(untitled) (European Journal of Pure and Applied Mathematics, 2023)
In a plain vanilla option, its holder is given the right, but not the obligation, to buy or sell the underlying stock at a specified price (strike price) at a predetermined date. If the exercise date is at maturity, the option is called a European; if the option is exercised anytime prior to maturity, it is called an American. In a British option, the holder can enjoy the early exercise feature of American option whereupon his payoff is the ‘best prediction’ of the European payoff given all the information up to exercise date under the hypothesis that the true drift of the stock equals a specified contract drift. In this paper, in contrast to the constant interest rate and constant volatility assumptions, we consider the British option by assuming that the economic state of the world is described by a finite state continuous-time Markov chain. Also, we provide a solution to a free boundary problem by using PDE arguments. However, closed form expression for the arbitrage-free price are not available in our setting.