sheet02 - J Wissel Financial Engineering with Stochastic...

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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 (Ω ∞ , F ∞ ,P ) with P [ { ω t = H } ] = 1 2 for all t ≥ 1, we define an infinite-time-horizon binomial model with stock price process S t = S Y 1 ··· Y t , where Y t ( ω ) = ( 3 / 2 ( ω t = H ) , 1 / 2 ( ω 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 self-financing trading strategy with initial value X = 0 by choosing Δ = 1 and Δ j ( ω ) = ( 4 j if ω 1 = ... = ω j = T, 0 if ω 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 ( ω ) = smallest integer n for which ω n is H . a) For any ω 6 = ( TTT... ), show that X N ( ω ) ( ω ) = 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 ω ∈ Ω X n ( ω ). 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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This note was uploaded on 01/25/2011 for the course ORIE 5600 at Cornell.

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sheet02 - J Wissel Financial Engineering with Stochastic...

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