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Unformatted text preview: Math 238, Financial Mathematics Homework 5 Solutions March 17, 2009 Problem 1 For the Vasicek short rate model, the price c ( t, T 1 , K, T 2 ) at time t of a call option maturing at time T 1 with strike price K on a zero coupon bond with maturing at time T 2 (with t < T 1 < T 2 ) is given by: c ( t, T 1 , K, T 2 ) = p ( t, r t , T 2 ) N ( d ) p ( t, r t , T 1 ) KN ( d σ p ) , (1) where d = 1 σ p log { p ( t, r t , T 2 ) p ( t, r t , T 1 ) K } + 1 2 σ p (2) σ p = 1 a { 1 e a ( T 2 T 1 ) } r σ 2 2 a (1 e 2 a ( T 1 t ) ) (3) Equivalently, this can be expressed as: c ( t, T 1 , K, T 2 ) = p ( t, r t , T 2 ) N ( d 1 ) Kp ( t, r t , T 1 ) N ( d 2 ) , (4) where d 2 = log ( p ( t, r t , T 2 ) /Kp ( t, r t , T 1 ) 1 2 Σ 2 √ Σ 2 (5) d 1 = d 2 + √ Σ 2 (6) Σ 2 = σ 2 2 a 3 { 1 2 ae 2 a ( T 1 t ) }{ 1 e a ( T 2 ( T 1 t )) } 2 . (7) This can be obtained using the framework of Bjork Chapter 24 by observing that the volatility σ z of the process Z ( t ) = p ( t,T 2 ) p ( t,T 1 ) is deterministic . This is done by applying Ito’s formula to the expression Z ( t ) = exp { A ( t, T 2 ) A ( t, T 1 ) [ B ( t, T 2 ) B ( t, T 1 )] r ( t ) } , where p ( t, T ) = e A ( t,T ) r ( t ) B ( t,T ) is the affine term structure. Upon applying Ito’s formula to Z ( t ), we obtain: 1 σ z ( t ) = σ [ B ( t, T 2 ) B ( t, T 1 )] (8) = σ a e at [ e aT 2 e aT 1 ] . (9) Thus σ z is indeed deterministic, so the conditions of Assumption 24.5.1 are satisfied. Thus, Proposition 24.12 applies, and we obtain the price of the call option with strike K and maturity T 1 on a zero coupon bond maturing at time T 2 in the Vasicek model. Problem 2 (a) Applying Ito’s formula to p ( t, T ) = exp { R T t f ( t, s ) ds } , we obtain: dp ( t, T ) = { r ( t ) + b ( t, T ) } dt + a ( t, T ) p ( t, T ) dW ( t ) , (10) where a ( t, T ) = Z T t σ ( t, s ) ds (11) b ( t, T ) = Z T t α ( t, s ) ds + 1 2 a 2 ( t, T ) (12) Next, note that the drift of any price process under the risk neutral martingale measure must be equal to the short rate. Thus, b ( t, T ) = 0. This gives: Z T t α ( t, s ) ds = 1 2 Z T t σ ( t, s ) ds 2 (13) Finally, taking the derivative of the previous expression with respect to T , we obtain the result: α ( t, T ) = σ ( t, T ) Z T t σ ( t, s ) ds (14) (b) The 5step HJM algorithm for pricing a bond is given by the following: (1) Specify the volatilities σ ( t, T ). (2) Use the HJM consistency relation α ( t, T ) = σ ( t, T ) R T t σ ( t, s ) ds to obtain the drift parameters of the forward rates....
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This note was uploaded on 11/08/2009 for the course MATH 238 taught by Professor Papanicolaou during the Winter '08 term at Stanford.
 Winter '08
 Papanicolaou
 Math

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