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Unformatted text preview: Lecture 22 More on Monte Carlo Methods in Finance Monte Carlo methods apply to several aspects in mathematical finance, which we will briefly describe in what follows. 1 Simulation of SDEs Consider an Ito process X defined as a solution to the SDE dX t = t dt + t dW t , t [0 ,T ] , with X = x. (22.1) For simplicity, assume t = a ( X t ) and t = b ( X t ) for some deterministic functions a ( ) and b ( ). The Euler discretization scheme (see [2] or [3]) provides a firstorder approximation to (22.1) as follows: For a small time increment > 0, suppose T/ = n (a positive integer). Let Y i = X i , i = 0 , 1 ,...,n . Then (22.1) is approximated by the (discrete) time series Y i = Y i 1 + a ( Y i 1 ) + b ( Y i 1 ) i , i = 1 ,...,n, with Y = x, (22.2) where 1 , 2 ,... are iid N (0 , 1) random variables. An obvious way to simulate the sequence { Y i } would start with generating the sequence { i } then follow the iterations in (22.2)....
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 Spring '08
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