Economic Modelling · Principles of option pricing

Early Exercise of American Options on Non-Dividend Shares

Updated 11 October 2026 · Fact-checked

An American option can be exercised at any time up to expiry; a European option only at expiry. For a non-dividend share, an American call is never optimally exercised early, so it equals the European call. An American put can be worth exercising early, so it is worth at least the European put.

Understand Early Exercise of American Options

An American option can be exercised at any time up to expiry. A European option can be exercised only at expiry. So the American option gives you every right the European one gives, plus extra. This means its value can never be lower: C_A ≥ C_E and P_A ≥ P_E.

Now take an American call on a share that pays no dividends. If you exercise early, you pay the strike K and get the share. Compare this with selling the call. The call is worth at least S − K·e^(−r(T−t)), which is more than S − K whenever r > 0 and t < T. So selling (or holding) the call beats exercising it. Early exercise also throws away the remaining time value, which is the protection against the share falling. Hence the American call equals the European call: C_A = C_E.

The argument has conditions: no dividends (or other income) before expiry, and interest rates that are positive (non-negative gives the same weak conclusion). If the share pays a dividend, exercising just before the ex-dividend date can be optimal, because you capture the dividend. The call then can be worth more than its European twin.

American puts behave differently. If you exercise a put, you receive K now. That cash earns interest. If the share price is very low, the most you can still gain from waiting is small, because the price cannot fall below zero. So taking K now and earning interest can beat waiting. Early exercise of a put can therefore be optimal, and P_A > P_E in general for r > 0.

In a binomial tree you test this at every node. You compare the value from continuing (the discounted risk-neutral expectation) with the immediate exercise value. The option value at the node is the larger of the two.

Key rules to remember

Ordering of values: C_A ≥ C_E and P_A ≥ P_E; The American option has all the rights of the European one, so it cannot be worth less.
American call, no dividends: C_A = C_E; Holds for a share with no dividends before expiry and r ≥ 0. Early exercise is not optimal (strictly so when r > 0).
Lower bound for a European call: C_E ≥ max(S − K·e^(−r(T−t)), 0); Used to show C > S − K, which proves early exercise of the call loses value.
Lower bound for an American put: P_A ≥ max(K − S, 0); Intrinsic value. If P_A would fall below K − S, exercise is better.
Put-call relation for American options (no dividends): S − K ≤ C_A − P_A ≤ S − K·e^(−rT); This is an inequality, not an equality. Exact put-call parity holds only for European options.
Binomial node value (American): V = max(exercise value, e^(−r·Δt)[q·V_u + (1 − q)·V_d]); q is the risk-neutral probability. Apply at every node before expiry.

How to solve Early Exercise of American Options questions

Use this method for any question on early exercise, whether it is conceptual or numerical.

1Identify the option type: call or put, American or European.
2Check for dividends or other income on the underlying before expiry, and note the sign of the interest rate.
3For a call with no dividends and r ≥ 0, state that early exercise is not optimal and C_A = C_E. Give the reason: selling beats exercising, since C ≥ S − K·e^(−r(T−t)) > S − K, and time value would be lost.
4For a put, or a call with dividends, compare at each date the immediate exercise value with the continuation value.
5In a tree, work backwards from expiry. At each node compute the continuation value using risk-neutral probability, then take the maximum with the exercise value.
6Mark the nodes where exercise is optimal. State the option value at the root.
7Compare with the European value if asked, and explain any difference as the early exercise premium.

Quickest way: Three-question check

When to use it: For multiple-choice questions and quick conceptual parts.

Is it a call on a non-dividend share? If yes, American = European. Never exercise early.
Is it a put? Early exercise may be optimal when the share is low and rates are positive. American put ≥ European put.
Is there a dividend? A call may be exercised just before the ex-dividend date.
For numbers in a tree, compute max(exercise, continuation) at each node, working backwards.

Common mistakes in Early Exercise of American Options

Saying an American call on a non-dividend share is worth more than the European call because it is more flexible.
Students assume extra rights always add value.
Fix: The extra right has no value here, since early exercise is never optimal. State C_A = C_E, with the condition of no dividends.
Saying American puts are never exercised early, by copying the call result.
The call argument is memorised without its reasoning.
Fix: For a put, exercising gives K now, which earns interest. Deep in-the-money puts can be exercised early, so P_A can exceed P_E.
Using exact put-call parity for American options.
Parity C − P = S − K·e^(−rT) is learned as a general rule.
Fix: It holds only for European options. For American options use the inequality S − K ≤ C_A − P_A ≤ S − K·e^(−rT).
Forgetting to compare with exercise value at intermediate tree nodes.
Students price the tree as if it were European, only comparing at the end.
Fix: At every node before expiry, take the maximum of the continuation value and the immediate payoff.
Ignoring the condition on dividends when stating the call result.
The rule is remembered as 'never exercise a call early'.
Fix: Always state: no dividends before expiry. With dividends, early exercise just before the ex-dividend date can be optimal.

Worked examples

Example 1

A share price is ₹100 and pays no dividends. An American call has strike ₹90 and expires in one year. The risk-free rate is 5% per year continuously compounded. Show that early exercise now is not optimal.

Show the solution

Exercise now gives S − K = 100 − 90 = ₹10.
The European call has lower bound S − K·e^(−rT) = 100 − 90·e^(−0.05).
e^(−0.05) = 0.951229, so 90 × 0.951229 = 85.61.
The lower bound is 100 − 85.61 = ₹14.39.
The American call is worth at least the European call, so it is at least ₹14.39.
₹14.39 > ₹10, so selling or holding beats exercising now.

Answer: The call is worth at least ₹14.39, but exercising gives only ₹10. Early exercise is not optimal, so the American call equals the European call.

Example 2

A one-period binomial tree has S₀ = ₹50, with u = 1.2 and d = 0.8. The risk-free rate is 5% per period (simple, so discount factor 1/1.05). Price an American put with strike ₹60 expiring after one period.

Show the solution

Share prices at the end: S_u = 60 and S_d = 40.
Put payoffs at expiry: V_u = max(60 − 60, 0) = 0 and V_d = max(60 − 40, 0) = 20.
Risk-neutral probability q = (1.05 − 0.8) ÷ (1.2 − 0.8) = 0.25 ÷ 0.4 = 0.625.
Continuation value = [0.625 × 0 + 0.375 × 20] ÷ 1.05 = 7.5 ÷ 1.05 = 7.1429.
Immediate exercise value at time 0 = 60 − 50 = 10.
Option value = max(10, 7.1429) = 10.

Answer: The American put is worth ₹10, and it should be exercised immediately. The European put is worth ₹7.14 (rounded), so the early exercise premium is about ₹2.86.

Exam tips

For conceptual questions, always give the reason, not just the result: selling beats exercising, and time value plus interest on the strike are lost.
Write the condition (no dividends, r ≥ 0) every time you state C_A = C_E.
In tree questions, show the continuation value and exercise value side by side at each node, and circle the larger.
In MCQs, watch for options that claim exact parity or that American puts equal European puts. Both are wrong in general.

Practice questions from Principles of option pricing

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.

FRM Part IAmerican Options and Early Exercise

Early Exercise of American Options: frequently asked questions

Why is an American call on a non-dividend share never exercised early?

Exercising gives S − K. The call is worth at least S − K·e^(−r(T−t)), which is larger when r > 0. Exercising early also loses the time value. So you gain more by selling or holding the call.

Is an American put ever exercised early?

Yes. If the share price is low enough, receiving K now and earning interest on it can beat waiting. This is why an American put can be worth more than the European put.

What is the difference between American and European option value?

The American option is worth at least as much as the European one. For a call on a non-dividend share, they are equal. For a put, or a call on a dividend-paying share, the American can be worth more.

Does dividend payment change the call result?

Yes. A dividend lowers the share price on the ex-dividend date. It can then be optimal to exercise the call just before that date to receive the share and the dividend.