Exponential GARCH-Itô Volatility Models

In finance practice, volatility plays a pivotal role. Low-frequency and high-frequency financial data are widely used to analyze volatility dynamics. For example, generalized auto-regressive conditional heteroskedasticity (GARCH) models are introduced to catch the low-frequency volatility dynamics, such as volatility clustering, by employing the squared low-frequency log-return as the innovation (Bollerslev,, 1986; Engle,, 1982). However, when the volatility changes rapidly, it is often difficult to catch the change using only the low-frequency log-returns as the innovations (Andersen et al.,, 2003). On the other hand, high-frequency financial data are available to construct the so-called realized volatility for estimating daily integrated volatility. Examples include two-time scale realized volatility (TSRV) (Zhang et al.,, 2005), multi-scale realized volatility (MSRV) (Zhang,, 2006), kernel realized volatility (KRV) (Barndorff-Nielsen et al.,, 2008), quasi-maximum likelihood estimator (QMLE) (Aït-Sahalia et al.,, 2010; Xiu,, 2010), pre-averaging realized volatility (PRV) (Jacod et al.,, 2009), and robust pre-averaging realized volatility (Fan and Kim,, 2018; Shin et al.,, 2021). These realized volatility estimators contain high-frequency information about the market volatility, and many studies show that incorporating high-frequency information helps account for low-frequency market dynamics (Corsi,, 2009; Hansen et al.,, 2012; Kim and Wang,, 2016; Shephard and Sheppard,, 2010). Several conditional volatility models have been developed to combine high-frequency and low-frequency data and enhance volatility estimation and predication by employing realized volatility as the volatility proxy. Examples include the realized volatility based modeling approaches (Andersen and Bollerslev, 1997a, ; Andersen and Bollerslev, 1997b, ; Andersen and Bollerslev, 1998a, ; Andersen and Bollerslev, 1998b, ; Andersen et al.,, 2003), the heterogeneous auto-regressive (HAR) models (Corsi,, 2009), the realized GARCH models (Hansen et al.,, 2012), the high-frequency based volatility (HEAVY) models (Shephard and Sheppard,, 2010), and the unified GARCH/SV-Itô models (Kim and Fan,, 2019; Kim and Wang,, 2016; Song et al.,, 2021). These models have been developed based on the linear auto-regressive structure of realized volatilities. However, we often observe that non-linear auto-regressive structures, such as exponential functions, better capture the volatility dynamics (Nelson,, 1991; Hansen and Huang,, 2016; Kawakatsu,, 2006). This may be because log-volatilities often have a stronger linear auto-regressive relationship. In fact, when variables are close to normal distributions, linear models work well. To check the normality of realized volatilities, we draw normal QQ-plots of realized volatilities and log realized volatilities for AAPL stock. Figure 1 shows that the log-transformation makes the realized volatilities closer to a normal distribution. Most of assets have the similar phenomena. Thus, we can conjecture that log-volatilities better explain volatility dynamics. To harness this feature, Hansen and Huang, (2016) employed the exponential GARCH structure with the log-realized volatility as the innovation, and their empirical study indicates that the non-linear GARCH structure helps account for market dynamics. Although, as discussed above, empirical studies support that incorporating high-frequency data with a non-linear auto-regressive structure better captures the market dynamics, the mathematical gap between the empirical low-frequency discrete-time non-linear volatility models, such as the exponential realized GARCH, and high-frequency-based continuous-time diffusion process is not well-studied. In fact, several studies have been conducted to fill the gap between the discrete-time volatility models and continuous-time diffusion process (Kim and Fan,, 2019; Kim and Wang,, 2016; Song et al.,, 2021). However, these studies are based on the linear auto-regressive structure, and the extension from linear to non-linear structures is not straightforward. This fact increases the demand for developing continuous-time diffusion process-based models that provide a rigorous mathematical formulation for the non-linear auto-regressive structure of realized volatilities.

In this paper, we develop a novel diffusion process to model high-frequency financial data, which can accommodate a non-linear GARCH structure of the realized volatilities. From the empirical study, we often observe that the log-realized volatility has a stronger auto-regressive structure than the original realized volatility. To reflect this, we employ the exponential GARCH structure as the non-linear function. Specifically, the log-realized volatility follows the realized GARCH structure. To connect this low-frequency volatility structure with the continuous-time diffusion process, we develop a novel continuous instantaneous volatility process. Since the volatility process has a non-linear structure, the linear structure of the unified GARCH-Itô (Kim and Wang,, 2016) is not applicable. Furthermore, usual log-diffusion processes for instantaneous volatility processes can not provide the solution. To tackle this issue, we propose a novel instantaneous volatility process, based on the average integrated volatility process. Then, the proposed instantaneous volatility process is continuous with respect to time, and its daily integrated volatility is decomposed into the exponential realized GARCH with the daily log-integrated volatility as the innovation and exponential martingale difference. We call it the exponential realized GARCH-Itô (ERGI) model. Unlike the linear realized GARCH model, the log-realized volatility can have negative values. Thus, we allow model parameters to be negative. To estimate the model parameter, we propose a quasi-maximum likelihood estimation procedure. Specifically, we adopt the Gaussian quasi-likelihood function and use the realized volatility as the proxy of the conditional expected value. Furthermore, we establish its asymptotic properties. To illustrate the benefit of the proposed model, we apply the ERGI model to real trading high-frequency data and find that the exponential structure helps account for the volatility dynamics.

The rest of paper is organized as follows. In Section 2, we propose the ERGI model and investigate its properties. In Section 3, we suggest the quasi-maximum likelihood estimation procedure and study its asymptotic behaviors. In Section 4, we conduct a simulation study to check the finite sample performance of the ERGI model. In Section 5, we apply the ERGI model to the top 50 high trading volume assets among the S&P 500 compositions. Section 6 contains the conclusions. All the technical proofs are collected in Section 7.

In this section, we develop an exponential realized GARCH-Itô (ERGI) model as follows.

We note that $\beta^{*}$ is an adjust term to handle some remaining terms. Specifically, we set $\beta^{*}=\frac{1+\beta\varrho_{2}}{\varrho_{2}-2\varrho_{3}}$, where $\varrho$’s are functions of $\beta$, which are defined in Theorem 1. The log-average integrated volatility, $\log\bar{\sigma}_{t}^{2}(\theta)$, provides the innovation term, and $Z_{t}$ is the random fluctuation. Since the process $b_{t}(\theta)$ is continuous with respect to time $t$, the instantaneous volatility process $\sigma_{t}^{2}(\theta)$ is also continuous. At the integer time points, we have

That is, $b_{n}(\theta)$ can be explicitly expressed by the past log-integrated volatilities, and $b_{t}(\theta)$ has a form of the interpolation between these values. Thus, when considering $b_{t}(\theta)$, the ERGI model has a similar structure of realized GARCH-Itô model (Song et al.,, 2021) with the log-integrated volatility as the innovations. However, unlike the realized GARCH-Itô model, to obtain the non-linear exponential realized GARCH form, the instantaneous volatility process has a non-linear structure, such as $\bar{\sigma}_{t}^{2}(\theta)\{1+(t-[t])b_{t}(\theta)\}$. The solution for this structure is

Details can be found in Lemma 1. Using the above solution, we can measure the integrated volatility $\int_{n-1}^{n}\sigma_{t}^{2}(\theta)dt$ with $\sigma_{n-1}^{2}(\theta)$ and $\int_{n-1}^{n}b_{s}(\theta)ds$, which has the realized GARCH form with the log-integrated volatilities. Specific properties of the integrated volatility are shown in the following theorem.

Theorem 1(a) shows that the log-integrated volatility is decomposed into the realized GARCH with the log-integrated volatility innovations, $h_{n}(\theta)$, and the martingale difference $D_{n}$. Thus, the log-realized GARCH, $h_{n}(\theta)$, is the conditional expected value of the log-integrated volatility, but it is not the conditional expected value of the original integrated volatility. In Theorem 1(b), we show that the integrated volatility is decomposed into the exponential function of the realized GARCH with the log-integrated volatility innovations, $H_{n}(\theta)$, which has the additional interceptor term from the martingale difference term $D_{n}$, and the exponential martingale difference $M_{n}$. Since the expectation is a linear function, the additional interceptor term does not appear in the linear realized GARCH. However, the ERGI is non-linear, and, thus, we have the additional interceptor term. The main purpose of this paper is to develop a model for analyzing the original integrated volatility, so we develop a statistical inference based on (2.2). Theorem 1 indicates that the proposed model has a non-linear exponential GARCH structure. From the empirical study, we find that this non-linear structure helps explain the volatility dynamics as compared to the usual linear realized GARCH. Details can be found in Section 5.

We conducted Monte-Carlo simulations to check the finite sample performance of the ERGI model. The log-prices were generated from the ERGI model given in Definition 1 for $n$ days with $m$ high-frequency observations. The model parameters were set to be $(\omega_{0},\gamma_{0},\beta_{0},\nu_{0})=(-0.1,0.3,0.5,2)$ and $\mu_{t}=0$. Then, the GARCH parameters $(\omega_{0}^{g},\gamma_{0},\beta_{0}^{g})=(0.3207,0.3,0.4405)$. For the jump part, we set the intensity $\lambda_{t}=10$ and the jump size $|J_{t}|=0.05$. The signs of the jump size were randomly generated. Let $t_{d,j}=d-1+j/m$ for $d=1,\ldots,n$ and $j=0,\ldots,m$. We generated the noisy observations as follows:

where $\epsilon_{t_{d,j}}$’s are i.i.d. normal random variables with mean zero and standard deviation $0.01\sqrt{\int_{d-1}^{d}\sigma_{t}^{2}(\theta^{g})dt}$. To generate the true process, we chose $m=11700$. We varied $n$ from 100 to 500 and $m$ from 390 to 11700, which corresponds to the number of minutes and 2-seconds during the open-to-close period, respectively. We used $Y_{t_{d,j}}$ as the high-frequency observations. To estimate the integrated volatilities, we used the jump robust pre-averaging method (Aït-Sahalia and Xiu,, 2016; Jacod et al.,, 2009) as follows:

where we take the weight function $g(x)=x\wedge(1-x)$, the bandwidth size $K=\lfloor m^{1/2}\rfloor$,

$\boldsymbol{1}_{\{\cdot\}}$ is an indicator function, and $\tau_{m}=c_{\tau}m^{-0.235}$ is a truncation level for the constant $c_{\tau}$. We chose $c_{\tau}$ as four times the sample standard deviation of the pre-averaged prices $m^{1/4}\bar{Y}(t_{d,k})$. We estimated the parameters using the procedure in Section 3. We repeated the whole procedure 500 times.

To check the performance of the realized volatility estimator, we calculated squared relative errors as follows:

Then, we calculated the sample average of squared relative errors over 500 simulations. We have the average errors 0.0117, 0.0463, and 0.10751 for $m=11700,1170,$ and $390$, respectively. As $m$ increases, the average errors decreases. This result supports the theoretical findings in the realized volatility estimator literature (Aït-Sahalia and Xiu,, 2016; Jacod et al.,, 2009).

Table 1 reports the mean squared errors (MSE) of the parameter estimates $\widehat{\theta^{g}}$ with $n=100,\,200,\,500$ and $m=390,\,1170,11700$. In Table 1, MSEs usually decrease as the number of high-frequency observations or daily observations increases. This result supports the theoretical findings in Section 3.

To check the asymptotic normality of the GARCH parameters $(\omega^{g},\gamma,\beta^{g})$, we calculated the Z-statistics defined in Section 3.3. In Figure 2, we draw the standard normal quantile-quantile plots (QQ-plots) of the Z-statistics estimates of $\omega^{g}$, $\gamma$, and $\beta^{g}$ for $m=390$ and $n=100,200,500$. In Figure 2, we find that the Z-statistics become close to the standard normal distribution as the sample period increases. This result supports the theoretical findings in Section 3. Thus, based on the proposed Z-statistics, we can conduct hypothesis tests using the standard normal distribution.

We examined the out-of-sample performance of estimating the one-day-ahead GARCH volatility $\exp(H_{n+1}(\theta^{g}_{0}))$. To estimate future GARCH volatility, we employed the proposed conditional ERGI estimator $\exp(\widehat{H}_{n+1}(\widehat{\theta^{g}}))$, realized GARCH volatility estimator (Hansen et al.,, 2012; Song et al.,, 2021), and PRV of the previous day. For example, the realized GARCH volatility estimator is estimated based on the following conditional volatility,

That is, the realized GARCH volatility estimator has the usual linear GARCH structure with the realized volatilities. We measured the mean squared errors with the one-day-ahead sample period over 500 samples as follows:

where $\widehat{\mbox{var}}_{n+1,i}$ is one of the above future volatility estimators at the $i$th sample path given the available information at time $n$. Figure 3 depicts the mean squared errors for the ERGI, realized GARCH, and PRV against varying the numbers of low- and high-frequency observations, $n$ and $m$. In Figure 3, we find that the ERGI models show the best performance. The interesting finding is that the realized GARCH model can also capture some volatility dynamics. This may be because even if the volatility dynamic structure is non-linear, it could have some linear dynamics. Especially, when the log-volatility quantities are small, by Taylor’s expansion, the linear model can capture some non-linear dynamics. However, using the only linear structure, it cannot fully explain the non-linear dynamic structure. From these results, we can conjecture that modeling appropriate dynamic structure helps account for market dynamics.

We applied the proposed ERGI model to real trading high-frequency data. We obtained the top 50 trading volume assets intra-day data from January 2010 to December 2016 from the TAQ database in the Wharton Research Data Services (WRDS) system. We used the log-prices and employed the jump robust PRV estimation procedure defined in Section 4 to estimate open-to-close integrated volatility. In the empirical study, we chose the tuning parameter $c_{\tau}$ as 10 times the sample standard deviation of pre-averaged prices $m^{1/4}\bar{Y}(t_{d,k})$. To check the accuracy of the PRV estimator, we calculated standard errors (SE) as follows. We first calculated the asymptotic variance, proposed by Aït-Sahalia and Xiu, (2016), and divided the square root of the asymptotic variance estimator by the square root of the number of high-frequency observations. We report data summary in Table 2. The number of high-frequency data is from 16,000 to 90,000 on average, and we find that the proportion of the jump variation is about 8% to 40% of the total variation on average. The standard error is less than 10% of realized volatility.

We first estimated the model parameters using the recent 1000 days data. From the estimated model parameters, we obtained the following conditional expected volatility for each asset

Table 3 reports the estimation results. From Table 3, we show that dynamic structures can be explained by the past log-PRV, and the coefficients of realized volatilities are statistically significant. Thus, the proposed exponential model is valid.

For comparison, we employed the realized GARCH (Hansen et al.,, 2012; Song et al.,, 2021), unified GARCH-Itô (UGARCH) (Kim and Wang,, 2016), and HAR (Corsi,, 2009) models. To measure the performance of the volatility, we used the mean squared prediction errors (MSPE) and relative mean squared prediction errors (RMSPE) as follows:

where $Vol_{i}$ is one of the ERGI, realized GARCH, HAR, and UGARCH. We used $RV_{i}$ as the nonparametric daily volatility estimator. Furthermore, we calculated the out-of-sample R-square (OSR) (Campbell and Thompson,, 2008) as follows:

where $Vol_{i}^{*}$ is the proposed ERGI, and $Vol_{i}$ is one of the realized GARCH, HAR, unified GARCH-Itô, and sample mean of the in-sample $RV_{i}$’s. We predicted the one-day-ahead conditional expected volatility by the ERGI, realized GARCH, HAR, and UGARCH models using the in-sample period data. We fixed the in-sample period as 500 days and used the rolling window scheme to estimate the parameters. The number of out-of-sample was 1,262. To check the period dependency, we split the period into two equal parts. We denote the two periods as Period 1 and Period 2. Table 4 reports the average rank and the number of the first rank of MSPEs and RMSPEs for the ERGI, realized GARCH, HAR and UGARCH for Period 1, Period 2, and the whole period over the 50 assets. Figure 4 depicts the relative MSPE and RMSPE for the realized GARCH, HAR, and UGARCH with respect to the ERGI for Period 1, Period 2, and the whole period. Figure 5 draws the OSR for the ERGI with respect to realized GARCH, HAR, UGARCH, and sample mean for Period 1, Period 2, and the whole period. From Table 4 and Figures 4–5, we find that the realized volatility-based model, such as the ERGI, realized GARCH, and HAR models, perform better than the UGARCH model, which incorporates the squared open-to-close returns as the innovation. That is, incorporating the realized volatility helps account for the volatility dynamics. When comparing the realized volatility-based models, the proposed ERGI model shows the best performance. From this result, we can conjecture that the non-linear exponential form with realized volatilities helps explain the market dynamics.

To further compare the predictive accuracy among the ERGI, realized GARCH, HAR, and UGARCH models, we conducted Diebold-Mariano tests (Diebold and Mariano,, 2002) as follows. We first calculated the residuals for the four models:

where $Vol_{i}$ is one of the ERGI, realized GARCH, HAR, and UGARCH, and $RV_{i}$ is the non-parametric realized volatility. We define

where $e_{i}^{*}$ is the residuals from the ERGI and $e_{i}$ is the residuals from one of realized GARCH, HAR, and UGARCH. Then, we conducted hypothesis tests for

The first alternative statement ($\mathbb{E}[d_{i}]<0$) is to test whether the ERGI is better, while the second alternative statement ($\mathbb{E}[d_{i}]>0$) is to test whether other model is better than the ERGI. We call them “less” and “greater” tests, respectively. Figure 6 depicts box plots of p-values of the less and greater DM tests for the ERGI versus one of the realized GARCH, HAR, and UGARCH for Period 1, Period 2, and the whole period. From Figure 6, the less tests show that p-values of 22, 16, and 46 assets for realized GARCH, HAR, and UGARCH models, respectively, were less than 10% over the whole period. In contrast, the greater tests indicate that a couple of assets for the HAR model have significant p-values over the whole period. From these results, although the ERGI does not give significant better predictive accuracy for all assets, we can conclude that for most assets, the ERGI is at least not worse than other models, and, for some assets, the ERGI shows significantly better performance than the other models.

To check the volatility persistence of the nonparametric volatility, we studied the residuals between the nonparametric volatility and estimated conditional volatilities, $Vol_{i}-RV_{i}$, where $Vol_{i}$ is the predicted one-day-ahead conditional expected volatility by ERGI, realized GARCH, HAR, and UGARCH using the in-sample period data. Then, we checked their autocorrelations. Table 5 reports the average rank and number of the first rank of the first order autocorrelation for the ERGI, realized GARCH, HAR, and UGARCH for Period 1, Period 2, and the whole period over the 50 assets. Figure 7 provides the box plots of the first order autocorrelation for the ERGI, realized GARCH, HAR, and UGARCH for Period 1, Period 2, and the whole period over the 50 assets. From Table 5 and Figure 7, we find that the ERGI has relatively small autocorrelations. That is, the ERGI model can reduce the volatility persistence. These numerical results provide evidence to conclude that the non-linear exponential auto-regressive structure helps explain the market dynamics in the volatility time series.

In this paper, we propose a novel jump diffusion process to model the non-linear auto-regressive structure of the realized volatility. We employ the exponential GARCH structure. By introducing a continuous instantaneous volatility process whose integrated volatility follows the exponential realized GARCH structure, we fill the gap between the empirical discrete-time non-linear volatility model with the realized volatility and high-frequency based continuous-time diffusion process. That is, this paper provides rigorous mathematical background to understand the exponential realized GARCH structure. To estimate the model parameter, we propose the quasi-maximum likelihood estimation procedure and establish its asymptotic properties. From the empirical study, we find the benefits of incorporating the non-linear exponential realized GARCH.

In this paper, we focus on the continuous part of log-return processes for the open-to-close period. However, it is important and interesting to study dynamic structures of the jump variation and close-to-open returns. We leave this for future study.