In this chapter we build some more sophisticated market models that track the evolution of stock prices over a succession of time periods. Over each individual time period, the market follows our simple binary model of Chapter 1. The possible trajectories of the stock prices are then encoded in a tree. A simple corollary of our work of Chapter 1 will allow us to price claims by taking expectation with respect to certain probabilities on the tree under which the stock price process is a discrete parameter martingale.
Definitions and basic properties of discrete parameter martingales are presented and illustrated in §2.3, and we see for the first time how martingale methods can be employed as an elegant computational tool. Then, §2.4 presents some important martingale theorems. In §2.5 we pave the way for the Black–Scholes analysis of Chapter 5 by showing how to construct, in the martingale framework, the portfolio that replicates a claim. In §2.6 we preview the Black–Scholes formula with a heuristic passage to the limit.
The multiperiod binary model
Our single period binary model is, of course, inadequate as a model of the evolution of an asset price. In particular, we have allowed ourselves to observe the market at just two times, zero and T. Moreover, at time T, we have supposed the stock price to take one of just two possible values.