受约束的粘性解框架下一揽子永久美式股票.doc
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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
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- 约束 粘性 框架 一揽子 永久 美式 股票
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