Fast Convergence of Regress-Later Estimates in Least Squares Monte Carlo

Abstract

Many problems in financial engineering involve the estimation of unknown conditional expectations across a time interval. Often Least Squares Monte Carlo techniques are used for the estimation. One method that can be combined with Least Squares Monte Carlo is the "Regress-Later" method. Unlike conventional methods where the value function is regressed on a set of basis functions valued at the beginning of the interval, the "Regress-Later" method regresses the value function on a set of basis functions valued at the end of the interval. The conditional expectation across the interval is then computed exactly for each basis function. We provide sufficient conditions under which we derive the convergence rate of Regress-Later estimators. Importantly, our results hold on non-compact sets. We show that the Regress-Later method is capable of converging significantly faster than conventional methods and provide an explicit example. Achieving faster convergence speed provides a strong motivation for using Regress-Later methods in estimating conditional expectations across time.