Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 30K / d

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- Solution d

Solution

At the optimal terminal wealth, normalized marginal utility defines an equivalent martingale measure

Q

and hence a marginal utility price

p_{Q} (Y) = \frac{1}{1 + r} E_{Q} Y .

This is an arbitrage-free price, so

p_{Q} (Y) \in I

. Concavity of utility implies that the finite-quantity indifference bid does not exceed its marginal price:

π (Y) \leq p_{Q} (Y) .

Consequently

π (Y) \leq sup {p : p \in I} .

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