American Options and Early Exercise

An American option may be exercised at any stopping time before expiry, so its value is the optimal expected discounted payoff rather than merely the value of its terminal payoff. The difference is an optimal-stopping problem. Early exercise is not automatically valuable: an American call on a non-dividend-paying stock should never be exercised early, whereas an in-the-money put or a call before an ex-dividend date may rationally be exercised.

Value as a Snell envelope

Let h(S_t) be intrinsic value. Under the pricing measure, the holder chooses a stopping time tau:

V_t = sup_(tau in [t,T]) E_Q[ exp(-integral_t^tau r_u du) * h(S_tau) | F_t ]

At an exercise date, compare immediate exercise with continuation:

V(t,S) = max(h(S), C(t,S))
C(t,S) = E_Q[discounted V(t+dt, S_(t+dt)) | S_t=S]

The resulting V is the Snell envelope. The set where h >= C is the exercise region, separated from continuation by a free boundary S*(t).

Contract	Early exercise under standard assumptions
Call, no dividends, non-negative rates	Never optimal
Put, positive rates	Often optimal when deeply ITM
Call, discrete dividends	Possible just before ex-date
Futures option	Depends on settlement and margin conventions

Why a non-dividend call waits

For a call, exercising gives up remaining time value and pays the strike early. With no dividends, retaining the call preserves upside while deferring K; the stock can be synthetically held through a call plus a bond. More formally:

C_american = C_european  when q = 0 and r >= 0

This statement is sensitive to assumptions. Securities-lending fees, negative rates, discrete dividends, settlement lags, and corporate actions must be represented in the carry. Do not apply it blindly to ETF options or hard-to-borrow single names.

Binomial trees: transparent baseline

The Cox-Ross-Rubinstein tree uses:

u = exp(sigma sqrt(dt))
d = 1/u
p = (exp((r-q)dt) - d) / (u-d)

Backward induction prices the continuation value and applies exercise:

import numpy as np

def american_put_crr(S0, K, r, q, sigma, T, steps):
    dt = T / steps
    u, d = np.exp(sigma*np.sqrt(dt)), np.exp(-sigma*np.sqrt(dt))
    p = (np.exp((r-q)*dt) - d) / (u-d)
    disc = np.exp(-r*dt)
    j = np.arange(steps + 1)
    spots = S0 * u**j * d**(steps-j)
    values = np.maximum(K - spots, 0.0)
    for n in range(steps - 1, -1, -1):
        values = disc * (p * values[1:n+2] + (1-p) * values[:n+1])
        j = np.arange(n + 1)
        spots = S0 * u**j * d**(n-j)
        values = np.maximum(values, K - spots)
    return values[0]

Convergence is not monotonic. Use many steps, test against a finite-difference solver, and align dividends to actual ex-dates rather than smearing a known cash dividend into a continuous yield.

PDE and free-boundary methods

In the continuation region, the option satisfies the Black-Scholes PDE:

V_t + 0.5 sigma^2 S^2 V_SS + (r-q) S V_S - rV = 0
V >= h

Together these form a linear complementarity problem. Projected SOR, penalty methods, or policy iteration enforce the inequality. PDEs are accurate and fast for one or two factors, and provide smooth Greeks near the exercise boundary. Boundary conditions and grid concentration around strike matter more than using an exotic solver.

Least-squares Monte Carlo

High-dimensional American claims need simulation. Longstaff-Schwartz regresses continuation value on basis functions of state variables, then exercises paths whose intrinsic value exceeds the estimated continuation.

# At each backward date, regress discounted future cashflows on ITM states:
X = np.column_stack([np.ones(itm.sum()), S[itm], S[itm]**2])
coef = np.linalg.lstsq(X, cashflow[itm] * discount, rcond=None)[0]
continuation = X @ coef
exercise = intrinsic[itm] > continuation

Use out-of-sample paths or cross-fitting to reduce look-ahead bias from using the same paths for regression and valuation. Add state variables sufficient for the payoff: an American option under stochastic volatility cannot be accurately exercised from spot alone.

Exercise risk and operations

The theoretical boundary is not an exercise instruction. Holders face commissions, funding, assignment uncertainty, and cut-off times. Writers face pin risk and random assignment, especially around dividends. A portfolio system should model exercise probabilities, corporate-action calendars, and position netting—not only a continuous exercise frontier.

Key takeaways

• American value is an optimal-stopping premium over European value.
• Non-dividend calls with non-negative rates have no early-exercise premium.
• Trees are auditable; PDEs solve the free boundary accurately; LSMC scales to many factors.
• Discrete dividends and financing conventions determine real early-exercise decisions.
• Validate boundaries and Greeks, not only prices, before relying on an exercise model.