FRM Exam Part I · The Black-Scholes-Merton Model

Warrants, Employee Stock Options and Dilution in Black-Scholes-Merton

Updated 11 October 2026 · Fact-checked

Warrants and employee stock options are call options issued by the company itself, so exercise creates new shares and dilutes existing holders. To value a warrant, price an equivalent call with Black-Scholes-Merton, then multiply by N ÷ (N + M), where N is existing shares and M is new shares from exercise. For employee options, use expected life, not contractual life.

Understand Warrants, Employee Stock Options and Dilution

An exchange-traded call is a contract between two investors. If it is exercised, one investor delivers an existing share to the other. The company is not involved and its share count does not change.

A warrant is different. The company issues it, and when the holder exercises, the company creates new shares and sells them at the strike price K. Employee stock options (ESOs) work the same way: they are call options the company grants to staff. In both cases exercise increases the number of shares, so the value of the firm is split among more shares. This is dilution.

Dilution lowers what the holder receives. Suppose there are N existing shares and M warrants, each for one share. If the firm is worth N × S just before exercise, then after exercise the firm has also received M × K in cash and has N + M shares. The share price becomes (N × S + M × K) ÷ (N + M). The holder pays K and gets that share, so the gain is N × (S − K) ÷ (N + M). That is N ÷ (N + M) times the gain on an ordinary call. This ratio is always below 1, so a warrant is worth less than an otherwise identical exchange-traded call.

ESOs add features that ordinary options do not have. They usually start at-the-money, have a vesting period, cannot be sold or transferred, and are normally forfeited if the employee leaves before vesting. Employees also tend to exercise early, because they cannot hedge or sell the option. All of this makes the effective life shorter than the contractual life.

In practice you value an ESO with Black-Scholes-Merton using the expected life as the time to maturity, or with a binomial tree that allows early exercise and employee departure. You then apply a dilution adjustment where the question asks for it. The grant-date fair value is the cost the firm recognises as an expense.

Key formulas to remember

Dilution factor: Dilution factor = N ÷ (N + M); N = existing shares, M = new shares created if all warrants or options are exercised. Always between 0 and 1.
Warrant value: Warrant value = [N ÷ (N + M)] × c; c is the Black-Scholes-Merton value of an otherwise identical ordinary European call, using the current stock price S0 as the input.
Share price after exercise: S after = (N × S + M × K) ÷ (N + M); S is the pre-exercise price per share. The firm receives M × K in cash and issues M new shares.
Warrant payoff at exercise: Payoff = N × (S − K) ÷ (N + M), if S > K; Equals S after − K. Zero if S ≤ K.
BSM call value: c = S0 × N(d1) − K × e^(−rT) × N(d2); d1 = [ln(S0 ÷ K) + (r + σ²÷2) × T] ÷ (σ × √T); d2 = d1 − σ × √T; For ESOs, set T equal to expected life, not contractual life.
Total cost of an option grant: Total cost = number of options × fair value per option; If dilution is to be reflected, use the diluted per-option value.

How to solve Warrants, Employee Stock Options and Dilution questions

Use this order for any question on warrants, ESOs or dilution.

1Identify the instrument. Is it issued by the company (warrant or ESO) or traded between investors? Only company-issued instruments dilute.
2List the inputs: S0, K, r, σ, the maturity, and any dividend yield. Note N and M if they are given.
3For an ESO, decide the maturity. Use the expected life if it is given. Do not use the contractual life unless told to.
4Compute or read the value c of the equivalent ordinary European call using Black-Scholes-Merton.
5Apply the dilution factor N ÷ (N + M). Check that you used existing shares in the numerator and the total after exercise in the denominator.
6Multiply the per-option value by the number of options if the question asks for total value or total expense.
7Sanity-check. The warrant value must be below c, and it must be positive. A shorter expected life must give a lower value for an at-the-money option.

Quickest way: Dilution shortcut: scale the call by N ÷ (N + M)

When to use it: Use it when the question gives you the ordinary call value, or when you can compute it quickly, and asks for the warrant or option value after dilution.

Write N and M. Compute N ÷ (N + M) as a decimal.
Take the call value c from the question or from your BSM calculation.
Multiply: warrant value = factor × c.
Eliminate options that equal c or are above it, and options that use M ÷ (N + M) instead of N ÷ (N + M).
For total cost, multiply by the number of options last.

Common mistakes in Warrants, Employee Stock Options and Dilution

Using M ÷ (N + M) as the dilution factor.
The word dilution points to the new shares, so students put M on top.
Fix: The factor scales the call value down to the holder's share of the gain. It is N ÷ (N + M), and it is always close to 1 when M is small.
Valuing an ESO using its full contractual life.
Students treat the ESO as an ordinary 10-year option.
Fix: ESOs are exercised early and forfeited on leaving. Use the expected life as T, or a tree with early exercise.
Applying a dilution adjustment to an exchange-traded option.
Students apply the formula to every option question once they have learned it.
Fix: Exchange-traded options are settled with existing shares. No new shares are created, so no dilution adjustment applies.
Valuing an ESO at intrinsic value, or at zero because it is at-the-money.
An at-the-money option has zero intrinsic value, so it looks worthless.
Fix: It has time value. The fair value from BSM or a tree is positive and is the grant-date cost.
Forgetting that the dilution factor must be less than 1, and getting a warrant value above the call value.
Inverting the fraction or adding M to the numerator.
Fix: Always check the answer against c. A warrant is worth less than the equivalent ordinary call.
Multiplying by the number of options before applying the dilution factor on a per-option basis, then applying it twice.
Mixing per-option and total values in one calculation.
Fix: Work per option first, apply the factor once, then scale by the number of options.

Worked examples

Example 1

A company has 10 million shares outstanding and issues 2 million warrants, each giving the right to buy one new share. The share price is $50 and the strike is $50. An ordinary European call with the same terms has a Black-Scholes-Merton value of $6.00. What is the value of each warrant, and what is the total value of the warrants?

Show the solution

N = 10 million, M = 2 million, so N + M = 12 million.
Dilution factor = N ÷ (N + M) = 10 ÷ 12 = 0.8333.
Warrant value = 0.8333 × $6.00 = $5.00.
Total value = 2 million × $5.00 = $10 million.
Check: $5.00 is below the ordinary call value of $6.00, as it must be.

Answer: Each warrant is worth $5.00 and the total value of the warrants is $10 million.

Example 2

A company with 49 million shares grants 1 million at-the-money employee stock options on one share each. S0 = $40, contractual life 10 years, expected life 6 years. Black-Scholes-Merton gives an ordinary call value of $19.00 for 10 years and $15.50 for 6 years. Reflecting dilution, what is the total grant-date cost? Options: A) $19.00 million B) $15.50 million C) $15.19 million D) $18.62 million.

Show the solution

For an ESO, use the expected life of 6 years, so the call value is $15.50. This rules out A and D, which are based on the 10-year value.
N = 49 million, M = 1 million, so N + M = 50 million.
Dilution factor = 49 ÷ 50 = 0.98.
Value per option = 0.98 × $15.50 = $15.19.
Total cost = 1 million × $15.19 = $15.19 million.
B ignores dilution, so it is not the answer.

Answer: C) $15.19 million.

Exam tips

Look at who issues the instrument. If the company issues it, think dilution. If it is a standard exchange-traded option, use plain Black-Scholes-Merton.
Memorise the factor as existing shares over total shares after exercise: N ÷ (N + M). Then check the answer is below the ordinary call value.
For ESOs, the key phrases are expected life, vesting, forfeiture and non-transferability. Each pushes the value below that of a comparable ordinary option.
Questions often give you the BSM call value. Do not recompute d1 and d2 when you only need to apply the factor.
Expect conceptual questions as well: why a warrant is worth less than a call, and why ESO fair value is based on expected life rather than contractual life.

Practice questions from The Black-Scholes-Merton Model

Warrants, Employee Stock Options and Dilution: frequently asked questions

What is the difference between warrants and call options?

A call option is a contract between two investors and is settled with existing shares. A warrant is issued by the company, so exercise creates new shares and dilutes existing shareholders. Warrants are also usually longer dated.

How does dilution affect warrant valuation?

The holder's payoff is N ÷ (N + M) times the payoff of an ordinary call. So you value an equivalent call with Black-Scholes-Merton and multiply by that factor. The warrant is worth less than the call.

How do you value employee stock options with Black-Scholes-Merton?

Use the expected life of the option as the time to maturity, because employees exercise early and forfeit options when they leave. A binomial tree with early exercise and departure rates is a more flexible alternative. The grant-date fair value is the cost the firm records.

Why can't employees sell their stock options?

ESOs are non-transferable and cannot be hedged easily by the employee. This often leads to early exercise, which is why the effective life is shorter than the contractual life.