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

Part of: Mathematical modeling, applications of mathematics Applications Markov processes

Published online by Cambridge University Press:  30 January 2018

Vassili N. Kolokoltsov
Vassili N. Kolokoltsov*
Affiliation:
The University of Warwick
*
∗ Postal address: Department of Statistics, The University of Warwick, Coventry CV4 7AL, UK. Email address: v.kolokoltsov@warwick.ac.uk
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Abstract

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We introduce a notion of kth 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 general kth 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 of kth order duality of Markov processes in terms of their generators, which is new even for the well-studied case of put-call symmetries.

Keywords

Stochastic monotonicitystochastic dualitygenerators of dual processesdual semigroupput-call symmetry and reversalpowered and digital optionsstraddle

MSC classification

Primary: 60J25: Continuous-time Markov processes on general state spaces
Secondary: 62P05: Applications to actuarial sciences and financial mathematics 97M30: Financial and insurance mathematics 60J60: Diffusion processes 60J75: Jump processes
Type
Research Article
Information
Journal of Applied Probability , Volume 52 , Issue 1 , March 2015 , pp. 82 - 101
Copyright
© Applied Probability Trust 

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