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Choosing Basis Functions for Least Squares Monte Carlo Option Pricing

Article Quant Q&A · Author: MrPefister
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Summary

The document discusses how to choose regression basis functions in the Longstaff–Schwartz least squares Monte Carlo method for valuing American-style options, including real options with multiple risk factors. The question is whether the few Laguerre polynomials shown in the original paper are generally suitable and what role the basis functions play. In LSMC, these functions approximate the continuation value from simulated state variables, helping determine whether exercise now is preferable to holding the option.

The answer says the original small set is illustrative rather than universal. It may fit poorly, and even a basic Black–Scholes case may require a richer set for a good approximation. With multiple stochastic factors, products of basis functions may be useful. The practical recommendation is to compare candidate bases for the particular model and problem. The discussion gives no selection procedure, error bounds, or numerical evidence, so basis choice and pricing accuracy remain problem-specific empirical questions.

Key ideas

  • LSMC basis functions approximate the option's continuation value from simulated states.
  • The Laguerre functions in the original paper are examples, not a universal prescription.
  • A richer basis may be needed even for a simple vanilla option problem.
  • Multiple risk factors can call for products of basis functions.
  • Candidate bases should be evaluated for the specific model and valuation task.

Tags

OptionsDerivatives pricingStatisticsBacktesting
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# Least Squares Monte Carlo Method for Option Pricing - Basis functions


# Least Squares Monte Carlo Method for Option Pricing - Basis functions












I am trying to implement a LSMC to value an american-style real option with an underlying project value that is exposed to several risk factors.

In the paper of Longstaff & Schwartz, they use the first three Laguerre polynomials to execute their regressions. Unfortunately, they don't provide any explanation about what these functions are and how they are used.

My question is, can we always use these same functions irrespective of the dynamics of the underlying/model setup? How do we choose these basis functions and what do they actually mean?

Obviously I am new to this, so besides the technical stuff, I would very much appreciate some intuitive explanation.

Thank you for your support!

## Answer by Yian Pap (score 3)

https://quant.stackexchange.com/a/37454

No, obviously LS chose 3 simple basis functions to illustrate their method initially. These will work poorly in general, even for a simple vanilla BS you probably need 5-6 of those for a good fit to the continuation value function (and thus the American price). But perhaps you haven't read the paper carefully. If I remember well they have like 7 different examples there where they use progressively more basis functions, products between them (when there are more that one stochastic factors) etc. In general the answer is no, you don't use what's in the paper, but you have to experiment to see what works better for your particular problem.

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