Suppose we model a stock price process S using a stochastic volatility model dS(t) = oS(t)dt + Vy (t)S(t)dW! (t), where e R and > 0 are constants. and W1 and W2 are two Brownian motions with correlation e (-1,1). Given an option with payoff (S(T)) at time T. where is a given deterministic function, we intend to derive the Black-Scholes-Merton PDE for the price of the option. Let X be the replicating portfolio for (S(T), and be the corresponding trading strategy process. (a) Take the ansatz X(t) = f(t,S(t), Y(t)) and (t) = (t, S(t), Y(t), for some determin- istic function f (t, x, y) to be determined. Re-write the self-financing condition dx(t) = (t)dS(t) + r(X(t)-(t)S(t))dt, using It's formula and (6) (Hint: You need the two-dimensional Ito's formula on p.167 of textbook Vol. II. Also you may use the fact that dW1(t) (t) = dt) (b) Write your answer in (a) in integral form, and take expectation of the whole thing. You should end up with an equation of the form The Black-Scholes-Merton PDE can then be read out from the part (...) above. What is the PDE? Suppose we model a stock price process S using a stochastic volatility model dS(t) = oS(t)dt + Vy (t)S(t)dW! (t), where e R and > 0 are constants. and W1 and W2 are two Brownian motions with correlation e (-1,1). Given an option with payoff (S(T)) at time T. where is a given deterministic function, we intend to derive the Black-Scholes-Merton PDE for the price of the option. Let X be the replicating portfolio for (S(T), and be the corresponding trading strategy process. (a) Take the ansatz X(t) = f(t,S(t), Y(t)) and (t) = (t, S(t), Y(t), for some determin- istic function f (t, x, y) to be determined. Re-write the self-financing condition dx(t) = (t)dS(t) + r(X(t)-(t)S(t))dt, using It's formula and (6) (Hint: You need the two-dimensional Ito's formula on p.167 of textbook Vol. II. Also you may use the fact that dW1(t) (t) = dt) (b) Write your answer in (a) in integral form, and take expectation of the whole thing. You should end up with an equation of the form The Black-Scholes-Merton PDE can then be read out from the part (...) above. What is the PDE