sol3 - ACTSC/STAT 446/846 – Assignment 3 – Solutions...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 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. 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 t-10-t 2 )( μdt + σdW t ) + 8( Y t-10-t 2 ) σ 2 dt = (2 t + ( Y t-10-t 2 )(4 μ + 8 σ 2 )) dt + 4 σ ( Y t-10-t 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 non-zero drift. 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. 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)( T-t ) σ √ T-t . (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 an investor who sells an option at time 0 and buys Δ units of asset at time 0, thus borrowing S Δ-C ( 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 } ....
View Full Document

This note was uploaded on 03/17/2012 for the course ACTSC 446 taught by Professor Adam during the Spring '09 term at Waterloo.

Page1 / 5

sol3 - ACTSC/STAT 446/846 – Assignment 3 – Solutions...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online