ESOs: Using the Binomial Model
By David Harper
On April 1, 2004, the Financial Accounting Standards Board (FASB) published a proposal on the new accounting treatment of employee stock options ESOs. The final rules will probably be issued sometime in the fall of 2004. But the final rules will most likely resemble the proposal: FASB has rejected - clearly to its own satisfaction - the most visible and obvious criticisms of the proposal to "expense" stock options.
Currently, most companies use the Black-Scholes options-pricing model to price their ESOs. The new rules, however, encourage - but do not require - companies to use the binomial model. We can therefore expect companies to shift to the binomial in the next annual report season. In this section, we explain the idea behind the binomial model.
The Binomial Builds a Tree of Future Stock Prices
The Black-Scholes is a closed-form model, which means it solves for, or 'deduces', an option's price from an equation. In contrast, the binomial is an open-form or lattice model. It creates a tree of possible future stock-price movements and 'induces' the option's price. Let's start with a single-step binomial. Assume we grant an option on a $10 stock that will expire in one year. We also assume there is a 50% chance that the price will jump 12% over the year and a 50% chance that the stock will drop 12%.
There are three basic calculations. First, we plot the two possible future stock prices. Second, we translate the stock prices into future options values: at the end of the year, this option will be worth either $1.20 or nothing. Third, we discount the future values into a single present value. In this case, the $1.20 discounts to $1.14 because we assume a 5% risk-less rate. After we weight each possible outcome by 50%, the single-step binomial says our option is worth $0.57 at grant.
A full-fledged binomial simply extends this one-step model into a random walk of many steps (or intervals). As such, calculating the binomial involves the same three basic actions. First, the tree of possible future stock prices is constructed, and the volatility input determines the magnitude of each up or down jump. Second, the future stock prices are translated into option values at each interval on the tree. Third, these future option values are discounted back to a single present value. This third step is called backward induction.
Backward induction simply starts with the final options values and works backward through a series of one-step mini-models. For example, the options value for Su4 above (the next-to-last value at the top of the tree) is just a weighted blend of the two final nodes that come after it. And Su3 becomes a weighted blend of the Su4 and Su2, and so on until the model converges to a single option value - in present-value terms - at the front of the tree.
The Binomial Tree Values an American-Style Option with Flexibility
A big advantage of the binomial is that it can value an American-style option, which can be exercised before the end of its term, and it is the style of option ESOs usually take. The model achieves this valuation capacity by comparing the calculated value at each node (as above) to the intrinsic value at that node. In the few cases where intrinsic value is greater, the model assumes the option is worth the intrinsic value at the node. This has the overall effect of increasing the value of the American-style option relative to a European-style option, as some of the nodes are increased.
You can see that the binomial is a brute-force model that can be constructed with almost unlimited flexibility. The FASB prefers the binomial model because it can build-in the unique features of an ESO.
Consider two key features that the FASB recommends companies build into the binomial model: vesting restrictions and early exercise.
The binomial tree above is the same as before, except with two differences. First, because the option is un-vested in the early years, the model does not assume any early exercises during these years (which would be done to redeem high intrinsic values in the upward jumping paths). Second - and this is a key difference - the binomial allows for an exercise factor. FASB calls this a "suboptimal exercise factor". An exercise factor of 2x, for example, allows the model to assume that employees will exercise the option if the stock price increases to double (2x) the exercise price. The idea behind this factor is simply to anticipate early exercise of in-the-money options under favorable circumstances. If the exercise factor is triggered, the option is assumed to be exercised, and the binomial tree basically stops on that node.
You can see these two features reduce the value of the option, all other things being equal. The un-vested section of the model limits the value at each node to the discounted value of the two future nodes (even where the intrinsic value is greater and would therefore be normally used instead). The exercise factor eliminates additional value that could accrue to the option if it were to continue to ride the upward trajectory.
The New Accounting Rule Favors the Binomial
The proposed accounting rule (amended SFAS 123) favors the binomial for pricing ESOs. As companies shift from the Black-Scholes to the binomial, there are four key differences in the valuation methods to note:
Keep in mind that ESOs are far less liquid than traded options as an employee cannot sell his or her option on a public exchange. You may recall that the Black-Scholes handles this with a band-aid solution: companies use a reduced 'expected life' instead of the full 10-year term as an input into the Black-Scholes. Because the binomial model already builds-in these illiquidity factors through the vesting restrictions and early exercise assumptions, the binomial accepts the full 10-year term as an input.
The binomial contains more assumptions than the Black-Scholes. Some have argued that the binomial will produce dramatically lower expense estimates than the Black-Scholes, but this is not necessarily the case.
Switching from Black-Scholes to binomial can slightly increase, maintain or decrease the options expense. Certainly if a company sets an aggressively low exercise factor like 1.25x (which would assume employees will exercise their options when the stock is 25% above the exercise price), then the binomial will produce a lower estimate of value. On the other hand, if all of the inputs are unchanged and the exercise factor is high, options' value under the binomial may increase because it incorporates the additional value of American-style ESOs, which can be exercised early.
Of course a company can also try to bring about a lower value by tweaking the inputs as it switches models. For example, shifting from 40% volatility under Black-Scholes to a volatility range of 20% to 40% under the binomial is likely to produce a lower options value. But, in this example, the real cause for a lower value is not a change in options-pricing models so much as reduction in average volatility from 40-30%.
Below we compare the Black-Scholes value to the binomial value for an option on a $100 stock. We've used the same volatility for both models, so the primary valuation difference is reduced to (1) the expected-life input used in the Black-Scholes compared to (2) the exercise factor used in the binomial. Other variables matter, of course, but this is the key difference between the models when the same volatility is used. You can see that, when you put everything together, the binomial could be higher, lower or similar to the Black-Scholes.
This and the previous section of this feature summarize two different approaches to estimating the fair value of an ESO at the time it is granted. Under the proposed rules, this fair value must be recognized as an expense on income statements with fiscal years starting after Dec 15, 2004.
If there were a public market or exchange for trading ESOs, the company could and would use market prices. Lacking that, the binomial model represents an attempt to fine-tune the theoretically correct fair value of an ESO given its unique features. However, it is just an attempt to capture fair value at grant, in light of future uncertainty. The ultimately realized cost of the option will depend on the future stock-price trajectory, which is likely to diverge from the fair value.
ESOs: Dilution - Part 1
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