A Class Of Perpetual Executive Stock Options With Unrestricted Exercise

Executive stock options (ESOs hereafter) are American call options granted bya firm to its managers or employees as a form of benefit in addition to salary. Sincethe mid1980s, ESOs have become an important part of executive compensation inboth foreign and domestic companies. ESOs can be seen as an expense to the firmbecause the firm is buying services from the executives. Since ESOs issued by a firmare so large, the cost to the firm is also very impressive. In order to give a reasonablevaluation for ESOs, it is necessary to have rational prediction of the future exercisestrategies of the executives.The meaning of block exercise is that the employees can exercise any more dur-ing the exercise time; While in the unrestricted exercise situation, the executive canexercise any copies of ESOs at arbitrary implementation moment. Song Liping&YuWanghui(2011)(2013)[1]research the model of perpetual American ESOs with blockexercise and unrestricted exercise from the perspective of expected utility maximiza-tion for the employees. They discussed the existence, uniqueness and smoothness ofthe corresponding solution of variational inequality and some other properties.By use of the executive’s wealth utility maximization method, we investigate thepricing model of the perpetual ESOs with unrestricted exercise and block exercise.The model we discuss is diferent from the model researched by Song Liping&YuWanghui(2011)(2013)[1]. In their models, They calculate the total wealth first and thencalculate the utility. However, in this paper, we mainly discuss the problem by calculateone option’s utility first and then sum them, the value function and the number ofoptions are linear relationship. We prove that the value function of the expected utilitymaximum is precisely the viscosity solution of the degenerated parabolic variationalinequality initial value problem. Our proving method can be easily extended to othertypes of executive stock options with unrestricted exercise, such as the problem (1.1).With this result, we also prove that our unrestricted exercise is the same as block exercise.