Continuous Time Finance, Spring 2004 – Homework 1Distributed 1/28/04, due 2/4/04(1) In the Section 1 notes, we proved that ifVsolves the Black-Scholes PDE with final-valuef, thenV(S0,0) =e-rTE[f(ST)] whereSsolves the SDEdS=rS dt+σS dwwith initialvalueS(0) =S0. Let’s do something similar for a stochastic interest rate. Suppose the spotratertsolves a diffusion of the formdr=α dt+β dwwithr(0) =r0, whereα=α(r, t) andβ=β(r, t) are fixed functions ofrandt. Consider the functionU(r, t) defined by solvingUt+αUr+12β2Urr-rU= 0 with final valueU(r, T) = 1. Show thatU(r0,0) =Ee-T0r(s)ds.[Comment: if the SDE forris the risk-neutral process, thenU(r0,0) is the value of a zero-coupon bond that pays one dollar at timeT. Hint: show thatU(r(t), t) exp-t0r(s)dsis a martingale.](2) Consider a non-dividend-paying stock whose share price satisfiesdS=μS dt+σS dw,and assume for simplicity that the risk-free rateris constant. Consider a European optionwith maturityTand payofff(ST).
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Mathematical finance, Black-Scholes PDE, Martingale Representation Theorem, St /Bt