The Mathematics of Financial Modelingand Investment Management

15-ArbPric-ContState/Time Page 468 Wednesday, February 4, 2004 1:08 PM

468 The Mathematics of Financial Modeling and Investment Management

WORKING WITH EQUIVALENT MARTINGALE MEASURES

The concepts established in the preceding sections of this chapter might

seem very complex, abstract, and scarcely useful. On the contrary, they

entail important simplifications in the computation of derivative prices.

We will see examples of these computations when we cover bond pric-

ing and credit derivatives in later chapters. Here we want to make a few

general comments on how these tools are used.

The key result of the arbitrage pricing theory is that, under the

equivalent martingale measure, all discounted price processes become

martingales and all price processes have the same drift. Therefore, all

calculations can be performed under the assumption that the change to

an equivalent martingale measure has been made. This environment

allows important simplifications. For example, as we have seen, the

option pricing problem becomes a problem of computing the present

value of simpler processes.

Obviously one has to go back to a real environment at the end of

the pricing exercise. This is essentially a calibration problem, as risk-

neutral probabilities have to be estimated from real probabilities.

Despite this complication, the equivalent martingale methodology has

proved to be an important tool in derivative pricing.

SUMMARY

■ A trading strategy is a vector-valued process that represents portfolio

weights at each moment.

■ Trading gains are defined as stochastic integrals.

■ A self-financing trading strategy is one whose value at every moment is

the initial value plus the trading gains at that moment.

■ An arbitrage is a self-financing trading strategy whose initial value is

either negative and the final value nonnegative or the initial value non-

negative and the final value positive.

■ The Black-Scholes option pricing formula can be established by repli-

cating self-financing trading strategies.

■ The Black-Scholes pricing argument is based on constructing a self-

financing trading strategy that replicates the option price in each state

and for each time.

■ Absence of arbitrage implies that a replicating self-financing trading

strategy must have the same price as the option.