On Financial Markets Based on Telegraph Processes

TL;DR

This paper introduces a novel class of financial market models based on generalized telegraph processes, deriving a hyperbolic differential equation and explicit option pricing formulas, expanding the theoretical framework of market modeling.

Contribution

It develops a new telegraph process-based market model that is arbitrage-free and complete, with explicit European option pricing formulas and a hyperbolic analog of Black-Scholes.

Findings

Model admits arbitrage but can be arbitrage-free under certain conditions

Derived a hyperbolic differential equation similar to Black-Scholes

Obtained explicit formulas for European option prices

Abstract

The paper develops a new class of financial market models. 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.