Quasi Monte Carlo – Fooling around with QuantLib

QuantLib 1.33 provides an implementation of Owen scrambling for Sobol sequences following Burley’s paper [1]. Compared to unscrambled sequences it provides faster convergence and removal of artifacts. We test the properties by performing the integration

$I_1 = \int_{[0,1]^d} \Pi_{i=1}^d (1 + 0.01 (x_i-0.5)) dx_i$

from [2] for dimensions d from 100 to 5000 using sequences of length 30031, see section 6.3 and Figure 9 in [2]. The following graph shows the order of the absolute integration error

$\log_{10} | I_1 - \hat{I_1} |$

for an unscrambled sequence using Kuo direction numbers vs. the Burley scrambled version. To use the sequence generator in OpenSourceRiskEngine select the sequence type “Burley2020SobolBrownianBridge” as in this example.

References:

[1] Brent Burley: Practical Hash-based Owen Scrambling, Journal of Computer Graphics Techniques, Vol. 9, No. 4, 2020

[2] Sobol, Asotsky, Kreinin, Kucherenko: Construction and Comparison of High-Dimensional Sobol’ Generators