Stochastic Volatility Models: A Survey with Applications to Option Pricing and Value at Risk

This chapter presents an introduction to the current literature on stochastic volatility models. For these models the volatility depends on some unobserved components or a latent structure. Given the time-varying volatility exhibited by most financial data, in the last two decades there has been a growing interest in time series models of changing variance and the literature on stochastic volatility models has expanded greatly. Clearly, this chapter cannot be exhaustive, however we discuss some of the most important ideas, focusing on the simplest forms of the techniques and models used in the literature. The chapter is organised as follows. Section 8.1 considers some motivations for stochastic volatility models: empirical stylised facts, pricing of contingent assets and risk evaluation. While Section 8.2 presents models of changing volatility, Section 8.3 focuses on stochastic volatility models and distinguishes between models with continuous and discrete volatility, the latter depending on a hidden Markov chain. Section 8.4 is devoted to the estimation problem which is still an open question, then a wide range of possibility is given. Sections 8.5 and 8.6 introduce some extensions and multivariate models. Finally, in Section 8.7 an estimation program is presented and some possible applications to option pricing and risk evaluation are discussed. Readers interested in the practical utilisation of stochastic volatility models and in the applications can skip Section 8.4.3 without hindering comprehension.