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

Baojun Bian; Shuntai  Hu; Quan  Yuan; Harry  Zheng

doi:10.3934/dcds.2015.35.5413

Article Contents

2015, Volume 35, Issue 11: 5413-5433. Doi: 10.3934/dcds.2015.35.5413

This issue Previous Article Backward stochastic Schrödinger and infinite-dimensional Hamiltonian equations Next Article Some linear-quadratic stochastic differential games for equations in Hilbert spaces with fractional Brownian motions

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
3.
Department of Mathematics, Imperial College, London SW7 2AZ

Received: September 30, 2013

Revised: October 31, 2014

Published: October 2015

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Abstract

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.

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Mathematics Subject Classification: Primary: 35J25, 35J70; Secondary: 35Q91, 35Q93.

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