The Importance of Being Scrambled: Supercharged Quasi Monte Carlo

Abstract

In many nancial applications Quasi Monte Carlo (QMC) based
on Sobol low-discrepancy sequences (LDS) outperforms Monte Carlo showing
faster and more stable convergence. However, unlike MC QMC lacks a practical
error estimate. Randomized QMC (RQMC) method combines the best
of two methods. Application of scrambled LDS allow to compute condence
intervals around the estimated value, providing a practical error bound. Randomization
of Sobol' LDS by two methods: Owen's scrambling and digital shift
are compared considering computation of Asian options and Greeks using hyperbolic
local volatility model. RQMC demonstrated the superior performance
over standard QMC showing increased convergence rates and providing practical
error bounds around the estimated values. Eciency of RQMC strongly depend
on the scrambling methods. We recommend using Sobol LDS with Owens
scrambling. Application of eective dimension reduction techniques such as the
Brownian bridge or PCA is critical to dramatically improve the eciency of
QMC and RQMC methods based on Sobol LDS.

Keywords: Quasi Monte Carlo, Randomized Quasi Monte Carlo, Sobol sequences, Monte Carlo option pricing, Skew hyperbolic local volatility model