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hw3 (1)

# hw3 (1) - Continuous Time Finance Spring 2004 Homework 3...

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Continuous Time Finance, Spring 2004 – Homework 3 Distributed 2/27/04, due 3/10/04 (1) Assume the Vasicek model dr = ( θ - ar ) dt + σ dw for the risk-neutral short rate process. Consider a call option with maturity T and strike K , on a zero-coupon bond with maturity S > T . Its payoff at time T is ( P ( T, S ) - K ) + . Show using Black’s formula that the value of this option at time t is P ( t, S ) N ( d 1 ) - KP ( t, T ) N ( d 2 ) where d 1 = 1 σ p log P ( t, S ) P ( t, T ) K + 1 2 σ p , d 2 = 1 σ p log P ( t, S ) P ( t, T ) K - 1 2 σ p with σ p = σ 1 - e - 2 a ( T - t ) 2 a 1 / 2 B ( T, S ) . (The function B ( T, S ) is the one from the representation P ( t, T ) = A ( t, T ) e - B ( t,T ) r ( t ) .) (2) Since Vasicek is a one-factor model, the call option of Problem 1 can be replicated by a self-financing trading strategy using any pair of tradeables. (a) What trading strategy produces a replicating portfolio using tradeables P ( t, T ) and P ( t, S )? (b) What trading strategy produces a replicating portfolio using tradeables P ( t, S ) and the money market fund B ( t )?
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