Risk measures for derivatives with Markov-modulated pure jump processes |
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Authors: | Robert J. Elliott Leunglung Chan Tak Kuen Siu |
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Affiliation: | (1) Haskayne School of Business, University of Calgary, Calgary, AB, Canada;(2) Department of Mathematics and Statistics, University of Calgary, Calgary, AB, Canada;(3) Department of Actuarial Mathematics and Statistics School of Mathematical and Computer Sciences and the Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh, UK |
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Abstract: | We consider a regime-switching HJB approach to evaluate risk measures for derivative securities when the price process of the underlying risky asset is governed by the exponential of a pure jump process with drift and a Markov switching compensator. The pure jump process is flexible enough to incorporate both the infinite, (small), jump activity and the finite, (large), jump activity. The drift and the compensator of the pure jump process switch over time according to the state of a continuous-time hidden Markov chain representing the state of an economy. The market described by our model is incomplete. Hence, there is more than one pricing kernel and there is no perfect hedging strategy for a derivative security. We derive the regime-switching HJB equations for coherent risk measures for the unhedged position of derivative securities, including standard European options and barrier options. For measuring risk inherent in the unhedged option position, we first need to mark the position into the market by valuing the option. We employ a well-known tool in actuarial science, namely, the Esscher transform to select a pricing kernel for valuation of an option and to generate a family of real-world probabilities for risk measurement. We also derive the regime-switching HJB-variational inequalities for coherent risk measures for American-style options. |
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Keywords: | Coherent risk measures Pure jump processes Esscher transform Jump risk American options Exotic options Regime-switching HJB equations Combined optimal stopping and control HJB-variational inequalities |
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