Annals of Finance

, Volume 4, Issue 1, pp 55–74 | Cite as

A PDE approach for risk measures for derivatives with regime switching

  • Robert J. Elliott
  • Tak Kuen Siu
  • Leunglung Chan
Research Article


This paper considers a partial differential equation (PDE) approach to evaluate coherent risk measures for derivative instruments when the dynamics of the risky underlying asset are governed by a Markov-modulated geometric Brownian motion (GBM); that is, the appreciation rate and the volatility of the underlying risky asset switch over time according to the state of a continuous-time hidden Markov chain model which describes the state of an economy. The PDE approach provides market practitioners with a flexible and effective way to evaluate risk measures in the Markov-modulated Black–Scholes model. We shall derive the PDEs satisfied by the risk measures for European-style options, barrier options and American-style options.


Risk measures Regime-switching PDE Regime-switching HJB equation Stochastic optimal control Esscher transform Delta-neutral hedging Jump risk American options Exotic options 

JEL Classification Numbers

G32 G13 


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Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  • Robert J. Elliott
    • 1
  • Tak Kuen Siu
    • 2
  • Leunglung Chan
    • 3
  1. 1.Haskayne School of Business, Scurfield HallUniversity of CalgaryCalgaryCanada
  2. 2.Department of Actuarial Mathematics and Statistics, School of Mathematical and Computer Sciences and the Maxwell Institute for Mathematical SciencesHeriot-Watt UniversityEdinburghUK
  3. 3.Department of Mathematics and StatisticsUniversity of CalgaryCalgaryCanada

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