}e^{aX+bX^{2}}$ $\displaystyle=$
$\displaystyle\int\tfrac{1}{\sqrt{2\pi}}e^{ax+bx^{2}}e^{-\tfrac{x^{2}}{2}}\mathrm{d}x=\int\tfrac{1}{\sqrt{2\pi}}e^{\tfrac{a}{\sqrt{1-2b}}u-\tfrac{u^{2}}{2}}\tfrac{1}{\sqrt{1-2b}}\mathrm{d}u$
$\displaystyle=$
$\displaystyle\tfrac{1}{\sqrt{1-2b}}\mathrm{E}e^{\tfrac{a}{\sqrt{1-2b}}U}=\tfrac{1}{\sqrt{1-2b}}e^{\tfrac{1}{2}\tfrac{a^{2}}{1-2b}}.$

∎

Proof of Theorem 1. From the risk neutral dynamics, we have $\log
h_{t+1}=\tilde{\omega}+\beta\log h_{t}+v_{t}^{\ast}$ where

$\displaystyle v_{t}^{\ast}$ $\displaystyle=$
$\displaystyle\tilde{\tau}_{1}z_{t}^{*}+\tau_{2}(z_{t}^{\ast
2}-1)+\gamma\sigma u_{t}^{\ast},$

so that $\log h_{t+k}=\beta^{k-1}\log
h_{t+1}+\sum_{i=0}^{k-2}\beta^{i}(v_{t+k-1-i}^{\ast}+\tilde{\omega})$. It
follows that

$\mathbb{E}_{t}^{Q}[h_{t+k}]=\mathbb{E}_{t}^{Q}[\exp\\{\beta^{k-1}\log
h_{t+1}+\sum_{i=0}^{k-2}\beta^{i}(v_{t+k-1-i}^{\ast}+\tilde{\omega})\\}]=h_{t+1}^{\beta^{k-1}}\prod_{i=0}^{k-2}F_{i},$

where
$F_{i}=\mathbb{E}_{t}^{\mathbb{Q}}\left[e^{\beta^{i}(v_{t+k-1-i}^{\ast}+\tilde{\omega})}\right]$.
Using the expression for $v_{t}^{\ast}$, and applying Lemma A.3, we have,

$\displaystyle F_{i}$ $\displaystyle=$
$\displaystyle\mathbb{E}_{t}^{\mathbb{Q}}\left[\exp\left\\{\beta^{i}(\tilde{\omega}-\tau_{2})+\beta^{i}\tilde{\tau}_{1}z+\beta^{i}\tau_{2}z^{2}+\beta^{i}\gamma\sigma
u\right\\}\right]$ $\displaystyle=$
$\displaystyle\exp\left\\{\beta^{i}(\tilde{\omega}-\tau_{2})\right\\}\mathbb{E}_{t}^{\mathbb{Q}}\left[\exp\left\\{\beta^{i}\tilde{\tau}_{1}z+\beta^{i}\tau_{2}z^{2}\right\\}\right]\mathbb{E}_{t}^{\mathbb{Q}}\left[\exp\left\\{\beta^{i}\gamma\sigma
u\right\\}\right]$ $\displaystyle=$
$\displaystyle\exp\left\\{\beta^{i}(\tilde{\omega}-\tau_{2})\right\\}(1-2\beta^{i}\tau_{2})^{-1/2}\exp\\{\tfrac{1}{2}\tfrac{\beta^{2i}\tilde{\tau}_{1}^{2}}{1-2\beta^{i}\tau_{2}}\\}\exp\left\\{\tfrac{1}{2}\beta^{2i}\gamma^{2}\sigma^{2}\right\\}$
$\displaystyle=$
$\displaystyle(1-2\beta^{i}\tau_{2})^{-1/2}\exp\left\\{\beta^{i}(\tilde{\omega}-\tau_{2})+\tfrac{1}{2}\beta^{2i}[\tfrac{\tilde{\tau}_{1}^{2}}{1-2\beta^{i}\tau_{2}}+\gamma^{2}\sigma^{2}]\right\\},$

where we suppressed the superscripts and subscripts on $z$ and $u$ to simplify
the exposition. $\square$

##  Appendix B Additional Empirical Results: Robustness Checks

This appendix provides additional empirical results. These results primarily
serve to demonstrate that the main empirical results are robust to a variety
of alternative approaches to the empirical analysis, such as alternative
measures of the market VRP, alternative objective functions, and different
choice for the realized measures.

###  B.1 Robustness Check: HAR type VRP results

The volatility risk premium used in the the main body of the paper used the
realized volatility over the past months as the predictor of volatility for
the following month, which implicitly use a martingale assumption for
volatility. Here we consider an alternative regression-based approach, which
is known as the HAR model, see Corsi (2009).171717For other ways other ways to
construct the expected volatility, see Bekaert and Hoerova (2014) With this
approach the monthly volatility is regressed on lagged daily, weekly, and
monthly volatility,

$\mathrm{RVcc}_{t+1:t+22}=\beta_{0}+\beta_{d}\mathrm{RVcc}_{t}+\beta_{w}\mathrm{RVcc}_{t-4:t}+\beta_{m}\mathrm{RVcc}_{t-21:t}+\epsilon_{t+1:t+22},$

where $\mathrm{RVcc}_{a:b}=\frac{1}{b-a+1}\sum_{t=a}^{b}\mathrm{RVcc}_{t}$. We
also estimate this model using a rolling window with 750 observations (and
lagged an additional 22 days to avoid look-ahead bias in our
estimates).181818For time $t$ the is estimated with data located in
$[t-750,t-22]$, otherwise the dependent variable will contain future
realizations of $\mathrm{RVcc}$. Table B.1 and table B.1 provide in-sample and
out-of-sample results, and all main results are identical to those obtained
with the martingale definition of the VRP. The Realized GARCH model continues
to outperform all other models.

Table B.1: In-Sample Statistics of Model Performance with Alternative VRP
Specification

Model | $\mathrm{RG}$ | $\mathrm{EG}$ | $\mathrm{G}$ | $\mathrm{HN}$ | $\mathrm{HN}_{\mathtt{vd}}$
---|---|---|---|---|---
Volatility Risk Premium |  |  |
Bias | 0.774 | -2.556 | -3.126 | -3.078 | -1.782
RMSE | 3.682 | 4.404 | 4.812 | 4.746 | 4.133
$\triangle\%$ |  | 16.4% | 23.5% | 22.4% | 10.9%
MAE | 2.184 | 3.070 | 3.514 | 3.458 | 2.750
$\triangle\%$ |  | 28.9% | 37.8% | 36.8% | 20.6%
Volatility Index (VIX) |  |  |
Bias | -0.048 | -0.203 | -0.318 | -0.203 | 0.164
RMSE | 2.504 | 2.898 | 3.012 | 3.565 | 3.504
$\triangle\%$ |  | 13.6% | 16.9% | 29.8% | 28.5%
MAE | 1.866 | 2.222 | 2.199 | 2.439 | 2.419
$\triangle\%$ |  | 16.0% | 15.1% | 23.5% | 22.9%
Annualized Volatility |  |  |
Bias | -0.823 | 2.353 | 2.808 | 2.875 | 1.946
RMSE | 3.783 | 4.753 | 4.547 | 5.893 | 5.516
$\triangle\%$ |  | 20.4% | 16.8% | 35.8% | 31.4%
MAE | 2.393 | 3.653 | 3.596 | 4.285 | 3.881
$\triangle\%$ |  | 34.5% | 33.4% | 44.1% | 38.3%

Note: Bias is defined as model-based quantity minus the corresponding market-
based quantity; The rows starting with $\triangle\%$ present the increase in
percentage of RMSE and MAE for the competing models relative to $\mathrm{RG}$.
In this table, the market VRP is defined by
$\mathrm{VRP}_{t}^{\mathtt{market}}=\mathrm{VIX}_{t}-\sqrt{\mathrm{\widehat{\mathrm{RVcc}}}_{t+1:t+22}}\times
100$, where the annualized volatility,
$\sqrt{\mathrm{\widehat{\mathrm{RVcc}}}_{t+1:t+22}}\times 100$, is the
predicted value from the HAR model.

Out-of-sample Statistics of Model Performance with Alternative VRP
Specification

| Volatility Risk Premium | Volatility Index (VIX) | Annualized Volatility
---|---|---|---
Full Sample Period: 20070103 - 20181230
Model | $\mathrm{RG}$ | $\mathrm{EG}$ | $\mathrm{G}$ | $\mathrm{HN}$ | $\mathrm{HN}_{\mathtt{vd}}$ | $\mathrm{RG}$ | $\mathrm{EG}$ | $\mathrm{G}$ | $\mathrm{HN}$ | $\mathrm{HN}_{\mathtt{vd}}$ | $\mathrm{RG}$ | $\mathrm{EG}$ | $\mathrm{G}$ | $\mathrm{HN}$ | $\mathrm{HN}_{\mathtt{vd}}$
Bias | 0.450 | -2.685 | -2.624 | -2.676 | -1.614 | 0.155 | -0.212 | 0.063 | -0.533 | 0.132 | -0.295 | 2.473 | 2.687 | 2.143 | 1.746
RMSE | 4.189 | 5.017 | 4.783 | 4.901 | 4.841 | 2.870 | 3.259 | 3.278 | 4.298 | 3.903 | 3.681 | 5.068 | 4.284 | 5.764 | 5.663
DM stat. |  | 3.168 | 2.621 | 2.931 | 3.112 |  | 3.976 | 3.579 | 2.659 | 1.887 |  | 4.285 | 1.377 | 3.180 | 2.898
$\triangle\%$ |  | 16.5% | 12.4% | 14.5% | 13.5% |  | 11.9% | 12.4% | 33.2% | 26.5% |  | 27.4% | 14.1% | 36.1% | 35.0%
MAE | 2.883 | 3.588 | 3.378 | 3.542 | 3.519 | 2.113 | 2.402 | 2.387 | 2.842 | 2.615 | 2.283 | 3.777 | 3.288 | 3.928 | 3.743
DM stat. |  | 3.127 | 2.370 | 3.095 | 3.951 |  | 3.995 | 3.333 | 4.222 | 3.137 |  | 6.273 | 4.484 | 7.322 | 6.394
$\triangle\%$ |  | 19.7% | 14.6% | 18.6% | 18.1% |  | 12.0% | 11.5% | 25.6% | 19.2% |  | 39.6% | 30.6% | 41.9% | 39.0%
Financial Crisis Period: 2007.01.03 - 2012.12.30
Model | $\mathrm{RG}$ | $\mathrm{EG}$ | $\mathrm{G}$ | $\mathrm{HN}$ | $\mathrm{HN}_{\mathtt{vd}}$ | $\mathrm{RG}$ | $\mathrm{EG}$ | $\mathrm{G}$ | $\mathrm{HN}$ | $\mathrm{HN}_{\mathtt{vd}}$ | $\mathrm{RG}$ | $\mathrm{EG}$ | $\mathrm{G}$ | $\mathrm{HN}$ | $\mathrm{HN}_{\mathtt{vd}}$
Bias | 0.257 | -3.043 | -3.172 | -3.048 | -2.109 | -0.294 | -0.824 | -0.484 | -1.427 | -0.321 | -0.551 | 2.219 | 2.688 | 1.621 | 1.788
RMSE | 4.981 | 5.867 | 5.744 | 5.649 | 5.779 | 3.383 | 3.972 | 3.944 | 5.497 | 4.866 | 4.668 | 6.087 | 4.985 | 7.258 | 7.318
DM stat. |  | 2.229 | 2.212 | 1.824 | 2.476 |  | 4.028 | 3.277 | 2.515 | 1.676 |  | 2.841 | 0.448 | 2.562 | 2.557
$\triangle\%$ |  | 15.1% | 13.3% | 11.8% | 13.8% |  | 14.8% | 14.2% | 38.5% | 30.5% |  | 23.3% | 6.4% | 35.7% | 36.2%
MAE | 3.340 | 4.341 | 4.220 | 4.210 | 4.383 | 2.557 | 3.069 | 2.946 | 3.798 | 3.327 | 2.849 | 4.552 | 3.868 | 4.908 | 4.954
DM stat. |  | 2.978 | 2.761 | 2.819 | 4.246 |  | 4.254 | 2.820 | 3.906 | 2.541 |  | 4.309 | 2.683 | 5.542 | 5.344
$\triangle\%$ |  | 23.1% | 20.8% | 20.6% | 23.8% |  | 16.7% | 13.2% | 32.7% | 23.1% |  | 37.4% | 26.3% | 41.9% | 42.5%
Post-Crisis Period: 2013.01.03-2018.12.30
Model | $\mathrm{RG}$ | $\mathrm{EG}$ | $\mathrm{G}$ | $\mathrm{HN}$ | $\mathrm{HN}_{\mathtt{vd}}$ | $\mathrm{RG}$ | $\mathrm{EG}$ | $\mathrm{G}$ | $\mathrm{HN}$ | $\mathrm{HN}_{\mathtt{vd}}$ | $\mathrm{RG}$ | $\mathrm{EG}$ | $\mathrm{G}$ | $\mathrm{HN}$ | $\mathrm{HN}_{\mathtt{vd}}$
Bias | 0.645 | -2.326 | -2.075 | -2.304 | -1.120 | 0.606 | 0.403 | 0.611 | 0.363 | 0.585 | -0.039 | 2.729 | 2.686 | 2.667 | 1.705
RMSE | 3.206 | 3.989 | 3.571 | 4.014 | 3.670 | 2.242 | 2.339 | 2.434 | 2.591 | 2.602 | 2.302 | 3.782 | 3.442 | 3.709 | 3.250
DM stat. |  | 2.696 | 1.567 | 2.819 | 2.281 |  | 1.260 | 1.704 | 2.882 | 2.848 |  | 5.427 | 4.927 | 5.640 | 4.597
$\triangle\%$ |  | 19.6% | 10.2% | 20.1% | 12.6% |  | 4.2% | 7.9% | 13.5% | 13.9% |  | 39.1% | 33.1% | 37.9% | 29.2%
MAE | 2.426 | 2.835 | 2.536 | 2.874 | 2.653 | 1.669 | 1.735 | 1.828 | 1.885 | 1.902 | 1.715 | 3.003 | 2.709 | 2.948 | 2.532
DM stat. |  | 1.389 | 0.426 | 1.543 | 1.206 |  | 0.990 | 1.819 | 2.683 | 2.751 |  | 4.903 | 4.168 | 5.177 | 4.337
$\triangle\%$ |  | 14.4% | 4.3% | 15.6% | 8.6% |  | 3.8% | 8.7% | 11.4% | 12.2% |  | 42.9% | 36.7% | 41.8% | 32.3%

Note: Let the “model error” be the difference between the model-implied
quantity and the market-based quantity. It sample average is denoted Bias and
the rows indicated with “$\triangle\%$” state who much larger the RMSE or MAE
is for each alternative models, measured relative to $\mathrm{RG}$. Diebold
and Mariano statistics (DM stat. ) are computed for the relative MSE or MAE
losses, where the standard errors are calculated with the Parzen kernel with
$H=42$ as bandwidth. The market VRP in this table is defined by
$\mathrm{VRP}_{t}^{\mathtt{market}}=\mathrm{VIX}_{t}-\sqrt{\mathrm{\widehat{\mathrm{RVcc}}}_{t+1:t+22}}\times
100$, where the annualized volatility,
$\sqrt{\mathrm{\widehat{\mathrm{RVcc}}}_{t+1:t+22}}\times 100$, is the
predicted value from the HAR model.

###  B.2 Robustness Check: Estimation with logarithmic VIX

The VIX index is occasionally very volatile and it reached very high values
during the financial crises. We inspect if some of our results are driven by
outliers. Rather than measuring pricing errors using the level of VIX, we
define VIX pricing errors for the logarithmically transformed VIX. This
transformation is a well known method for dimming the influence of outliers.
Table B.2 provides parameters estimated with logVIX error specifications while
in-sample pricing performances are summarized in B.3. Again, we do not find
significant changes in those results.

Table B.2: Full Sample Parameter Estimation with Alternative Pricing Errors
for log-VIX

Model |  $\mathrm{RG}$ |  $\mathrm{EG}$ |  $\mathrm{G}$ |  $\mathrm{HN}$ |  $\mathrm{HN}_{\mathtt{vd}}$
---|---|---|---|---|---
$\lambda$ |  0.014 |  0.137 |  0.303 |  4.673 |  4.969
|  (0.036) |  (0.010) |  (0.021) |  (0.339) |  (0.796)
$\omega$ |  -0.148 |  -0.120 |  1.60E-06 |  -6.08E-07 |  -9.63E-07
|  (0.014) |  (0.009) |  (8.88E-08) |  (1.54E-08) |  (2.73E-08)
$\beta$ |  0.985 |  0.986 |  0.930 |  0.955 |  0.950
|  (0.001) |  9.09E-04 |  (0.005) |  (0.001) |  (0.000)
$\alpha$ |  |  |  0.064 |  1.70E-06 |  1.33E-06
|  |  |  (0.004) |  (1.84E-08) |  (2.87E-09)
$\delta$ |  |  |  |  150.120 |  182.711
|  |  |  |  (2.206) |  (0.988)
$\tau_{1}$ |  -0.082 |  -0.052 |  |  |
|  (0.005) |  (0.004) |  |  |
$\tau_{2}$ |  0.018 |  0.103 |  |  |
|  (0.003) |  (0.003) |  |  |
$\gamma$ |  0.117 |  |  |  |
|  (0.010) |  |  |  |
$\kappa$ |  0.303 |  |  |  |
|  (0.230) |  |  |  |
$\phi$ |  1.063 |  |  |  |
|  (0.025) |  |  |  |
$\delta_{1}$ |  -0.088 |  |  |  |
|  (0.012) |  |  |  |
$\delta_{2}$ |  0.127 |  |  |  |
|  (0.010) |  |  |  |
$\sigma^{2}$ |  0.308 |  |  |  |
|  (0.009) |  |  |  |
$-\xi$ |  0.790 |  |  |  |  1.191
|  (0.086) |  |  |  |  (0.022)
$\pi^{\mathbb{P}}$ |  0.985 |  0.986 |  0.994 |  0.994 |  0.995
$\ell_{r}$ |  12895.03 |  12660.02 |  12522.64 |  12517.70 |  12651.56
$\ell_{x}$ |  -3129.50 |  |  |  |
$\ell_{\mathtt{vix}}$ |  2360.77 |  1703.01 |  1883.67 |  1435.09 |  1574.06
$\ell_{r,x}$ |  9765.52 |  |  |  |
$\ell_{r,\mathtt{vix}}$ |  15255.796 |  14363.024 |  14406.315 |  13952.796 |  14225.623
$\ell_{r,x,\mathtt{vix}}$ |  12126.292 |  |  |  |

Note: Robust standard errors are in parenthesis. The persistence parameter
$\pi^{\mathbb{P}}$ is measured by $\beta+\phi\gamma$ for
$\mathrm{RG}_{\mathtt{r}}$ and $\beta$ for all other models. For the model
$\mathrm{HN}_{\mathtt{vd}}$ we report the value of $(1+2\alpha\xi)^{-1}$ in
place of $-\xi$ (the implied value for $\xi$ is here -121098.46).

Table B.3: In-sample Statistics of Model Performance with Alternative Error
Specification

Model | $\mathrm{RG}$ | $\mathrm{EG}$ | $\mathrm{G}$ | $\mathrm{HN}$ | $\mathrm{HN}_{\mathtt{vd}}$
---|---|---|---|---|---
Volatility Risk Premium |  |  |
Bias | -0.561 | -4.092 | -4.326 | -4.627 | -3.697
RMSE | 3.664 | 5.268 | 5.456 | 5.679 | 4.971
$\triangle\%$ |  | 30.5% | 32.9% | 35.5% | 26.3%
MAE | 2.547 | 4.509 | 4.712 | 4.944 | 4.105
$\triangle\%$ |  | 43.5% | 45.9% | 48.5% | 37.9%
Volatility Index (VIX) |  |  |
Bias | -0.202 | -0.434 | -0.398 | -0.856 | 0.052
RMSE | 2.729 | 3.134 | 3.077 | 4.334 | 3.754
$\triangle\%$ |  | 12.9% | 11.3% | 37.0% | 27.3%
MAE | 1.958 | 2.287 | 2.191 | 2.528 | 2.386
$\triangle\%$ |  | 14.4% | 10.6% | 22.5% | 17.9%
Annualized Volatility |  |  |
Bias | 0.359 | 3.658 | 3.928 | 3.771 | 3.748
RMSE | 3.531 | 4.608 | 4.524 | 5.984 | 5.665
$\triangle\%$ |  | 23.4% | 22.0% | 41.0% | 37.7%
MAE | 2.340 | 4.139 | 4.101 | 5.009 | 4.601
$\triangle\%$ |  | 43.5% | 42.9% | 53.3% | 49.1%

Note: Bias is defined as model generated value minus their market counterpart;
The rows starting with $\triangle\%$ present the increase in percentage of
RMSE and MAE for the competing models relative to $\mathrm{RG}$.

###  B.3 Robustness Check: Alternative Realized Measures

A wide range of realized measures of volatility have been proposed since the
use of the realized variance was popularized by Andersen and Bollerslev
(1998). As alternative measures we use the realized variance based on both
5-minute and 10-minute returns, denoted $\mathrm{RV}{}_{5\mathrm{m}}$ and
$\mathrm{RV}{}_{10\mathrm{m}}$, respectively. We also consider the bi-power
variation, $\mathrm{BV}{}_{5\mathrm{m}}$, by Barndorff-Nielsen (2004), which
is computed from 5-minute returns, and three different realized kernels,
denoted $\mathrm{RK}_{\mathrm{P}}$, $\mathrm{RK}_{\mathrm{TH}}$, and
$\mathrm{RK}_{\mathrm{B}}$, that are based on different kernel function.
$\mathrm{RK}_{\mathrm{P}}$ is based on the Parzen kerne,
$\mathrm{RK}_{\mathrm{TH}}$, on the modified Tukey-Hanning, which is denoted
$\mathrm{TH_{2}}$ in Barndorff-Nielsen et al. (2008), and
$\mathrm{RK}_{\mathrm{B}}$ is the realized kernel estimator based on the
Bartlett kernel function. Finally, the median-based realized variance., by
Andersen et al. (2012) is denoted $\mathrm{MedRV}$. The realized measures were
obtained from the Realized Library at Oxford-Man institute (version 0.3).

Table B.4 provides in-sample pricing performances of the Realized GARCH model
based on different realized measures. The results are quite similar across
with other realized measures. For the VRP we observed some minor improvements
over $\mathrm{RV}{}_{5\mathrm{m}}$, but the relative differences are quite
small. Interestingly, the two jump robust estimators,
$\mathrm{BV}{}_{5\mathrm{m}}$ and $\mathrm{MedRV}$, are somewhat wors at
fitting the VIX and Annualize Volatility, which indicates that including the
jump component is useful in this context. However, for the VRP both jump-
robust measures perform similarly to other realized measures. The
$\mathrm{RV}{}_{5\mathrm{m}}$ seems to be adequate for the present modeling
problem, as previously argued in Liu et al. (2015) for a range of problems.

Table B.4: In-sample Statistics of Model Performance with Alternative Realized
Measures

Model | $\mathrm{RV}{}_{5\mathrm{m}}$ | $\mathrm{RV}{}_{10\mathrm{m}}$ | $\mathrm{BV}{}_{5\mathrm{m}}$ | $\mathrm{RK}_{\mathrm{P}}$ | $\mathrm{RK}_{\mathrm{TH}}$ | $\mathrm{RK}_{\mathrm{B}}$ | $\mathrm{MedRV}$
---|---|---|---|---|---|---|---
Volatility Risk Premium |  |  |  |  |
Bias | -0.498 | -0.497 | -0.442 | -0.514 | -0.486 | -0.482 | -0.520
RMSE | 3.825 | 3.838 | 3.818 | 3.843 | 3.764 | 3.764 | 3.844
$\triangle\%$ |  | 0.34% | -0.18% | 0.48% | -1.62% | -1.60% | 0.49%
MAE | 2.635 | 2.646 | 2.613 | 2.649 | 2.586 | 2.585 | 2.657
$\triangle\%$ |  | 0.42% | -0.81% | 0.55% | -1.90% | -1.91% | 0.84%
Volatility Index (VIX) |  |  |  |  |
Bias | -0.048 | -0.047 | -0.036 | -0.032 | -0.047 | -0.046 | -0.037
RMSE | 2.504 | 2.520 | 2.668 | 2.575 | 2.548 | 2.547 | 2.808
$\triangle\%$ |  | 0.63% | 6.15% | 2.77% | 1.75% | 1.72% | 10.84%
MAE | 1.866 | 1.896 | 1.985 | 1.920 | 1.917 | 1.912 | 2.078
$\triangle\%$ |  | 1.59% | 6.00% | 2.80% | 2.64% | 2.38% | 10.20%
Annualized Volatility |  |  |  |  |
Bias | 0.449 | 0.450 | 0.406 | 0.481 | 0.439 | 0.436 | 0.483
RMSE | 2.942 | 2.911 | 3.066 | 2.909 | 3.171 | 3.165 | 3.048
$\triangle\%$ |  | -1.08% | 4.02% | -1.16% | 7.22% | 7.02% | 3.47%
MAE | 2.002 | 1.977 | 2.130 | 2.013 | 2.190 | 2.185 | 2.134
$\triangle\%$ |  | -1.28% | 6.02% | 0.53% | 8.57% | 8.40% | 6.17%

Note: Empirical results based on alternative realized volatility measures.
$\mathrm{RV}{}_{5\mathrm{m}}$ and $\mathrm{RV}{}_{10\mathrm{m}}$ are the
realized variances based on 5 minute returns and 10 minute returns,
respectively, $\mathrm{BV}{}_{5\mathrm{m}}$ is the bipower variation based on
five-minute returns,$\mathrm{RK}_{\mathrm{P}}$, $\mathrm{RK}_{\mathrm{TH}}$,
and $\mathrm{RK}_{\mathrm{B}}$ are realized kernels estimator based the the
Parzen, a modified Tukey-Hanning, and the Bartlett kernel functions. Finally,
$\mathrm{MedRV}$ is the median based realized variance. The rows starting with
$\triangle\%$ present the increase in percentage of RMSE and MAE for the
competing models relative to the one using 5min RV.