61 consider now an option that pays o v1 h or v1 t

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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: Pm (i) There is an investment allocation xi so that i=1 xi ai,j 0 for each j and Pm i=1 xi ai,k > 0 for some k . Pn (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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