In this study, we investigate the effect of expiration on the outputs of an option game problem. For this purpose, we solve a problem with finite expiration. Many previous studies have investigated the option game approach, and most of them consider infinite expiration in order to enable the problem to be solved analytically. However, there exist several situations in which it is necessary to consider finite expiration, e. g. a patent licensing agreement. Therefore, in this study, we focus on a model based on the case of a patent licensing agreement for a licenser and perform calculations for optimizing the agreement under the condition of finite expiration. We also determine the effect of sudden death risk on the outputs of an option game problem, which, in this case, are optimal terms for the licenser. By numerical experiments, we find that both expiration and sudden death risk have a strong influence on the optimal terms.