Employee Stock Options: An Up-and-Out Protected Barrier Call

Abstract

A well-known numerical lattice model, widely used to value employee stock options (ESOs), can be interpreted as a variation on the up-and-out protected barrier call, a version of which is valued in closed form by Carr (1995). We clarify that valuation formula and extend it to take account of the reality of possible vesting date exercise by employees.

Key Words:

• Employee stock options
• up-and-out protected barrier call

Acknowledgements

Brisley acknowledges support from Social Sciences and Humanities Research Council of Canada standard research grant #410-2007-1564.

Notes

¹ Heynen and Kat (1994) address the valuation of barrier options when the barrier is not active over the option's entire life.

² Henderson (2005) is one of several papers to examine ESO valuation and incentives from the employee's point of view and contains a discussion and further references.

³ Financial Accounting Standards Board Statement FAS123R makes compulsory the expensing of all ESOs in the US, effective for fiscal years commencing after 15 June 2005. Similar rules have been implemented by the International Accounting Standards Board. FAS123R permits several methodologies but specifically cites the Hull and White (2004) model and illustrates its use with numerical examples, as does Securities Exchange Commission Staff Accounting Bulletin #107.

⁴ The error originates from the original proof in Appendix A3 (p. 200) of Carr (1995). There, a probability decomposition is written However, the right-hand side should read As a result the third line, should read Equation (A3) should end which results in (A4) becoming resulting in