Valuation and Optimal Decision for Perpetual American Employee Stock Options under a Constrained Viscosity Solution Framework

BIAN Baojun; YUAN Quan; HU Shuntai

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Valuation and Optimal Decision for Perpetual American Employee Stock Options under a Constrained Viscosity Solution Framework

BIAN Baojun 1 *,YUAN Quan 2,HU Shuntai 2

1.Department of Mathematics, Tongji University, Shanghai 200092

2.Department of Mathematics, Tongji University, ShangHai 200092

*Correspondence author

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Funding: This work was supported by National Science Foundation（No.No.11071189,No.71090404）

Opened online:11 January 2013

Accepted by: none

Citation: BIAN Baojun,YUAN Quan,HU Shuntai.Valuation and Optimal Decision for Perpetual American Employee Stock Options under a Constrained Viscosity Solution Framework[OL]. [11 January 2013] http://en.paper.edu.cn/en_releasepaper/content/4511967

This paper is concerned with the constrained viscosity solution of the Hamilton-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 fluid model with a capped exercise rate. The value of the ESOs defined as the maximal expectation of the discounted exercise benefits satisfies 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 infinity is also studied followed by a numerical simulation model. The corresponding optimal exercise decision is determined by the value function of this optimization problem.

Keywords:HJB equation; constrained viscosity solution; comparison principal; Employee stock options; optimal decision; numerical simulation

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