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Binomial Treesadvanced

American Early Exercise Logic

At every node of the binomial tree, the holder of an American option compares the continuation value (the discounted risk-neutral expectation of the next two nodes) against the immediate exercise value (intrinsic payoff). The node value is the larger of the two. This max⁡\max at every node is the only structural difference between American and European pricing on the same tree.

Why it matters

Holding an option means owning the future right to exercise. Exercising now means converting that right into cash today. Sometimes the cash today, reinvested at rr, beats waiting. For a deep in-the-money American put, the intrinsic value K−SK - S is bounded above by KK, and earning interest on KK from today rather than at expiry can outweigh whatever extra upside is left in the put. For an American call on a non-dividend stock, time value is always positive and early exercise is never optimal.

Formulas

American node value
fi=max⁡(intrinsici,  e−rΔt[pfi,u+(1−p)fi,d])f_i = \max\bigl(\text{intrinsic}_i,\; e^{-r\Delta t}[p f_{i,u} + (1-p) f_{i,d}]\bigr)
Intrinsic value
intrinsic=max⁡(Si−K,0) (call),max⁡(K−Si,0) (put)\text{intrinsic} = \max(S_i - K, 0) \text{ (call)},\quad \max(K - S_i, 0) \text{ (put)}

Worked examples

Scenario

American put with strike K=K = A$50 at a node where the stock S=S = A$35 and the continuation value (backward-induction estimate) is A$12.

Solution

Intrinsic value = \max(50 - 35, 0) = A\15$. Continuation value = A\12$. Node value = \max(15, 12) = A\15$. The holder exercises now. The early-exercise premium of A$3 over the European value will propagate to the root and lift the American put's price above the European put's price.

Common mistakes

  • ✗American calls on non-dividend stocks should sometimes be exercised early. They should never be exercised early. Hull (2022) §11.5 proves C≥max⁡(S−Ke−rT,0)>S−KC \geq \max(S - Ke^{-rT}, 0) > S - K, so selling the call always beats exercising it.
  • ✗Early exercise is rare for puts. Deep in-the-money American puts are routinely exercised early, especially when interest rates are high or the put is far below strike. The early-exercise boundary is a key output of any production American option pricer.
  • ✗Dividends only matter for puts. Discrete dividends actually make early exercise of American calls potentially optimal just before the ex-dividend date, because the stock drops on the ex-date and the call holder forfeits the dividend by not exercising.

Revision bullets

  • •Compare intrinsic against continuation value at every node
  • •American call on non-dividend stock: never exercise early
  • •American put when deep ITM: often exercise early
  • •Discrete dividends can trigger early exercise of calls
  • •Early-exercise boundary depends on SS, rr, σ\sigma, dividends

Quick check

An American call on a non-dividend-paying stock is exercised early. This is:

Connected topics

Am. vs Eu.2-Step Tree

In learning paths

Pricing Models Path

Sources

  1. Hull (2022), §13.6
    Hull, John C. Options, Futures, and Other Derivatives. 11th ed. Pearson, 2022. ISBN 978-0-13-693997-9.
    Sets out the $\max$ rule at each node for American payoffs on the binomial tree and proves early exercise is suboptimal for non-dividend calls.
  2. Cox, Ross & Rubinstein (1979)
    Cox, John C., Stephen A. Ross and Mark Rubinstein. "Option Pricing: A Simplified Approach." Journal of Financial Economics 7(3), 1979, pp. 229–263.
    Original binomial framework that handles American-style payoffs by inserting the immediate-exercise check at every node.
  3. Merton (1973)
    Merton, Robert C. "Theory of Rational Option Pricing." Bell Journal of Economics and Management Science 4(1), 1973, pp. 141–183.
    Classical proof that early exercise of an American call on a non-dividend-paying stock is never optimal.
  4. ASX Equity Options
    Australian Securities Exchange. ASX equity options are American-style. ASX product disclosure, accessed 2026.
    Confirms that single-stock options on ASX trade as American-style, making early-exercise logic directly relevant for Australian student examples.
How to cite this page
Dr. Phil's Quant Lab. (2026). American Early Exercise Logic. Derivatives Atlas. https://phucnguyenvan.com/concept/american-early-exercise
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