The Black-Scholes model is a formula for the fair value of a European-style option, meaning one that can only be exercised at the end of its life. It uses six inputs: the current share price, the exercise price, the time to expiry, expected volatility, the risk-free interest rate and expected dividends. HMRC's Corporate Finance Manual calls it "the most widely known" option pricing model. In UK practice it matters most when a company accounts for employee share options under FRS 102 Section 26 or IFRS 2. Both standards require options to be expensed at their fair value at grant date.
The formula
For a call option on a share paying a continuous dividend yield q:
C = S·e−qT·N(d1) − K·e−rT·N(d2)
where d1 = [ln(S/K) + (r − q + σ²/2)·T] ÷ (σ·√T), and d2 = d1 − σ·√T.
- S. The current share price;
- K. The exercise (strike) price;
- T. The time to expiry in years. For employee options this is the expected life, not the contractual term;
- σ. The expected annual volatility of the share price;
- r. The continuously compounded risk-free interest rate;
- q. The expected dividend yield;
- N( ). The cumulative standard normal distribution.
In plain terms, the first term is the share value you expect to receive if the option ends in the money. The second is the present value of the exercise price you expect to pay. N(d2) is the risk-neutral probability that the option is exercised.
Intrinsic value and time value
HMRC's manual splits an option's value into two parts. Intrinsic value is the profit from exercising immediately, which can never be negative. Time value is the value of the chance that the share price moves in the holder's favour before expiry. Time value depends on how long the option has to run, how volatile the underlying share is, and market interest rates. An out-of-the-money option has only time value. Black-Scholes prices both parts together. This is why an option granted at the money, with an exercise price equal to today's share price, still has a significant fair value even though its intrinsic value is nil.
Worked example: valuing an employee option grant
An unquoted company grants 10,000 options with an exercise price of £2.00 when the shares are valued at £2.00. The figures below are illustrative assumptions, not market data: expected life 4 years, expected volatility 40%, risk-free rate 4% and no dividends.
| Step | Result |
|---|---|
| d1 = [ln(1) + (0.04 + 0.08) × 4] ÷ (0.40 × 2) | 0.60 |
| d2 = 0.60 − 0.80 | −0.20 |
| N(d1) and N(d2) | 0.7257 and 0.4207 |
| Present value of exercise price: £2.00 × e−0.16 | £1.7043 |
| Option value: £2.00 × 0.7257 − £1.7043 × 0.4207 | £0.734 (about 73p) |
| Total fair value of the grant | about £7,344 |
If the options vest after 3 years of service and all are expected to vest, the company recognises about £2,448 a year as an employee expense, with a matching credit to equity. The model is sensitive to volatility. On the same assumptions, 30% volatility gives about 60p per option and 50% gives about 87p. Volatility is the input most worth documenting.
How FRS 102 and IFRS 2 use it
Under FRS 102, equity-settled options granted to employees are measured at the fair value of the equity instruments at grant date (paragraph 26.8). Where the options vest over a service period, the expense is recognised over that period (paragraph 26.6). Paragraph 26.11 sets a three-tier hierarchy:
- Tier 1. An observable market price for the options, if one exists;
- Tier 2. Entity-specific observable market data, such as a recent transaction in the options;
- Tier 3. An alternative valuation method such as an option pricing model. Its inputs are the share price, exercise price, expected volatility, option life, expected dividends and the risk-free rate, using market data as far as possible.
Almost every private company ends up at tier 3. FRS 102 requires the volatility estimate to be consistent with the method used to value the shares. The basis for conclusions accepts that option pricing models can be hard for unlisted entities to apply, so directors use judgement in choosing a model appropriate to the company. IFRS 2 likewise requires share-based payments, including employee share options, to be recognised in profit or loss.
Vesting conditions affect the calculation differently. Service conditions and non-market performance conditions, such as a profit target, are left out of the fair value. Instead they are reflected by estimating how many options will vest, and that estimate is updated each year. Market conditions, such as a share price target, are built into the fair value and are not trued up later (paragraph 26.9). Black-Scholes cannot easily model a market condition, so a Monte Carlo or binomial model is often used instead.
Limitations to keep in mind
- European exercise only. Employee options are usually exercisable over a window, which is why an expected life is used in place of the full term;
- Constant volatility and interest rate. Real markets do not behave this way, and an unquoted company has no traded share price history to measure volatility from;
- Tradeable options assumed. Employee options cannot be sold, and early exercise and leavers have to be dealt with through the assumptions;
- Garbage in, garbage out. The formula is exact, but the inputs are estimates. Auditors will focus on the share valuation and the volatility.
Black-Scholes and tax
The accounting fair value is not the figure used for tax. For example, the £250,000 limit on unexercised EMI options per employee is measured using the market value of the shares at the date of grant, not the Black-Scholes value of the options. A company running an EMI scheme usually needs both figures: an HMRC-facing share valuation for the tax rules, and an option valuation for the accounts.
Sources: HMRC's CFM13210 Valuing options, the FRC's FRS 102 (paragraphs 26.6 to 26.11 and B26.1), the IFRS Foundation's IFRS 2 Share-based Payment and ITEPA 2003 Schedule 5 paragraph 5. See also our guides to accounting for share options and EMI share options.
Acumon prepares share and option valuations through our valuations service. We also account for share-based payments in FRS 102 accounts.