A stochastic volatility model with random level shifts and its applications to S&P 500 and NASDAQ return indices

Abstract

This paper proposes a framework for the modelling, inference and forecasting of volatility in the presence of level shifts of unknown timing, magnitude and frequency. First, we consider a stochastic volatility model comprising both a level shift and a short‐memory component, with the former modelled as a compound binomial process and the latter as an AR(1). Next, we adopt a Bayesian approach for inference and develop algorithms to obtain posterior distributions of the parameters and the two latent components. Then, we apply the model to daily S&P 500 and NASDAQ returns over the period 1980.1–2010.12. The results show that although the occurrence of a level shift is rare, about once every 2 years, this component clearly contributes most to the variation in the volatility. The half‐life of a typical shock from the AR(1) component is short, on average between 9 and 15 days. Interestingly, isolating the level shift component from the overall volatility reveals a stronger relationship between volatility and business cycle movements. Although the paper focuses on daily index returns, the methods developed can potentially be used to study the low‐frequency variation in realized volatility or the volatility of other financial or macroeconomic variables.