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Unformatted text preview: ACTSC/STAT 446/846 Tutorial #1 January 21, 2009 Let W = { W t ,t } be a standard Brownian motion. 1. Let Z 1 ,Z 2 ,... be positive iid random variables with mean 1 . Define X = x and X n = Q n j =1 Z j for all n 1 . Show that X = { X n ,n } is a martingale with respect to the filtration generated by the Z n s and find its mean. 2. The price of a stock at time n is modelled by S n = S e 1 + + n , where S is a positive constant and where the k s are i.i.d. random variables with common distribution ( a with probability q ; b with probability 1 q . The continuously compounded riskfree rate is given by r . Find the value of q for which the discounted price D n = e rn S n is a martingale with respect to F = {F n } n , the filtration generated by the k s. If you need any restrictions on a,b and r , state and justify them. 3. Let B = { B t ,t } be the Brownian motion with drift and diffusion coefficient obtained as a linear transformation of...
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 Spring '09
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