American Options and Optimal Stopping

Motivation: why this matters in quant finance

European options are exercised only at expiry; their pricing reduces to evaluating an expectation

V_0 = e^{-rT}\mathbb{E}^{\mathbb{Q}}[X]

. American options can be exercised at any time before expiry — the holder chooses when to exercise, optimally. This makes pricing an optimal stopping problem: maximise the expected discounted payoff over all stopping times.

Almost all listed equity options in the US are American. Index options (SPX) are European; ETF options (SPY) are American. Most commodity and futures options are American. Pricing them properly is therefore not a niche problem — it's the standard.

The mathematical framework (optimal stopping) is also a gateway to:

Real options (when to invest, when to abandon a project).
Convertible bonds (whose pricing involves compound stopping).
Bermudan swaptions (stopping at a discrete set of dates).
Snell envelopes in stochastic control.

This note develops the framework: the optimal stopping problem, the value function, the early-exercise boundary, and the standard numerical methods (binomial, Longstaff-Schwartz, FDM with PSOR).

The informal idea

For a European call: the holder waits to expiry

T

and gets

(S_T - K)^+

. For an American call: at every time

\tau \in [0, T]

, the holder can choose to exercise and receive

(S_\tau - K)^+

(the intrinsic value), or to hold and continue.

The fundamental question: when is it optimal to exercise?

Hold value $H_t$ : the value of continuing to hold the option, given it has not been exercised yet. Computed by backward induction.

Exercise value $X_t = (S_t - K)^+$ for a call.

Optimal rule: exercise the first time

H_t < X_t

The early-exercise boundary $S^*(t)$ partitions $(S, t)$ -space into two regions:

Hold region ( $S < S^*(t)$ for a put, $S > S^*(t)$ for some calls): $V(S, t) > X(S)$ , hold.
Exercise region ( $S \ge S^*(t)$ for a put, $S \le S^*(t)$ for a call with dividends): $V(S, t) = X(S)$ , exercise.

Pricing reduces to computing

V(S, t)

on a grid, with the constraint

V \ge X

everywhere — a free-boundary problem.

Formal statement

Let $\mathcal{T}$ be the set of stopping times $\tau \le T$ adapted to the filtration $(\mathcal{F}_t)$ . The American option price is

V_0 = \sup_{\tau \in \mathcal{T}} \mathbb{E}^{\mathbb{Q}}\!\left[e^{-r\tau} X(S_\tau)\right].

Snell envelope. The value process is the smallest supermartingale dominating the discounted payoff:

V_t = \mathrm{ess\,sup}_{\tau \ge t}\, \mathbb{E}^{\mathbb{Q}}\!\left[e^{-r(\tau - t)} X(S_\tau) \,\big|\, \mathcal{F}_t\right].

In discrete time, this satisfies the dynamic programming recursion:

V_T = X(S_T), \quad V_t = \max\!\big(X(S_t),\, e^{-r\Delta t}\mathbb{E}^{\mathbb{Q}}[V_{t+\Delta t} \mid \mathcal{F}_t]\big).

Two cases:

$X(S_t) > e^{-r\Delta t}\mathbb{E}[V_{t+\Delta t}]$ : exercise.
Otherwise: hold.

In continuous time,

V

satisfies a variational inequality (the obstacle problem):

\min\!\left(rV - \mathcal{L} V,\, V - X\right) = 0,

where $\mathcal{L}$ is the Black-Scholes infinitesimal generator. Equivalently:

In the hold region: $rV - \mathcal{L} V = 0$ (BS PDE) and $V > X$ .
In the exercise region: $V = X$ .
On the free boundary $S = S^*(t)$ : smooth-pasting condition $\partial V/\partial S |_{S^*} = \partial X/\partial S |_{S^*}$ ensures the solution is $C^1$ .

When is early exercise optimal?

American call on non-dividend stock. Never optimal to exercise early (proven via put-call parity bound).

V^{Am} = V^{Eur}

American put (on any underlying). Always potentially optimal — the exercise boundary

S^*(t)

is finite and below

K

. Deep ITM puts may be optimally exercised early to capture the time value of the strike

K

American call on dividend-paying stock. Sometimes optimal — exercise just before a dividend to capture it.

Heuristic. Early exercise is optimal when the present value of foregone cash flows (dividends, interest on

K

) exceeds the present value of remaining optionality.

Numerical methods

Binomial tree (CRR)

Backward induction on a binomial tree. At each node, the value is

V_t^{(i)} = \max\!\big(X_t^{(i)},\, e^{-r\Delta t}(p V_{t+\Delta t}^{(i+1)} + (1-p)V_{t+\Delta t}^{(i)})\big).

Simple, robust, slow ( $O(N^2)$ for $N$ time steps).

Longstaff-Schwartz (LSM) — for path-dependent or high-dim

Monte Carlo with regression-based estimation of the continuation value:

Simulate $N$ paths.
Backward induction: at each time $t_k$ , fit a regression of $V_{t_{k+1}}$ on basis functions of $S_{t_k}$ over the in-the-money paths.
Use the regression as an estimator of $\mathbb{E}[V_{t_{k+1}} | S_{t_k}]$ . Compare to exercise value, decide.

Standard for high-dimensional problems (basket Americans, callable swaptions).

FDM with PSOR

Discretise the BS PDE, but at each time step solve the projected problem

V \ge X

using projected SOR or a primal-dual LCP solver. Same complexity as European FDM (

O(M)

per time step), but with a constraint check per node per iteration.

Worked example: American put via binomial tree

$S_0 = 100$ , $K = 100$ , $T = 1$ , $r = 0.05$ , $\sigma = 0.2$ . Compute via 100-step binomial tree.

import numpy as np

def american_put_binomial(S0, K, T, r, sigma, N):
    dt = T / N
    u = np.exp(sigma * np.sqrt(dt))
    d = 1.0 / u
    p = (np.exp(r*dt) - d) / (u - d)
    discount = np.exp(-r*dt)
    
    # Stock prices at maturity
    j = np.arange(N+1)
    S = S0 * u**j * d**(N - j)
    V = np.maximum(K - S, 0)  # put payoff
    
    # Backward induction
    for n in range(N - 1, -1, -1):
        j = np.arange(n + 1)
        S = S0 * u**j * d**(n - j)
        V_hold = discount * (p * V[1:n+2] + (1-p) * V[0:n+1])
        V_exercise = np.maximum(K - S, 0)
        V = np.maximum(V_hold, V_exercise)
    return V[0]

p_am = american_put_binomial(100, 100, 1, 0.05, 0.2, 1000)
# Compare with European put: 5.5735
print(f"American put: {p_am:.4f}")  # ~ 6.0902

European put on the same parameters: $\$ 5.57 $. American put:$ $6.09 $. Early-exercise premium:$ $0.52$.

Key properties

Always at least the European value. $V^{Am} \ge V^{Eur}$ , with strict inequality when early exercise has positive probability.
Free boundary moves with time. $S^*(t)$ is monotone in $t$ : deeper ITM near expiry, shallower ITM far from expiry (more option value to preserve).
Smooth pasting. At the boundary, the value function and its first derivative match the payoff. This determines $S^*(t)$ uniquely.
No closed form for finite-maturity American put. Even in BS, no analytic formula. Approximations: Bjerksund-Stensland, Geske-Johnson, Carr-Jarrow-Myneni.
Perpetual American put ( $T = \infty$ ): closed form. $V(S) = (K - S^*)(S/S^*)^{-\gamma}$ with $\gamma$ a model constant.

Common confusions and pitfalls

"American calls don't have early exercise — same as European." True only on non-dividend stocks. With dividends, early exercise can be optimal just before ex-div date.
Smooth pasting is not optional. Without it, the free boundary is undetermined and numerical methods produce nonsense.
PSOR convergence. The projected SOR iteration converges if the matrix is M-matrix (typically true for BS PDE with reasonable parameters). For aggressive PDE setups (high vol, low strike, near-zero rates), convergence can be slow.
LSM regression bias. The continuation value is estimated from the same paths used to evaluate the exercise rule. This introduces a bias of order $1/\sqrt{N}$ . Standard practice: use one set of paths to fit the regression, and another independent set to estimate the price (in-sample vs out-of-sample).
Curse of dimensionality. FDM scales badly in dimension; LSM scales gracefully but requires careful basis selection. Above 5-10 dimensions, both struggle.
American spread/butterfly puzzle. Early exercise of a single-leg American put can be sub-optimal if the holder is hedging a multi-leg position; the joint early-exercise of a spread is more complex than treating each leg independently.

Where this goes next

Monte Carlo pricing — extends to LSM for Americans.
FDM implicit/Crank-Nicolson — extends to PSOR for the obstacle problem.
Barrier options — a different breed of path-dependence.
Convertible bonds and real options — applications of optimal stopping beyond pure derivatives.