Journal of Applied Mathematics and Stochastic Analysis 
Volume 2007 (2007), Article ID 72326, 19 pages
doi:10.1155/2007/72326
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

Abstract

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.