Summary

Having applied our basic Black–Scholes model to the pricing of some exotic options, we now turn to more general market models.

In §7.1 we replace the (constant) parameters that characterised our basic Black–Scholes model by previsible processes. Under appropriate boundedness assumptions, we then repeat our analysis of Chapter 5 to obtain the fair price of an option as the discounted expected value of the claim under a martingale measure. In general this expectation must be evaluated numerically. We also make the connection with a generalised Black–Scholes equation via the Feynman–Kac Stochastic Representation Theorem.

Our models so far have assumed that the market consists of a single stock and a riskless cash bond. More complex equity products can depend on the behaviour of several separate securities and, in general, the prices of these securities will not evolve independently. In §7.2 we extend some of the fundamental results of Chapter 4 to allow us to manipulate systems of stochastic differential equations driven by correlated Brownian motions. For markets consisting of many assets we have much more freedom in our choice of ‘reference asset’ or numeraire and so we revisit this issue before illustrating the application of the ‘multifactor’ theory by pricing a ‘quanto’ product.

We still have no satisfactory justification for the geometric Brownian motion model. Indeed, there is considerable evidence that it does not capture all features of stock price evolution.