The Mathematics of Financial Modelingand Investment Management

14-Arbitrage Page 425 Wednesday, February 4, 2004 1:08 PM

Arbitrage Pricing: Finite-State Models 425

EXHIBIT 14.3 Binomial Model: The Figure Illustrates a Binomial Tree with Three

Dates, Three Final Prices, and Four States: uu,ud,du,dd

Real probabilities of states are typically constructed from the proba-

bilities of a movement up or down. Call p the probability of a move-

ment up; 1 – p is thus the probability of a movement down. Suppose

that the state s, which is identified by a price path, has k movements up

and T – k movements down. The probability of the state s is

p Tk–

s = p

k( 1 – p)

Consider the final date T. Each of the possible final prices ST = ukdT – kS 0 ,

k = 0,1,...,T can be obtained through

T T!

= -------------------------

 k k!(Tk– )!

paths with k movements up and T – k movements down. The probabil-

ity distribution of final prices is therefore a binomial distribution:

( –

T Tk–

PST = ukdTkS 0 )= pk( 1 – p)

k

Following the same reasoning, one can demonstrate that at any interme-

diate date the probability distribution of prices is a binomial distribu-

tion as follows: