Pricing through the Choquet integral

The classical no-arbitrage pricing theory allows to price assets through a linear pricing rule, by assuming a frictionless and competitive market. Moreover, completeness of the market assures that the pricing rule is defined as a discounted expected value with respect to a unique equivalent martingale 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 suggests to work with non-linear pricing rules that can allow frictions in the market. A generalized pricing rule can be achieved by replacing additive measures with non-additive measures such as convex capacities and belief functions in Dempster-Shafer theory. The paper recaps results on non-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 rule can be expressed as a Choquet expectation with respect to a convex capacity. In the trinomial market model the lower probability is a belief function, but it cannot be used to reach the lower expectation through the Choquet integral. Nevertheless it can avoid a generalized Dutch book condition in the framework of partially resolving uncertainty.