American Option Calculator — Longstaff-Schwartz Pricing
Value American options with Longstaff-Schwartz least-squares Monte Carlo: exercise boundary, early-exercise premium, standard error, and a CRR tree benchmark.
Longstaff–Schwartz American Put Engine
A clear path from inputs to results.
For traders and analysts who need the value of early-exercise flexibility: American puts, callable structures, or any right to act before expiry. The engine simulates risk-neutral price paths with a seeded random number generator, learns an exercise rule at each date by regressing continuation values on Laguerre basis functions, and reports the American price with its Monte Carlo standard error, cross-checked against a Cox-Ross-Rubinstein binomial tree and the Black-Scholes European value.
How to use
- Pick American put or call, and keep the Laguerre basis (or switch to a plain polynomial basis in moneyness S/K).
- Set spot, strike, rate, dividend yield, volatility, and maturity in years.
- Choose the simulation budget: paths (1,000 to 20,000) and exercise dates, i.e. time steps (5 to 200), plus a PRNG seed so the run is exactly reproducible.
- Read the four headline cards: LSMC price with standard error, CRR binomial benchmark, Black-Scholes European value, and the early-exercise premium.
- Move the "Selected step" slider to inspect the continuation regression at that date: the R-squared of the fit, the intrinsic-versus-continuation scatter, and the implied exercise boundary.
- Check the exercise rate and the exercise-boundary-by-time chart to see when and where the policy chooses to exercise.
The formula
Reference: Longstaff & Schwartz (2001), "Valuing American Options by Simulation: A Simple Least-Squares Approach," Review of Financial Studies 14(1); Cox, Ross & Rubinstein (1979) binomial benchmark; Hull, Options, Futures, and Other Derivatives.
Worked example
Two-year American put, at the money, 25% volatility
- S0 = 100, K = 100, r = 4%, dividend yield 1.5%, volatility 25%, T = 2 years, 5,000 paths, 50 exercise dates, Laguerre basis, seed 1234567.
- 1
The engine draws 5,000 GBM paths with the seeded generator, then walks backward from maturity: at each of the 49 exercise dates it regresses discounted future payoffs on four Laguerre functions of moneyness using in-the-money paths only, and exercises whenever intrinsic value beats the fitted continuation. With this seed the LSMC estimate is 11.6037 with a standard error of 0.1730, against a Black-Scholes European put of 11.0496 and a 70-step Cox-Ross-Rubinstein American benchmark of 11.6643. The gap to the tree is about 0.35 standard errors, so the two agree within noise.
American put 11.6037 (standard error 0.1730), European value 11.0496, early-exercise premium 0.5541, CRR(70) benchmark 11.6643; 46.26% of paths exercised early, at an average time of 1.63 years.
Interpreting the result
The American value must sit at or above the European value, and the difference is the early-exercise premium: here about half a point, the price of being able to lock in intrinsic value before expiry. The CRR tree is deterministic and converges to the true American price, so treat it as the anchor and read the LSMC-to-CRR gap in units of the standard error. The exercise boundary tells a holder where to act: for a put, exercise becomes optimal once spot falls below the boundary level plotted for that date.
Assumptions & limitations
- Risk-neutral geometric Brownian motion with constant rate, dividend yield, and volatility; no jumps or stochastic volatility.
- Exercise is only tested on the chosen grid dates (50 by default), not continuously, which makes the price a slight underestimate of the continuous-exercise value.
- The continuation function is approximated by four Laguerre (or polynomial) basis functions fitted by ordinary least squares on in-the-money paths, so basis and regression error enter the price.
- Monte Carlo standard error shrinks like 1/sqrt(N); at the 5,000-path default it is roughly 0.17 on this example.
- The seeded PRNG makes every run exactly reproducible; changing the seed moves the estimate by about one standard error.
Frequently asked questions
How many Monte Carlo paths are enough?
Accuracy improves like the standard error, sigma-hat over sqrt(N): quadrupling the paths halves it. At the 5,000-path default the standard error here is about 0.17 on an 11.6 price, roughly 1.5% relative. Judge each run against the CRR benchmark: if the LSMC price sits more than two or three standard errors below the tree, raise the path count or the number of exercise dates.
What is the early exercise premium?
It is the American price minus the European price: the extra value of the right to exercise before maturity. For a put it is almost always positive, since locking in the strike early can beat waiting; for a call on a non-dividend underlying it is zero in theory. Here it is about 0.55 on an 11.05 European base.
What do the basis functions do?
The Longstaff-Schwartz trick needs E[future payoff | S_t], which has no closed form, so the engine approximates it with a cubic-style Laguerre polynomial in moneyness S/K (or plain powers) fitted by ordinary least squares, where h_t is the intrinsic payoff and the fitted sum is the continuation value C_t. Four basis functions keep the regression stable for vanilla payoffs; more complex payoffs would need more.
Why include a binomial tree at all?
The Cox-Ross-Rubinstein tree has no simulation noise, so it is an independent check on the whole pipeline: path generation, regressions, and the exercise policy. If the two disagree by more than a couple of standard errors, either the path count is too low, the exercise dates are too coarse, or the basis is too small.
What does the exercise boundary mean?
At each date it is the spot level where the fitted continuation value equals intrinsic value, so exercising is at least that good. For an American put the boundary lies below the strike and rises toward it as expiry nears; the chart plots this level over time, extracted from the regression cross-sections.
Does the seed change the price?
Yes, within sampling error. The seed fixes the random numbers so the whole run, paths and regressions and boundary, replays exactly, which is what makes the numbers in the worked example verifiable. A different seed shifts the estimate by roughly one standard error, about 0.17 in this example.
Methodology last reviewed 2026-10-06. Results are informational and not financial or engineering advice — see our disclaimer.