Jump telegraph processes and a volatility smile

Abstract

We continue to study financial market models based on generalized telegraph processes with alternating velocities. The model is supplied with jumps occurring at the times of velocity switchings. This model 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 rates. A risk-neutral measure and arbitrage-free formulae for a standard call option are constructed. A new version of convergence under suitable scaling to the Black-Scholes model is proved, and the explicit limit is obtained. Next, we examine numerically the explicit formulae for call prices to obtain the behaviour of implied volatilities. Moreover, this model has some features of models with memory. The historical volatility of jump telegraph model is similar to historical volatility of the moving average type model.