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# handout00_pre - Reviews of Martingale approach for...

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1 Stats243 Summer 2007 Reviews of Martingale approach for Black-Scholes Models ¾ Brownian motion ¾ Ito calculus ¾ Change of measure ¾ Martingale representation theorem ¾ Martingale approach for Black-Scholes model Stats243 Summer 2007 1. Brownian Motion Definition: The process W = (W t : t ¸ 0) is a P-Brownian motion if and only if (i) W t is continuous and W 0 = 0; (ii) The value of W t is distributed, under P, as N(0,t); (iii) The increment W s+t – W s is distributed as N(0,t), under P, and is independent of F s , the history of what the process did up to time s. Odd properties of Brownian motion (BM) BM is continuous but nowhere differentiable. BM could hit any real value, but, with probability one, it will be back down again to zero. BM is self-similar. BM is also called a Wiener process .

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