a3 - ACTSC/STAT 446/846 Assignment 3 Due on December 5 at...

Info iconThis preview shows pages 1–2. 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 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 t-t ), 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 Black-Scholes 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 Black-Scholes model, determine an expression for t as a function of the risk-free 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) Delta-Hedging 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....
View Full Document

Page1 / 3

a3 - ACTSC/STAT 446/846 Assignment 3 Due on December 5 at...

This preview shows document pages 1 - 2. Sign up to view the full document.

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