Skip to content

Economic Modelling · Binomial option-pricing model

American Options and Early Exercise in Binomial Trees

Updated 11 October 2026 · Fact-checked

An American option can be exercised at any time up to expiry. In a binomial tree you work backwards. At each node, compute the continuation value (discounted risk-neutral expectation of the next two nodes), compare it with the exercise value, and take the larger. That larger value is the option value at the node.

Understand American Options and Early Exercise in Binomial Trees

A European option can only be exercised at expiry. An American option can be exercised at any time up to expiry. That extra right can only add value, so an American option is worth at least as much as the European option with the same terms.

On a binomial tree you price a European option by working backwards from expiry. At each node the value is the discounted risk-neutral expectation of the two next values. For an American option you add one check at every node. The continuation value is what you get by holding the option one more step. The exercise value is the payoff if you exercise now. The option value at the node is the larger of the two.

This check matters most for puts. If the share price falls far, the put is deep in the money. Exercising early gives cash K − S now, which can then earn interest. Waiting only adds the chance that the price rebounds. So early exercise of an American put can be optimal, especially with a high interest rate and a low share price.

For a call on a share that pays no dividends, early exercise is never optimal. The continuation value is always at least the exercise value, so the American call has the same value as the European call. Dividends change this: an early exercise can be worthwhile just before a dividend is paid.

The tree does not tell you the exercise strategy separately. It comes out of the comparison. The nodes where the exercise value is larger form the exercise region. Any option value must also never be below zero or below the exercise value.

Key rules to remember

Risk-neutral probability of an up move
q = (e^(rΔt) − d) ÷ (u − d)
Uses a continuously compounded rate r and a step length Δt. If the question gives a simple annual rate per step, use (1 + i) in place of e^(rΔt). Needs d < e^(rΔt) < u for no arbitrage.
Continuation value at a node
C = e^(−rΔt) × [q × V_up + (1 − q) × V_down]
V_up and V_down are the option values at the next two nodes. For an American option these are already the larger-of values.
American put value at a node
V = max( K − S, C )
K − S is the exercise value when positive. If K − S is negative, exercise value is zero, so V is at least C.
American call value at a node
V = max( S − K, C )
For a share with no dividends, C is never below S − K, so early exercise is not optimal.
Value at expiry
Put: max(K − S, 0). Call: max(S − K, 0)
Same starting point as for European options.
Ordering of values
American value ≥ European value
Equality holds for a non-dividend call. For a put the American value can be strictly higher.

How to solve American Options and Early Exercise in Binomial Trees questions

Use the same backward method for any American option question. Keep the tree tidy and write both values at every node.

  1. 1Write down S0, K, u, d, the step length, the interest rate and the option type. Note any dividends.
  2. 2Build the share price tree forward. Label each node with its share price.
  3. 3Calculate q = (e^(rΔt) − d) ÷ (u − d), or the simple-interest version if the rate is given that way. Check 0 < q < 1.
  4. 4At expiry, set each node value to the payoff max(K − S, 0) for a put or max(S − K, 0) for a call.
  5. 5Move back one step. At each node compute the continuation value C = e^(−rΔt) × [q × V_up + (1 − q) × V_down].
  6. 6At the same node compute the exercise value. Set the node value to the larger of the two. Mark nodes where you exercise.
  7. 7Repeat until you reach time 0. The value at the first node is the option price.
  8. 8State the exercise strategy if asked: the nodes where exercise value exceeds continuation value.

Quickest way: Compare and carry the maximum

When to use it: Use for any two- or three-step tree where you are short of time.

  1. Compute q and the discount factor once. Reuse them at every node.
  2. Fill expiry payoffs first, then work down one column at a time.
  3. At each node write C and exercise value side by side, and circle the larger.
  4. Check early: a call on a non-dividend share never needs early exercise, so price it as European.
  5. For a put, the lowest-price nodes are the likeliest exercise points. Check these first.
  6. Finish by checking that the American value is at least the European value.

Common mistakes in American Options and Early Exercise in Binomial Trees

  • Using the exercise value only at expiry.

    Students copy the European method and forget the extra check.

    Fix: Compare continuation and exercise values at every node before expiry, including the first node.

  • Carrying the continuation value forward instead of the larger value.

    Students compute C at a node, note the exercise value separately, but use C in the next step back.

    Fix: Overwrite the node with max(exercise, C). Only that value is used at the earlier node.

  • Using real-world probabilities instead of q.

    The question may mention a probability of an up move, and students use it.

    Fix: Always use the risk-neutral q when pricing. Real-world probabilities do not enter the option price.

  • Forgetting to discount by one step only.

    Students discount from expiry to time 0 at every node.

    Fix: Discount by e^(−rΔt) at each step back, since values at the next nodes are already values at that time.

  • Saying early exercise is optimal for a non-dividend call.

    Students assume an American option is always exercised early if it is in the money.

    Fix: For a non-dividend share the call's continuation value is at least its exercise value. Exercise early only if the numbers show it.

  • Taking the exercise value as negative when the option is out of the money.

    Students use K − S without a floor.

    Fix: Exercise value is max(K − S, 0) for a put. Out-of-the-money nodes have exercise value zero.

Worked examples

Example 1

A share has price ₹100. Over each of two one-year steps it rises by a factor u = 1.2 or falls by a factor d = 0.8. The annual risk-free rate is 5% per year, effective (use a discount factor of 1/1.05 per step). Price a two-year American put with strike ₹100.

Show the solution
  1. Share prices: time 1 are 120 and 80. Time 2 are 144, 96 and 64.
  2. q = (1.05 − 0.8) ÷ (1.2 − 0.8) = 0.25 ÷ 0.4 = 0.625. So 1 − q = 0.375.
  3. Expiry put payoffs: at 144 it is 0, at 96 it is 4, at 64 it is 36.
  4. Node 120: C = (0.625 × 0 + 0.375 × 4) ÷ 1.05 = 1.5 ÷ 1.05 = 1.4286. Exercise value = max(100 − 120, 0) = 0. Value = 1.4286.
  5. Node 80: C = (0.625 × 4 + 0.375 × 36) ÷ 1.05 = (2.5 + 13.5) ÷ 1.05 = 16 ÷ 1.05 = 15.2381. Exercise value = 100 − 80 = 20. Since 20 > 15.2381, exercise early. Value = 20.
  6. Node 100 at time 0: C = (0.625 × 1.4286 + 0.375 × 20) ÷ 1.05 = (0.89286 + 7.5) ÷ 1.05 = 8.39286 ÷ 1.05 = 7.9932. Exercise value = 0. Value = 7.9932.

Answer: The American put is worth about ₹7.99. It is optimal to exercise early at the time-1 down node (share price ₹80).

Example 2

Using the same tree and rate as the previous example, price the European put and state how much extra value the early exercise right adds.

Show the solution
  1. Use q = 0.625 and 1 − q = 0.375, with the same expiry payoffs 0, 4 and 36.
  2. Node 120: value = 1.4286, as before.
  3. Node 80: no early exercise, so value = C = 15.2381.
  4. Node 100 at time 0: value = (0.625 × 1.4286 + 0.375 × 15.2381) ÷ 1.05 = (0.89286 + 5.71429) ÷ 1.05 = 6.60715 ÷ 1.05 = 6.2925.
  5. Difference = 7.9932 − 6.2925 = 1.7007.

Answer: The European put is worth about ₹6.29. The early exercise right adds about ₹1.70, so the American put is worth more.

Exam tips

  • Show both the continuation value and the exercise value at every node. Marks are given for the comparison, not only the final price.
  • State q and check it lies between 0 and 1. Show the discount factor you use.
  • If asked when early exercise is optimal, give the nodes and explain in words: the put is deep in the money, and cash now can earn interest.
  • Know the standard result: a non-dividend call is not exercised early, so the American and European values are equal. Use it to save time and to check answers.
  • In computer-based questions, build the tree in a grid and use a MAX formula at each node, so the early exercise check is automatic.

Practice questions from Binomial option-pricing model

American Options and Early Exercise in Binomial Trees in other exams

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

American Options and Early Exercise in Binomial Trees: frequently asked questions

What is the difference between American and European option pricing in a binomial tree?

The tree and the risk-neutral probability are the same. For a European option you only discount expected values back. For an American option you also compare with the exercise value at each node and take the larger.

When is early exercise optimal for an American put?

It tends to be optimal when the share price is low, so the put is deep in the money, and when interest rates are high. Exercising gives cash now that can earn interest. The tree shows exactly which nodes qualify.

Is early exercise ever optimal for an American call?

Not on a share that pays no dividends, because the continuation value is never below the exercise value. With dividends, early exercise can be optimal just before a dividend is paid.

Do I use real-world probabilities for American options?

No. You price with the risk-neutral probability q, as for European options. Real-world probabilities do not affect the price.