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Unformatted text preview: ACTSC/STAT 446/846 Assignment 3 Due on December 5 at 11:30am Problem 1: Itos Lemma [7pts] Let X t be dened by the SDE dX t = dt + dW t , where > , > 0 and { W t ,t } is a standard Brownian motion. (1) Use Itos formula to write the following processes Y t in the form of a stochastic integral ( dY t = u ( t ) dt + v ( t ) dW t or an equivalent integral form): (a) Y t = X 3 t , t (b) Y t = 10 + t 2 + e 4 X t , t (c) Y t = exp( X 2 tt ), t 0. (2) Which of the processes in (a)(c) are martingales with respect to the ltration generated by { W t ,t } ? Explain. Problem 2: Delta and Gamma Hedging [10pts] In class, we saw that the replicating strategy for a call option in the BlackScholes model was such that the position at time t in the risky asset should be equal to the options delta t = C ( S t ,t ) S t where C ( S t ,t ) is the value of the call option at time t when the risky assets price is S t . (a) For a call option in the BlackScholes model, determine an expression for t as a function of the riskfree rate r , the underlying assets volatility , the strike price K , and the maturity T of the option. (Note: you can nd this in McDonald chap. 12 or somewhere else: just state your source when giving the expression. You do not need to prove the formula.) (b) DeltaHedging a call option refers to the process by which an investor sells a call at time t , buys t units of the underlying asset (by borrowing some money) so that overall, the positions value and its delta are 0 at time t . However, at time t + t , the value of the position may not be 0. Consider , the value of the position may not be 0....
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 Spring '09
 Adam

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