Geometric Brownian Motion question

February 18, 2023February 18, 2023 Discipline: Finance
Learning Goal: I’m working on a finance question and need guidance to help me learn.

Problem 6Let 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, 2where qi is the annual dividend yield rate, σi is the annual volatility, and mi − qi is theexpected continuously compounded rate at which the mean price of the ith securityincreases. 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 a2-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,whereδij =1 if i = j0 if i 6= j.

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