WHD Problem 2.5

There are n assets S_i,...,S_n satisfying the following stochastic differential equations dS_i = σ_i S_i dX_i + μ_i S_i dt, for i = 1, ..., n

The Wiener processes dX _i satisfy E[dX_i] = 0 and E[dX_i²] = dt, as usual, but the asset price changes are correlated with E[dX_idX_j] = ρ_ijdt where -1 ≤ ρ_ij ≤ 1.

Derive Ito's Lemma for a function f(S_i,...,S_n) of the n assets.

Actually, I already know most or all of the content of The Mathematics of Financial Derivatives; I am reading it as a review in preparation for more advanced materials I am hoping to get to in the near future.

This exercise is pretty trivial, mostly a matter of keeping track of all the variables.

The Taylor series expansion is
df = (∂f/∂t)dt + Σ(∂f/∂S_i)dS_i + ½Σ(∂²f/∂S_i∂S_j)dS_idS_j

dS_idS_j = (σ_i S_i dX_i + μ_i S_i dt)(σ_j Sj dXj + μj Sj dt) = σ_i σ_j S_i Sj dX_i dXj = ρ_ij σ_i σ_j S_i Sj dt
dropping all higher order terms and then substituting in the correlation assumption.

So finally we obtain
df = [ (∂f/∂t) + Σ μ_i S_i (∂f/∂S_i) + ½ Σ ρ_ij σ_i σ_j S_i Sj (∂²f/∂S_i∂S_j) ] dt + Σ σ_i S_i (∂f/∂S_i) dX_i