Optimal Exercise Strategies Of The Perpetual Executive Stock Options With Block Exercise And Unrestricted Exercise

Executive stock options are called ESOs for short, which are American call optionsthat the company granted to the managers or employees as a compensation. Sincethe mid1980s, ESOs have become an important part of executive compensation inthe United States and other countries. 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. Because company executives can not short sell companystock (illegal), the executives can not construct the portfolio composed of companystocks, ESOs and risk-free bonds to hedge the risk of ESOs. In fact, it is an incompletemarket, which makes the no-arbitrage pricing method invalid for the pricing of ESOs.In this paper, we mainly discuss the definite problem of a parabolic variational in-equality which comes from the study of the optimal exercise strategies for the perpetualESOs in financial market. The existence, uniqueness and regularity of the correspond-ing solution as well as some properties of the free boundary and its limit behaviorare established in a rigorous theoretical framework. Our research is divided into twoaspects: perpetual ESOs with block exercise and perpetual ESOs with unrestrictedexercise. Block exercise is that the executive can exercise none of the ESOs or exerciseall of the ESOs at the same time. While in the unrestricted exercise situation, theexecutive can exercise any copies of ESOs at arbitrary implementation moment.First of all, by use of the executive’s wealth utility maximization method, weinvestigate the pricing model of the perpetual ESOs with block exercise. It is a freeboundary problem of the perpetual American call options with a utility function. Thecorresponding value function and the optimal exercise policy are not only related tothe stock price, but also to the number of ESOs. The value function is the solution tothe definite problem of a degenerate parabolic variational inequality. There has been a lot of interest in the problem of ESOs with block exercise. However, most of theliterature only give a qualitative analysis or numerical calculation without a rigoroustheoretical research. Kadam et.al(2005)[14]give the analytical solution to the value ofthe perpetual ESOs in the utility-based model. However, they only consider one ESOs,and the corresponding value function is only related to the stock price. Because theutility function is a nonlinear function, the total value of the ESOs that the executiveholds is not a linear function of the number of ESOs. Thus, the assumption that theexecutive only holds one ESOs is unreasonable. We establish a optimal stopping modelfor the perpetual ESOs with block exercise based on the wealth utility maximizationmethod. The corresponding value function is a binary function of the number of ESOsand the stock price. By the theory of optimal stopping, we derive that the valuefunction satisfies a degenerate parabolic variational inequality. And then by usingslicing method, i.e. the variable τ (the number of ESOs multiplied by the risk aversioncoefcient) discrete method, we study the definite problem of the parabolic variationalinequality. We get the existence, uniqueness and regularity of the solution and provethe continuity, monotonicity and limit behavior of the free boundary (the optimalexercise boundary).Secondly, we study the model of the perpetual ESOs with unrestricted exercise.By use of the method of maximizing the expected utility of the executive’s wealth,Rogers and Scheinkman(2007)[49]have established a stochastic optimal control modelfor the ESOs with finite-horizon and obtained the definite problem of the correspondingparabolic variational inequality. However, they did not give a rigorous theoreticalstudy on this definite problem, and only provided the executive’s optimal strategyand the corresponding properties through numerical analysis. Based on the model ofRogers and Scheinkman (2007)[49], we study the corresponding model of the ESOs withinfinite-horizon. By stochastic optimal control theory, we get the value function whichsatisfies a definite problem of a degenerate parabolic variational inequality. Becausethe obstacle condition of the variational inequality contains the partial derivative of thevalue function to the variable τ (the number of ESOs multiplied by the risk aversioncoefcient), it makes the theoretical study of the definite problem of the the variationalinequality very difcult. We also turn to the slicing method to investigate this problem,and prove the existence, regularity and some other properties of the solution.Finally, numerical analysis is given for both block exercise and unrestricted ex- ercise. By use of the numerical solution method of partial diferential equations, weanalyze and compare the optimal exercise strategies of the executive in two exercisecases.