The Mathematics of Financial Modelingand Investment Management

15-ArbPric-ContState/Time Page 452 Wednesday, February 4, 2004 1:08 PM

452 The Mathematics of Financial Modeling and Investment Management

stochastic process as the stochastic process of the trading strategy. As

the two equations describe the same process, they must coincide. Equat-

ing the coefficients in the deterministic and stochastic terms, we can

determine the trading strategy and write a deterministic partial differen-

tial equation (PDE) that the pricing function of the option must satisfy.

The latter PDE together with the boundary conditions provided by the

known value of the option at the expiry date uniquely determine the

option pricing function.

Note that the above is neither a demonstration that there is an

option pricing function, nor a demonstration that there is a replicating

trading strategy. However, if both a pricing function and a replicating

trading strategy exist, the above process allows one to determine both

by solving a partial differential equation. After determining a solution

to the PDE, one can verify if it provides a pricing function and if it

allows the creation of a self-financing trading strategy. Ultimately, the

justification of the existence of an option’s pricing function and of a rep-

licating self-financing trading strategy resides in the possibility of actu-

ally determining both. Absence of arbitrage assures that this solution is

unique.

Generalizing the Pricing of European Options

We can now generalize the above pricing methodology to a generic

European option and to more general price processes for the bond and

for the underlying stock. In the most general case, the process underly-

ing a derivative need not be a stock price process. However, we suppose

that the underlying is a stock price process so that replicating portfolios

can be formed. We generalize in three ways:

■ The option’s payoff is an arbitrary finite-variance random variable.

■ The stock price process is an Itô process.

■ The short-rate process is stochastic.

Following the definition given in the finite-state setting, we define a

European option on some underlying process St as an asset whose pay-

off at time T is given by the random variable YT = g(ST) where g(x), x ∈

R is a continuous real-valued function. In other words, a European

option is defined as a security whose payoff is determined at a given

expiry date T as a function of some underlying random variable. The

option has a zero payoff at every other date t ∈ [0,T]. This definition

clearly distinguishes European options from American options which

yield payoffs at random stopping times.