Longstaff–Schwartz Monte Carlo: Scaling and Basis-Function Limits

Summary

The Longstaff–Schwartz method prices American options by simulating possible asset paths and using regression to estimate continuation values. The discussion notes that simulation can accommodate options whose value depends on multiple factors, so the method is applicable beyond a single underlying dimension.

Its main limitations are Monte Carlo sampling error and the challenge of choosing an effective regression basis and its size. A basis that is too simple may approximate continuation values poorly, while adding terms does not guarantee better performance. The answer points to diagnostic tests in the original paper for assessing parameter choices on a particular problem. It does not provide benchmarks, complexity estimates, or evidence about how efficiency changes as dimension grows, so it offers no quantitative guidance on high-dimensional performance.

Key ideas

• Longstaff–Schwartz uses Monte Carlo simulation and regression to estimate continuation values for American options.
• Simulation can handle options that depend on multiple factors.
• Monte Carlo estimates have sampling error.
• Regression performance depends on selecting suitable basis functions and their number.
• Diagnostic tests can help evaluate parameter choices for a given problem.

Tags

# Longstaff Schwartz algorithm


# Longstaff Schwartz algorithm












I am new in finance, I have implemented the Longstaff Schwartz algorithm for pricing american otion - one asset (dimension = 1).

My questions :

Does this algorithm still efficient for a high dimension ? What are the other drawbacks of this algorithm ?

## Answer by Bob Jansen (score 1)

https://quant.stackexchange.com/a/45577

As Alex C says in the comments, Longstaff and Schwarz did consider multiple factors and mention it as one of the advantages (page 114 in the journal):

> By its nature, simulation is a promising alternative to traditional finite difference and binomial techniques and has many advantages as a framework for valuing, risk managing, and optimally exercising American options. For example, simulation is readily applied when the value of the option depends on multiple factors.

Emphasis mine. Disadvantages that spring to mind are that the method

- Is a Monte Carlo method which implies that results will have a simulation error;

- It is a priori unclear what kind of basis function work best and how many are needed for performing the least squares regression with success.

On the upside, the diagnostic test described on pages 127 and 128 can be used when testing whether the parameters one chooses work well for the problem at hand.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.