How many inner simulations to compute conditional expectations with least-square Monte Carlo?

TL;DR

This paper investigates how to optimally choose the number of inner simulations when estimating conditional expectations using least-square Monte Carlo, aiming to maximize computational efficiency.

Contribution

It derives the optimal number of inner simulations K for a given computational budget and provides methods to estimate this optimal value.

Findings

Optimal K depends on the relative costs of sampling Y and X.

Sampling more Y per X can significantly improve computational efficiency.

Numerical examples demonstrate the practical benefits of the proposed approach.

Abstract

The problem of computing the conditional expectation E[f (Y)|X] with least-square Monte-Carlo is of general importance and has been widely studied. To solve this problem, it is usually assumed that one has as many samples of Y as of X. However, when samples are generated by computer simulation and the conditional law of Y given X can be simulated, it may be relevant to sample K $\in$ N values of Y for each sample of X. The present work determines the optimal value of K for a given computational budget, as well as a way to estimate it. The main take away message is that the computational gain can be all the more important that the computational cost of sampling Y given X is small with respect to the computational cost of sampling X. Numerical illustrations on the optimal choice of K and on the computational gain are given on different examples including one inspired by risk management.