# 3 concrete examples 64 0 32 h 0 hh h 0 hh 16 16 h h 0

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Unformatted text preview: he solution of the one period problems. Let a be a string of H ’s and T ’s of length n 1 which represents the outcome of the ﬁrst n 1 events. The value of the option at time n after the events in a have occurred, Vn (a), and the amount of stock we need to hold in this situation, n (a), in order to replicate the option payo↵ satisfy: ✓ ◆ 1 1 Sn+1 (aH ) Sn (a) = Vn+1 (aH ) (6.8) Vn (a) + n (a) 1+r 1+r ✓ ◆ 1 1 Vn (a) + n (a) Sn+1 (aT ) Sn (a) = Vn+1 (aT ) (6.9) 1+r 1+r 185 6.2. BINOMIAL MODEL Deﬁne the risk neutral probability p⇤ (a) so that n Sn (a) = 1 [p⇤ (a)Sn+1 (aH ) + (1 1+r n p⇤ (a))Sn+1 (aT )] n (6.10) A little algebra shows that p⇤ (a) = n (1 + r)Sn (a) Sn+1 (aT ) Sn+1 (aH ) Sn+1 (aT ) (6.11) In the binomial model one has p⇤ (a) = (1 + r d)/(u d). However, stock n prices are not supposed to follow the binomial model, but are subject only to the no arbitrage restriction that 0 < p⇤ (a) < 1. Notice that these probabilities n depend on the time n an...
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## This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).

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