ROI Question on Lattice vs Black-Scholes option expensing

Hi Achaessa,

 This is a very good question, and I have encountered it a lot throughout my career. Actually, I have written a book about options (investment options rather than employee stock options), and have published a few articles about this topic. This makes the option valuation models a topic that is quite close and dear to my heart.

Now, before answering your question, I would like to look a bit at the history. When Mr. Black, Mr. Scholes and Mr. Merton came up with the Black-Scholes model (BS model), they took the lattice model, aka, the binomial model, and made some assumptions associated with it. Some of these assumptions are that:

(1) The stock price movement is a lognormal distribution based on the expected volatility, expected interest rate and expected dividends.

(2) The time increments in the binomial model will be infinitismally small, which is the ultimate increment.

(3) The only restriction on the binomial tree will be the expiry date of the option, for example, the option holder will not exercise the option until its expiry date.

(4) The only estimations that are needed for the model are the expected interest rate, the expected volatility and expected dividend.

In other words, the BS model is simply a standard binomial model with above mentioned assumptions, and it was specifically designed for market-traded options.

So, how do we use the BS model for employee stock options? The answer for that is simply introducing a factor that will modify the term of the BS standard binomial model (I will call it the BS model going forward) to accommodate the potential exercising or cancellation of the option prior to its nominal expiry date. We can easily calculate this factor (Time to Option Expriy Factor, or TOE Factor) from the historical data. Statistical data from OPTRACK has proved that the correlation of the TOE factors for a single company year over year is statistically significant.

Now, why someone should use a customized binomial model instead of the BS model would be one of three reasons:

• The stock price movement would not follow a lognormal distribution. For example, there are corporate structures that restrict the price of the stock to an upper limit, and mandates distributing dividends beyond a certain income level to maintain the stock price. Another example is related to a company that knows with certainty that the potential increase of the stock is significantly different from the potential decrease of the stock beyond the impact of the expected dividend and interest rate. For such situations, the BS model would not succeed, and we would need a binomial model. Personally, the only time I was able to see such a situation was in the case of Income Trusts, a corporate structure in Canada that allows distributing the earnings prior to the taxes rather than distributing them as dividends.


• The company knows for sure what the blackout periods are in the future throughout the life of the contract. This is quite unlikely because many blackout periods are related to corporate actions that cannot be anticipated.


• The company can estimate with a high level of accuracy a term called: "The Suboptimal Exercise Value"; this is the percentage increase in the stock price at which the majority of the option holders would exercise their options. Most companies using the customized binomial model calculate this value from the historical exercise pattern. Having said that, based on the statistical data in OPTRACK, the correlation between the historical suboptimal exercise values year over year is statistically insignificant. In other words, if a company calculates the suboptimal exercise value based on historical data, the historical data will not be more accurate than a non-scientific guess.

There is a misnomer out there that the binomial model would result in a lower valuation compared to the BS model. Actually because the customized binomial model takes on more parameters than the BS model and these additional parameters cannot be accurately estimated, the value can be either higher or lower based on the selection of these parameters. In other words, the customized binomial model can be either higher or lower based on the selection of the parameters.

Actually, if someone thinks of using the customized binomial model instead of the BS model just for the purpose of reducing their expense, they can achieve that objective by selecting a lower TOE Factor in the BS model, as long as they can justify their selection to the auditors.  

There is also an additional cost associated with the binomial model which is related to the audit fees, which is usually forgotten.

Just a final statistic: Although OPTRACK supports both the customized binomial model and the BS model, over 98% of our clients are using the BS model.

Please feel free to call me or e-mail me to further discuss or if you have any questions; my direct line is 416.258.1376.

Ramy R. Taraboulsi, P.Eng, M.Sc., MBA, CFA

CEO, SyncBASE Inc.
Text/Mobile: 4162581376@pcs.rogers.com
Tel: 213.232.2727 x 101

RTaraboulsi@OPTRACK.net

http://www.SyncBASE.com

Skype ID: syncbase

Hi John,

Yes, this is absolutely correct. Let me try to address this via a modified BS model which uses the TOE Factor for estimating the expected term:

If the participants fully understand the mechanics of options investing, they would not exercise their options, unless the company is distributing dividends, before the nominal expiry date of the option; If they exercise it prior to that date, again, assuming that the company does not distribute dividends, then they would be throwing away the time value of money associated with the option.

As participants hold on to their grants for a longer time, the historical TOE Factor will be going up, and accordingly the expected term in the BS calculation will also be going up. This will result in a higher cost to the company.

Nothing comes from nothing: The knowedgeable employee gets a higher value by holding on to their options, and this value ultimately comes from the company.

I hope that this addresses your question.

Thanks.

Ramy.