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up
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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 shortlived) 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
 DURRETT
 The Land

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