Abstract:
A new class of financial market models is proposed. These models are based on continuous time random motions with alternating constant velocities c± (so called "telegraph" process) and with jumps h± 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. In the framework of this model we capture bullish and bearish trends in a market evolution. Values h± describes sizes of possible crashes, jumps and spikes. Thus, we study a model that is both realistic and general enough to enable us to incorporate different trends and extreme events. We construct financial market model based on the random processes with finite velocities which possess a simplicity of Black-Scholes model. Replicating strategies for European options are constructed in detail. Explicit formulae for option prices are obtained. Some peculiarities as memory effects and a detailed description of volatility are discussed also.