3.6 American Options & Optimal Stopping

Basic Prerequisites: Stochastic Calculus & Itô’s Lemma and Black–Scholes–Merton: Hub. (these are the folder-level prerequisites for pages 02–06; page 01 states its own, smaller, entry requirements)

1. Intuition & Practical Objective

A European option fixes one decision date: hold to $T$ , then collect. An American option lets the holder choose when to stop. That single change - from “at maturity” to “at any time” - converts a pricing formula into an optimal-stopping problem: the price is the maximum over all exercise strategies of the discounted payoff, and the exercise strategy is itself unknown.

The whole topic rests on one structural fact. For a European option the price solves a linear PDE with known boundary data. For an American option the boundary is a free boundary $S^{*} (t)$ - the price level below which it is optimal to exercise a put (above it, for a call) - and it is part of the unknown. The PDE holds only on the “continue” region; on the “exercise” region the price equals intrinsic value. This is a variational inequality / linear-complementarity problem, and it is why American options have no Black–Scholes-style closed form (except in the perpetual, infinite-horizon limit).

This folder is the single owner of the optimal-stopping theory and the American analytic-approximation methods. Three facts orient everything:

When is early exercise optimal? Never for a dividend-free call (the discounted payoff is a submartingale). For puts always possible (dividends/interest make waiting costly); for calls only with dividends.
How much is early exercise worth? The early-exercise premium - the American value minus the European value. It is 0 for the no-dividend call and strictly positive otherwise.
How do you price it? A free-boundary problem: solve analytically by approximation (perpetual closed forms, Barone–Adesi–Whaley, Bjerksund–Stensland) or numerically (trees, finite differences with PSOR/penalty, Monte Carlo LSM + duality - see Numerical Methods: Hub).

The one-sentence essence. “An American option is worth the discounted payoff under the best stopping rule; finding that rule is a free-boundary problem, and the price is the smallest supersolution of the BSM operator that dominates the intrinsic payoff.”

2. Mathematical Ground Truth & Derivations

Quick-Reference Lookup (job #1). Formulas from Shreve II Ch 8, Björk Ch 21 and Haug Ch 3; the check column was re-executed and reproduced exactly from the verified corpus (§3).

Notation: $S$ spot, $K$ (or $X$ ) strike, $T$ maturity, $r$ rate, $b$ cost of carry, $q$ dividend yield, $σ$ vol, $N (\cdot)$ standard normal CDF, $γ = 2 r / σ^{2}$ , $L^{*}$ exercise boundary.

Quantity	Formula	Verified check
Perpetual put, no div (boundary)	$L^{*} = \frac{γ K}{1 + γ} = \frac{2 rK}{2 r + σ ^{2}}$	$K = 100, r = 10%, σ = 25% \Rightarrow L^{*} = 76.1905$
Perpetual put, no div (value)	$V (x) = K - x$ ( $x \leq L^{}$ ); $(K - L^{}) (x / L^{})^{- γ}$ ( $x \geq L^{}$ )	$V (90) = 13.9719$ ; argmax over $L$ gives $76.1900$ ✓
Perpetual put, carry $b$ (Haug 3.5)	$p = \frac{X}{1 - γ _{2}} (\frac{γ _{2} - 1}{γ _{2}} \cdot \frac{S}{X})^{γ_{2}}$ , $γ_{2} = (\frac{1}{2} - \frac{b}{σ ^{2}}) - (\frac{b}{σ ^{2}} - \frac{1}{2})^{2} + \frac{2 r}{σ ^{2}}$	$X = 100, r = 10%, b = 2%, σ = 25%, S = 90 \Rightarrow 20.7939$ (Haug)
Perpetual call, carry $b$ (Haug 3.5)	$c = \frac{( γ _{1} - 1 ) ^{γ_{1} - 1}}{γ _{1}^{γ_{1}}} (\frac{S}{X})^{γ_{1}} X$ , $γ_{1} = (\frac{1}{2} - \frac{b}{σ ^{2}}) + (\frac{b}{σ ^{2}} - \frac{1}{2})^{2} + \frac{2 r}{σ ^{2}}$	$b \geq r$ : never exercise early
BAW call (Haug 3.1)	$c_{BSM} + A_{2} (S / S^{})^{q_{2}}$ for $S < S^{}$ , else $S - X$ ; $A_{2} = \frac{S ^{}}{q _{2}} [1 - e^{(b - r) T} N (d_{1} (S^{}))]$	$X = 100, r = 10%, b = 0$ : $1.8769$ (Haug 1.8771), $15.5684$ (Haug 15.5689)
BS-1993 call (Haug 3.2)	$α S^{β} - α ϕ (S, T, β, I, I) + \dots$ , $β = (\frac{1}{2} - \frac{b}{σ ^{2}}) + (\frac{b}{σ ^{2}} - \frac{1}{2})^{2} + \frac{2 r}{σ ^{2}}$	$S = 42, X = 40, T = .75, r = 4%, b = - 4%, σ = 35% \Rightarrow 5.2704$ (Haug)
Put–call transformation (Haug 3.4)	$P (S, X, T, r, b, σ) = C (X, S, T, r - b, - b, σ)$	reproduces all puts in Table 3-2 ✓
Early-exercise premium	$V^{Am} - v^{Eu} \geq 0$	put: $4.6921 - 4.4496 = 0.2425$
LCP / VI characterisation (Shreve II 8.3.18–20)	$min (V - intrinsic, L V) = 0$ , $L V := r V - r S V_{S} - \frac{1}{2} σ^{2} S^{2} V_{SS}$	residual $min (V - p, L V) = 0$ at every $x$ ✓
Early exercise of a call	never if $b \geq r$ ; else optimal once $S \geq S^{*} (t)$	no-div: 0 exercise nodes; $q = 5%$ : 55 769 nodes ✓

Sign convention caveat (Shreve II 8.3.18). The operator is written $L V = r V - r S V_{S} - \frac{1}{2} σ^{2} S^{2} V_{SS}$ , the negative of the usual BSM operator. It is $0$ in the continuation region and $= rK$ in the exercise region of the perpetual put. Confusing the sign flips the inequality.

3. Computational Implementation - the closed forms, checked

Everything below runs on the standard library only and reproduces the verified numbers in §2.

4. Failure Modes & First-Principles Breakdowns

Hub signposts - the full analysis is in 05 · Failure Modes & Practice. In one line each:

Applying the European formula to an American put. Understates value by the early-exercise premium ( $0.2425$ here), which grows with $r$ , $σ$ and maturity.
Analytic approximations have maturity-dependent error. BAW is excellent for short maturities ( $+ 0.45%$ ) and drifts to $+ 2%$ by $T = 3$ - always validate against a tree.
Discrete exercise is not continuous exercise. A Bermudan exercised 4×/year is worth $\approx 0.13$ less than the continuous American; the free-boundary theory gives the continuous value, which is an upper bound for any real (discrete) contract.

5. References

Shreve, Steven E.: Stochastic Calculus for Finance II
Björk, Tomas: Arbitrage Theory in Continuous Time (3rd ed.)
Haug, Espen Gaarder: The Complete Guide to Option Pricing Formulas (2nd ed.)
Glasserman, Paul: Monte Carlo Methods in Financial Engineering
Shreve, Steven E.: Stochastic Calculus for Finance I

6. Connected Graph Bridges

Foundational base: Stochastic Calculus & Itô · Calculus & Optimization
Upstream: Black–Scholes–Merton: Hub (the PDE being constrained) · No-Arbitrage & Binomial Trees (the discrete seed of the recursion)
Numerical implementation detail lives elsewhere (linked, not duplicated): FDM: penalty & PSOR · Monte Carlo · Numerical Methods · Advanced
Sub-pages (in-folder): 01 From Zero · 02 Optimal-Stopping Theory · 03 Analytic Approximations · 04 Free Boundary & Complementarity · 05 Failure Modes · 06 Advanced Extensions

Beginner: start at 01 · Practitioner: start at 05

Kwant Atlas

Explorer

3.6 American Options & Optimal Stopping

1. Intuition & Practical Objective

2. Mathematical Ground Truth & Derivations

3. Computational Implementation - the closed forms, checked

4. Failure Modes & First-Principles Breakdowns

5. References

6. Connected Graph Bridges

3.6.1 American Options from Zero

3.6.2 Optimal-Stopping Theory

3.6.3 Analytic Approximations

3.6.4 The Free Boundary & the Linear-Complementarity Problem

3.6.5 Failure Modes & Real-World Practice

3.6.6 Advanced Extensions