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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
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).
- Spring '10
- The Land