Pricing Options under Telegraph Processes

In this paper we introduce a financial market model based on continuous time random motions with alternating constant velocities and jumps, which occur with velocity switches. Given that jump directions match velocity directions of the underlying random motion properly in relation to interest rates, in this setting will be free of arbitrage. Additionally, we suppose also the interest rate depending on the market state. The replicating strategies for options are constructed in detail, and closed form formulas for option prices are obtained.