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Unformatted text preview: STAT/ACTSC 446/846 Assignment #4 (due November 30, 2007)446 students: the first 6 problems are required. The two last ones are bonus but you cant get more than 40 in total.846 students: All problems are required. It will be marked over 50 pts and not more than 50 in total. Problem 1: [6 pts] Practice of Itos Formula Let X t be defined by the SDE dX t = dt + dW t , where W 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 ( Y t , t ) dt + v ( Y t , 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 t ), 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: [4 pts] Vasicek Model (1) Using Itos lemma, find the expression of the solution of the general OrnsteinUhlenbeck process: dr t = ( r r t ) dt + dW t , r = r . Hint: you might need the auxiliary function f ( x, t ) = xe t . (2) The drift term is equal to ( r r t ). How can and r be interpreted? (3) It is a famous model to model stochastic interest rates, called Vasicek Model. A major drawback of this model is that the interest rate can take negative values with a very small probability. Which property of the riskless asset is not satisfied if the interest rate can take negative values? Problem 3: [8 pts] Delta and Gamma Hedging 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 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) 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 an investor who sells an option at time 0 and buys units of asset at time 0, thus borrowing S  C ( S , 0). Using the values S = 100, = 0 . 2, T = 1 year, r = 0 . 04, K = 90), compute the difference between the value of the investors replicating portfolio at time 1/365 (consisting of units of the asset and a loan) and the options value at time 1/365, for every value of the underlying asset S 1 / 365 in { 95 , 96 , . . ., 105 } ....
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 Fall '09
 Adam

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