FRM Exam Part I · Properties of Options

Early Exercise of American Options: When Is It Optimal?

Updated 11 October 2026 · Fact-checked

An American option can be exercised any time before expiry. An American call on a non-dividend-paying stock is never optimally exercised early, so it is worth the same as the European call. An American put can be optimal to exercise early when deep in the money. Dividends can make early call exercise optimal.

Understand Early Exercise of American Options

A European option can be exercised only at expiry. An American option can be exercised at any time up to expiry. That extra right can never make the option worth less, so an American option is worth at least as much as the European option with the same terms.

The key question is whether the extra right has any value. For a call on a stock with no dividends, it does not. Exercising early means you pay the strike K now and receive the stock. But holding the call instead keeps two things: you delay paying K, so you earn interest on it, and you keep the option's time value and the insurance against the stock falling below K. Selling the call in the market is always better than exercising it. So early exercise is never optimal, and American call = European call (no dividends, positive interest rates assumed).

A simple proof: the European call value satisfies c ≥ S₀ − K·e^(−rT). Because K·e^(−rT) < K, we get c > S₀ − K whenever r > 0. Exercising early only gives S₀ − K, which is less than the value of the live call.

For an American put, the logic reverses. Exercising gives you K in cash now. You can earn interest on that K. If the put is deep in the money, the most it can still gain is limited, because the stock cannot fall below zero. Then the interest on K outweighs the small remaining upside, and early exercise can be optimal. Higher interest rates and lower volatility make early put exercise more likely. The American put is worth more than the European put.

With dividends, the stock price drops when it goes ex-dividend. That hurts call holders and helps put holders. An American call may be optimal to exercise early, but only just before an ex-dividend date, to capture the dividend. For puts, dividends make early exercise less attractive.

Key formulas to remember

American vs European values: C ≥ c and P ≥ p; The American option is worth at least as much as the European option with the same strike and expiry.
No-dividend call result: C = c; Holds for a non-dividend-paying stock with positive interest rates. Never exercise early.
Lower bound on European call (no dividends): c ≥ max(S₀ − K·e^(−rT), 0); Since K·e^(−rT) < K, this exceeds the exercise value S₀ − K when r > 0.
Lower bound on European put (no dividends): p ≥ max(K·e^(−rT) − S₀, 0); The American put must be worth at least K − S₀, its exercise value.
Bounds for American options (no dividends): S₀ − K ≤ C − P ≤ S₀ − K·e^(−rT); Put-call parity holds only as an inequality for American options.
Early exercise payoff: Call: S − K; Put: K − S; Early exercise is optimal only if this value exceeds the value of holding the option.

How to solve Early Exercise of American Options questions

Use this method for any question on whether an American option should be exercised early.

1Identify the option type: call or put, and whether the underlying pays dividends or income.
2If it is a call on a non-dividend-paying stock, answer: do not exercise early. C = c. Sell it instead if you want out.
3If it is a call with dividends, check the timing. Early exercise can only be optimal just before an ex-dividend date.
4For that call, compare the dividend with the interest earned on the strike plus lost time value. Exercise only if the dividend is large enough.
5If it is a put, check how deep in the money it is. Deep in the money, high interest rate, low volatility favor early exercise.
6If the question gives numbers, compare the exercise value (S − K or K − S) with the value of holding, such as the European value or the binomial continuation value.
7State the conclusion and the reason in one line: interest on K, time value, insurance value, or dividend.

Quickest way: Three-question shortcut

When to use it: Use for conceptual multiple-choice questions where you must pick the true statement.

Call and no dividends? Never early. American call equals European call.
Call and dividends? Only just before ex-dividend date, and only if the dividend is big.
Put? Early exercise can be optimal when deep in the money, especially with high rates and low volatility. The American put is worth more than the European put.

Common mistakes in Early Exercise of American Options

Saying an American call on a non-dividend stock should be exercised early because it is deep in the money.
Students focus on locking in intrinsic value and forget time value and interest on the strike.
Fix: Remember that selling the call captures at least S − K plus time value. Exercising throws the extra value away.
Saying American and European puts have the same value.
The no-dividend call result is wrongly applied to puts.
Fix: Only the non-dividend call is equal. The American put is worth more than the European put.
Believing early exercise of a dividend-paying stock's call can occur at any time.
Students forget that the stock drops only on the ex-dividend date.
Fix: The optimal time is immediately before an ex-dividend date. Between dividends, waiting is better.
Using put-call parity as an equality for American options.
The European formula c + K·e^(−rT) = p + S₀ is memorized without its condition.
Fix: For American options use the inequality S₀ − K ≤ C − P ≤ S₀ − K·e^(−rT), for non-dividend stocks.
Thinking higher volatility makes early put exercise more likely.
Students mix it up with higher option value.
Fix: Higher volatility raises the value of waiting, so it makes early exercise less likely. Higher rates make it more likely.

Worked examples

Example 1

A stock pays no dividends and trades at $60. A 6-month American call has strike $50 and the risk-free rate is 4% continuously compounded. The call trades at $11.20. Should you exercise early to capture the gain, and what is the lower bound on the European call?

Show the solution

Exercise value = S − K = 60 − 50 = $10.
Lower bound on European call = S₀ − K·e^(−rT) = 60 − 50·e^(−0.04 × 0.5).
e^(−0.02) = 0.980199, so 50 × 0.980199 = 49.01.
Lower bound = 60 − 49.01 = $10.99.
The American call equals the European call, so it is worth at least $10.99, which is above $10.
The market price of $11.20 is also above the $10 exercise value.

Answer: Do not exercise early. Exercising gives $10, but the call is worth at least $10.99 and trades at $11.20. Sell it instead. The lower bound is about $10.99.

Example 2

A non-dividend-paying stock has fallen to $2. An American put has strike $50, 1 year to expiry, and the risk-free rate is 5% continuously compounded. What is the exercise value, the lower bound on the European put, and is early exercise sensible?

Show the solution

Exercise value = K − S = 50 − 2 = $48.
European put lower bound = K·e^(−rT) − S₀ = 50·e^(−0.05) − 2.
e^(−0.05) = 0.951229, so 50 × 0.951229 = 47.56.
Bound = 47.56 − 2 = $45.56.
The European put is worth at least $45.56, which is below the $48 from exercising now.
A European put cannot be worth more than K·e^(−rT) = $47.56 (stock cannot go below zero), so it is below $48. The American put is worth at least $48.

Answer: Exercise value is $48 and the European lower bound is about $45.56. The European put can never reach $48, so exercising the deep in-the-money American put early is sensible.

Exam tips

The most tested fact is that an American call on a non-dividend stock is never exercised early. Memorize it with its condition.
Questions often ask which factor makes early put exercise more likely. Choose higher interest rates, deeper in the money, or lower volatility.
Check whether the question mentions dividends. If so, the only early exercise date for a call is just before ex-dividend.
Expect to compare S − K with S₀ − K·e^(−rT). Use your calculator's e^x key and keep four decimals.
Do not use put-call parity as an equality for American options. Look for inequality answers.

Practice questions from Properties of Options

Early Exercise of American Options 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

Early Exercise of American Options: frequently asked questions

Why is it never optimal to exercise an American call early on a non-dividend stock?

Exercising pays the strike early and gives up the option's time value and its downside protection. The call is always worth more than S − K when interest rates are positive. So you do better by selling the call or holding it.

When should you exercise an American put early?

Early exercise can be optimal when the put is deep in the money. Then the interest you earn on the strike received outweighs the small extra gain from further falls. Higher interest rates and lower volatility make this more likely.

Do dividends change early exercise of American options?

Yes. A dividend lowers the stock price on the ex-dividend date, which hurts call holders. An American call may be exercised early, but only just before an ex-dividend date. Dividends make early put exercise less attractive.

Is an American option always worth more than a European option?

It is worth at least as much, since it gives extra rights. For a call on a non-dividend stock the two values are equal. For puts, and for calls on dividend-paying stocks, the American option can be worth more.