Claim pricing and hedging under market incompleteness and ‘mean–variance’ preferences

This paper derives bid and ask prices of a general contingent claim in an incomplete market for an investor who can also trade in a given asset and in bonds. The investor's preferences are modelled by a quadratic derived utility function. The ask (resp. bid) price is determined in such a way that the investor is indifferent between the two following policies: (i) selling (resp. buying) the claim at this price and investing optimally in the asset and in bonds; (ii) investing optimally in the asset and in bonds, without trading the contingent claim. The ask (resp. bid) price thus derived is the expectation of the final payoff of the claim under the signed martingale measure that M. Schweizer called ‘minimal martingale measure’, plus (resp. minus) a quantity that is directly proportional to: (i) the number of claims being traded; (ii) a coefficient of risk-aversion; (iii) the smallest variance of the error of replication due to hedging one single claim. The results we derive apply to very general claims, hence to portfolios of claims.