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FEbook1MAIN - THE FACTOR APPROACH TO DERIVATIVE PRICING The...

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Unformatted text preview: THE FACTOR APPROACH TO DERIVATIVE PRICING The BIG Picture in a LITTLE Book James A. Primbs January 20, 2009 2 Contents 1 Basic Building Blocks and Stochastic Differential Equation Models 1 1.1 Brownian Motion and Poisson Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Gaussian Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.2 Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.3 Poisson Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.4 Poisson Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.5 Increments of Brownian Motion and Poisson Processes . . . . . . . . . . . . . . . . . . 3 1.2 Stochastic Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.1 Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.2 The Differential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.3 Compound Poisson Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.4 Ito Stochastic differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.5 Poisson Driven Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2 Ito’s Lemma 11 2.1 Ito’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1.1 The chain rule of ordinary calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1.2 Ito’s lemma for Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.1.3 Replacing dz 2 by dt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1.4 Discussion of Ito’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Ito’s lemma for Poisson Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2.1 Interpretation of Ito’s lemma for Poisson . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 More versions of Ito’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3.1 Ito’s Lemma for Compound Poisson Processes . . . . . . . . . . . . . . . . . . . . . . . 16 2.3.2 Ito’s Lemma for Brownian and Compound Poisson Processes . . . . . . . . . . . . . . 16 2.3.3 Ito’s Lemma for vector processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4 Ito’s lemma, the product rule, and a rectangle . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Summary ....
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This note was uploaded on 06/17/2010 for the course MS&E 345 taught by Professor Jimprimbs during the Winter '10 term at Stanford.

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FEbook1MAIN - THE FACTOR APPROACH TO DERIVATIVE PRICING The...

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