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Home / Financial Tools / American Option Pricing via Longstaff-Schwartz Least-Squares Monte Carlo

Graduate-Level Modeling · Numerical Methods & American Option Pricing

American Option Pricing via Longstaff-Schwartz Least-Squares Monte Carlo

A binomial tree prices an American option fine in one dimension. This is the algorithm that took over the moment a real desk needed more than one.

How To Use This Model

Reading This Tool

A binomial lattice prices American options fine in one dimension, but breaks down fast for path-dependent or multi-factor payoffs. This tool prices a plain American option with the technique that actually scales: Longstaff and Schwartz's least-squares Monte Carlo.

Simulate the underlying forward, then work backward from maturity, at every step regressing the discounted continuation value on the stock price across only the in-the-money paths, and exercise wherever that regression says continuation is worth less than exercising now. The early exercise boundary that falls out of this is the same one option theory has described analytically for a century.

Option Contract

Simulation Engine

Without dividends, an American call is never optimally exercised early and its price should match the European Black-Scholes call almost exactly, this tool will show you that live.

Option Values

Put

American Price (LSM)

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European Price (Black-Scholes)

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Early Exercise Premium

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Standard Error (LSM)

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Estimated Early Exercise Boundary

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Monte Carlo Convergence: Running Price Estimate vs. Paths Simulated

Sample Of Simulated Underlying Paths

American Premium Over European Across A Range Of Strikes

Detail By Strike

Why Regression, Not Just Simulation, Is The Whole Trick

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Why Only In-The-Money Paths Get Regressed

Restricting the regression to in-the-money paths isn't just an efficiency shortcut, out-of-the-money paths would never be exercised anyway, and including them would let the regression waste its fit on a region of the curve that never actually matters for the exercise decision, degrading the accuracy of the fit exactly where it counts.

Reading The Early Exercise Boundary

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Why Dividends Change Everything For American Calls

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Numerical Methods & American Option Pricing

The Core Algorithm

1. Simulate paths: St+Δt = St·exp[(r−q−σ²/2)Δt + σ√Δt·Z]
2. From t=T−Δt backward: regress discounted future cash flow on (1, St, St²) using in-the-money paths only
3. Exercise if immediate payoff > fitted continuation value, else continue
4. Price = mean of all paths' cash flows, each discounted from its own optimal exercise time

When To Actually Use This Method

  • Teaching numerical methods for American-style derivatives, especially the regression-based alternative to lattice methods, in a computational finance or derivatives course.
  • Pricing American or Bermudan options where a lattice becomes impractical, multiple underlying factors, path-dependent features, or high-dimensional problems.
  • Cross-checking a lattice-based American option price with an independent numerical method before relying on either.

Key Assumptions & Limitations

  • The regression basis (quadratic in the stock price) is a simplification, real implementations often add more basis functions or use Laguerre polynomials as in the original paper for better accuracy at the boundary.
  • LSM produces a biased-low estimate in finite samples, since the regression's own noise gets exploited by the pathwise exercise decision, real implementations often pair it with an upper-bound dual method to bracket the true price.
  • This tool uses a single underlying factor, LSM's real advantage emerges with multiple correlated underlyings where lattice methods become computationally infeasible.

Foundational Reference

Longstaff, F. A., & Schwartz, E. S. (2001). Valuing American Options by Simulation: A Simple Least-Squares Approach. The Review of Financial Studies, 14(1), 113-147.

Want the lattice-based version for comparison?

Open The Binomial Option Pricing Lattice →