The Longstaff–Schwartz (LS) algorithm is a popular least square Monte Carlo method for American option pricing. We prove that the mean squared sample error of the LS algorithm with quasi-regression is equal to [Formula: see text] asymptotically, a where [Formula: see text] is a constant, [Formula: see text] is the number of simulated paths. We suggest that the quasi-regression based LS algorithm should be preferred whenever applicable. Juneja & Kalra (2009) and Bolia & Juneja (2005) added control variates to the LS algorithm. We prove that the mean squared sample error of their algorithm with quasi-regression is equal to [Formula: see text] asymptotically, where [Formula: see text] is a constant and show that [Formula: see text] under mild conditions. We revisit the method of proof contained in Clément et al. [E. Clément, D. Lamberton & P. Protter (2002) An analysis of a least squares regression method for American option pricing, Finance and Stochastics, 6 449–471], but had to complete it, because of a small gap in their proof, which we also document in this paper.
