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We introduce a notion of $k$th order stochastic monotonicity and duality thatallows one to unify the notion used in insurance mathematics (sometimesrefereed to as Siegmund's duality) for the study of ruin probability and theduality responsible for the so-called put - call symmetries in option pricing.Our general $k$th order duality can be financially interpreted as put - callsymmetry for powered options. The main objective of the present paper is todevelop an effective analytic approach to the analysis of duality leading tothe full characterization of $k$th order duality of Markov processes in termsof their generators, which is new even for the well-studied case of put -callsymmetries.

Stochastic monotonicity and duality of $k$th order with application to put-call symmetry of powered options