This assignment involves using Itˆo formula to establish a 2-dimensional formula for a non-random function involving two securities following geometric Brownian motions.

Problem 6
Let S1(t) and S2(t) be the prices of 2 securities obeying geometric Brownian motions:
dSi(t) = (mi − qi) Si(t)dt + σi Si(t)dBi(t), i = 1, 2
where qi is the annual dividend yield rate, σi is the annual volatility, and mi − qi is the
expected continuously compounded rate at which the mean price of the ith security
increases. Suppose that B1(t) and B2(t) are are independent standard Brownian motions.
Let f (t, x, y) be a twice continuously differentiable (non-random) function. Establish a
2-dimensional Itˆo formula for f (t, S1, S2).
Hint: Start with Taylor’s formula for f (t, x, y) and use the multiplication rules:
dBi(t) dBj(t) = δijdt, dBi(t)dt = 0, (dt)
a = 0 for a > 1,
where
δij =



1 if i = j
0 if i 6= j.