Optimal Strategy Of ESO

In this dissertation, we systematically study the optimal exercise strategy of per-petual Executive Stock Options (ESOs) under continuous-exercise model. First of all,based on the model of Rogers and Scheinkman [70], under a utility function, we estab-lish a stochastic control model, regarding the remaining number of ESOs as the control,for the ESOs with fnite maturity. Then a variational inequality (or HJB equations)can be derived in viscosity sense from a standard stochastic control theory. Next, wenaturally defne the perpetual ESOs as the limit of fnite horizon case. We fnd thatthe limit exists when α <r+σ2/2, where r, α and σ are discount rate, the company’sstock expected return and volatility, respectively. Then we focus on the perpetualESOs model under exponential utility (i.e. U(x)=e γxwhere γ>0is risk aver-sion). Based on the solution of the corresponding variational inequality, we constructthe optimal exercise strategy of the perpetual ESOs. Furthermore, using the optimalexercise strategy, the ESOs’ cost to the company can be approximatively calculated.As the beginning, we consider the following two special cases:(1) stock sales, i.e.,the strike price K=0;(2) discount rate r=0, in which optimal exercise strategy aswell as the cost to the frms can be presented in closed-form.In addition, according to the construction of the optimal exercise strategy, whichonly depends on the free boundary arising from the variational inequality, we turn tostudy the corresponding variational inequality in the sequel. Due to the relationshipbetween the three parameters r, α and σ, we split the problem into three scenarios:(1) r> α,(2) r=α,(3) r <α <r+σ2/2.For the case r> α, based on the idea in [Q4], by diferentiating the value functionwith respect to option’s number twice, we can convert the original variational inequalityinto a Stefan type free boundary problem. To avoid the degenerate issue at the origin,we study the free boundary problem for ε>0by carefully picking the initial andboundary values to satisfy the compatibility conditions. Then by sending ε to zero, weget a function sequence which converges to the unique classical solution to the originalproblem. Additionally, we get the C∞regularity of the free boundary.Next, we use a semi-discretization method to deal with the rest two cases. Diferent from the method used by Song and Yu (2011)[74], we semi-discrete the equation ofthe marginal value function. Since the problem degenerates at the origin, we need tocarefully choose the initial time and values such that the semi-discretization problemadmits a unique solution and bounds the free boundary at every time layer. Then wecan obtain the solution to the original problem by taking the limit of the sequence.Also, the decreasing and continuous properties of the free boundary are derived.Finally, using the strict “convexity” of the variational inequality at the origin, theasymptotic behavior of the optimal exercise strategy is studied when the number ofESOs goes to zero. In addition, the formal asymptotic expansion of the free boundaryis obtained with the help of some key transformations.