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Unformatted text preview: Using our new
terminology we can say that the only price for the option which is consistent
with absence of arbitrage is c = 5, so that must be the price of the option.
To ﬁnd prices in general, it is useful to look at things in a di↵erent way. Let
ai,j be the proﬁt for the ith security when the j th outcome occurs.
Theorem 6.1. Exactly one of the following holds:
(i) There is an investment allocation xi so that i=1 xi ai,j 0 for each j and
i=1 xi ai,k > 0 for some k .
(ii) There is a probability vector pj > 0 so that j =1 ai,j pj = 0 for all i. Here an x satisfying (i) is an arbitrage opportunity. We never lose any
money but for at least one outcome we gain a positive amount. Turning to (ii),
the vector pj is called a martingale measure since if the probability of the j th
outcome is pj , then the expected change in the price of the ith stock is equal
to 0. Combining the two interpretations we can restate Theorem 6.2 as:
Theorem 6.2. There is no arbitrage if and only if ther...
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- Spring '10
- The Land