hw6 (1) - dy = μdt + σdw + JdN , starting at y (0) = 0....

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Continuous Time Finance, Spring 2004 – Homework 6 Distributed 4/14/04, due 4/28/04 (1) We studied the Dupire equation, for calls on a non-dividend-paying stock. This problem asks you to derive the analogous equation for calls on a foreign currency rate. Since the letter C will be used for the call price, we use S for the foreign currency rate. Recall that under the (domestic investor’s) risk-free measure it evolves by dS = ( r - q ) S dt + σ ( S, t ) S dw where r is the domestic risk-free rate and q is the foreign risk-free rate. Assume r and q are constant, and assume σ ( S, t ) is a deterministic function of S and t . Let C ( K, T ) = e - rT E [( S T - K ) + ] be the time-zero value of a call with strike K and maturity T under this model. Show that it solves C T = 1 2 K 2 σ 2 ( K, T ) C KK + ( q - r ) KC K - qC for T > 0 and K > 0, with initial condition C ( K, 0) = ( S 0 - K ) + and boundary condition C (0 , T ) = e - qT S 0 where S 0 is the time-zero spot exchange rate. (2) Consider scaled Brownian motion with drift and jumps:
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Unformatted text preview: dy = μdt + σdw + JdN , starting at y (0) = 0. Assume the jump occurences are Poisson with rate λ , and the jump magnitudes J are Gaussian with mean 0 and variance δ 2 . Find the probability distribution of the process y at time t . ( Hint : don’t try to solve the forward Kolmogorov PDE. Instead observe that you know, for any n , the probability that n jumps will occur before time t ; and after conditioning on the number of jumps, the distribution of y is a Gaussian whose mean and variance are easy to determine. Assemble these ingredients to give the density of y as an infinite sum.) [ Comment : Using essentially the same idea, Merton gave an explicit formula for the value of an option when y is the logarithm of the stock price under the subjective measure.] 1...
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This note was uploaded on 01/02/2012 for the course FINANCE 347 taught by Professor Bayou during the Fall '11 term at NYU.

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