Pricing through the Choquet integral

The classical no-arbitrage pricing theory allows to price assets through a linear pricing rule, byassuming a frictionless and competitive market. Moreover, completeness of the market assuresthat the pricing rule is defined as a discounted expected value with respect to a unique equivalentmartingale measure. On the other hand, under no-arbitrage assumption, incomplete models,such as the trinomial model, lead to a set of equivalent martingale measures. This suggeststo work with non-linear pricing rules that can allow frictions in the market. A generalizedpricing rule can be achieved by replacing additive measures with non-additive measures suchas convex capacities and belief functions in Dempster-Shafer theory. The paper recaps results onnon-additive measures and Choquet expectation as non-linear functional to be used in pricing.In the literature it has been proved that, under suitable conditions, a non-linear pricing rulecan be expressed as a Choquet expectation with respect to a convex capacity. In the trinomialmarket model the lower probability is a belief function, but it cannot be used to reach the lowerexpectation through the Choquet integral. Nevertheless it can avoid a generalized Dutch bookcondition in the framework of partially resolving uncertainty