hw4 (1) - Continuous Time Finance, Spring 2004 Homework 4...

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Continuous Time Finance, Spring 2004 – Homework 4 Posted 3/19/04, due 3/31/04 (1) To solve Problem 5 of HW3 you needed to know that if dr = ( θ - ar ) dt + σ dw then the function v ( x, t ) defined by v ( x, t ) = E r ( t )= x ± e - R T t r ( s ) ds f ( r ( T )) ² (1) solves v t + ( θ - ax ) v x + 1 2 σ 2 v xx - xv = 0 for t < T , with final-time condition v ( x, T ) = f ( x ). This is a special case of the Feynman- Kac formula. Give a self-contained proof, using the method of HW1, problem 1. (You should assume that the PDE has a unique solution with this final-time condition; your task is to prove that the solution of the PDE satisfies (1).) (2) The Section 6 notes explain how a trinomial tree can be used to approximate the random walk dx = σ dw , and how working backward in this tree amounts to a standard finite-difference scheme for solving the backward Kolmogorov equation u t + 1 2 σ 2 u xx = 0. Let’s try to do something similar for the “geometric Brownian motion with drift” process dy = μy dt + σy dw , whose backward Kolmogorov equation is v t + μyv y + 1 2 σ 2 y 2 v yy = 0. Assume the time interval is Δ t , and at time t = n Δ t the tree has nodes at - n Δ y, . . . , n Δ y . The process on the tree goes from ( y, t ) to ( y y, t t ) with probability p u , to ( y, t t ) with probability p m , and to ( y - Δ y, t + Δ t ) with probability p d . (a) How must
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hw4 (1) - Continuous Time Finance, Spring 2004 Homework 4...

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