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Finm345A09Homework3

Finm345A09Homework3 - FINM345/STAT390 Stochastic Calculus...

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FINM345/STAT390 Stochastic Calculus Hanson Autumn 2009 Lecture 3 Homework: Stochastic Jump and Diffusion Processes (Due by Lecture 4 in Chalk FINM345 Digital Dropbox) You must show your work, code and/or worksheet for full credit. There are 10 points per question if correct answer. Corrections are in Red , October 14, 2009 1. Finance Oriented Martingales. A martingale in continuous time satisfies the essential property that E[ M ( t ) | M ( s )] = M ( s ) , for all 0 s < t provided its absolute value has finite expectation, i.e., E[ | M ( t ) | ] < for all t 0, plus some other technical properties. (a) Show that M 1 ( t ) = ln( X ( t )) - E[ln( X ( t ))] is a martingale where Y ( t ) = ln( X ( t )) symbolically satisfies the solution to the general linear diffusion SDE transformed to state-independent SDE form (E.g., (3.3) on L3-54 or (4.25) in text). 2. Exponential-Martingale Counterexample to Simple Notion that Martin- gales are always Driftless, a common stochastic finance legend.

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Finm345A09Homework3 - FINM345/STAT390 Stochastic Calculus...

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