# This is no accident what we have shown is that with

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Unformatted text preview: le up down stock 30 10 179 option 20 c c 180 CHAPTER 6. MATHEMATICAL FINANCE Suppose we buy x units of the stock and y units of the option, where negative numbers indicate that we sold instead of bought. One possible strategy is to choose x and y so that the outcome is the same if the stock goes up or down: 30x + (20 c)y = 10x + ( c)y Solving, we have 40x + 20y = 0 or y = 2x. Plugging this choice of y into the last equation shows that our proﬁt will be ( 10 + 2c)x. If c > 5, then we can make a large proﬁt with no risk by buying large amounts of the stock and selling twice as many options. Of course, if c < 5, we can make a large proﬁt by doing the reverse. Thus, in this case the only sensible price for the option is 5. A scheme that makes money without any possibility of a loss is called an arbitrage opportunity. It is reasonable to think that these will not exist in ﬁnancial markets (or at least be short-lived) since if and when they exist people take advantage of them and the opportunity goes away....
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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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