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November  2015, 35(11): 5413-5433. doi: 10.3934/dcds.2015.35.5413

## Constrained viscosity solution to the HJB equation arising in perpetual American employee stock options pricing

 1 Department of mathematics, Tongji University, Shanghai 200092 2 Department of Mathematics, Tongji University, Shanghai 200092, China, China 3 Department of Mathematics, Imperial College, London SW7 2AZ, United Kingdom

Received  October 2013 Revised  November 2014 Published  May 2015

We consider the valuation of a block of perpetual ESOs and the optimal exercise decision for an employee endowed with them and with trading restrictions. A fluid model is proposed to characterize the exercise process. The objective is to maximize the overall discount returns for the employee through exercising the options over time. The optimal value function is defined as the grant-date fair value of the block of options, and is then shown by the dynamic programming principle to be a continuous constrained viscosity solution to the associated Hamilton-Jacobi-Bellman (HJB) equation, which is a fully nonlinear second order elliptic partial differential equation (PDE) in the plane. We prove the comparison principle and the uniqueness. The numerical simulation is discussed and the corresponding optimal decision turns out to be a threshold-style strategy. These results provide an appropriate method to estimate the cost of the ESOs for the company and also offer favorable suggestions on selecting right moments to exercise the options over time for the employee.
Citation: Baojun Bian, Shuntai Hu, Quan Yuan, Harry Zheng. Constrained viscosity solution to the HJB equation arising in perpetual American employee stock options pricing. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5413-5433. doi: 10.3934/dcds.2015.35.5413
##### References:
 [1] B. Bian, M. Dai, L. Jiang, Q. Zhang and Y. Zhong, Optimal decision for selling an illiquid stock, J. Optim. Theory Appl., 151 (2011), 402-417. doi: 10.1007/s10957-011-9897-0. [2] J. Carpenter, The exercise and valuation of executive stock options, J. Financial Economics, 48 (1998), 127-158. [3] M. G. Crandall, H. Ishii and P. L. Lions, A user's guide to viscosity solutions, Bulletin Amer. Math. Soc., 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5. [4] W. H. Fleming and R. Rishel, Deterministic and Stochastic Optimal Control, Applications of Mathematics, No. 1, Springer-Verlag, Berlin-New York, 1975. [5] B. J. Hall and K. J. Murphy, Stock option for undiversified executives, J. Accounting Economics, 33 (2002), 3-42. doi: 10.1016/S0165-4101(01)00050-7. [6] J. Ingersoll, The subjective and objective evaluation of incentive stock options, J. Business, 79 (2006), 453-487. [7] A. Jain and A. Subramanian, The intertemporal exercese and valuation of employee options, Accounting Review, 79 (2004), 705-743. doi: 10.2308/accr.2004.79.3.705. [8] L. Jiang, Mathematical Modeling and Methods of Option Pricing, World Scientific Publishing Co., Inc., River Edge, NJ, 2005. doi: 10.1142/5855. [9] R. Lambert, D. Larchker and R. Verrecchia, Portfolio considerations in valuing executive compensation, J. Accounting Research, 29 (1991), 129-149. doi: 10.2307/2491032. [10] T. Leung and R. Sircar, Accounting for risk aversion, vesting, job termination risk and multiple exercises in valuation of employee stock options, Math. Finance, 19 (2009), 99-128. doi: 10.1111/j.1467-9965.2008.00359.x. [11] L. C. G. Rogers and J. Scheinkman, Optimal exercise of executive stock options, Finance Stoch., 11 (2007), 357-372. doi: 10.1007/s00780-007-0041-9.

show all references

##### References:
 [1] B. Bian, M. Dai, L. Jiang, Q. Zhang and Y. Zhong, Optimal decision for selling an illiquid stock, J. Optim. Theory Appl., 151 (2011), 402-417. doi: 10.1007/s10957-011-9897-0. [2] J. Carpenter, The exercise and valuation of executive stock options, J. Financial Economics, 48 (1998), 127-158. [3] M. G. Crandall, H. Ishii and P. L. Lions, A user's guide to viscosity solutions, Bulletin Amer. Math. Soc., 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5. [4] W. H. Fleming and R. Rishel, Deterministic and Stochastic Optimal Control, Applications of Mathematics, No. 1, Springer-Verlag, Berlin-New York, 1975. [5] B. J. Hall and K. J. Murphy, Stock option for undiversified executives, J. Accounting Economics, 33 (2002), 3-42. doi: 10.1016/S0165-4101(01)00050-7. [6] J. Ingersoll, The subjective and objective evaluation of incentive stock options, J. Business, 79 (2006), 453-487. [7] A. Jain and A. Subramanian, The intertemporal exercese and valuation of employee options, Accounting Review, 79 (2004), 705-743. doi: 10.2308/accr.2004.79.3.705. [8] L. Jiang, Mathematical Modeling and Methods of Option Pricing, World Scientific Publishing Co., Inc., River Edge, NJ, 2005. doi: 10.1142/5855. [9] R. Lambert, D. Larchker and R. Verrecchia, Portfolio considerations in valuing executive compensation, J. Accounting Research, 29 (1991), 129-149. doi: 10.2307/2491032. [10] T. Leung and R. Sircar, Accounting for risk aversion, vesting, job termination risk and multiple exercises in valuation of employee stock options, Math. Finance, 19 (2009), 99-128. doi: 10.1111/j.1467-9965.2008.00359.x. [11] L. C. G. Rogers and J. Scheinkman, Optimal exercise of executive stock options, Finance Stoch., 11 (2007), 357-372. doi: 10.1007/s00780-007-0041-9.
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