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1、精品论文受约束的粘性解框架下一揽子永久美式股票期权的定价及最优策略研究 边保军 ，袁泉 ，胡顺泰 同济大学理学院，上海 200092摘要：本文研究了一个 Hamilton-Jacobi-Bellman (HJB) 方程的受约束的粘性解，该 HJB 方程来自于给一揽子永久美式股票期权 (ESOs) 的定价模型。首先假定员工的瞬时实施率有上 限，定义 ESOs 的价值为累积贴现实施收益的最大期望，推导出它满足的 HJB 方程，在验证 了比较原理之后，证明了该 HJB 方程具有唯一受约束的粘性解，且由这个解可以确定对应的 最优实施策略。之后讨论了当实施率上限趋向无穷时的极限情况，最后利用数值计算方法
2、求出 了近似解。关键词：HJB 方程，受约束的粘性解，比较原理，员工股票期权，最优策略，数值模拟中图分类号： O29Valuation and Optimal Decision for Perpetual American Employee Stock Options under a Constrained Viscosity Solution FrameworkBIAN Baojun , YUAN Quan , HU ShuntaiDepartment of Mathematics, Tongji University, Shanghai 200092Abstract: This paper
3、 is concerned with the constrained viscosity solution of theHamilton-Jacobi-Bellman (HJB) equation arising from the valuation of a block of perpetual American employee stock options (ESOs). The exercise process of the employee is described by a uid model with a capped exercise rate. The value of the
4、 ESOs dened as the maximal expectation of the discounted exercise benets satises the HJB equation. The existence and uniqueness of the constrained viscosity solution is obtained after the proof of the associated comparison principal. The limit case with the cap value approaching innity is also studi
5、ed followed by a numerical simulation model. The corresponding optimal exercise decision is determined by the value function of this optimization problem.Key words: HJB equation, constrained viscosity solution, comparison principal, Employeestock options, optimal decision, numerical simulation.基金项目：
6、 This work was supported by National Science Foundation(No.11071189, No.71090404)作者简介： Bian Baojun （1962-），male，professor，major research direction：partial dierential equation, nancialmathematics. Correspondence author：Yuan Quan（1985-），female，PhD，major research direction：nancial mathematics. Hu Shunt
7、ai（1987-），male，PhD，major research direction：nancial mathematics.- 32 -0 IntroductionIn recent years, employee stock options (ESOs in short) have been extensively used by companies as a form of compensation or reward to the employees globally. An employee stock option is usually a call option issued
8、by a company on its common stock, granting the holder a right to buy a certain number of shares of the underlying stock at a predetermined price, often called the strike price during a certain period of time. In most cases, this period lasts several years. When the stock price goes up, the holder ca
9、n exercise the options to buy stock at the strike price and then sell the shares at the market price, thereby keeping the dierence as prot. Obviously the employee stock options serve as an incentive, encouraging the employees to strive for the benets of the company, boosting the stock price so that
10、they can get more prot from exercising these options.With the cost of ESOs becoming increasingly signicant to the companies in the past decades, since 2004 it has been required by the Financial Accounting Standards Board (FASB) that all the companies should estimate and report the grant-date fair va
11、lue of the ESOs issued, which gives rise to a desire for a reasonable method to evaluate the ESOs. Meanwhile the employees need directions in exercising so as to make the maximal prots. Consequently the discussion about the valuation and related optimal strategy has become a focus in mathematical re
12、search of nance, thereby covered by an extensive literature.Furthermore, its worth pointing out that compared to the standardized exchange-traded options, ESOs have several unique features in dierent aspects (see 1). In general, ESOs are American-style call options, i.e. they can be exercised at any
13、 time before expiration, with a long maturity ranging from 5 to 10 years, which much exceeds that of the standardized options. In addition, for the most part they involve a vesting period from the grant date, during which employees are prohibited from exercising any of the options, in order to maint
14、ain their incentive eect for the nancial benets of the company. On top of that, the transfer and hedging restrictions are also remarkable features which need handling with care. In most cases, employees are forbidden either to transfer ESOs, or to short sell the company stock to hedge against his po
15、sitions in those options. Hence they should exercise the ESOs before expiration or just leave them worthless at expiration, leading to an appealing for instructions on how to work out the optimal strategy in order to maximize the returns through exercising over time. Besides, other prominent feature
16、s include job termination risk, i.e. the risk of getting red or leaving the company voluntarily in the duration of the ESOs, and a list of exible contract items. In conclusion, all these features result in the non-standardized ESOs operating in an incomplete market, which causes the failure of the c
17、ommon valuation methods dealing with pricing options in a complete market.So far a variety of approaches have been proposed to get insight into this problem and have gained fruitful results in this eld.Earlier researches(see 2345) are devoted to studying the optimal exercise strategy under the assum
18、ption that the employee would exercise the whole block of options at a single date. In this case, the optimal strategy is independent of the quantity of options she holds, which turns out to contradict the empirical evidence in which employees prefer distributed exercising over time, rather than at
19、a single date. By virtue of utility function measuring personal risk preference, 6 established a multi-period model to examine the exercise policy for a risk-averse employee under the discrete time framework. In 7, Rogers made use of numerical examples based on utility-based models to illustrate the
20、 optimal exercise boundary which relies on a group of factors, particularly the number of options being held.In this paper, we consider an employee who is granted a block of perpetual ESOs, namely she can exercise the options at any time from the grant date on. Further we suppose she is prohibited f
21、rom trading on the underlying stock and impose a restriction on her instant exercise rate, which make her face an incomplete market. The stochastic optimal control approach is applied to evaluate the block of ESOs and accordingly nd the optimal exercise policy for the employee.Treating the number of
22、 options as continuous, we adopt a uid model to characterize the exercise process and restrict the exercise rate not to exceed an upper bound. Its justied by the common perspective of companies that if the employee exercised a large quantity of options in a short period of time, the market stock pri
23、ce would probably be depressed thus doing harm to the company.We set our goal as maximizing the expected overall discount returns through exercising the options over time for the employee. This desired optimum value denes the value function in our optimization problem.To our knowledge, all existing
24、literature concerning ESOs including the aforementioned aim at maximizing the employees expected accumulated utility attained by exercising the options, thereby leading to the associated optimal exercise policy based on the employees risk preference. However, the unique feature of our model is that
25、instead of pursuing utility maximization as in most literature, we target at maximizing the overall discount exercise returns, which naturally can be regarded as the grant-date fair value of the block of ESOs. As a result, with this optimization problem solved, the value of the ESOs and the correspo
26、nding optimal exercise strategy can be determined at the same time.In fact, the optimization process is terminated once the employee has exercised all her options, which means the state of this process has constraint. So we shall study the value function under the constrained viscosity solution fram
27、ework.The rest of the paper is organized as follows. In section 1, we shall formulate our model to characterize the valuation process as a stochastic optimal control problem, give denition to the value function and obtain the associated HJB equation. In section 2, the value function isshown to be th
28、e constrained viscosity solution of the HJB equation. The comparison principle enables us to further conrm the uniqueness of the solution. Later in section 3, we exploit a numerical simulation method to obtain the approximation of the value function and thereby determine the optimal policy which eme
29、rges in threshold style. Afterwards, more numerical examples are presented to illustrate the impact of varying parameters on the optimal policy, accompanied by some reasonable nancial explanations. By use of these results, we provide the company an appropriate estimated cost of the ESOs and meanwhil
30、e favorable suggestions for the employee on how to get most out of the ESOs through wise exercising over time. Section5 concludes this paper with our future interests and possible extensions of this model.1 Problem FormulationLet Xt denote the stock price of the company at time t, following a geomet
31、ric Brownian motiondXt = Xt dt + Xt dWt ,X0 = x (1)where positive constants , represent the expected stock return rate and volatility respectively, and Wt is a standard Brownian motion.Consider an employee who is granted a total number N shares of perpetual AmericanESOs with the strike price K at ti
32、me 0.We use a uid model to describe her exercise process. Let Yt denote the aggregated number of options she has exercised up to time t, which is driven by the following dierential equationdYt = ut dt, Y0 = 0(2)where the exercise rate ut is our control variable, restricted in the control set = 0, wi
33、thconstant 0. Obviously, Yt t0 is a non-negative non-decreasing right-continuous process. Let S = (0, ) (0, N ). Then at any time t 0, ), the pair of state variables (Xt , Yt ) shouldbelong to the state space S.Denition 1: We say a control u() is admissible with respect to the initial value (x, y) S
34、if and only if (i) u() is Ft = Xs : s t adapted; (ii) u(t) for all t 0; (iii)thecorresponding state process (Xt , Yt ) S for all t 0. Denote by A = A(x, y) the set of alladmissible controls.The admissibility requires that for all t 0, the control can only take values in the controlset , depending on
35、 the available information up to time t, rather than the indeterminatefuture, and meanwhile guarantees the state (Xt , Yt ) S, especially Yt N . In fact, once Yattains N at time , it results in ut = 0 for all t .+Throughout this paper, we assume the employee is prohibited from trading the companys s
36、tock which disables her from hedging the risk involved in the options held, which puts her in an incomplete market.In the sequel, we turn to the stochastic optimal control approach, aiming at maximizing the employees expected overall discounted-to-zero returns through exercising the options overtime
37、. Hence the objective function is dened byJ (x, y; u.) = E 0et (Xt K )ut dt | Xt = x, Yt = y G, G 0(3)where is the discount rate, satisfying 0 and G+ =.0, G 0Here serves as a time scale factor, aecting the time horizon of exercising the whole blockof options. Moreover, larger encourages quicker exer
38、cise actions with permissible exercise rate. Thereby it gives the value functionv(x, y) =supu()A(x,y)J (x, y; u(). (4)By the standard arguments in stochastic analysis theory, seeing 8 for more details, we derive the HJB equation governing the value function v(x, y),which is equivalent to,Lv + max(uB
39、v) = 0,(x, y) 0, ) 0, N (5)uLv + (Bv)+ = 0,(x, y) 0, ) 0, N (6)where 2 2Lv = xvx +x vxx v,(7)2Bv = vy + (x K )+ .(8)The Dirichlet boundary condition naturally follows,v(0, y) = 0,0 y N. (9)2 Value FunctionIn this section, we focus on the value function of this optimization problem. The value functio
40、n is shown to be the constrained viscosity solution of the given HJB equation by the rigorous denition given. Then we use the comparison principle to verify the uniqueness of the viscosity solution. Moreover, we determine the optimal exercise strategy for the employee by virtue of the value function
41、.2.1 Value Function as the Constrained Viscosity SolutionFirst we illustrate some properties of the value function to prepare for the further study.Lemma 1. The fol lowing assertions hold.(i) For each x 0, ), v(x, y) is non-increasing in y;(ii) For each y 0, N , v(x, y) is non-decreasing in x;(iii)
42、v(x, y) is Lipschitz continuous in (x, y). More precisely, we haveX i|v(x1 , y1 ) v(x2 , y2 )| min, 2(N y1 )|x1 x2 | + (2x2 + 1)|y1 y2 |,(10)for any (xi , yi ) 0, ) 0, N (i = 1, 2).Proof. Recall the denition of the value function in (4), +v(x, y) =supEu.A(x,y)et (Xt K )0ut dt.(11)Results in (i)(ii)
43、state the monotonicity of v(x, y) with respect to x, y.In fact, for a certain x 0, ), suppose 0 y1 y2 N and we have A(x, y2 ) A(x, y1 ). Let u. A(x2 , y), then u. A(x1 , y) implyingv(x, y1 ) J (x, y1 ; u.) = J (x, y2 ; u.).Since u. is arbitrary, it follows v(x, y1 ) v(x, y2 ) which means (i) holds.t
44、Similarly, for a certain y 0, N , suppose 0 x1 x2 and we have A(x1 , y) =A(x2 , y). Denote by X i the solution of (1) with initial values X0 = xi for i = 1, 2. Then2( )t+Wt = xi e2with expectations E(X i ) = xi et , i = 1, 2.For any u. A(x1 , y) = A(x2 , y), we havet ,(12)J (x1 , y; u.) = E t0et (X
45、1 K )tut dt = E0 et( 2+x1 e+( 2 )t+W)+ Kt( 2 )t+W)ut dt E0 etx2 e 2 Kut dtt K )= E et (X 2 +0ut dt= J (x2 , y; u.) (13) Due to the arbitrariness of u. , taking the supreme on both sides yields v(x1 , y) v(x2 , y)which justies (ii).We proceed to prove (iii) by showing that v(x, y) is Lipschitz contin
46、uous in both x and y.On one hand, given y 0, N , let x1 , x2 0, ). Without loss of generality, we supposex1 x2 . Then from (13), for any u. A(x1 , y) = A(x2 , y), it follows J (x1 , y; u.) J (x2 , y; u.).Dene a stopping time = inft 0 : t u ds = N y. In fact, u= 0 for all s . Thus0 st (st)+ ()+ J (x2 , y; u.) J (x1 , y; u.) = E0 etX 2 K X 1 Kut dt
