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Backward stochastic difference equations for dynamic convex risk measures on a binomial tree

Part of: Mathematical finance Stochastic analysis

Published online by Cambridge University Press:  30 March 2016

Robert J. Elliott ,
Tak Kuen Siu  and
Samuel N. Cohen
Robert J. Elliott*
Affiliation:
University of Adelaide and University of Calgary
Tak Kuen Siu*
Affiliation:
Macquarie University and City University London
Samuel N. Cohen*
Affiliation:
University of Oxford
*
∗ Postal address: School of Mathematical Sciences, University of Adelaide, SA 5005, Australia. Email address: [email protected]
∗∗ Postal address: Department of Applied Finance and Actuarial Studies, Faculty of Business and Economics, Macquarie University, Sydney, NSW 2109, Australia.
∗∗∗ Postal address: Mathematical Institute, University of Oxford, Andrew Wiles Building, Woodstock Road, Oxford OX2 6GG, UK.
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Abstract

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Using backward stochastic difference equations (BSDEs), this paper studies dynamic convex risk measures for risky positions in a simple discrete-time, binomial tree model. A relationship between BSDEs and dynamic convex risk measures is developed using nonlinear expectations. The time consistency of dynamic convex risk measures is discussed in the binomial tree framework. A relationship between prices and risks is also established. Two particular cases of dynamic convex risk measures, namely risk measures with stochastic distortions and entropic risk measures, and their mathematical properties are discussed.

Keywords

Dynamic convex risk measureconditional nonlinear expectationbinomial treebackward stochastic difference equationstochastic distortion probability

MSC classification

Primary: 91G20: Derivative securities
Secondary: 91G80: Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems) 60H07: Stochastic calculus of variations and the Malliavin calculus
Type
Research Papers
Information
Journal of Applied Probability , Volume 52 , Issue 3 , September 2015 , pp. 771 - 785
Copyright
Copyright © 2015 by the Applied Probability Trust 

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