Quantitative Finance

10Stochastic Volatility Models

10.1 Introduction

In Chapter 3 of this book, we discussed the implied volatility and we observed that it is hardly constant for any given stock. This is a violation of the assumption of constant volatility in the model. Further evidence may be provided by calculating histograms of the returns and testing for normality. Most of such tests fail especially when the time interval for calculating returns is less than a day. Thus, the entire class of High Frequency models needs to be more complex than a geometric Brownian motion, hence the need for stochastic volatility (SV) model.

10.2 Stochastic Volatility

We begin this section with the following definition.

Definition 10.2.1

We call any model of the following type:

(10.1)

a continuous time SV model. The functions and are twice differentiable in all variables, and with continuous second partial derivatives, is a standard Brownian motion and is a stochastic process.

The stochastic factor is typically ...