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Unformatted text preview: sset Pricing Model In this section, we will explore an approach to option pricing which is more
in keeping with traditional economic thinking. Our hidden agenda here is to
show how ideas from Section 6.2 can be used to simplify a complicated looking
problem.
Each investor is assumed to have a utility function that nondecreasing and
concave. If 0 1 and x < y then
U ( x + (1 )y ) U (x) + (1 )U (y ) (6.18) 192 CHAPTER 6. MATHEMATICAL FINANCE
3.5
3
2.5
2
1.5
1
0.5
0
0 5 10 15 20 25 Geometrically, the line segment from (x, U (x)) to (y, U (y )) lies below the graph
of the function.
In economic terms investors are risk averse. They prefer a sure payo↵ of
x + (1
)y to a lottery ticket that pays x with probability and pays y with
probability 1
.
Lemma 6.7. If U is smooth then U is concave if U 00 0.
Proof. U 00 0 implies that the U 0 is decreasing, so if x > y
1
(x y) Z x+(1 )y U (z ) dz
0 y 1
x y Z x U 0 (z ) dz y In words, the average slope over [y, x + (1
)y ], and intervalof length (x y )
is larger than that over [y,...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell.
 Spring '10
 DURRETT
 The Land

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