The Binomial Option Pricing Model

The simplest model for pricing d erivative securities is the binomial model. It generalizes the o n e period \up-down" model of Chapte r 1 t o a m ulti-period setting, assuming t hat t he price of the u nderlying asset follows a random walk. In the binomial model, there are N trading periods and N+1 trading d ates, t 0 t 1 ::: t N when it is possible to i n vest in a risky security with p r i c e S n n = 0 1 ::: N, a n d a riskless bond with o n e-period yield R. T h e price varies according t o t he rule S n+1 = S n H n+1 0 n N ; 1 (1) where H n+1 is a random variable such t hat H n+1 = 8 > < > : U with probability p D with probability q (2) with p + q = 1. The s i t uation can be visualized in terms of a binomial tree, s h own in Figure 1. Each node o f t he t r e e i s l a beled by a p a i r o f i n tegers (n n j), n = 0 : : : N , a n d j = 0 ::: n. We u s e t he c o n vention that n o d e (n n j) leads to n o d es (n + 1 j) a n d (n + 1 j+ 1) a t t he next trading d ate with t he \up" move corresponding t o (n+ 1 j + 1)a n d t he \down" move to (n + 1 j). The price of the u nderlying asset at t he n o d e (n n j) i s t herefore