Pricing Option And EIAs When Discrete Dividends Follow A Markov-modulated Jump Diffusion Model

In recent years,there has been considerable interests in applications of regime switching models driven by a hidden Markov chain to various financial problems.The regime switching can be interpreted as substantial changes in the state or condition of the underlying economy,which are modeled by a continuous-time Markov process,including the adjustment in the economical structure,the evolution of the market mode as well as the cycle changes in the economy and business.Then the parameters of the risk assets's value process are functions of the Markovian chain.If the economical state is fixed,the value process just follows one model.Otherwise,it will follow different models as the state of the economy changes.Meanwhile,stock prices are typically modeled directly,without referring to the economical value of the dividend payments by possessing the stock.Even though the dividends were taken into account,they are always considered to be paid continuously.However,in reality,the dividend payments are paid discretely.This is a very real problem,but there has so far been no entirely satisfactory way to handle it,especially when the dividend payments are stochastic and discrete.From this point of view,following the dividend discounted model and arbitrage pricing theory(APT)which implies that the price of a stock should be equal to the net present value of its all future dividend payments,this article proposes a Markov-modulated jump diffusion model for dividend process.Using the generalized Ito for-mula,we obtain the explicit solution for dividend payments of the model.From ar-bitrage pricing theory(APT),the stock price process is then deduced.Since the dis-counted stock price process is a martingale between the adjacent dividend payment time,via the Risk-neutral pricing method,we obtain the explicit solutions of the Eu-ropean option and the point-to-point EIA valuation.