The magazine of the Institute & Faculty of Actuaries
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# Calculating fair value

P lacing a value on share-based compensation is a
relatively new area for actuaries but is one that
is ideally suited to our core skills applying financial and statistical theory to model uncertainty. So, where for a pension scheme valuation, assumptions need to be made about investment return, salary increases, and pension increases, for a share plan valuation read risk-free rate of interest, share price volatility, and dividend yield. Similarly, whereas mortality rates affect how long we expect a pension to be paid, with share plans it is exercise behaviour that will determine how long an employee holds onto an option.
In both their paper and SIAS presentation, the authors make it clear that the starting point for their work in this area is the Black-Scholes option pricing formula, which dates back to the early 1970s. This is a simple-to-use formula that places a value on an option to buy or sell a share at a given price (the exercise price) on a given date. Although this formula is still used by option traders today, it does have certain restrictions when it comes to valuing share-based compensation for employees. In particular, it does not allow for performance conditions and exercise windows, and so much of the authors’ work has concentrated on extending the Black-Scholes approach to allow for these features, both of which are commonly seen in employee share plans.
It is important to realise that the models that the authors have developed are an extension of Black-Scholes rather than a replacement, since they are based on the same core mathematical principles. An analogy here might be the Mini of the 1970s the authors have kept the same chassis and engine but added in power steering, four-wheel drive, and ABS. So, while the basic model is unchanged, and will get you to the same destination on a straight road in good driving conditions, the revamped version allows you to go off-road and cope with ice.

Performance conditions
Unlike a short-term traded option, a large number of employee share plans have performance conditions that need to be satisfied in order for an award to vest. For example, an award will only vest if the company’s total shareholder return beats the median in the FTSE100. A simplistic (but incorrect) way to value such a share plan would be to multiply the Black-Scholes option value by the probability of passing the test. So, if the Black-Scholes value were £30, the value in the above example would be £15, since there is a 50/50 chance of the company beating the median and, therefore, passing the performance test. The problem with this approach is that it ignores the fact that the chance of passing the test is correlated to the option payout the models which the authors have developed therefore incorporate this feature (the mathematics is set out in the paper) and, in this example, might lead to a value of, say, £20 to £25 as opposed to £15.
Perhaps a slightly different example might better illustrate the impact of correlation. Imagine you asked a bookmaker at the start of the football season for the odds on the team who finishes third at the end of the season having a positive goal difference. In such circumstances, while it would be reasonable to believe there will be a 50/50 chance that any of the teams in the division will have a positive or negative goal difference, the team finishing third are much more likely to have a positive goal difference. (This logic does, of course, not apply to the Scottish Premier League.) Scotland aside, I doubt that you will find many bookmakers who would be generous or stupid enough to offer even money odds on this particular bet (or, if there are, they will probably have gone out of business by the time you try and pick up your winnings at the end of the season).

Exercise behaviour
Most employee option plans give the holder the choice over when to exercise their options. For a save-as-you-earn scheme there will typically be a six-month window in which exercise can occur whereas, for an executive option plan, the window may be as long as seven years. Once again, this is something that the Black-Scholes formula is unable to cope with because it assumes that exercise occurs on a fixed date. The authors have, therefore, developed a lattice model. This is a tree diagram of future share price movements, with each branch splitting to represent upwards or downwards share price movements, which enables future exercise behaviour to be modelled. Historical analysis suggests that, while a proportion of option holders will always exercise each year irrespective of share price, exercise behaviour does tend to be closely correlated with the gain made (ie a high share price will lead to a large number of employees cashing their options in). The lattice that the authors have developed therefore enables modelling to be carried out based on the share price relative to exercise price. For example, if the share price is 33% (say) above the exercise price then one-half of option holders are assumed to exercise each year.