A Class of Nonlinear Stochastic Volatility Models

This paper proposes a class of nonlinear stochastic volatility models. The proposed class encompasses many parametric stochastic volatility models that have appeared in the literature, including the well known lognormal stochastic volatility model. The new class is based on the Box-Cox transformation and offers an alternative to the one introduced in Andersen (1994). An advantage of our proposed class is the ease with which different specifications on stochastic volatility can be easily tested. Furthermore, as a byproduct of this general way of modeling stochastic volatility, one obtains the functional form of transformation which induces marginal normality of volatility. The efficient method of moments approach is used to estimate model parameters. Empirical applications are performed using an individual stock price series and a stock index series. Empirical results reveal that the lognormal stochastic volatility model is rejected for the index returns but not for the stock returns. Moreover, our results suggest that all the stochastic volatility models previously used in the literature are rejected for the index returns. As a consequence, the stock volatility can be well described by the lognormal distribution as its marginal distribution, consistent with the result found in a recent literature (cf Andersen et al (2001a)). However, the index volatility does not follow the lognormal distribution as its marginal distribution. JEL classification: C14, C22, G12