Journal of Applied Mathematics and Stochastic Analysis
Volume 2007 (2007), Article ID 72326, 19 pages
Research Article

Jump Telegraph Processes and Financial Markets with Memory

Nikita Ratanov

Faculty of Economics, Universidad del Rosario, Calle 14, No.4-69, Bogotá, Colombia

Received 21 November 2006; Revised 22 April 2007; Accepted 9 August 2007

Copyright © 2007 Nikita Ratanov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The paper develops a new class of financial market models. These models are based on generalized telegraph processes with alternating velocities and jumps occurring at switching velocities. The model under consideration is arbitrage-free and complete if the directions of jumps in stock prices are in a certain correspondence with their velocity and with the behaviour of the interest rate. A risk-neutral measure and arbitrage-free formulae for a standard call option are constructed. This model has some features of models with memory, but it is more simple.