10 Problem 5. The Black-Scholes differential equation and options. (30 points, 6 points each) Suppose the time t price S of a non-dividend paying stock follows geometric Brownian motion, i.e., dS = S,dt + o S,dz where u is the expected return of the stock, o is the volatility of the stock, and dz = vdt with ~ N(0,1). As we saw in class, under some additional assumptions (which you may assume are satisfied) any derivative written on this stock must satisfy the following differential equation: af af +rSt t = rf aS? 2 where t is time, f is the price of the derivative at time t, and r is the risk-free rate. This is the Black-Scholes differential equation. In particular, a European style call option written on this stock must satisfy the Black- Scholes differential equation. The formula for the time 0