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Unformatted text preview: 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) = e- rT E [ f ( S T )] where S solves the SDE dS = rS dt + S dw with initial value S (0) = S . Lets do something similar for a stochastic interest rate. Suppose the spot rate r t solves a diffusion of the form dr = dt + dw with r (0) = r , where = ( r, t ) and = ( r, t ) are fixed functions of r and t . Consider the function U ( r, t ) defined by solving U t + U r + 1 2 2 U rr- rU = 0 with final value U ( r, T ) = 1. Show that U ( r , 0) = E e- R T r ( s ) ds . [Comment: if the SDE for r is the risk-neutral process, then U ( r , 0) is the value of a zero- coupon bond that pays one dollar at time T . Hint: show that U ( r ( t ) , t ) exp- R t r ( s ) ds is a martingale.] (2) Consider a non-dividend-paying stock whose share price satisfies dS = S dt + S dw , and assume for simplicity that the risk-free rate...
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This note was uploaded on 01/02/2012 for the course FINANCE 347 taught by Professor Bayou during the Fall '11 term at NYU.
- Fall '11