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Risk measures for derivatives with Markov-modulated pure jump processes

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  • Robert Elliott
  • Leunglung Chan
  • Tak Siu

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. Copyright Springer Science+Business Media, LLC 2006

Suggested Citation

  • Robert Elliott & Leunglung Chan & Tak Siu, 2006. "Risk measures for derivatives with Markov-modulated pure jump processes," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 13(2), pages 129-149, June.
    • Handle: RePEc:kap:apfinm:v:13:y:2006:i:2:p:129-149
    DOI: 10.1007/s10690-007-9038-9
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    References listed on IDEAS

    as
    1. Robert J. Elliott & Leunglung Chan & Tak Kuen Siu, 2005. "Option pricing and Esscher transform under regime switching," Annals of Finance, Springer, vol. 1(4), pages 423-432, October.
    2. X. Guo, 2001. "Information and option pricings," Quantitative Finance, Taylor & Francis Journals, vol. 1(1), pages 38-44.
    3. Tak Kuen Siu & Howell Tong & Hailiang Yang, 2001. "Bayesian Risk Measures for Derivatives via Random Esscher Transform," North American Actuarial Journal, Taylor & Francis Journals, vol. 5(3), pages 78-91.
    4. Robert Elliott & Carlton-James Osakwe, 2006. "Option Pricing for Pure Jump Processes with Markov Switching Compensators," Finance and Stochastics, Springer, vol. 10(2), pages 250-275, April.
    5. Robert J. Elliott & John van der Hoek, 1997. "An application of hidden Markov models to asset allocation problems (*)," Finance and Stochastics, Springer, vol. 1(3), pages 229-238.
    6. Philippe Artzner & Freddy Delbaen & Jean‐Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228, July.
    7. Robert J. Elliott & Carlton-James U. Osakwe, 2006. "Option Pricing for Pure Jump Processes with Markov Switching Compensators," Finance and Stochastics, Springer, vol. 10(2), pages 250-275, April.
    8. Robert Elliott & Tak Siu & Leunglung Chan, 2008. "A PDE approach for risk measures for derivatives with regime switching," Annals of Finance, Springer, vol. 4(1), pages 55-74, January.
    9. Elliott, R. J. & Malcolm, W. P. & Tsoi, Allanus H., 2003. "Robust parameter estimation for asset price models with Markov modulated volatilities," Journal of Economic Dynamics and Control, Elsevier, vol. 27(8), pages 1391-1409, June.
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    Cited by:

    1. Robert Elliott & Tak Siu, 2010. "On risk minimizing portfolios under a Markovian regime-switching Black-Scholes economy," Annals of Operations Research, Springer, vol. 176(1), pages 271-291, April.
    2. Siu, Tak Kuen, 2016. "A functional Itô’s calculus approach to convex risk measures with jump diffusion," European Journal of Operational Research, Elsevier, vol. 250(3), pages 874-883.
    3. Siu, Tak Kuen, 2008. "A game theoretic approach to option valuation under Markovian regime-switching models," Insurance: Mathematics and Economics, Elsevier, vol. 42(3), pages 1146-1158, June.

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