Case 2 the stock drops to 50 your stock is worth

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Unformatted text preview: ck that if not, there would be a non-zero vector in O that would be in H. To apply Theorem 6.2 to our simplified example, we begin by noting that in this case ai,j is given by stock i = 1 option i = 2 j=1 j=2 30 10 20 c c By Theorem 6.2 if there is no arbitrage, then there must be an assignment of probabilities pj so that 30p1 10p2 = 0 (20 c)p1 + ( c)p2 = 0 From the first equation we conclude that p1 = 1/4 and p2 = 3/4. Rewriting the second we have c = 20p1 = 20 · (1/4) = 5 To prepare for the general case note that the equation 30p1 10p2 = 0 says that under pj the stock price is a martingale (i.e., the average value of the change in price is 0), while c = 20p1 + 0p2 says that the price of the option is then the expected value under the martingale probabilities. Two-period binary tree. Suppose that a stock price starts at 100 at time 0. At time 1 (one day or one month or one year later) it will either be worth 120 or 90. If the stock is worth 120 at time 1, then it might be worth 140 or 115 at time 2. If the price is 90 a...
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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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