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I'm studying for an exam in financial models and came across this question:

"An agent with $C^2$ strictly increasing concave utility $U$ has wealth $w_0$ at time 0, and wishes to invest his wealth in shares and bonds so as to maximise his expected utility of wealth at time 1. Explain how the solution to his optimization problem generates an equivalent martingale measure."

The Fundamental Theorem of Asset Pricing states that (roughly) there is no arbitrage iff there's an measure $Q$ such that the discounted stock price process becomes a martingale. So , in our (ideal) market, we have such a measure Q. So I think the question is asking how this measure arises in the solution.

Thank you - I've been confused about this for a while...

(Please don't knock this question for being unrealistic. As I said, I'm revising for an exam!)

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Could you give more details about the probability laws of the "bonds" and "shares" that you are dealing with in your exercise ? Regards PS : Tagging "Homework" this question would be natural. –  TheBridge Apr 13 '11 at 12:42
    
No information on the distribution of the shares is given. Usually, we consider single-period models with d shares (assumed dependent) and a riskless bond of rate r. –  Ben Apr 13 '11 at 13:08
    
Using all provided information maybe it's possible to go utility function$\to$non-arbitrage$\to$martingale measure. What can you conclude from the form of the utility function? –  Ilya Apr 13 '11 at 20:06
    
I'm not sure how the form of the utility function is useful. This is part of a longer question, so perhaps it doesn't play a role just yet. –  Ben Apr 13 '11 at 22:52
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1 Answer 1

This is very standard financial theory. The answer to the question is given in the first chapter of Duffie's Dynamic Asset Pricing Theory and in Cochrane's Asset Pricing. The latter is more elementary, but you have to read more to get to the answer.

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