# 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!

• Traditionally, LS is indeed implemented using weighted Laguerre polynomials. This is because this set of functions forms a complete orthogonal system of the $L^2(0,\infty)$ space (see math.stackexchange.com/questions/100461/…). Intuitively, this means that any function $f : \Omega \to \Bbb{R}$ whose square is integrable over $\Omega=[0,\infty[$ can be expressed as a linear combinations of weighted laguerre polynomials, which is precisely your goal in LS (estimate the continuation value as a combo of elementary functions). – Quantuple Dec 21 '17 at 11:50
• See also this paper which seems to focus on some of the questions you ask concerning the choice of basis functions for real options' pricing (fep.up.pt/conferencias/pfn2006/Conference%20Papers/540.pdf) – Quantuple Dec 21 '17 at 11:53

## 1 Answer

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.