The Mathematics of Financial Modelingand Investment Management

14-Arbitrage Page 424 Wednesday, February 4, 2004 1:08 PM

424 The Mathematics of Financial Modeling and Investment Management

because it gives a simple and mathematically tractable model of stock

price behavior that tends, in the limit of a zero time step, to a Brownian

motion. We introduce a market populated by one risk-free asset and by

one or more risky assets whose price(s) follow(s) a binomial or trino-

mial model. In the next section we will see how to compute the price of

derivative instruments in this market.

In the binomial model of stock prices, we assume that at each time

step the stock price will assume one of two possible values. This is a

restriction of the general multiperiod finite-state model described in the

previous sections and in Chapter 6 on probability. The latter is, as we

have seen in the previous section, a hierarchical structure of partitions

of the set of states. The number of sets in any partition is arbitrary, pro-

vided that partitions grow more refined with time.

The binomial model assumes that there are two positive numbers, d

and u, such that 0 < d < u and such that at each time step the price St of

the risky asset changes to dSt or to uSt. In general one assumes that 0 < d

< 1 < u so that d represents a price decrease (a movement down) while u

represents a price increase (a movement up). It is often required that

d =     1

---

u

In this case an equal number of movements up and down leave prices

unchanged. The binomial model is a Markov model as the distribution

of St clearly depends only on the value of St – 1.

A binomial model can be graphically represented by a tree. For

example, Exhibit 14.3 shows a binomial model for three periods. A

binomial model over T time steps, from 0 to T, produces a total of 2T

paths. Therefore, the corresponding space of states has 2T states. How-

ever, the number of different final prices ST = ukdT – kS 0 , k = 0,1,...,T is

determined solely by the number of u and d in each path and increases

by 1 at each time step; there are as many final prices as dates. For exam-

ple, the model in Exhibit 14.3 shows three final prices and four states.

Note that there is a simple relationship between the numbers d and

u and returns. In fact, we can write,

St + 1 – St uSt – St

Rt(up) = ---------------------- = ------------------- = u – 1

St St

Rt(down) = d – 1