hw5 (1)

# hw5 (1) - dC = μ C C dt σ C Cdw C where w C is a Brownian...

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Continuous Time Finance, Spring 2004 – Homework 5 Posted 4/3/04, due 4/14/04 ( typo in the SDE for C corrected 4/7/04 ) In problem 3 of HW1 we considered options on a foreign exchange rate, assuming the interest rate in each currency was constant. Now we have more sophisticated interest rate models; let’s see how they work in this setting. Let C ( t ) be the exchange rate in dollars/pound, and consider an option that gives a dollar investor the right to buy pounds at exchange rate K at time T ; its payoﬀ (to the dollar investor) is ( C ( T ) - K ) + dollars at time T. (1) Use subscripts D , P , and C to distinguish analogous dollar, pound, and exchange-rate objects: for example P D ( t, T ) = dollar value at time t of a zero-coupon bond worth one dollar at time T. Use Hull-White models for the dollar and pound short rates: dr D = ( θ D ( t ) - a D r D ) dt + σ D dw D where w D is a Brownian motion under the dollar investor’s risk-neutral measure; and dr P = ( θ P ( t ) - a P r P ) dt + σ P dw P where w P is a Brownian motion under the pound investor’s risk-neutral measure. Assume the exchange rate has constant drift and volatility:
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Unformatted text preview: dC = μ C C dt + σ C Cdw C where w C is a Brownian motion under some (subjective) probability. The Brownian motions may be correlated: assume dw D dw P = ρ DP dt, dw D dw C = ρ DC dt, dw P dw C = ρ PC dt, where ρ DP , ρ DC , and ρ PC are constant. (a) What is the value (to the dollar investor, at time t < T ) of the payoﬀ (1)? (Make your answer as explicit as possible.) (b) Describe a trading strategy for the dollar investor that replicates this payoﬀ. (Again, be as explicit as possible.) (c) Is a similar analysis possible if we use one-factor HJM models for the interest rates rather than Hull-White? [Extra credit: consider the analogous question for quanto call, whose value to the dollar investor is ( S ( T )-K ) + at time T , where S is the price of a stock in pounds. This is of course the stochastic-interest-rate analogue of our discussion of quantos, in Section 3 and problem 4 of HW2.] 1...
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