Derivatives pricing in a Markov chain jump-diffusion setting
In this work we develop a Markov Chain Jump-Diffusion (MCJD) model, where we have a financial market in which there are several possible states. Asset prices in the market follow a generalised geometric Brownian motion, with drift and volatility depending on the state of the market. So for example, one state may represent a bull market where drifts are high, whilst another state may represent a bear market where where drifts are low. The state the market is in is governed by a continuous time Markov chain. We add to this diffusion process jumps in the asset price which occur when the market changes state, and the jump sizes are dependent on the states the market is transiting to and transiting from. We also allow the market to transit to the same state, which corresponds to a jump in the asset price with no change to the drift or volatility. We will develop conditions of no arbitrage in such a market, and methods for pricing derivatives of assets whose prices follow MCJD processes. We will also consider Term-Structure models where the short rate (or forward rate) follows an MCJD process.