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Unformatted text preview: PDE for Finance, Spring 2011 Homework 3 Distributed 2/28/11, due 3/21/11 . 1) Consider the linear heat equation u t- u xx = 0 on the interval 0 < x < 1, with boundary condition u = 0 at x = 0 , 1 and initial condition u = 1. (a) Interpret u as the value of a suitable double-barrier option. (b) Express u ( t,x ) as a Fourier sine series, as explained in Section 3. (c) At time t = 1 / 100, how many terms of the series are required to give u ( t,x ) within one percent accuracy? 2) Consider the SDE dy = f ( y ) dt + g ( y ) dw . Let G ( x,y,t ) be the fundamental solution of the forward Kolmogorov PDE, i.e. the probability that a walker starting at x at time 0 is at y at time t . Show that if the infinitesimal generator is self-adjoint, i.e.- ( fu ) x + 1 2 ( g 2 u ) xx = fu x + 1 2 g 2 u xx , then the fundamental solution is symmetric, i.e. G ( x,y,t ) = G ( y,x,t ). 3) Consider the stochastic differential equation dy = f ( y,s ) ds + g ( y,s ) dw , and the associated backward and forward Kolmogorov equations u t + f ( x,t ) u x + 1 2 g 2 ( x,t ) u xx = 0 for t < T , with u = at t = T and s + ( f ( z,s ) ) z- 1 2 ( g 2 ( z,s ) ) zz = 0 for s > 0, with ( z ) = ( z ) at s = 0 ....
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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