In this chapter we explore the simplest option pricing models. We assume that there is a single trading period, from the present date (time 0) to a fixed future time T. We examine mathematical models for the behaviour of the prices of one or more underlying stocks, beginning with the simplest case of a single stock whose price at time T takes one of just two possible values. This analysis is extended to a model with two stocks, models with three possible prices for each stock, and finally to a general pricing model with d > 1 stocks, each with m possible outcomes for its price at T.

The principal economic requirement, arising from the assumption that the financial markets being modelled are efficient is that, provided all market participants share the same information about the random evolution of the stock price, an investor in this market should not be able to make a profit without incurring some level of risk. This assumption is cast in mathematical terms and underlies the methods we develop to establish how to determine the value of financial instruments whose price depends only on the price of the underlying stock at time T: the so-called European derivative securities, whose study is the principal topic of this and the next two chapters.