An option pricing model based on jump telegraph processes

Abstract

A new class of financial market models is developed. These models are based on generalized telegraph processes: Markov random flows with alternating velocities and jumps occurring when the velocities are switching. While such markets may admit an arbitrage opportunity, the model under consideration is arbitrage-free and complete if directions of jumps in stock prices are in a certain correspondence with their velocity and interest rate behaviour. An analog of the Black-Scholes fundamental differential equation is derived, but, in contrast with the Black-Scholes model, this equation is hyperbolic. Explicit formulas for prices of European options are obtained using perfect and quantile hedging.

A jump telegraph model for option pricing

“…Appendix contains the exact formulas for the distributions of the underlying processes, which are necessary for the call option price formula. This paper exploits the ideas presented by the author on the 2nd Nordic-Russian Symposium on Stochastic Analysis [14] and continues the author's previous paper devoted to the homogeneous telegraph model [15]. known [6], [7]- [8] and they are called the telegraph and integrated telegraph processes respectively.…”

Option pricing model based on a Markov-modulated diffusion with jumps

“…However, a presence of jumps and/or diffusion components has not been estimated. Nevertheless, an implied volatility with respect to a moneyness variable in stochastic volatility models of the Ornstein-Uhlenbeck type (see Nicolato and Venardos (2003)) looks very similar to the volatility smile in jump telegraph model (see Ratanov (2007b)).…”