Cost-Efficient Contingent Claims with Choquet Pricing

Abstract

We examine a problem, in which an investor seeks the cheapest contingent claim that achieves a minimum performance subject to a maximum allowed risk exposure. Specifically, our problem minimizes a non-linear cost functional, subject to both a minimum performance measure and a maximum risk measure, where all expectations are taken in the sense of Choquet. Solutions to our problem are called cost-efficient claims, and possess a desirable monotonicity property; the claims are anti-comonotonic with respect to the underlying asset, and therefore a hedge against its risk. By viewing our problem in the context of convex optimization, we apply a Karush-Kuhn-Tucker theorem to give necessary and sufficient conditions for cost efficiency. Such conditions also hold when the distortion functions are assumed to be absolutely continuous, but not necessarily continuously differentiable. This allows us to consider a broader set of risk measures, including the popular conditional value at risk (a.k.a. the expected shortfall). Under some additional assumptions, we explicitly characterize cost- efficient claims in closed-form, thereby extending the results of Ghossoub. Finally, a numerical example is provided to illustrate our results in full detail.