Canonical Least-Squares Monte Carlo Valuation of American Options: Convergence and Empirical Pricing Analysis

Table 2

Polynomial families used in the numerical experiment.

	Constant term	First term	Second term	𝑛th term
Monomials	𝑀0(𝑥)=1	𝑀1(𝑥)=𝑥	𝑀2(𝑥)=𝑥^2	𝑀𝑛(𝑥)=𝑥^𝑛
Weighted Laguerre	𝑊0(𝑥)=1	𝑊1(𝑥)=1−𝑥	𝑊21(𝑥)=𝑥2!^2−4𝑥+2	𝑊𝑛𝑒(𝑥)=^𝑥𝑑𝑛!𝑥^𝑛𝑛(𝑥^𝑛𝑒^−𝑥)
Shifted Legendre	𝑆0(𝑥)=1	𝑆1(𝑥)=2(2𝑥−1)	𝑆2(𝑥)=46𝑥^2−6𝑥+1	𝑆𝑛(𝑥)=(−1)^𝑥2^𝑛𝑑𝑛!𝑥^𝑛𝑛[1−(2𝑥−1)^2]^𝑛

Note. This table gives three orthogonal polynomial families. The first three polynomials plus the constant term of each family are used in this numerical experiment.