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Unformatted text preview: ACTSC/STAT 446/846 – Assignment 3 – Solutions 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 ≥ 0; (b) Y t = 10 + t 2 + e 4 X t , t ≥ 0; (c) Y t = exp( X 2 tσt ), t ≥ 0. (2) Which of the processes in (a)(c) are martingales with respect to the ﬁltration generated by { W t ,t ≥ } ? Explain. Solution: (a) dY t = 0 + 3 X 2 t dX t + 1 2 6 X t ( dX t ) 2 = 3 X 2 t ( μdt + σdW t ) + 3 X t σ 2 dt = (3 Y 2 3 t μ + 3 Y 1 3 t σ 2 ) dt + 3 σY 2 3 t dW t (b) (b) dY t = 2 tdt + 4 e 4 X t dX t + 1 2 16 e 4 X t ( dX t ) 2 = 2 tdt + 4( Y t10t 2 )( μdt + σdW t ) + 8( Y t10t 2 ) σ 2 dt = (2 t + ( Y t10t 2 )(4 μ + 8 σ 2 )) dt + 4 σ ( Y t10t 2 ) dW t (c) dY t =σY t dt + 2 Y t X t dX t + ( Y t )(1 + 2 X 2 t )( dX t ) 2 =σY t dt + 2 Y t X t ( μdt + σdW t ) + Y t (1 + 2 X 2 t ) σ 2 dt = (σY t + 2 Y t X t μ + Y t (1 + 2 X 2 t ) σ 2 ) dt + 2 Y t X t σdW t (2) None of the three process are martingales as they all have nonzero drift. 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 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 BlackScholes model, determine an expression for Δ t as a function of the riskfree rate r , the underlying asset’s volatility σ , the strike price K , and the maturity T of the option. 1 (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.) Solution: We have that Δ t = N ( d 1 ) where d 1 = ln( S t /K ) + ( r + σ 2 / 2)( Tt ) σ √ Tt . (b) DeltaHedging 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 an investor who sells an option at time 0 and buys Δ units of asset at time 0, thus borrowing S ΔC ( S , 0). Using the values S = 100, σ = 0 . 2, T = 1 year, r = 0 . 04, K = 90), compute the diﬀerence between the value of the investor’s replicating portfolio at time 1/365 (consisting of Δ units of the asset and a loan) and the option’s value at time 1/365, for every value of the underlying asset S 1 / 365 in { 95 , 96 ,..., 105 } ....
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This note was uploaded on 03/17/2012 for the course ACTSC 446 taught by Professor Adam during the Spring '09 term at Waterloo.
 Spring '09
 Adam

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