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Unformatted text preview: MATH 238 WINTER 2009 PROBLEM SET 2 - SOLUTIONS Problem 1 Derive the put-call parity relation using the Black-Scholes equation and then give a no-arbitrage interpretation of it. Solution Let C ( x,t ) and P ( x,t ) denote the solutions of the Black-Scholes equation with terminal condition C ( x,T ) = ( x- K ) + and P ( x,T ) = ( K- x ) + , respectively. Consider the function f ( x,t ) = C ( x,t )- P ( x,t ) Since C ( x,t ) and P ( x,t ) both satisfy the Black-Scholes PDE, f ( x,t ) also satisfies the Black-Scholes PDE. The terminal condition is f ( x,T ) = C ( x,T )- P ( x,T ) = x- K . Now consider the function g ( x,t ) = x- Ke − r ( T − t ) Computing the partial derivatives, we have ∂g ∂t =- rKe − r ( T − t ) ∂g ∂x = 1 ∂ 2 g ∂x 2 = 0 Thus ∂g ∂t + 1 2 σ 2 ( x,t ) x 2 ∂ 2 g ∂x 2 + rx ∂g ∂x- rg = 0 , so g ( x,t ) also satisfies the Black-Scholes PDE. The terminal condition is g ( x,T ) = x- K . Thus f ( x,t ) and g ( x,t ) both satisfy the Black-Scholes PDE with the same terminal condition. So by the uniqueness of solutions of the Black-Scholes PDE, and setting x = S ( t ), the stock price at time t , we conclude that f ( S,t ) = g ( S,t ). This results in the put-call parity: C ( S,t )- P ( S,t ) = S- Ke − r ( T − t ) . A no-arbitrage interpretation of this relation may be given as follows: Consider two portfolios where the first portfolio is long one European call option and short one European put option with the same strike K and maturity T . The second portfolio consists of the underlying asset S ( t ) (long) and is short an amount of cash which equals Ke − r ( T − t ) at time t . Then both portfolios have the same payoff S ( T )- K at time T . Thus in the absence of arbitrage their values must be the same at all earlier times t ≤ T , for otherwise there would be an arbitrage opportunity from buying the portfolio with the smaller value at time t and short-selling the portfolio with the larger value, which would result in an immediate positive cash flow. Liquidation of the positions at expiration time T generates zero cash flow, hence there would have been a riskless profit contradicting the no-arbitrage assumption. Problem 2 (a) Use the formula for the explicit solution of the BS equation to price one call option today, where S (0) = $850, the strike time is three months, the strike price is K = $950, the volatility is 40% and the interest rate is 1%. (b) Plot this price as a function of the volatility from 15% to 60%. (c) Calculate the hedge ratio one month before the strike time for (a). (d) What is the price of a put with the same parameters as (a)? (e) What is the price of a put with parameters as in (a) except that K = $750?...
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This note was uploaded on 11/08/2009 for the course MATH 238 taught by Professor Papanicolaou during the Winter '08 term at Stanford.
- Winter '08