•Deriving the buyer’s and seller’s exercise regions.•Using American style derivatives with a stochastic maturity.•Deriving the equations for the exercise boundaries.•Finding the fair game call option price.•Giving some numerical examples. The purpose of this paper is to examine the problem of pricing discounted perpetual game call options. In addition to the properties of the American options, the game options give the seller the right to cancel the contract at some chosen from him moment. As a compensation for this, he has to pay some amount above the usual payment. We assume that this penalty payment is a constant. We examine the case without maturity – the exercise can be made in every future moment. We first derive the optimal exercise regions for the buyer and the seller and then calculate the fair option price. Our approach is based on some American style derivatives with a stochastic maturity date.