12
Financial Applications
In this chapter we discuss several financial applications of GARCH models. In connecting these models with those frequently used in mathematical finance, one is faced with the problem that the latter are generally written in continuous time. We start by studying the relation between GARCH and continuous-time processes. We present sufficient conditions for a sequence of stochastic difference equations to converge in distribution to a stochastic differential equation as the length of the discrete time intervals between observations goes to zero. We then apply these results to GARCH(1, 1)-type models. The second part of this chapter is devoted to the pricing of derivatives. We introduce the notion of the stochastic discount factor and show how it can be used in the GARCH framework. The final part of the chapter is devoted to risk measurement.
12.1 Relation between GARCH and Continuous-Time Models
Continuous-time models are central to mathematical finance. Most theoretical results on derivative pricing rely on continuous-time processes, obtained as solutions of diffusion equations. However, discrete-time models are the most widely used in applications. The literature on discrete-time models and that on continuous-time models developed independently, but it is possible to establish connections between the two approaches.
12.1.1 Some Properties of Stochastic Differential Equations
This first section reviews basic material from diffusion processes, which will ...
