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.
Highlights
IntroductionIn 1951, Goldstein and Kac (see Goldstein (1951); Kac (1974) and Kac (1950)). proposed an interesting random motion model for the movement of a particle on the line (or one dimension)
Accepted: 5 August 2021In 1951, Goldstein and Kac (see Goldstein (1951); Kac (1974) and Kac (1950))proposed an interesting random motion model for the movement of a particle on the line
The novelty of the paper consists of new results for transformed classical telegraph process, new models for stock prices, and new applications of these models to option pricing
Summary
In 1951, Goldstein and Kac (see Goldstein (1951); Kac (1974) and Kac (1950)). proposed an interesting random motion model for the movement of a particle on the line (or one dimension). Variety of transformations of telegraph process and its association with many areas were studied by Orsingher; see Orsingher (1985); Orsingher and Beghin (2009); Orsingher and De Gregorio (2007); Orsingher and Ratanov (2002); Orsingher and Somella (2004); Orgingher (1990) They include hyperbolic equations, fractional diffusion equations, random flights, planar and cyclic random motions, among others. The main idea of application of telegraph process in finance is the following one: Instead of the geometric Brownian motion (GBM) we propose the following model for the price St of a stock at time t : Ste = S0 exp( x (t)), Rt where x (t) = 0 v(s)ds, v(t) is a continuous-time Markov chain with state space (v1 , v2 ), and with λi being the rates of the exponential waiting times, i = 1, 2.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
