6 - General models

Summary

We generalise the Black-Scholes model in two directions: several assets and general coefficients in the equations describing the stock price dynamics. First we stick to the simple case where the equations for stock prices are linear. It seems pretty clear that all new features will be captured by the case of two assets. The extension from two to more assets is not likely to surprise us so we begin with a detailed discussion of some effects arising from one added dimension. Then we prepare the grounds for more general models and our objective here is to prove the Girsanov Theorem which enables us to find a risk-neutral probability. The celebrated Lévy Theorem, which characterises Wiener processes among continuous martingales, is used, together with a multi-dimensional version of the Itô formula, to prove this important result. Finally, we briefly consider some applications of these theorems to a multi-stock market.

Two assets

First we need a probability space (Ω, ℱ, P) on which two independent Wiener processes W₁, W₂ are defined. For its construction it is natural to consider two probability spaces, (Ω, ℱ_i, P_i), i = 1, 2, each accommodating a Wiener process, and take the product Ω = Ω₁ × Ω₂ as the sample space, the product σ-field ℱ = ℱ₁×ℱ₂ (the smallest σ-field containing all rectangles A₁×A₂, A_i ∈ ℱ_i) and the product probability P = P₁×P₂ (the extension of P(A₁ × A₁) = P₁(A₁)P₂(A₂) from rectangles to the whole ℱ).