(a) What is the Black-Scholes partial differential equation (PDE) for the price f(t, S) at time...

 

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(a) What is the Black-Scholes partial differential equation (PDE) for the price f(t, S₁) at time t of a European derivative security on a stock with price S,? What does each term in the equation mean? (b) The Black-Scholes PDE does not specify whether the derivative is a call option, a put option, or some other derivative. How do you incorporate the derivative payoff when using Black-Scholes PDE to price a derivative? (c) (Not graded, optional for those interested) There is an exotic European-style derivative whose price at time t equals Zekt S2 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: Jas² ast =2aSt. • If a does not depend on St, then second derivative of as? with respect to S+ is 2a: Pas as = 2a. If b does not depend on t, then, derivative of bet with respect to t is bkekt. Obekt Ət = bkekt (a) What is the Black-Scholes partial differential equation (PDE) for the price f(t, S₁) at time t of a European derivative security on a stock with price S,? What does each term in the equation mean? (b) The Black-Scholes PDE does not specify whether the derivative is a call option, a put option, or some other derivative. How do you incorporate the derivative payoff when using Black-Scholes PDE to price a derivative? (c) (Not graded, optional for those interested) There is an exotic European-style derivative whose price at time t equals Zekt S2 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: Jas² ast =2aSt. • If a does not depend on St, then second derivative of as? with respect to S+ is 2a: Pas as = 2a. If b does not depend on t, then, derivative of bet with respect to t is bkekt. Obekt Ət = bkekt

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