Volatility plays a crucial role in theory and applications of asset pricing, optimal portfolio allocation, and risk management. This paper proposes a combined model of autoregressive moving average (ARFIMA), generalized autoregressive conditional heteroscedasticity (GRACH), and skewed-
Volatility plays a crucial role in the theory and application of asset pricing, optimal portfolio allocation, and risk management. It is therefore important to introduce a model addressing the key features of volatility data sets and develop a valid estimation method. The most dominant features of the usual volatilities, the historic volatilities, the implied volatilities, and the realized volatilities are long memories, conditional heteroscedasticity and skewed error distribution.
The fractionally integrated autoregressive moving average (ARFIMA) model (Granger and Joyeux, 1980; Hosking, 1981) and the generalized autoregressive conditional heteroscedasticity (GARCH) model (Bollerslev, 1986) have been widely used to explain the long memory and the time-varying conditional variance features of volatilities, respectively. Park (2016) used a skewed student
However, none of the currently available methods propose a model incorporating all the above three features simultaneously. For example, Baillie
In this paper, we propose a model which combines ARFIMA, GARCH, and skewed-
A Bayesian approach is used to estimate the parameters of the model. We employ Just Another Gibbs Sampler (JAGS, http://mcmc-jags.sourceforge.net/) to generate Markov chain Monte Carlo (MCMC) samples from the joint posterior distributions of the parameters. A key advantage of JAGS is that the software runs MCMC automatically once a user specifies a model (the distribution of data and the prior distributions of parameters); therefore, practitioners who are not experts in MCMC may also easily implement Bayesian inference using JAGS.
Given the posterior samples, several features of interest, such as the estimated marginal posterior densities, posterior moments, quantiles, scatter plots exhibiting interesting relations between parameters, may be derived in a straightforward manner. In addition, any restrictions on parameters, such as inequality restrictions on coefficients in the GARCH model to ensure stationarity, can be implemented in a straightforward way. Finally, useful prior information, which may be available from the data context or previous data analysis, can be used in Bayesian approaches (Czado and Min, 2011).
This paper is organized as follows. Section 2 presents the ARFIMA + GARCH + skewed-
In this section, we describe the proposed model, the priors for the parameters, and a Markov chain Monte Carlo method for Bayesian inference.
We consider a model that captures the features of conditional heteroscedasticity, long-memory, asymmetry, and fat tails via a combination of ARFIMA, GARCH, and skewed-
where
From Bollerslev
where
Asymmetry and fat tails are featured for the zero-mean-unit-variance error
where
Now we consider a model consisting of ARFIMA(1,
with constraints
We assume prior independence among
A diffuse normal prior is used for
Clearly, the posterior distribution of the parameters Θ = (
Implementation of the MH algorithm is often complicated for practitioners who are not experts in MCMC and/or coding. We get around this difficulty by generating MCMC samples by using user-friendly JAGS software. Given the model specification, it employs either Gibbs sampler when available or MH algorithm to generate posterior samples.
To use JAGS, the distribution of data needs to be specified as one of the distributions built in JAGS. However, the distributions in the ARFIMA(1,
For the likelihood of Θ, we first derive the joint pdf of
where
where
In usual financial volatility analysis, since data sets are recorded daily or more frequently and the series length is large, this approximation would be good. Now the likelihood function of Θ in the ARFIMA(1,
The likelihood function (
We consider a daily closing price for the volatility index (VIX), which is a measure of market expectations on volatility over the next 30 days conveyed by S&P 500 stock index option prices. The data set can be obtained from the Chicago Board Options Exchange (CBOE) web site, http://www.cboe.com/micro/vix/historical.aspx. We use data from January 3, 2006 to November 30, 2016, which contains 2,748 observations.
Time series plot of the VIX is presented in Figure 1. From the figure, we can see apparent conditional heteroscedastic feature of the VIX data; that high peaks from August 2007 to March 2009 due to the financial crisis of 2007–2008 cause large volatilities and that the relatively lower index values between 2013 and 2015 have smaller volatilities.
Table 1 shows basic summary statistics of the VIX. Figure 2 displays the histogram (with a normal approximation) and the autocorrelation function (ACF) plot of the VIX. The values of skewness, median, and mean (Table 1) and the histogram (Figure 2) show that the VIX is skewed to the right and has highly fat tails. The ACF plot of the VIX shows the feature of long memory which can be seen that the VIX is skewed to the right and has highly fat tails. The ACF plot of the VIX shows the feature of long memory which can be also supported by the GPH (Geweke and Porter-Hudak, 1983) test statistics of 8.663.
Table 2 presents the
We applied the proposed Bayesian method to the first differences of the VIX data and generated posterior samples of parameters using JAGS. The first 3,000 iterations are discarded as burn-in and we obtained 15,000 samples after the burn-in from 3 different chains. Convergence of MCMC is diagnosed by Gelman-Rubin shrink factors and trace plots of the MCMC samples.
Table 3 displays the estimated posterior mean, standard error, and an approximate 95% credible interval of each parameter. All parameters are significant in that their 95% credible intervals do not contain zero. The estimated posterior mean −0.332 of the fractional integration parameter
The estimates satisfy the stationary condition
Figure 4 shows the estimated marginal posterior distributions of the parameters (solid line) with normal approximations (dashed line). The figure shows that marginally the posterior distributions of the parameters seem to be well approximated by normal distributions, except for
For comparison, we applied a model ARFIMA + GARCH + skewed-
The most notable differences are the estimates of the parameters of the ARFIMA model. In the ARFIMA + GARCH + skewed-
We also applied ARFIMA + GARCH +
In this paper, we have proposed a fully Bayesian implementation of the ARFIMA + GARCH + skewed-
Bayesian method naturally produce estimates that satisfy the stationary constraints in GARCH (1, 1), by incorporating constraints in the prior. The scatter plot of the samples of (
In this Bayesian implementation, since the likelihood function of the parameters of the ARFIMA + GARCH + skewed-
This research was supported by the Basic Science Research Program of the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2016R1A2 B4008914) (Rosy Oh and Man-Suk Oh) and (2016R1A2B4008780) (Dong Wan Shin).
Summary statistics of the volatility index data
Min | Median | Mean | Max | SD | Skewness | Kurtosis |
---|---|---|---|---|---|---|
9.890 | 17.230 | 20.060 | 80.860 | 9.574 | 2.392 | 7.546 |
Test for unit root
ADF test | KPSS test | |
---|---|---|
0.0422 | < 0.01 |
ADF = augmented Dicky-Fuller; KPSS = Kwiatkowski-Phillips-Schmidt-Shin.
Estimation results for the first difference of volatility index from ARFIMA + GARCH + skewed-
Estimated posterior mean | Standard error | 95 % credible interval | ||
---|---|---|---|---|
Lower | Upper | |||
−0.008 | 0.002 | −0.013 | −0.004 | |
0.186 | 0.036 | 0.118 | 0.256 | |
−0.332 | 0.029 | −0.387 | −0.276 | |
0.068 | 0.014 | 0.046 | 0.099 | |
0.213 | 0.023 | 0.170 | 0.260 | |
0.778 | 0.022 | 0.732 | 0.818 | |
1.376 | 0.037 | 1.304 | 1.450 | |
4.419 | 0.317 | 3.871 | 5.113 |
ARFIMA = autoregressive moving average; GARCH = generalized autoregressive conditional heteroscedasticity.
Estimation results for the first difference of volatility index from various models
95 % credible interval | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
ARFIMA + GARCH + skewed- | ARFIMA + GARCH + | ARFIMA + GARCH + normal | ||||||||||
Est. | SE | Lower | Upper | Est. | SE | Lower | Upper | Est. | SE | Lower | Upper | |
0.001 | 0.010 | −0.018 | 0.020 | −0.021 | 0.004 | −0.029 | −0.015 | −0.013 | 0.004 | −0.021 | −0.006 | |
0.028 | 0.030 | −0.031 | 0.088 | 0.189 | 0.044 | 0.105 | 0.278 | 0.155 | 0.046 | 0.070 | 0.253 | |
−0.181 | 0.024 | −0.227 | −0.133 | −0.305 | 0.037 | −0.379 | −0.236 | −0.263 | 0.038 | −0.343 | −0.192 | |
3.504 | 0.096 | 3.320 | 3.697 | - | - | - | - | - | - | - | - | |
0.095 | 0.017 | 0.065 | 0.132 | 0.097 | 0.019 | 0.062 | 0.136 | 0.143 | 0.018 | 0.110 | 0.180 | |
0.267 | 0.025 | 0.223 | 0.319 | 0.249 | 0.026 | 0.193 | 0.297 | 0.253 | 0.024 | 0.208 | 0.304 | |
0.727 | 0.024 | 0.675 | 0.769 | 0.743 | 0.026 | 0.696 | 0.798 | 0.718 | 0.022 | 0.675 | 0.760 | |
1.287 | 0.030 | 1.230 | 1.346 | - | - | - | - | - | - | - | - | |
4.471 | 0.315 | 3.915 | 5.156 | 3.921 | 0.238 | 3.505 | 4.435 | - | - | - | - |
ARFIMA = autoregressive moving average; GARCH = generalized autoregressive conditional heteroscedasticity; Est. = Estimated posterior mean; SE = standard error.
Deviance information criterion
skew- | Normal | |
---|---|---|
8886.260 | 9033.838 | 9508.023 |