# Stochastic

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Unformatted text preview: tion is 4 1 80 + 120 + 100 1 · 40 + · 15 + · 20 = = 10 15 15 6 30 The last derivation may seem a little devious, so we will now give a second derivation of the price of the option based on absence of arbitrage. In the scenario described above, our investor has four possible actions: A0 . Put \$1 in the bank and end up with \$1 in all possible scenarios. A1 . Buy one share of stock at time 0 and sell it at time 1. A2 . Buy one share at time 1 if the stock is at 120, and sell it at time 2. A3 . Buy one share at time 1 if the stock is at 90, and sell it at time 2. These actions produce the following payo↵s in the indicated outcomes X1 120 120 90 90 X2 140 115 120 80 A0 1 1 1 1 A1 20 20 10 10 A2 20 5 0 0 A3 0 0 30 10 option 40 15 20 0 Noting that the payo↵s from the four actions are themselves vectors in fourdimensional space, it is natural to think that by using a linear combination of these actions we can reproduce the option exactly. To ﬁnd the coe cients zi for the actions Ai we write four equations in four unknowns, z0 + 20z1 + 20z2 = 40 z0 + 20z1 5z2 = 15 z0 10z1 + 30z3...
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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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