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# HO5n - Math 238 Financial Mathematics Final Problem set...

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Math 238, Financial Mathematics Final Problem set March 3, 2009 These problems are due on Tuesday March 10. You can give them to me in class, drop them in my office, or put them in my mailbox outside the mathematics department. Problem 1 Discuss the pricing of options on bonds, that is, using the Vasicek short rate model let P ( t, r t , T 2 ) be the price at time t < T 2 of a zero coupon bond maturing at time T 2 . Price a call option on this bond maturing at time T 1 with t < T 1 < T 2 , with strike price K . (Bjork Chapter 22.) You may use the framework of Chapter 24 in Bjork, sections 24.5, 24.6 and 24.7, if you wish, provided you explain how you are using this framework. Problem 2 In the HJM theory (Bjork Chapter 23) the forward rate f ( t, T ) is modeled by a family of SDEs f ( t, T ) = f (0 , T ) + integraldisplay t 0 α ( s, T ) ds + integraldisplay t 0 σ ( s, T ) dW ( s ) for 0 < t < T . The price of a zero coupon bond is given by p (0 , T ) = exp {- integraldisplay T 0 f (0 , s ) ds } and the short rate r ( t ) = f ( t, t ). (a) Using Ito’s formula formally (without worrying about differentiability conditions) derive the consistency relation α ( t, T ) = σ ( t, T ) integraldisplay

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HO5n - Math 238 Financial Mathematics Final Problem set...

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