Pricing with non-smooth utility function

Abstract

We study the utility based pricing of arbitrary contingent claims in a semi-martingale model of incomplete financial market. We use the super-differential concept. Then we generalize the marginal rate of substitution. So, we adapt the Davis approach to define a family of ‘reasonable’ prices of a contingent claim. Moreover, we prove that each ‘reasonable’ price is an arbitrage price and we prove that the family of ‘reasonable’ prices is bounded. More precisely, we define explicitly the lower (respectively upper) bound. In particular, if the agent's utility function is differentiable or if the agent's optimal wealth is a random variable with density and if the set of reasonable prices is not empty then it is reduced to the unique based utility price. One does ask: for which kind of contingent claim does the investor have a null general marginal rate of substitution? We note that, when the price process is a special semi-martingale in and when the contingent claim is in L 2 and possesses a Fôllmer–Schweizer decomposition, we prove, by using a duality concept, that, when the contingent claim is attainable, the answer of the previous question is yes, but in the general case the problem is still open.