Convex Pricing by a Generalized Entropy Penalty

In an incomplete Brownian-motion market setting, we propose a convex monotonic pricing functional for nonattainable bounded contingent claims which is compatible with prices for attainable claims. The pricing functional is defined as the convex conjugate of a generalized entropy penalty functional and an interpretation in terms of tracking with instantaneously vanishing risk can be given. 1. Introduction. Given an incomplete market, one of the problems of mathematical finance is to price nonattainable contingent claims. One way to do this is (expected or robust) utility indifference pricing. Typically, pricing functionals are desired to be convex, monotonic, (weakly) continuous and translation invariant or monetary. Furthermore, for attainable claims, the pricing functional should lead to the price of a replicating self-financing hedging strategy. However, having sold a nonattainable contingent claim for such a utility indifference price, it is not clear whether their exists a good way to hedge the claim from a (market) risk management point of view. In our approach, the risk stemming from not being able to perfectly repli-cate a nonattainable claim, measured at an instantaneous level, directly enters the pricing functional via an instantaneous penalty. The pricing functional can be represented using its convex conjugate which can be interpreted as a generalized relative entropy functional. The total penalty turns out to be the generalized entropy of an equivalent martingale measure depending on the claim, relative to the minimal martingale measure, introduced in Föllmer and Schweizer (1990). Similarly, as for expected exponential utility indifference pricing func-tionals [see Rouge and El Karoui (2000), Lazrak and Quenez (2003) and