Stochastic Monotonicity and Duality ofkth Order with Application to Put-Call Symmetry of Powered Options

Abstract

We introduce a notion ofkth order stochastic monotonicity and duality that allows us to unify the notion used in insurance mathematics (sometimes refereed to as Siegmund's duality) for the study of ruin probability and the duality responsible for the so-called put-call symmetries in option pricing. Our generalkth order duality can be interpreted financially as put-call symmetry for powered options. The main objective of this paper is to develop an effective analytic approach to the analysis of duality that will lead to the full characterization ofkth order duality of Markov processes in terms of their generators, which is new even for the well-studied case of put-call symmetries.