Mathematical Modeling in Finance with Stochastic Processes

82 CHAPTER 2. BINOMIAL OPTION PRICING MODELS

Figure 2.2: A binomial tree

We model a limited market where a trader can buy or short-sell a risky
security (for instance a stock) and lend or borrow money at a riskless rater.
For simplicity we assumeris constant over [0,T]. This assumption of con-
stantris not necessary, takingrto berion [ti,ti− 1 ] only makes calculations
messier.
Sndenotes the price of the risky security at timetnforn= 0, 1 ,...N.
This price changes according to the rule

Sn+1=SnHn+1, 0 ≤n≤N− 1

whereHn+1is a Bernoulli (two-valued) random variable such that

Hn+1=

{

U, with probabilityp

D, with probabilityq= 1−p.

Again for simplicity we assumeU and D are constant over [0,T]. This
assumption of constantris not necessary, for example, takingUto beUifor
i= 0, 1 ,...,Nonly makes calculations messier. A binomial tree is a way to
visualize the multiperiod binomial model, as in the figure:
A pair of integers (n,j), withn = 0,...N andj = 0,...,nidentifies
each node in the tree. We use the convention that node (n,j) leads to nodes