# sheet01 - J. Wissel Financial Engineering with Stochastic...

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J. Wissel Financial Engineering with Stochastic Calculus I Fall 2010 Assignment Sheet 1 1. Consider a one-period ( N = 1) binomial model for one bond and one stock with 0 < d < u and 0 < p < 1. At t = 0 we set up a portfolio of η 0 bonds and Δ 0 stocks and hold these positions until t = 1. We allow η 0 , Δ 0 R , i.e. we allow long and short positions in bond and stock. a) Show that if 1+ r u , we can choose η 0 , Δ 0 such that our initial portfolio value is X 0 = 0, and the value X 1 at time 1 is nonnegative with probability 1 and positive with positive probability. b) Show that if 1+ r d , we can choose η 0 , Δ 0 such that our initial portfolio value is X 0 = 0, and the value X 1 at time 1 is nonnegative with probability 1 and positive with positive probability. c) Now suppose d < 1 + r < u . Show that if our initial portfolio value is X 0 = 0, then either X 1 = 0 with probability 1, or X 1 is negative with positive probability. Remark:

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## This note was uploaded on 01/25/2011 for the course ORIE 5600 at Cornell University (Engineering School).

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sheet01 - J. Wissel Financial Engineering with Stochastic...

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