Handout00_pre - Reviews of Martingale approach for Black-Scholes Models Brownian motion Ito calculus Change of measure Martingale representation

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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.
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2 Stats243 Summer 2007 1. Brownian Motion • Brownian motion with drift – What is the covariance function? • Geometric Brownian motion (GBM) – GBM with drift is a model often used for stock prices. – Assume that μ is a drift factor and σ is a noise factor. – What is the covariance function? Stats243 Summer 2007 Consider functions of Brownian motion, can we establish certain calculus rules? Recall that, in
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This note was uploaded on 10/30/2011 for the course AMS 517 taught by Professor Xinghaipeng during the Spring '11 term at SUNY Stony Brook.

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Handout00_pre - Reviews of Martingale approach for Black-Scholes Models Brownian motion Ito calculus Change of measure Martingale representation

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