Problem 3.1. (a) What is the Black-Scholes Partial Differential Equation for the price f(t, St) at time t of a European derivative security on a stock with price St? Specify the meaning of the terms or symbols in the equation. (6) Note that the Black-Scholes Partial Differential Equation does not specify whether the derivative is a call option, a put option, or some other derivative. How do you incor- porate the derivative payoff when using Black-Scholes PDE to price a derivative? (c) (Optional, no points) There is an exotic European-style derivative whose price at time t equals Zekt s if the stock price at time t is St, with Z and k two constants. The stock price follows a geometric Brownian motion with volatility o equal to 40% per annum. The risk-free rate is 5% per annum continuously compounded. Determine k. The following calculus facts will be useful: If a does not depend on St, then, derivative of as with respect to St is 2aSt: das= 2aS. ast If a does not depend on St, then second derivative of as with respect to St is 2a: a as a s2 = 2a. If b does not depend on t, then, derivative of bekt with respect to t is bkekt: abekt at = bkekt Problem 3.1. (a) What is the Black-Scholes Partial Differential Equation for the price f(t, St) at time t of a European derivative security on a stock with price St? Specify the meaning of the terms or symbols in the equation. (6) Note that the Black-Scholes Partial Differential Equation does not specify whether the derivative is a call option, a put option, or some other derivative. How do you incor- porate the derivative payoff when using Black-Scholes PDE to price a derivative? (c) (Optional, no points) There is an exotic European-style derivative whose price at time t equals Zekt s if the stock price at time t is St, with Z and k two constants. The stock price follows a geometric Brownian motion with volatility o equal to 40% per annum. The risk-free rate is 5% per annum continuously compounded. Determine k. The following calculus facts will be useful: If a does not depend on St, then, derivative of as with respect to St is 2aSt: das= 2aS. ast If a does not depend on St, then second derivative of as with respect to St is 2a: a as a s2 = 2a. If b does not depend on t, then, derivative of bekt with respect to t is bkekt: abekt at = bkekt