Employee Stock Option Valuation – Black Scholes vs. Monte Carlo Simulation

December 2018 – Abhishek Mathews, CBV, CFA

We have done, countless, stock option valuations for both public and private companies. In the vast majority of cases, we have used “simple” black Scholes model as opposed to the more complex Monte Carlo Simulation. As such, we discuss a case where Black Scholes, in our experience, was not applicable and how we deviated based on best practices of a couple of CBV’s and ASA’s, we spoke to, to model out the results.

We term black scholes as “simple” as any finance student, in university, should be able to produce a stock option value so why would the courts or private entities require a business appraisal firm or a business valuation firm to value stock options?

FASB 123R states that when options are granted, they have to be valued using an option pricing model. Business valuators & business appraisers can pick between binomial lattice models, Black Scholes and Monte Carlo simulations to value these options. The models can be adjusted to reflect the specific characteristics of employee options and a company can use different option pricing models to value different option grants.

As it relates to employee stock options, Appendix B of FAS 123 discusses the value of employee stock options in some detail. It proposes a three-step valuation procedure:

  1. Estimate the expected life of the option.
  2. Use either of:
    1. the Black and Scholes (1973) model; or
    2. the Cox, Ross, and Rubinstein (1979) binomial tree to value the option, with the expected life as the time to maturity.
  3. Adjust the value to allow for the possibility of the employee leaving the company during the vesting period.

Any Canadian or U.S. Stock Option Valuation, which we have been involved with, definitely considered the three above steps.

Based on our experience, we believe that projecting forward risk free rates and volatility and exercise behavior five to ten years out, as in the case of the Binomial model, is difficult to predict say the least. Accordingly, we, typically, revert to the Black-Scholes model, relative to the Binomial model, if the valuation time frame is greater than 2 years. Furthermore, with lower inputs and estimates, Black-Scholes model has a lower risk than the Binomial model over a longer duration, all things equal.

The conventional Black Scholes model is designed to value European options on traded assets and does not explicitly factor in the dilution inherent in employee options or the illiquidity / vesting issues specific to these options.

The Black-Scholes model has the following key inputs:

  1. Exercise price;
  2. Market price;
  3. The risk free rate;
  4. The dividend yield;
  5. Volatility; and
  6. Expected term or life of the option.

The Black-Scholes model is based on the following assumptions:

  1. There are no riskless arbitrage situations;
  2. The stock price follows a constant Brownian motion (with m and s constant);
  3. There are no transaction costs or taxes and all securities are perfectly divisible;
  4. There are no dividends during the life of the option or warrant;
  5. Short selling with full use of proceeds is permitted;
  6. Security traded is continuous for both the option and the stock; and
  7. The risk-free rate of return is constant and the same for all maturities.

We note that none of these assumption hold perfectly in real-world situations, yet, these work well enough for liquid stock options on actively traded stocks that many market participants including hedge funds, traders, and investment bankers.

In our situation, we were asked to value an employee stock option for a wrongful dismissal / wrongful termination claim; We will not get into the nuances, of a shortened tranche and impact on volatility, thereon. However, we will focus on our results and how we had to deviate away from a traditional Black Scholes model to the more complicated Monte Carlo Simulation to arrive at a more logical conclusion for the wrongful dismissal / wrongful termination matter.

The Black – Scholes formula assumes that log share prices follow a continuous normal distribution. Share prices follow a lognormal distribution – as a consequence log share prices will follow a normal distribution. Equivalently we can say that log share returns follow a normal distribution as a consequence share returns will follow a lognormal distribution. Hence it doesn’t matter if we talk of lognormal share prices or normal log share returns, as these are just two sides of the same assumption.

The Black-Scholes model also punishes a deeply in the money stock option, as it assume that there is a negative time value of money. Without getting into too much theoretical jargon, the Black-Scholes, while simple to use, is not always the best valuation model to use when valuing employee stock options.

Our, proprietary, Monte Carlo Simulations allowed us the flexibility of adding the following, in addition to the standard variables in the Black Scholes model, for the atypical valuation/damages assignments:

  1. Distribution type;
  2. Simulation trial runs (e.g. 1,000 etc.); and
  3. Steps in Place.

 

At Minerva Valuation Advisors, we are trained, as business valuators, appraisers, bankers and financial analysts, in a variety of settings including in tax planning valuations, family law valuations, shareholder disputes valuations and merger & acquisitions valuations and advisory. We are not your typical business valuator or appraiser, we attempt to connect the dots using a variety of perspectives to give you some of our best insights.

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