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

American Options and Early Exercise: When to Exercise Early

Updated 11 October 2026 · Fact-checked

An American option can be exercised at any time up to expiry; a European option only at expiry. Without dividends, never exercise an American call early. Early exercise of a call can pay only just before an ex-dividend date. A put can be optimal early, when deep in the money and rates are positive.

Understand American Options and Early Exercise

An American option gives its holder the right to exercise at any time up to expiry. A European option can be exercised only at expiry. Because the American holder has every right the European holder has, plus more, an American option is never worth less than its European twin: C ≥ c and P ≥ p.

Start with the American call on a non-dividend-paying stock. Exercising early gives you S − K now. Holding instead keeps the time value of the option and defers paying K. You also earn interest on K, and the call keeps insurance against the stock falling below K. Both are lost if you exercise. So early exercise is never optimal, and C = c. The lower bound shows why: c ≥ S − K·e^(−rT) > S − K when r > 0 and T > 0. The market price is above the exercise value, so selling the option beats exercising it.

Dividends change this. The stock price falls on the ex-dividend date, which hurts call holders, and option holders do not receive the dividend. So it may be optimal to exercise a call just before an ex-dividend date, to capture the dividend. It is never optimal at other times. Early exercise is more likely when the dividend is large, the call is deep in the money, and little time value remains. For a European call, you subtract the present value of the dividends from S before using the Black-Scholes-Merton formula.

The American put is different. If the stock falls toward zero, the put's payoff is capped at K. Exercising now gets K immediately, and you can earn interest on that cash. Waiting only risks losing that interest. So early exercise can be optimal for a deep in-the-money put. Higher interest rates, lower volatility and a lower stock price make early exercise more likely. A put on a non-dividend stock still has P > p in general. Dividends make early put exercise less attractive, since a dividend lowers the stock price anyway.

Black-Scholes-Merton has no closed form for American options that may be exercised early. The standard tools are binomial trees, checking at each node whether the intrinsic value exceeds the continuation value, and analytical approximations. The FRM expects you to know the logic and the bounds, not to derive the approximations. Put-call parity holds only as bounds for American options: S − K ≤ C − P ≤ S − K·e^(−rT) for a non-dividend stock.

Key formulas to remember

American vs European value: C ≥ c and P ≥ p; The early exercise right can only add value. It is never negative.
No-dividend call: C = c; Never optimal to exercise early when the stock pays no dividend and r ≥ 0. Strictly, c > S − K when r > 0.
Lower bound for a European call, no dividends: c ≥ max(S − K·e^(−rT), 0); This bound is above S − K, which proves early exercise is not optimal.
Lower bound for a European put, no dividends: p ≥ max(K·e^(−rT) − S, 0); The American put can be worth K − S, which can exceed this bound. That is why early exercise may pay.
American put-call bounds, no dividends: S − K ≤ C − P ≤ S − K·e^(−rT); Parity is an inequality for American options, not an equality.
Bounds with dividends: S − D − K ≤ C − P ≤ S − K·e^(−rT), where D is the present value of dividends; Use this form when the stock pays known dividends during the option life.
Dividend-adjusted European call: Use S* = S − PV(dividends) in the Black-Scholes-Merton formula; Applies to European options with known discrete dividends.
Early exercise of a call before ex-dividend date: Exercise if D > K·(1 − e^(−r·Δt)) for the last ex-dividend date, roughly; A common rule: the dividend must exceed the interest earned on K until expiry. Treat it as a rule of thumb, since time value also matters.

How to solve American Options and Early Exercise questions

Use this approach for any question on early exercise or on comparing American and European values.

1Identify the option type (call or put), the underlying (dividend or no dividend), and the interest rate.
2If it is a call with no dividends, answer that early exercise is not optimal and C = c.
3If it is a call with dividends, look only at the moments just before ex-dividend dates. Compare the dividend with the interest saved on K and the time value lost.
4If it is a put, check whether it is deep in the money, whether rates are high and volatility low. These favor early exercise.
5For a numerical question, compute the intrinsic value (exercise now) and compare it with the continuation value from the tree or the given European price.
6At each tree node, set option value = max(intrinsic value, discounted risk-neutral expected value) for an American option.
7Check the result against the bounds: C ≥ c, P ≥ p, and the American put-call inequality.
8State the answer with the reason, not just the number.

Quickest way: Four-line early exercise check

When to use it: Use it for conceptual MCQs with option statements asking whether early exercise is optimal.

Call, no dividend: never exercise early.
Call, dividend: exercise only just before an ex-dividend date, and only if the dividend is large relative to interest on K and remaining time value.
Put: early exercise can pay when deep in the money, with high r and low volatility.
Numbers: at a node, take the larger of exercise value and continuation value.

Common mistakes in American Options and Early Exercise

Saying American calls on non-dividend stocks should be exercised early when deep in the money.
Students focus on intrinsic value and ignore time value and the interest earned on the strike.
Fix: Remember that you can sell the option for more than S − K. Exercising throws away time value and insurance.
Treating put-call parity as an equality for American options.
The equation c + K·e^(−rT) = p + S is memorized without its conditions.
Fix: Use parity only for European options. For American options use the inequality bounds.
Exercising a call at any time during the dividend period.
Students think the holder 'earns' the dividend by holding.
Fix: Only the stock holder gets the dividend. Early exercise of a call can only be optimal immediately before an ex-dividend date.
Claiming dividends make early put exercise more likely.
Students apply the call logic to puts.
Fix: A dividend lowers the stock price, which helps put holders, so they gain by waiting. Dividends make early put exercise less attractive.
Forgetting to compare with continuation value in a tree.
Students discount the expected payoff and stop there, as for a European option.
Fix: At every node of an American option tree, take max(intrinsic, continuation).

Worked examples

Example 1

A stock trades at $40. A one-year American put has strike $50. Interest is 5% continuously compounded and the stock pays no dividends. A European put with the same terms is priced at $9.20. Is it ever possible the American put is worth more than the European put, and what is its minimum value now?

Show the solution

Intrinsic value if exercised now: K − S = 50 − 40 = $10.
The American put is worth at least its intrinsic value, so P ≥ $10.
The European lower bound is K·e^(−rT) − S = 50 × e^(−0.05) − 40.
e^(−0.05) = 0.951229, so 50 × 0.951229 = 47.561, and the bound is 7.561.
The European price of $9.20 is below the exercise value of $10, so the American put is worth strictly more than the European put.

Answer: Yes. The American put is worth at least $10, above the European put at $9.20, because early exercise is valuable here.

Example 2

A one-step binomial tree: S0 = $50, u = 1.2, d = 0.8, r = 5% per period (simple discounting factor e^(−0.05)), strike K = $52 for an American put. Find the American put value at time 0, and say whether to exercise at time 0.

Show the solution

Up price: 50 × 1.2 = 60. Down price: 50 × 0.8 = 40.
Put payoffs: up = max(52 − 60, 0) = 0; down = max(52 − 40, 0) = 12.
Risk-neutral probability: p = (e^(0.05) − d) ÷ (u − d). e^(0.05) = 1.051271.
p = (1.051271 − 0.8) ÷ 0.4 = 0.628178.
Continuation value = e^(−0.05) × [0.628178 × 0 + 0.371822 × 12] = 0.951229 × 4.461864 = 4.2443.
Exercise value now = 52 − 50 = 2.
Since 4.2443 > 2, holding is better. The American put value is the larger of the two.

Answer: The American put is worth about $4.24. Do not exercise at time 0, because continuation value ($4.24) exceeds the exercise value ($2).

Exam tips

Expect conceptual MCQs: which option is never exercised early, and in which situation early exercise is plausible.
For calls with dividends, remember the only candidate time is just before an ex-dividend date.
In tree problems, write the exercise value next to the continuation value at each node. Missed comparisons are the main source of lost marks.
If an answer choice says American calls on non-dividend stocks may be exercised early, it is almost certainly wrong.
Use your calculator's e^x key for discount factors, and keep at least four decimal places until the final step.

Practice questions from The Black-Scholes-Merton Model

American Options and Early Exercise in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Actuarial Core Principles (IAI)Early Exercise of American Options

American Options and Early Exercise: frequently asked questions

When should you exercise an American call early?

On a non-dividend stock, never. With dividends, only just before an ex-dividend date, and only if the dividend is large enough to outweigh the interest earned on the strike and the time value given up.

Why can an American put be exercised early?

Its maximum payoff is the strike, so a deep in-the-money put has little more to gain. Exercising gives you the cash now, which earns interest. The higher the interest rate, the stronger this effect.

How much more is an American option worth than a European option?

It is worth at least as much and often more. For calls on non-dividend stocks the two are equal. For puts, and for calls with dividends, the American value can exceed the European value by the value of the early exercise right.

Does Black-Scholes-Merton price American options?

The standard formula prices European options. For American options you typically use a binomial tree or an approximation. You can use the formula for an American call on a non-dividend stock, since it equals the European call.