Stochastic

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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 pay-o↵ 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.

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