Recent Developments in Seasonal Volatility Models

Abstract

It is well-known that many financial time series such as stock returns exhibit leptokurtosis and time-varying volatility (Bollerslev, 1986; Engle, 1982; Nicholls & Quinn, 1982). The generalized autoregressive conditional heteroscedasticity (GARCH) and the random coefficient autoregressive (RCA) models have been extensively used to capture the time-varying behavior of the volatility. Studies using GARCH models commonly assume that the time series is conditionally normally distributed; however, the kurtosis implied by the normal GARCH tends to be lower than the sample kurtosis observed in many time series (Bollerslev, 1986). Thavaneswaran et al. (2005a) use an ARMA representation to derive the kurtosis of various classes of GARCHmodels such as power GARCH, non-Gaussian GARCH, non-stationary and random coefficient GARCH. Recently, Thavaneswaran et al. (2009) have extended the results to stationary RCA processes with GARCH errors and Paseka et al. (2010) further extended the results to RCA processes with stochastic volatility (SV) errors. Seasonal behavior is commonly observed in financial time series, as well as in currency and commodity markets. The opening and closure of the markets, time-of-the-day and day-of-the-week effects, weekends and vacation periods cause changes in the trading volume that translates into regular changes in price variability. Financial, currency, and commodity data also respond to new information entering into the market, which usually follow seasonal patterns (Frank & Garcia, 2009). Recently there has been growing interest in using seasonal volatility models, for example Bollerslev (1996), Baillie & Bollerslev (1990) and Franses & Paap (2000). Doshi et al. (2011) discuss the kurtosis and volatility forecasts for seasonal GARCH models. Ghysels & Osborn (2001) review studies performed on seasonal volatility behavior in several markets. Most of the studies use GARCH models with dummy variables in the volatility equation, and a few of them have been extended to a more flexible form such as the periodic GARCH. However, even though much research has been performed on volatility models applied to market data such as stock returns, more general specifications accounting for seasonal volatility have been little explored. First, we derive the kurtosis of a simple time seriesmodel with seasonal behavior in the mean. Then we introduce various classes of seasonal volatility models and study the moments, forecast error variance, and discuss applications in option pricing. We extend the results for non-seasonal volatility models to seasonal volatility models. For the seasonal GARCH Recent Developments in Seasonal Volatility Models