A non‐linear filtering approach to stochastic volatility models with an application to daily stock returns

Abstract

This paper develops a new method for the analysis of stochastic volatility (SV) models. Since volatility is a latent variable in SV models, it is dicult to evaluate the exact likelihood. In this paper, a non-linear ®lter which yields the exact likelihood of SV models is employed. Solving a series of integrals in this ®lter by piecewise linear approximations with randomly chosen nodes produces the likelihood, which is maximized to obtain estimates of the SV parameters. A smoothing algorithm for volatility estimation is also constructed. Monte Carlo experiments show that the method performs well with respect to both parameter estimates and volatility estimates. We illustrate the method by analysing daily stock returns on the Tokyo Stock Exchange. Since the method can be applied to more general models, the SV model is extended so that several characteristics of daily stock returns are allowed, and this more general model is also estimated. have investigated option pricing when volatility changes following a continuous stochastic process such as the Ornstein±Uhlenbeck process, and SV models are discrete approximations to such models.The maximum likelihood method is not easy to implement in SV models, but several alternative methods are now available (Ghysels, et al., 1996, andShephard, 1996 provide good reviews of the

Empirical Studies on Volatility in International Stock Markets

“…For some of the series d is however very close to being statistically significant. A similar SVM model was estimated by Fridman and Harris (1998), who studied daily returns on the Standard & Poor's index over the period 1980, and Watanabe (1999, who examined the daily Topix series over the eight year period 1990-1997. Both studies reported significant positive values for the contemporaneous relationship 10 .…”

“…7 Jacquier, Polson and Rossi (2001) estimate corr(εt, ηt) and observe a convincing negative relationship between contemporaneous unexpected stock index returns and unexpected volatility. Harvey and Shephard (1996), on the other hand, estimate corr(εt, η t+1 ) and Watanabe (1999) develops an SV model which includes the lagged shock to the return process as an explanatory variable in the variance equation allowing for an asymmetric response. Both studies report negative coefficients for the relation between current unexpected returns and future volatility.…”

“…Danielsson (1994) introduced the use of importance sampling techniques for the estimation of SV models. Other methods of estimating SV models are discussed in Kim, Shephard and Chib (1998), Fridman and Harris (1998) and Watanabe (1999). Bayesian approaches to estimating SV models have been explored by Jacquier, Polson and Rossi (1994) and Shephard and Pitt (1997).…”

Asymmetric International Transmission in the Conditional Mean and Volatility to the Japanese Market From the Us: Egarch Versus Sv Models

“…We apply the non-linear maximum likelihood method (MLE) proposed byWatanabe (1999) for estimating the SV models throughout this paper. This method is applicable for many types of SV models.…”

“…This method is applicable for many types of SV models. The number of grids for the numerical integration is set to 50 in this paper as recommended byWatanabe (1999).…”