Limit results for discretely observed stochastic volatility models with leverage e¤ect

Abstract

In this note we generalize the limit results in [Genon-Catalot, Jeantheau, Laredo, 2000, Bernoulli ] for simple stochastic volatility models to the case where a non zero correlation is allowed between the Brownian mo- tion driving the main di¤usion process and the Brownian motion driving the dymaics of the instantaneous variance. We also extend the results to the case where the main di¤usion admits a non zero drift which is linear in the variance process. The main motivation for such an extension is the application of these limit results in order to perform statistical infer- ence in some of the stochastic volatility models introduced in the ?nancial mathematics literature. In this framework it is of relevance the so called "leverage e¤ect" between the stock log-price and its volatility, which is indeed explained by a negative correlation between the Brownian motions driving the log-price process and its instantaneous variance. Moreover a linear term in the variance appears in the drift of the log-price diffusion.