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Hidden Markov Models in Finance pp 45–68Cite as

Pricing Options and Variance Swaps in Markov-Modulated Brownian Markets

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Part of the book series: International Series in Operations Research & Management Science ((ISOR,volume 104))

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.

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Author information

Authors and Affiliations

  1. Haskayne School of Business, University of Calgary, 2500, University Drive NW, Calgary, Alberta, Canada, T2N 1N4

    Robert J. Elliott

  2. Haskayne School of Business, University of Calgary, 2500, University Drive NW, Calgary, Alberta, Canada, T2N 1N4

    Anatoliy V. Swishchuk

Authors
  1. Robert J. Elliott
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  2. Anatoliy V. Swishchuk
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Correspondence to Robert J. Elliott .

Editor information

Editors and Affiliations

  1. University of Western Ontario, London, Canada

    Rogemar S. Mamon

  2. University of Calgary, Calgary, Canada

    Robert J. Elliott

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© 2007 Springer Science+Business Media, LLC

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Elliott, R.J., Swishchuk, A.V. (2007). Pricing Options and Variance Swaps in Markov-Modulated Brownian Markets. In: Mamon, R.S., Elliott, R.J. (eds) Hidden Markov Models in Finance. International Series in Operations Research & Management Science, vol 104. Springer, Boston, MA. https://doi.org/10.1007/0-387-71163-5_4

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