INTRODUCTION

Financial models rarely rely on functions that depend on a single variable. Generally, functions which themselves are functions of two or more variables are used. It?'s Lemma, which is regarded as the fundamental instrument in stochastic calculus, allows functions of this type to be differentiated. Stochastic calculus is the type of calculus that operates on stochastic processes. A stochastic process is a random variable that evolves over time.

We begin this chapter with a discussion on the stochastic process. Our starting point is an extremely simple discrete-time process and then we give an introduction to the Brownian motion (or Weiner process), which is a fundamental instrument to model the building of stock prices. These issues are discussed in Sections 10.2 and 10.3, respectively. We then indicate in Section 10.4 how the Weiner process can be generalized to a broader class of continuous-time processes, known as the Itō process. In Section 10.4 we also show how Itō's Lemma can be employed to differentiate functions of stochastic processes. A derivation of Itō's Lemma is presented in the Appendix and two illustrations of the lemma are provided in Section 10.5.