Accounting for Employee Stock Options: Accelerating Convergence

Abstract

Hull and White (2004) have developed a lattice pricing model that makes explicit reference to parameters that are not available in Black Scholes (1973) yet are important for the valuation of Employee Stock Options (ESOs). Cvitanic, Wiener and Zapatero (2008) point out that a key weakness of the lattice approach, when applied to valuing ESOs, is the sluggish convergence not generally experienced in trees configured to estimate plain vanilla options. Cvitanic, Wiener and Zapatero (2008) propose a very useful closed form solution which may or may not be endorsed by a whole host of regulatory and professional bodies. In this paper, we propose a small refinement to Hull and White (2004), based on insights developed by Boyle and Lau (1994) which ensures faster convergence in lattice estimation when barriers occur. Our model provides estimates consistent with Cvitanic, Wiener and Zapatero (2008). The proposed model should also neatly fit into the rubric currently prescribed by the American Financial Accounting Standards Board (FASB) and endorsed by the Securities and Exchange Commission (SEC). We also apply a number of truncation techniques to the lattice which removes the zero region of the tree. We truncate the ESO lattice above the early exercise boundary, peculiar to Hull White (2004), and dynamically specify the lattice array to spare memory and reduce estimation time.