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.
The closed form expression for the price of the British put and call options have longbeen established where both interest rate and volatility are assumed to be constant. In reality,these assumptions do not fully reflect the variable nature of the financial markets. In this paper, wederived a closed form expression for the arbitrage-free price of the British call option by assumingstochastic interest rate which follows the Cox-Ingersoll-Ross model and constant volatility as
where the first term is the arbitrage-free price of the European call option under stochastic interestrate and the second term is the early-exercise premmium. We have also shown that the pricefunction of the British call option satisfies the partial differential equation given by
Moreover, we have shown that the contract drift satisfies μc < rt+ρσ1σ2√rtλ(0, t+u) for u ∈ [0, τ ] and t ∈ [0, T].
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