Transformations of Telegraph Processes and Their Financial Applications

Anatoliy A. Pogorui, Anatoliy Swishchuk and Ramón M. Rodríguez-Dagnino
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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

Risks, 2021, vol. 9, issue 8, 1-21

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.

Keywords: classical telegraph equation; transformations of telegraph equation; asymmetric telegraph equation; Black–Scholes formula; European call and put options (search for similar items in EconPapers)
JEL-codes: C G0 G1 G2 G3 K2 M2 M4 (search for similar items in EconPapers)
Date: 2021
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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