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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.
- Fall '11