Valuation Techniques: Discounted Cash Flow, Earnings Quality, Measures of Value Added, and Real Options

COMPARISON OF L, M, AND μ MODELS

To carry out our comparison of how the three model types perform in valuing ESOs, we begin with how the use of a “representative investor” approach affects the models’ performances. We then use a numerical example to illustrate the potentially different ordering of prices obtained from the models, present a more general analysis, and measure the accuracy of the model outputs against an objective option value.

Calculation of Model Prices for a Representative Investor

ESO models commonly use the notion of a representative investor, whose characteristics (e.g., the parameter L or M) are then assumed to hold for all the employees to whom a particular option tranche is granted. This approach produces biases in the L model and the M model, however, because they are concave in, respectively, L and M.

In the context of an L model, for heterogeneous employees (i.e., those with different L’s), the model price of the average (i.e., representative) employee is strictly greater than the average of the model prices of each employee. The price of a four-year option and an eight-year option combined is not the same as the price of two six-year options, even though many ESO-pricing methodologies implicitly assume that it is.

This same effect is even more striking, quantitatively, when employee behaviors are described by M. Consider the following scenario: The parameters are S0 = X = 1, r = 5 percent, T = 10, d = 0, q = 0, v = 0, and σ = 40 percent. Employee A has ...