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Unformatted text preview: ck that if not, there would be a nonzero 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.
Twoperiod 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).
 Spring '10
 DURRETT
 The Land

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