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Transformations of Telegraph Processes and Their Financial Applications

Author

Listed:
  • Anatoliy A. Pogorui

    (Department of Mathematics, Zhytomyr Ivan Franko State University, Valyka Berdychivska St. 40, 10008 Zhytomyr, Ukraine)

  • Anatoliy Swishchuk

    (Department of Mathematics & Statistics, University of Calgary, Calgary, AB T2N 1N4, Canada)

  • Ramón M. Rodríguez-Dagnino

    (School of Engineering and Sciences, Tecnologico de Monterrey, Av. Eugenio Garza Sada 2501 Sur. C.P., 64849 Monterrey, Mexico)

Abstract

In this paper, we consider non-linear transformations of classical telegraph process. The main results consist of deriving a general partial differential Equation (PDE) for the probability density (pdf) of the transformed telegraph process, and then presenting the limiting PDE under Kac’s conditions, which may be interpreted as the equation for a diffusion process on a circle. This general case includes, for example, classical cases, such as limiting diffusion and geometric Brownian motion under some specifications of non-linear transformations (i.e., linear, exponential, etc.). We also give three applications of non-linear transformed telegraph process in finance: (1) application of classical telegraph process in the case of balance, (2) application of classical telegraph process in the case of dis-balance, and (3) application of asymmetric telegraph process. For these three cases, we present European call and put option prices. The novelty of the paper consists of new results for non-linear transformed classical telegraph process, new models for stock prices based on transformed telegraph process, and new applications of these models to option pricing.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jrisks:v:9:y:2021:i:8:p:147-:d:616353
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    References listed on IDEAS

    as
    1. Nikita Ratanov, 2007. "A jump telegraph model for option pricing," Quantitative Finance, Taylor & Francis Journals, vol. 7(5), pages 575-583.
    2. Orsingher, Enzo, 1985. "Hyperbolic equations arising in random models," Stochastic Processes and their Applications, Elsevier, vol. 21(1), pages 93-106, December.
    3. Nikita Ratanov, 2008. "Option Pricing Model Based on a Markov-modulated Diffusion with Jumps," Papers 0812.0761, arXiv.org.
    4. Nikita Ratanov, 2008. "Jump Telegraph-Diffusion Option Pricing," UNIMI - Research Papers in Economics, Business, and Statistics unimi-1070, Universitá degli Studi di Milano.
    5. Nikita Ratanov & Alexander Melnikov, 2007. "On Financial Markets Based on Telegraph Processes," Papers 0712.3428, arXiv.org.
    6. Nikita Ratanov, 2004. "Option Pricing Model Based on Telegraph Processes with Jumps," Borradores de Investigación 4330, Universidad del Rosario.
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    Cited by:

    1. Anatoliy A. Pogorui & Anatoliy Swishchuk & Ramón M. Rodríguez-Dagnino, 2022. "Asymptotic Estimation of Two Telegraph Particle Collisions and Spread Options Valuations," Mathematics, MDPI, vol. 10(13), pages 1-14, June.

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