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Unformatted text preview: J. Wissel Financial Engineering with Stochastic Calculus I Fall 2010 Assignment Sheet 2 1. On the infinite independent coin toss space ( , F ,P ) with P [ { t = H } ] = 1 2 for all t 1, we define an infinitetimehorizon binomial model with stock price process S t = S Y 1 Y t , where Y t ( ) = ( 3 / 2 ( t = H ) , 1 / 2 ( t = T ) . We assume that the interest rate is r = 0, so the bond price is B t = 1 for all t 0. We now define a selffinancing trading strategy with initial value X = 0 by choosing = 1 and j ( ) = ( 4 j if 1 = ... = j = T, 0 if i = H for some i { 1 ,...,j } for each j 1. Let X n , n 0 be the value process of this strategy, and define a random variable N ( ) = smallest integer n for which n is H . a) For any 6 = ( TTT... ), show that X N ( ) ( ) = 1 2 S . b) Conclude that lim n X n = 1 2 S almost surely. c) Show that E [ X n ] = 0 for every integer n 0. Hint: Write X n = X n I { n N } + X n I { n<N } and use linearity of the expectation. d) For fixed n 1, compute min X n ( ). e) Note that b) and c) imply E lim n X n 6 = lim n E [ X n ]. Explain why this does not violate the dominated convergence theorem.does not violate the dominated convergence theorem....
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 '09
 J.WISSEL

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