A share has an expected return of 10% per annum (with continuous compounding) and a volatility of 24% per annum. Changes in the share price satisfy

dS = S dt + S dX.

Set the current share price is $50, over a year, using a time interval of one week.

a) What is the distribution of the price increase for the share movement described?

b) Use Its lemma to find the stochastic differential equation satisfied by

• fS=log(S)
• g(S)= Sⁿ
• h(S,t)= Sⁿe^mt

c) Write the integral form of f(S) and find the value of S(t).

d) If there are two shares follow geometric Brownian motions, i.e.

dS₁ = ₁S₁ dt + ₁S₁ dX₁

dS₂ = ₂S₂ dt + ₂S₂ dX₂

The share price changes are correlated with the correlation coefficient . Find the stochastic differential equation satisfied by a function f(S₁, S₂).