A Model Of The Executive Stock Options With The Dilution Effect

In the paper, we mainly discuss a model of the executive stock options (ESOs) with the dilution effect. First of all, we use Rogers and Scheinkman(2007)’s framework through the executive’s utility maximization, we have established a stochastic optimal control model for the perpetual executive stock options considering the dilution effect and obtain the corresponding parabolic variational inequality,then give their financial significance.Secondly, we give the expression of the optimal exercise strategy and discuss the property of the strategy. Then we prove the verification theorem:the solution of the variational inequality problem is the value function of the stochastic optimal control model under some smooth conditions. Besides, we prove the uniqueness of the optimal exercise strategy.Finally, we consider a special situation:the risk-free interest rate r=0. We intro-duce some properties of the free boundary and the fixed solution problem of ordinary differential equation with the free boundary satisfied. Then we give the expression of the solution of the above parabolic variational inequalities.