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Option Pricing and Filtering with Hidden Markov-Modulated Pure-Jump Processes

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  • Robert J. Elliott
  • Tak Kuen Siu

Abstract

This article discusses the pricing of derivatives in a continuous-time, hidden Markov-modulated, pure-jump asset price model. The hidden Markov chain modulating the pure-jump asset price model describes the evolution of the hidden state of an economy over time. The market model is incomplete. We employ a version of the Esscher transform to select a price kernel for valuation. We derive a valuation formula for European options using a Fourier transform and the correlation theorem. This formula depends on the hidden Markov chain. It is then estimated using a robust filter of the chain.

Suggested Citation

  • Robert J. Elliott & Tak Kuen Siu, 2013. "Option Pricing and Filtering with Hidden Markov-Modulated Pure-Jump Processes," Applied Mathematical Finance, Taylor & Francis Journals, vol. 20(1), pages 1-25, March.
    • Handle: RePEc:taf:apmtfi:v:20:y:2013:i:1:p:1-25
    DOI: 10.1080/1350486X.2012.655929
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    Citations

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    Cited by:

    1. Gradojevic Nikola, 2016. "Multi-criteria classification for pricing European options," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 20(2), pages 123-139, April.
    2. Godin, Frédéric & Trottier, Denis-Alexandre, 2021. "Option pricing in regime-switching frameworks with the Extended Girsanov Principle," Insurance: Mathematics and Economics, Elsevier, vol. 99(C), pages 116-129.
    3. Donatien Hainaut & Yang Shen & Yan Zeng, 2018. "How do capital structure and economic regime affect fair prices of bank’s equity and liabilities?," Annals of Operations Research, Springer, vol. 262(2), pages 519-545, March.
    4. Tak Kuen Siu & Robert J. Elliott, 2019. "Hedging Options In A Doubly Markov-Modulated Financial Market Via Stochastic Flows," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(08), pages 1-41, December.
    5. Shen, Yang & Siu, Tak Kuen, 2013. "Stochastic differential game, Esscher transform and general equilibrium under a Markovian regime-switching Lévy model," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 757-768.
    6. Fan, Kun & Shen, Yang & Siu, Tak Kuen & Wang, Rongming, 2015. "Pricing annuity guarantees under a double regime-switching model," Insurance: Mathematics and Economics, Elsevier, vol. 62(C), pages 62-78.
    7. Dong-Mei Zhu & Jiejun Lu & Wai-Ki Ching & Tak-Kuen Siu, 2019. "Option Pricing Under a Stochastic Interest Rate and Volatility Model with Hidden Markovian Regime-Switching," Computational Economics, Springer;Society for Computational Economics, vol. 53(2), pages 555-586, February.
    8. Godin, Frédéric & Lai, Van Son & Trottier, Denis-Alexandre, 2019. "Option pricing under regime-switching models: Novel approaches removing path-dependence," Insurance: Mathematics and Economics, Elsevier, vol. 87(C), pages 130-142.
    9. Jun Yu, 2014. "Optimal Asset-Liability Management for an Insurer Under Markov Regime Switching Jump-Diffusion Market," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 21(4), pages 317-330, November.
    10. Mauricio Junca & Rafael Serrano, 2014. "Utility maximization in pure-jump models driven by marked point processes and nonlinear wealth dynamics," Papers 1411.1103, arXiv.org, revised Sep 2015.
    11. Robert Elliott & Tak Siu, 2015. "Asset Pricing Using Trading Volumes in a Hidden Regime-Switching Environment," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 22(2), pages 133-149, May.
    12. Oscar Lopez & Rafael Serrano, 2014. "Martingale approach to optimal portfolio-consumption problems in Markov-modulated pure-jump models," Papers 1406.3112, arXiv.org.
    13. Lin, Shih-Kuei & Peng, Jin-Lung & Chao, Wei-Hsiung & Wu, An-Chi, 2016. "The extension from independence to dependence between jump frequency and jump size in Markov-modulated jump diffusion models," The North American Journal of Economics and Finance, Elsevier, vol. 37(C), pages 217-235.
    14. Wei-han Liu, 2023. "Attaining stochastic optimal control over debt ratios in U.S. markets," Review of Quantitative Finance and Accounting, Springer, vol. 61(3), pages 967-993, October.
    15. Siu, Tak Kuen, 2023. "European option pricing with market frictions, regime switches and model uncertainty," Insurance: Mathematics and Economics, Elsevier, vol. 113(C), pages 233-250.
    16. Donatien Hainaut & Yan Shen & Yan Zeng, 2016. "How do capital structure and economic regime affect fair prices of bank's equity and liabilities?," Post-Print hal-01394133, HAL.
    17. Farzad Fard & Ning Rong, 2014. "Pricing and managing risks of ruin contingent life annuities under regime switching variance gamma process," Annals of Finance, Springer, vol. 10(2), pages 315-332, May.
    18. Robert J. Elliott & Tak Kuen Siu, 2023. "Hedging options in a hidden Markov‐switching local‐volatility model via stochastic flows and a Monte‐Carlo method," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 43(7), pages 925-950, July.
    19. Siu, Tak Kuen, 2016. "A self-exciting threshold jump–diffusion model for option valuation," Insurance: Mathematics and Economics, Elsevier, vol. 69(C), pages 168-193.

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