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On Financial Markets Based on Telegraph Processes

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  • Nikita Ratanov
  • Alexander Melnikov

Abstract

The paper develops a new class of financial market models. These models are based on generalized telegraph processes: Markov random flows with alternating velocities and jumps occurring when the velocities are switching. While such markets may admit an arbitrage opportunity, the model under consideration is arbitrage-free and complete if directions of jumps in stock prices are in a certain correspondence with their velocity and interest rate behaviour. An analog of the Black-Scholes fundamental differential equation is derived, but, in contrast with the Black-Scholes model, this equation is hyperbolic. Explicit formulas for prices of European options are obtained using perfect and quantile hedging.

Suggested Citation

  • Nikita Ratanov & Alexander Melnikov, 2007. "On Financial Markets Based on Telegraph Processes," Papers 0712.3428, arXiv.org.
    • Handle: RePEc:arx:papers:0712.3428
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    File URL: http://arxiv.org/pdf/0712.3428
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    References listed on IDEAS

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    1. Nikita Ratanov, 2007. "A jump telegraph model for option pricing," Quantitative Finance, Taylor & Francis Journals, vol. 7(5), pages 575-583.
    2. Tomas Björk & Henrik Hult, 2005. "A note on Wick products and the fractional Black-Scholes model," Finance and Stochastics, Springer, vol. 9(2), pages 197-209, April.
    3. Brennan, Michael J. & Schwartz, Eduardo S., 1976. "The pricing of equity-linked life insurance policies with an asset value guarantee," Journal of Financial Economics, Elsevier, vol. 3(3), pages 195-213, June.
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    Cited by:

    1. Bogachev, Leonid & Ratanov, Nikita, 2011. "Occupation time distributions for the telegraph process," Stochastic Processes and their Applications, Elsevier, vol. 121(8), pages 1816-1844, August.
    2. Antonio Di Crescenzo & Barbara Martinucci, 2013. "On the Generalized Telegraph Process with Deterministic Jumps," Methodology and Computing in Applied Probability, Springer, vol. 15(1), pages 215-235, March.
    3. Oscar Lopez & Rafael Serrano, 2014. "Martingale approach to optimal portfolio-consumption problems in Markov-modulated pure-jump models," Papers 1406.3112, arXiv.org.
    4. Anatoliy A. Pogorui & Anatoliy Swishchuk & Ramón M. Rodríguez-Dagnino, 2021. "Transformations of Telegraph Processes and Their Financial Applications," Risks, MDPI, vol. 9(8), pages 1-21, August.

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