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Pricing Options and Variance Swaps in Markov-Modulated Brownian Markets

In: Hidden Markov Models in Finance

Author

Listed:
  • Robert J. Elliott

    (University of Calgary)

  • Anatoliy V. Swishchuk

    (University of Calgary)

Abstract

Summary A Markov-modulated market consists of a riskless asset or bond, B, and a risky asset or stock, S, whose dynamics depend on Markov process x. We study the pricing of options and variance swaps in such markets. Using the martingale characterization of Markov processes, we note the incompleteness of Markov-modulated markets and find the minimal martingale measure. Black-Scholes formulae for Markov-modulated markets with or without jumps are derived. Perfect hedging in a Markov-modulated Brownian and a fractional Brownian market is not possible as the market is incomplete. Following the idea proposed by Föllmer and Sondermann [13] and Föllmer and Schweizer [12]) we look for the strategy which locally minimizes the risk. The residual risk processes are determined in these situations. Variance swaps for stochastic volatility driven by Markov process are also studied.

Suggested Citation

  • Robert J. Elliott & Anatoliy V. Swishchuk, 2007. "Pricing Options and Variance Swaps in Markov-Modulated Brownian Markets," International Series in Operations Research & Management Science, in: Rogemar S. Mamon & Robert J. Elliott (ed.), Hidden Markov Models in Finance, chapter 4, pages 45-68, Springer.
    • Handle: RePEc:spr:isochp:978-0-387-71163-8_4
    DOI: 10.1007/0-387-71163-5_4
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    Citations

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    Cited by:

    1. Zhengyuan Gao & Christian M. Hafner, 2019. "Looking Backward and Looking Forward," Econometrics, MDPI, vol. 7(2), pages 1-24, June.
    2. Subhojit Biswas & Diganta Mukherjee & Indranil SenGupta, 2020. "Multi-asset Generalised Variance Swaps in Barndorff-Nielsen and Shephard model," Papers 2011.13474, arXiv.org.
    3. Subhojit Biswas & Diganta Mukherjee, 2019. "A Proposal for Multi-asset Generalised Variance Swaps," Papers 1908.03899, arXiv.org.
    4. Yonit Barron, 2022. "A probabilistic approach to the stochastic fluid cash management balance problem," Annals of Operations Research, Springer, vol. 312(2), pages 607-645, May.
    5. Anatoliy Swishchuk, 2013. "Modeling and Pricing of Swaps for Financial and Energy Markets with Stochastic Volatilities," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 8660, January.
    6. David Saunders & Luis Seco & Markus Senn, 2020. "Price of liquidity in the reinsurance of fund returns," Papers 2011.13268, arXiv.org.
    7. Shen, Yang & Siu, Tak Kuen, 2013. "Pricing bond options under a Markovian regime-switching Hull–White model," Economic Modelling, Elsevier, vol. 30(C), pages 933-940.
    8. Barron, Yonit, 2023. "A stochastic card balance management problem with continuous and batch-type bilateral transactions," Operations Research Perspectives, Elsevier, vol. 10(C).

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