Stochastic

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Unformatted text preview: t time 1, then the possibilities at time 2 are 120 and 80. The last three sentences can be simply summarized by the following tree. 140 ⇠⇠⇠ ⇠⇠⇠ ⌘ 120 XXX XXX ⌘ 115 ⌘ ⌘ 100 Q Q Q Q ⇠⇠⇠ 90 XX X ⇠⇠ ⇠ 120 XXX 80 Using the idea that the value of an option is its expected value under the probability that makes the stock price a martingale, we can quickly complete the computations in our example. When X1 = 120 the two possible scenarios lead to a change of +20 or 5, so the probabilities of these two events should be 1/5 and 4/5. When X1 = 90 the two possible scenarios lead to a change of +30 or 10, so the probabilities of these two events should be 1/4 and 3/4. When X0 = 0 the possible price changes on the first step are +20 and 10, so their probabilities are 1/3 and 2/3. Making a table of the possibilities, we have 182 CHAPTER 6. MATHEMATICAL FINANCE X1 120 120 90 90 probability (1/3) · (1/5) (1/3) · (4/5) (2/3) · (1/4) (2/3) · (3/4) X2 140 115 120 80 (X 2 100)+ 40 15 20 0 so the value of the op...
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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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