Henri Pages's Optimal Consumption and Portfolio Policies When Markets Are Incomplete (Classic Reprint)
Excerpt from Optimal Consumption and Portfolio Policies When Markets Are Incomplete
The first question arises from the fact that when M is stictly included in X, only the marketed commodities have their price determined by arbitrage. There is an abundance of price functionals 45 that extend 7r over all of X, and one could choose a priori any one of them. However, one candidate is of special Interest
to us: it is the (unique) one which is measurable with respect. To the price system, i.e., such that the shadow price of consumption is itself in the price information set. With this particular valuation, it turns out. That a solution 6 to the extended maximization program can always be chosen to be price measurable. And thus marketed. To see this, we have to recall a result from option pricing Theory
which states that the price of any contingent claim can be written as its expectation under some probability. Let then Q be the probability associated with our choice of the price measurable valuation, and take the conditional expectation of 1? Under Q with respect to the price information set. The new consumption plan is price measurable by construction. In addition, it can be shown that it satisfies the same budget constraint and that it is as least as preferred as 6. But c? Is optimal by assumption, so that it. Should be clear that the two solutions are in fact indifferent, or even identical when the utility function is strictly concave.
Thus. As far as the optimization problem is concerned. There is nothing more than the securities prices themselves that the agent. Needs to observe to devise his optimal strategy. Even in a World
endowed with a very rich information structure, only that. Which is contained in the capital markets is relevant to manufacture his Optimal consumption plan. The crux of the argument is that an investor can loose nothing in terms of his expected utility by projecting the whole problem onto the information generated by prices. An intuitive account of how this works can tentatively be given as follows. Since there are infinitely many valuations which give the prices of the contingent claims outside M, the investor may be seen as maximizing his expected utility over the whole consumption space subject to an infinite number of budget constraints. Simply pick the particular valuation which is in the price information set and observe that the corresponding solution (3 is price measurable and thus marketed. Now expected utility can be improved upon only if some of the constraints are slack. But all the valuations agree on M so that all constaints are saturated.
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