Realized Stochastic Volatility Model with Skew-t Distributions for Improved Volatility and Quantile Forecasting

TL;DR

The paper develops Realized Stochastic Volatility (RSV) models that fuse realized volatility with flexible skewed-$t$ return innovations to improve volatility and tail-risk forecasting. It introduces three skew-$t$ variants (GH, Azzalini, Fernández–Steel) and two RSV variants (RSV-T and RSV-GH-ST, RSV-AZ-ST, RSV-FS-ST) within a Bayesian MCMC framework, and benchmarks them against SV, EGARCH, and REGARCH using DJIA and Nikkei 225 data. One-day-ahead forecasts of volatility, VaR, and ES are evaluated with robust loss functions (QLIKE, FZ0) and predictive-ability tests (GW) plus model confidence sets. Across multiple realized-measure proxies and out-of-sample periods, RSV models consistently outperform SV and EGARCH, with RSV-AZ-ST often delivering the strongest overall performance and skewed-t innovations improving tail forecasts. The findings highlight the practical value of incorporating realized volatility and distributional flexibility for reliable volatility and tail-risk forecasting in financial markets.

Abstract

Accurate forecasting of volatility and return quantiles is essential for evaluating financial tail risks such as value-at-risk and expected shortfall. This study proposes an extension of the traditional stochastic volatility model, termed the realized stochastic volatility model, that incorporates realized volatility as an efficient proxy for latent volatility. To better capture the stylized features of financial return distributions, particularly skewness and heavy tails, we introduce three variants of skewed t-distributions, two of which incorporate skew-normal components to flexibly model asymmetry. The models are estimated using a Bayesian Markov chain Monte Carlo approach and applied to daily returns and realized volatilities from major U.S. and Japanese stock indices. Empirical results demonstrate that incorporating both realized volatility and flexible return distributions substantially improves the accuracy of volatility and tail risk forecasts.

Figures (18)

• Figure 1: Density of the standardized GH skew-$t$ distribution in equation \ref{['eqn:rsv-gh-st-eps']}. (i) Varying $\beta = 0, -1, -2$ with fixed $\nu = 10$. (ii) Varying $\nu = 24, 16, 8$ with fixed $\beta = -1$.
• Figure 2: Density of the standardized Azzalini skew-$t$ distribution in equation \ref{['eqn:rsv-az-st-eps']}. (i) $\delta = 0, -0.6, -0.9$ with $\nu = 10$ fixed. (ii) $\nu = 15, 10, 5$ with $\delta = -0.9$ fixed.
• Figure 3: Density of the standardized FS skew-$t$ distribution in equation \ref{['eqn:dens-fsst']}. (i) $\gamma = 1, 0.8, 0.6$ with $\nu = 10$. (ii) $\nu = 20, 15, 10$ with $\gamma = 0.5$.
• Figure 4: Time series plots (top) and histograms (bottom) of daily returns (in percentage points) for the DJIA and N225.
• Figure 5: Time series plots of 5-minute RVs (top), their logarithms (middle), and corresponding histograms (bottom) for the DJIA and N225.