# Why were Laguerre polynomials a good choice of basis functions for American Monte Carlo?

I am implementing LSMC to price American options based on a custom model. I now need to make a choice of basis functions, so I am looking for the theoretical justification for using Laguerre polynomials in the Longstaff Schwartz paper.

The section "8.3 Choice of basis functions" in the Longstaff Schwartz paper seems to provide (in words-- not math-) some justification for this choice that might be understandable to a statistician, but does not sufficiently explain to me the preference for Laguerre polynomials:

Finally, the choice of basis functions also has implications for the statistical significance of individual basis functions in the regression. In particular, some choices of basis functions are highly correlated with each other, resulting in estimation difficulties for individual regression coefficients akin to the multicolinearity problem in econometrics.

In the paper "The Valuation of Real Options with the Least Squares Monte Carlo Simulation Method", a justification for Laguerre polynomials is given that these produce better numerical results:

In our analysis, we have compared eleven polynomial families, used as basis functions to estimate the continuation value, and we have analysed the convergence of the method increasing the number of basis functions. The numerical results suggest that the weighted Laguerre polynomials provide more accurate results, particularly for the case of compound and mutually exclusive options.

Is there a better mathematical explanation for why Laguerre polynomials are a good choice of basis functions?

In their paper, Francis Longstaff and Eduardo Schwartz found that using Laguerre, Hermite, Legendre or simple powers made very little difference in the results. Some time ago, I also played around with the various choices of polynomials and came to the same conclusion.

There are many more significant choices in the regression techniques:

1. the overall power and the choice of variables. For example, do you include the payoff? Do you scale the variables?
2. there are slight variations in the technique, such as Tsitsiklis and Van Roy (2001), Glasserman and Yu (2004). Would those be more appropriate for the problem at hand?
3. which paths? do you include all paths or just the in-the-money paths?
4. how do you compute the regression? Cholesky, QR, SVD?
5. are polynomials a good choice at all? Should you consider other kinds of regressions?

I cover those points in my book with concrete examples of the differences.

Now, I the choice of Laguerre polynomials may facilitate the point (4) above. Laguerre polynomials are orthogonal with regards to the exponential function. This emphasizes the data at a particular sample and de-emphasizes more distant data. It is possible that the resulting linear system is better behaved then. Another candidate would be Chebyshev polynomials but this implies to map the data to the [-1, 1] interval, while the Laguerre polynomials work directly on [0, infty]. Furthermore, the regression against discrete Laguerre polynomials can be implemented recursively in a stable manner, see Morrison (1967)