Valuing American Options by Simulation: A Simple Least-Squares Approach

Source: Longstaff, F. A. & Schwartz, E. S. (2001) · Review of Financial Studies 14(1), 113–147 · doi:10.1093/rfs/14.1.113

TL;DR

Introduces Least-Squares Monte Carlo (LSM), a simple and general way to value American and Bermudan options by simulation. At each potential exercise date, the continuation value is estimated by least-squares regression of realized discounted future payoffs on basis functions of the current state; the holder exercises whenever immediate payoff exceeds the estimated continuation value. Because it is simulation-based, LSM handles path-dependence and high dimensionality where finite-difference and lattice methods break down — the article values, e.g., an American swaption in a 20-factor string model of the term structure.

What it prices

American-style (early-exercisable) and Bermudan derivatives, including path-dependent and multi-factor payoffs — equity, commodity, FX, energy, mortgage, swap, and real options. The core difficulty is the optimal-exercise/dynamic-programming problem, which finite-difference and binomial techniques cannot handle once more than one or two factors drive the value.

Setup & assumptions

Simulate many paths of the underlying state under the risk-neutral measure.

No-arbitrage valuation: the continuation value at a date equals the conditional expectation of discounted future cash flows, which LSM approximates with a regression on (a finite set of) basis functions of the current state. The paper uses (weighted) Laguerre polynomials as one choice of basis.

Key result

Working backward from maturity, at each exercise date: (1) among in-the-money paths, regress the realized discounted continuation cash flows on basis functions of the state to obtain the estimated continuation function; (2) exercise on a path when immediate payoff > estimated continuation value, else continue; (3) repeat back to t=0 and average the resulting cash flows, discounted, to get the price. The article walks through this with a worked example: an American put, strike 1.10, exercisable at t = 1, 2, 3, riskless rate 6%, illustrated on 8 sample paths. Accuracy is benchmarked against finite-difference solutions (e.g., an implicit scheme with 40,000 time steps/year).

Inputs & implementation

Number and shape of basis functions (polynomials, Laguerre, etc.); number of paths; set of exercise dates.

Regress only on in-the-money paths for efficiency and stability.

Produces both the price and an estimated exercise rule.

Limitations

Price depends on the choice/number of basis functions: too few biases the boundary, too many overfits.

Regressing and exercising on the same in-sample paths makes the standard estimator low-biased; dual/upper-bound methods (Andersen–Broadie, Rogers) bracket the true value.

Computationally heavy with many exercise dates or for high accuracy.

Key references

Longstaff, F. & Schwartz, E. (2001) — Valuing American Options by Simulation — Review of Financial Studies

Tsitsiklis, J. & Van Roy, B. (2001) — Regression Methods for Pricing Complex American-Style Options — IEEE Trans. Neural Networks

Boyle, P. (1977) — Options: A Monte Carlo Approach — Journal of Financial Economics

Provenance: verified/generated from the paper's full text.