The pricing of callable perpetual American options

In this paper we deal with game option introduced by Kifer which is a contract that the buyer and the seller have both rights to exercise and to cancel it at any time, respectively. Since game option can be canceled or called by the seller, it is a callable American option. First, we review some results on the pricing of non-callable perpetual American options to emphasize a comparison with our results. Secondly, we derive the value function of the callable perpetual option by solving differential equation and investigate the optimal boundaries of the seller and the buyer. The value function is not differentiable at the stock price which equals strike price. Finally, when the stock pays continuously dividends with a positive rate, we can obtain the pricing formula of callable American perpetual options by applying first hitting time approach of Brownian motion. Also some numerical results are presented to demonstrate analytical properties of the value function.