Quantitative error estimates for a least-squares Monte Carlo algorithm for American option pricing

We prove new error estimates for the Longstaff–Schwartz algorithm. We establish an <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$O(\log^{\frac{1}{2}}(N)N^{-\frac{1}{2}})$</EquationSource> </InlineEquation> convergence rate for the expected L <Superscript>2</Superscript> sample error of this algorithm (where N is the number of Monte Carlo sample paths), whenever the approximation architecture of the algorithm is an arbitrary set of L <Superscript>2</Superscript> functions with finite Vapnik–Chervonenkis dimension. Incorporating bounds on the approximation error as well, we then apply these results to the case of approximation schemes defined by finite-dimensional vector spaces of polynomials as well as that of certain nonlinear sets of neural networks. We obtain corresponding estimates even when the underlying and payoff processes are not necessarily almost surely bounded. These results extend and strengthen those of Egloff (Ann. Appl. Probab. 15, 1396–1432, <CitationRef CitationID="CR6">2005</CitationRef>), Egloff et al. (Ann. Appl. Probab. 17, 1138–1171, <CitationRef CitationID="CR7">2007</CitationRef>), Kohler et al. (Math. Finance 20, 383–410, <CitationRef CitationID="CR17">2010</CitationRef>), Glasserman and Yu (Ann. Appl. Probab. 14, 2090–2119, <CitationRef CitationID="CR10">2004</CitationRef>), Clément et al. (Finance Stoch. 6, 449–471, <CitationRef CitationID="CR3">2002</CitationRef>) as well as others. Copyright Springer-Verlag Berlin Heidelberg 2013