On the error rate of conditional quasi-Monte Carlo for discontinuous functions

TL;DR

This paper analyzes the convergence rate of conditional quasi-Monte Carlo methods for discontinuous functions, demonstrating significant error reduction and near-optimal rates in applications like options pricing.

Contribution

It establishes that conditional QMC can smooth discontinuous integrands and achieve near-optimal convergence rates, improving over plain QMC in relevant financial problems.

Findings

Conditional QMC smooths discontinuous functions, enabling higher convergence rates.

Error rate of O(n^{-1+ε}) achieved for options pricing and Greeks estimation.

The rate applies to all but the highest order ANOVA terms of the integrand.

Abstract

This paper studies the rate of convergence for conditional quasi-Monte Carlo (QMC), which is a counterpart of conditional Monte Carlo. We focus on discontinuous integrands defined on the whole of $R^d$, which can be unbounded. Under suitable conditions, we show that conditional QMC not only has the smoothing effect (up to infinitely times differentiable), but also can bring orders of magnitude reduction in integration error compared to plain QMC. Particularly, for some typical problems in options pricing and Greeks estimation, conditional randomized QMC that uses $n$ samples yields a mean error of $O(n^{-1+\epsilon})$ for arbitrarily small $\epsilon>0$. As a by-product, we find that this rate also applies to randomized QMC integration with all terms of the ANOVA decomposition of the discontinuous integrand, except the one of highest order.