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hw1 (1)

hw1 (1) - Continuous Time Finance Spring 2004 Homework 1...

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Continuous Time Finance, Spring 2004 – Homework 1 Distributed 1/28/04, due 2/4/04 (1) In the Section 1 notes, we proved that if V solves the Black-Scholes PDE with final-value f , then V ( S 0 , 0) = e - rT E [ f ( S T )] where S solves the SDE dS = rS dt + σS dw with initial value S (0) = S 0 . Let’s do something similar for a stochastic interest rate. Suppose the spot rate r t solves a diffusion of the form dr = α dt + β dw with r (0) = r 0 , where α = α ( r, t ) and β = β ( r, t ) are fixed functions of r and t . Consider the function U ( r, t ) defined by solving U t + αU r + 1 2 β 2 U rr - rU = 0 with final value U ( r, T ) = 1. Show that U ( r 0 , 0) = E e - T 0 r ( s ) ds . [Comment: if the SDE for r is the risk-neutral process, then U ( r 0 , 0) is the value of a zero- coupon bond that pays one dollar at time T . Hint: show that U ( r ( t ) , t ) exp - t 0 r ( s ) ds is a martingale.] (2) Consider a non-dividend-paying stock whose share price satisfies dS = μS dt + σS dw , and assume for simplicity that the risk-free rate r is constant. Consider a European option with maturity T and payoff f ( S T ).
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