# 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.

• 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). Dec 21 '17 at 11:50
• @Quantuple, from the LS paper, is not clear to me that all the payoffs they consider (americans, Asians, Bermudans) belong to $\mathcal L^2$ . They quickly direct us to some references that don't cover all their cases. Moreover, they never show how the stock prices they use as regressors form a closed subspace of the payoff space (which is the necessary condition for ortho projection). I believe LS should just be proved on finite spaces, not for infinite dimensional ones. Feb 27 '20 at 13:47