hw4 (1)

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

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 01/02/2012 for the course FINANCE 347 taught by Professor Bayou during the Fall '11 term at NYU.

### Page1 / 3

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online