# sheet04 - J. Wissel Financial Engineering with Stochastic...

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J. Wissel Financial Engineering with Stochastic Calculus I Fall 2010 Assignment Sheet 4 1. Consider the N -period binomial model with d < 1 + r < u . For each T = 0 ,...,N and K 0 let C 0 ( K,T ) be the fair price at time 0 of a European call option with payoﬀ ( S T - K ) + at time T . Show that for T 1 T 2 and K 0 we have C 0 ( K,T 2 ) C 0 ± K (1+ r ) T 2 - T 1 ,T 1 ² . Hint: Use assignment 3 problem 2b), and Corollary 2.44 of the lecture notes. 2. We revisit the one-period trinomial model on a space Ω = { ω 1 2 3 } as discussed in the recitations. We assume B 0 = B 1 = 1 (zero interest rate), S 0 = 1, and S 1 and the real-world probability measure P is deﬁned by S ( ω 3 ) = 2 , P [ ω 3 ] = 1 4 , S ( ω 2 ) = 1 , P [ ω 2 ] = 1 2 , S ( ω 1 ) = 1 2 , P [ ω 1 ] = 1 4 . By the FTAP, the market B,S is arbitrage-free if and only if E ˜ P [ S 1 ] = S 0 where ˜ P is a probability measure on Ω which is equivalent to P . Any such ˜ P is called a risk-neutral measure. Let ˜ p = ˜ P

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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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sheet04 - J. Wissel Financial Engineering with Stochastic...

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