The Mathematics of Financial Modelingand Investment Management

14-Arbitrage Page 427 Wednesday, February 4, 2004 1:08 PM

Arbitrage Pricing: Finite-State Models 427

u – 1 – r

1 – q = ---------------------

ud–

The binomial model is complete and arbitrage free.

Suppose that there is more than one risky asset, for example two

risky assets, in addition to the risk-free asset. At each time step each

risky asset can go either up or down. Therefore there are four possible

joint movements at each time step: uu,ud,du,dd that we identify with

the states 1,2,3,4. Four probabilities must be determined at each time

step; four equations are therefore needed. Two equations are provided

by the martingale conditions:

1 1 1 1

S^1 q^1 uSt + q^2 uSt + q^3 uSt + q^4 uSt

t = ----------------------------------------------------------------------------------

1 + r

2 2 2 2

S^2 q^1 uSt + q^3 uSt + q^2 uSt + q^4 uSt

t = ----------------------------------------------------------------------------------

1 + r

A third equation is provided by the fact that probabilities must sum to

1. The fourth condition, however, is missing. The model is incomplete.
   The problem of approximating price processes when there are two
   stocks and one bond and where the stock prices follow two correlated
   lognormal processes has long been of interest to financial economists.
   As seen above, with two stocks and one bond available for trading, mar-
   kets cannot be completed by dynamic trading. This is not the case in the
   continuous-time model, in which markets can be completed by continu-
   ous trading in the two stocks and the bond. Different solutions to this
   problem have been proposed in the literature.^1

VALUATION OF EUROPEAN SIMPLE DERIVATIVES

Consider a market formed by a risky asset (a stock) that follows the

binomial model plus a risk-free asset. As we have seen in the previous

section, this market is complete and its risk-neutral probabilities are

(^1) Hua He, “Convergence from Discrete- to Continuous-Time Contingent Claims
Prices,” Review of Financial Studies 3, no. 4 (1990), pp. 523–546.