# QMC

## Convergence analysis of quasi-Monte Carlo sampling for quantile and expected shortfall

Quantiles and expected shortfalls are usually used to measure risks of stochastic systems, which are often estimated by Monte Carlo methods. This paper focuses on the use of the quasi-Monte Carlo (QMC) method, whose convergence rate is asymptotically …

## An Integrated Quasi-Monte Carlo Method for Handling High Dimensional Problems with Discontinuities in Financial Engineering

Quasi-Monte Carlo (QMC) method is a useful numerical tool for pricing and hedging of complex financial derivatives. These problems are usually of high dimensionality and discontinuities. The two factors may significantly deteriorate the performance …

## On the Error Rate of Conditional Quasi-Monte Carlo for Discontinuous Functions

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 $\mathbb{R}^d$, which can be unbounded. Under …

## An importance sampling-based smoothing approach for quasi-Monte Carlo simulation of discrete barrier options

Handling discontinuities in financial engineering is a challenging task when using quasi-Monte Carlo (QMC) method. This paper develops a so-called sequential importance sampling (SIS) method to remove multiple discontinuity structures sequentially …

## Asymptotic normality of extensible grid sampling

Recently, He and Owen (J R Stat Soc Ser B 78(4):917–931, 2016) proposed the use of Hilbert’s space filling curve (HSFC) in numerical integration as a way of reducing the dimension from d1 to d=1. This paper studies the asymptotic normality of the …

## Quasi-Monte Carlo for Discontinuous Integrands with Singularities along the Boundary of the Unit Cube

This paper studies randomized quasi-Monte Carlo (QMC) sampling for discontinuous integrands having singularities along the boundary of the unit cube $[0,1]^d$. Both discontinuities and singularities are extremely common in the pricing and hedging of …

## Efficient Computation of Option Prices and Greeks by Quasi-Monte Carlo Method with Smoothing and Dimension reduction

Discontinuities and high dimensionality are common in the problems of pricing and hedging of derivative securities. Both factors have a tremendous impact on the accuracy of the quasi--Monte Carlo (QMC) method. An ideal approach to improve the QMC …

## An Auto-Realignment Method in Quasi-Monte Carlo for Pricing Financial Derivatives with Jump Structures

Discontinuities are common in the pricing of financial derivatives and have a tremendous impact on the accuracy of quasi-Monte Carlo (QMC) method. While if the discontinuities are parallel to the axes, good efficiency of the QMC method can still be …

## Extensible Grids -- Uniform Sampling on a Space-Filling Curve

We study the properties of points in $[0,1]^d$ generated by applying Hilbert's space-filling curve to uniformly distributed points in $[0,1]$. For deterministic sampling we obtain a discrepancy of $O(n^{-1/d})$ for $d\ge2$. For random stratified …

## On the Convergence Rate of Randomized Quasi-Monte Carlo for Discontinuous Functions

This paper studies the convergence rate of randomized quasi--Monte Carlo (RQMC) for discontinuous functions, which are often of infinite variation in the sense of Hardy and Krause. It was previously known that the root mean square error (RMSE) of …