ex_6 - b) Now consider an HJM model, driven by the...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
ORIE 569 Problem set 6 April 3, 2008 Homework must be turned in by Thursday, April 17, at 10 a.m. Solutions turned in at a later time will not be graded. Place homework in the 569 homework box on the bridge between Upson and Rhodes. Exercise 1. a) Suppose that under the risk-neutral measure e P the short rate process R t is given by dR t = θ dt + σ, d f W t for a e P -Bownian motion f W interest rate model). Show that at time t the price B ( t, T ) of a zero-coupon bond with maturity T is equal to B ( t, T ) = exp σ 2 6 ( T t ) 3 θ 2 ( T t ) 2 ( T t ) R t ¥ . Use this formula to compute the forward rates and their dynamics, and check that these dynamics are consistent with the Heath-Jarrow-Morton (HJM) framework.
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: b) Now consider an HJM model, driven by the risk-neutral Brownian motion f W , in which the volatility ( t, T ) is of the form ( t, T ) = g ( t ) h ( T ) for two bounded and strictly positive deterministic functions g and h . Write down the dynamics of the forward rates f ( t, T ) and of the short rate R t = f ( t, t ) under the risk-neutral measure. Then Fnd deterministic functions ( t ), ( t ), and ( t ) such that R t satisFes dR t = ( t )( ( t ) R t ) dt + ( t ) d f W t . That is, R t solves the SDE of a general one-factor Hull & White model. Exercise 2. Exercise 10.9 on page 458 in Steve Shreves book. 1...
View Full Document

This note was uploaded on 08/05/2008 for the course ORIE 569 taught by Professor Alexanderschied during the Spring '08 term at Cornell University (Engineering School).

Ask a homework question - tutors are online