FRM Exam Part I · Binomial Trees

American Options and Early Exercise on a Binomial Tree

Updated 11 October 2026 · Fact-checked

An American option can be exercised at any time up to expiry. On a binomial tree you work backward from maturity. At each node the option value is the larger of intrinsic value (exercise now) and continuation value (discounted risk-neutral expected value of the next two nodes). Early exercise is optimal where intrinsic value is higher.

Understand American Options and Early Exercise

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 matching European option.

On a binomial tree you value both types by working backward from the final nodes. At expiry the value is the payoff: max(S − K, 0) for a call and max(K − S, 0) for a put. For a European option you then step back by discounting the risk-neutral expected value. For an American option you add one extra test at every earlier node.

At each earlier node you compare two numbers. The continuation value is what the option is worth if you hold it one more step. The intrinsic value is what you receive if you exercise now. The node value is the larger of the two. If intrinsic value wins, early exercise is optimal at that node.

This test matters mainly for puts. Exercising a deep in-the-money put gives you the strike in cash now, and that cash earns interest. If the put is so far in the money that waiting can add little extra payoff, exercising early is better. For a call on a stock that pays no dividends, with interest rates not negative, early exercise is never optimal, so the American call equals the European call. Dividends can make early exercise of a call worthwhile, just before an ex-dividend date.

The node-by-node maximum feeds forward. A higher value at a late node raises the continuation value at earlier nodes. You must carry the American values back, not the European ones.

Key formulas to remember

Risk-neutral probability of an up move: p = (e^(rΔt) − d) ÷ (u − d); For a non-dividend stock. With a per-period simple rate use (1 + r) in place of e^(rΔt). With a continuous yield q, use e^((r − q)Δt).
Continuation value at a node: f_cont = e^(−rΔt) × [p × f_up + (1 − p) × f_down]; f_up and f_down are the option values at the next two nodes. For American options these are the already-maximised values.
American node value: f = max(intrinsic value, f_cont); Apply at every node before expiry. At expiry the value is just the payoff.
Intrinsic values: Call: max(S − K, 0). Put: max(K − S, 0); Use the stock price at that node.
Early exercise premium: American value − European value ≥ 0; Build both trees if asked for the premium. The premium is zero for an American call on a non-dividend stock when rates are not negative.
Ordering of values: American option ≥ European option ≥ 0; Use this as a quick check on your answer.

How to solve American Options and Early Exercise questions

Use this backward-induction method for any American option question on a binomial tree.

1Write down S0, K, u, d, r, the step length Δt, option type and number of steps. Check whether a dividend or yield applies.
2Build the stock price tree forward: each node is S0 × u^j × d^(i − j).
3Compute p = (e^(rΔt) − d) ÷ (u − d). Check 0 < p < 1.
4Fill the final column with payoffs: max(S − K, 0) for a call or max(K − S, 0) for a put.
5Step back one column. At each node compute f_cont = e^(−rΔt) × [p × f_up + (1 − p) × f_down].
6At the same node compute intrinsic value from that node's stock price. Set the node value to the larger of the two. Mark nodes where intrinsic value wins.
7Repeat until the root. The root value is the American option price. Early exercise at the root only occurs if intrinsic value exceeds continuation value there.
8If asked for the premium, run the same tree without the max test and subtract.

Quickest way: Compare and carry back the maximum

When to use it: Use it when the tree has two or three steps and you must answer fast, especially for puts.

Check the option type first. An American call on a non-dividend stock with non-negative rates: price it as a European call and skip the max test.
For a put, compute every node. At out-of-the-money nodes intrinsic value is 0, so the node value equals the continuation value. Early exercise is only a possibility at in-the-money nodes.
At the lowest nodes, test exercise first. Deep in-the-money nodes are the likeliest early exercise points.
Keep full-precision p and the discount factor in the calculator memory. Round only at the end.
Sanity check: American price ≥ European price, and American price ≥ intrinsic value at the root.

Common mistakes in American Options and Early Exercise

Carrying the European values back instead of the American values
You find continuation value at a node, forget to replace it with intrinsic value, and move on.
Fix: After computing f_cont at every node, write down max(intrinsic, f_cont) before moving to the next column.
Applying the exercise test at expiry or ignoring it at the first step
Students are unsure which nodes count as decision points.
Fix: At expiry the value is the payoff. Test at every node before expiry. Also compare at the root to see if exercising today beats holding.
Using real-world probabilities instead of risk-neutral p
The question may mention an expected return or a probability of an up move.
Fix: Always compute p from (e^(rΔt) − d) ÷ (u − d). Ignore any real-world return for pricing.
Thinking an American call on a non-dividend stock should be exercised early
Students assume the extra right must always be used somewhere.
Fix: Remember that holding the call keeps the interest on the strike and the option's time value. The American call equals the European call when no dividends are paid and rates are not negative.
Using intrinsic value from the wrong node
With several nodes in a column, the stock prices are mixed up.
Fix: Label each node with its stock price before computing anything. Use that node's own S in max(K − S, 0).
Discounting with the wrong period or the wrong rate form
The step length is not one year, or the rate is given as simple per period.
Fix: Use e^(−rΔt) when r is a continuously compounded annual rate and Δt is in years. Use 1 ÷ (1 + r) when the rate is stated per period.

Worked examples

Example 1

A stock trades at ₹50 (treat the unit as given). A two-step tree has each step of one year, u = 1.2 and d = 0.8. The continuously compounded risk-free rate is 5% per year. Price a two-year American put with strike 52.

Show the solution

Prices: S_u = 60, S_d = 40. S_uu = 72, S_ud = 48, S_dd = 32.
p = (e^0.05 − 0.8) ÷ (1.2 − 0.8) = (1.051271 − 0.8) ÷ 0.4 = 0.628178. So 1 − p = 0.371822. Discount factor e^(−0.05) = 0.951229.
Final payoffs: max(52 − 72, 0) = 0, max(52 − 48, 0) = 4, max(52 − 32, 0) = 20.
Node u (S = 60): continuation = 0.951229 × (0.628178 × 0 + 0.371822 × 4) = 0.951229 × 1.487288 = 1.4147. Intrinsic = max(52 − 60, 0) = 0. Value = 1.4147.
Node d (S = 40): continuation = 0.951229 × (0.628178 × 4 + 0.371822 × 20) = 0.951229 × 9.949152 = 9.4638. Intrinsic = 52 − 40 = 12. Since 12 > 9.4638, exercise early. Value = 12.
Root: continuation = 0.951229 × (0.628178 × 1.4147 + 0.371822 × 12) = 0.951229 × (0.8887 + 4.4619) = 0.951229 × 5.3506 = 5.0897. Intrinsic = max(52 − 50, 0) = 2. Value = 5.09.
Check: a European put on this tree uses 9.4638 at node d, giving 0.951229 × (0.8887 + 0.371822 × 9.4638) = 0.951229 × 4.4075 = 4.19. The American value 5.09 is higher, as it must be.

Answer: The American put is worth about 5.09. Early exercise is optimal at the down node after one year (S = 40), where exercising gives 12 against a continuation value of about 9.46. The European put is worth about 4.19, so the early exercise premium is about 0.90.

Example 2

A stock is at 80. A one-step tree has u = 1.25 and d = 0.75. The risk-free rate for the step is 4% (simple, so discount by 1 ÷ 1.04). Value a one-step American put with strike 100 and decide whether to exercise now.

Show the solution

Prices after one step: S_u = 80 × 1.25 = 100 and S_d = 80 × 0.75 = 60.
Payoffs at expiry: max(100 − 100, 0) = 0 and max(100 − 60, 0) = 40.
p = (1.04 − 0.75) ÷ (1.25 − 0.75) = 0.29 ÷ 0.5 = 0.58. So 1 − p = 0.42.
Continuation value = [0.58 × 0 + 0.42 × 40] ÷ 1.04 = 16.8 ÷ 1.04 = 16.1538.
Intrinsic value now = max(100 − 80, 0) = 20.
Since 20 > 16.1538, exercise now. The American put value is 20.
The European put is worth 16.1538, so the early exercise premium is 20 − 16.1538 = 3.8462.

Answer: The American put is worth 20, and it should be exercised immediately. The European put is worth about 16.15, so the early exercise premium is about 3.85.

Exam tips

Expect a two-step put. The question often asks for the price, the node where early exercise occurs, or the early exercise premium. Compute the nodes in order and mark any exercise point.
Test the lowest nodes first for puts. They are the deepest in the money and the likeliest early exercise points.
For calls, read the question for dividends. With no dividend and non-negative rates, you can skip the exercise test and save time.
Use the ordering American ≥ European and American ≥ intrinsic value to reject wrong answer options quickly.
Store p and the discount factor in the calculator memory. Keep four or five decimals until the final answer.

Practice questions from Binomial Trees

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

How do you price an American option using a binomial tree?

Build the stock tree, compute the risk-neutral probability p, and fill the final nodes with payoffs. Then work backward. At each node take the larger of the intrinsic value and the discounted risk-neutral expected value of the next two nodes. The value at the root is the price.

What is the difference between European and American option valuation on a tree?

For a European option you only discount the expected values back through the tree. For an American option you also compare with the intrinsic value at every node before expiry and keep the larger. The tree and the probabilities are the same.

When is early exercise of an American put optimal?

Early exercise is more likely when the put is deep in the money, interest rates are high or volatility is low. On a tree it is optimal at any node where the intrinsic value K − S is larger than the continuation value. Exercising gives you the strike in cash, which can then earn interest.

Is early exercise ever optimal for an American call?

For a stock that pays no dividends and with non-negative interest rates, it is never optimal, so the American call equals the European call. If the stock pays dividends, early exercise can be optimal just before an ex-dividend date.