9867

Robustness of the Longstaff-Schwartz LSM Method of Pricing American Derivatives

This Demonstration shows the Longstaff–Schwartz least squares Monte Carlo method of computing the value of an American put option. We approximate the American option by a Bermudan option with 50 exercise times. The key point of the method is the estimation of conditional expectations by least squares regression on data obtained from Monte Carlo simulation using a basis for the space of -functions. In this Demonstration we use three different kinds of bases. We show the graphs of the functions that estimate conditional expectations at the exercise points on a graph, below the value of the option computed by means of the Longstaff–Schwartz method, and the value computed by means of the binomial tree method with 1000 time steps.

SNAPSHOTS

  • [Snapshot]
  • [Snapshot]
  • [Snapshot]

DETAILS

The Longstaff–Schwartz least-squares Monte Carlo method of valuing American type options is one of the most popular ones due to its flexibility. It can be used with many models of stock movements, but here we use the classical Black–Scholes model.
This Demonstration was inspired by the paper of Moreno and Navas, in which the robustness of the method with respect to a change of basis and the maximum degree of basis polynomials was investigated. Here we only consider three types of bases: monomial , Laguerre (used in the original Longstaff–Schwartz paper), and Chebyshev. For each choice of basis we allow the degree to be changed as well as the number of paths in a Monte Carlo sample. In addition you can choose whether the computations should be performed with scaled data or not. Scaling produces more accurate results with a Laguerre basis because the presence of exponential terms can lead to significant numerical errors.
For responsiveness we are forced to use much smaller sample sizes than the ones used in the Moreno–Navas article. Nevertheless comparison with option values computed with the Cox, Ross, Rubinstein binomial method with 1000 steps (the same as used by Moreno and Navas) shows that even with small sample sizes the accuracy of the Longstaff–Schwartz method is quite impressive.
F. A. Longstaff and E. S. Schwartz, "Valuing American Options by Simulation: A Simple Least-Squares Approach," Review of Financial Studies, 14(1), 2001 pp. 113–147.
M. Moreno and J. F. Navas, "On the Robustness of Least-Squares Monte Carlo (LSM) for Pricing American Derivatives," Review of Derivatives Research, 6(2), 2003 pp. 107–127.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.









 
RELATED RESOURCES
Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2014 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+