Risk measures for derivatives with Markov-modulated pure jump processes

Robert J. Elliott, Leunglung Chan, Tak Kuen Siu

Research output: Contribution to journalArticlepeer-review

7 Citations (Scopus)


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. © Springer Science+Business Media, LLC 2007.

Original languageEnglish
Pages (from-to)129-149
Number of pages21
JournalAsia-Pacific Financial Markets
Issue number2
Publication statusPublished - Jun 2006


  • American options
  • Coherent risk measures
  • Combined optimal stopping and control
  • Esscher transform
  • Exotic options
  • HJB-variational inequalities
  • Jump risk
  • Pure jump processes
  • Regime-switching HJB equations


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