Stochastic Calculus

1. Consider a basket of N assets each following the Geometric Brownian Motion, so the Stochastic
Differential Equation for each asset. is given by
(153‘ = Sillidt + SiO’idXi fOI‘ 1 S1 S [V
The price changes are correlated as measured by the linear correlation coefficients pij. Invoke the
multi-dimensional Ito Lemma to write down the SDE for F (Sl, SQ, . . . , S N) in the most compact
form possible (with clear drift. and diffusion terms). Apply dXide -> pijdt.
2. Construct an SDE for the process Y(t) = e”X(t)‘%“2t and show that the process is, in fact, an
Exponential Martingale of the form dY(t) = Z (1‘) g(t) dX (t). Identify the terms g(t) and Z (t)
A diffusion process Y(t) is a martingale if its SDE has no drift term. The SDE can be constructed
by evaluating partial derivatives of a function F(t, X) = Y(t) and substituting as follows:
oF 1 82F 8F
dF= – – dt -dXt.
(8t+2oX2> +oX