# a3 - ACTSC/STAT 446/846 – Assignment 3 – Due on...

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Unformatted text preview: ACTSC/STAT 446/846 – Assignment 3 – Due on December 5 at 11:30am Problem 1: Ito’s Lemma [7pts] Let X t be deﬁned by the SDE dX t = μdt + σdW t , where μ > ,σ > 0 and { W t ,t ≥ } is a standard Brownian motion. (1) Use Ito’s 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 t-σt ), t ≥ 0. (2) Which of the processes in (a)-(c) are martingales with respect to the ﬁltration 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 Black-Scholes model was such that the position at time t in the risky asset should be equal to the option’s delta Δ 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 asset’s price is S t . (a) For a call option in the Black-Scholes model, determine an expression for Δ t as a function of the risk-free rate r , the underlying asset’s volatility σ , the strike price K , and the maturity T of the option. (Note: you can ﬁnd 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) Delta-Hedging a call option refers to the process by which an investor sells a call at time t , buys Δ t units of the underlying asset (by borrowing some money) so that overall, the position’s value and its delta are 0 at time t . However, at time t + Δ t , the value of the position may not be 0. Consider , the value of the position may not be 0....
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a3 - ACTSC/STAT 446/846 – Assignment 3 – Due on...

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