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

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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 (this is a particularly simple case of the Hull & White 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.
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Unformatted text preview: b) Now consider an HJM model, driven by the risk-neutral Brownian motion f W , in which the volatility σ ( t, T ) is of the form σ ( 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 θ ( t ), κ ( t ), and ξ ( t ) such that R t satisFes dR t = κ ( t )( θ ( t ) − R t ) dt + ξ ( 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 Shreve’s book. 1...
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