Skip to content
All library documents

Bias, Uncertainty, and Practical Tradeoffs in QMC Simulation

Article Quant Q&A · Author: Twilight Sparkle
Read the original

Summary

This discussion compares pseudo-random number generators (PRNGs) with quasi-Monte Carlo (QMC) sequences. Although QMC can converge faster, low-discrepancy points may have uneven coverage at practical sample sizes. A convergence plot can look reassuring while failing to reveal bias, and uniform low-dimensional marginals may hide gaps that matter to the simulated payoff or integrand. The risks depend on how the sequence interacts with the problem being integrated.

PRNGs are presented as easier to use and as providing unbiased estimates under standard assumptions. Randomized QMC, such as using independently shifted point sets, is offered as a compromise: it can support an unbiased estimate and an error estimate, while generally reducing some of QMC’s convergence advantage. The discussion gives conceptual guidance rather than a benchmark or proof applicable to every sequence and model. Sequence design, randomization, dimensionality, and the integrand still determine whether QMC is appropriate.

Key ideas

  • QMC sequences can have uneven finite-sample coverage despite favorable-looking convergence plots.
  • Problem-relevant gaps in a quasi-random sequence can introduce bias into an estimate.
  • PRNGs are generally simpler to apply and yield unbiased estimates under standard assumptions.
  • Randomized QMC can provide an error estimate while retaining some low-discrepancy benefits.
  • The choice depends on understanding how the sequence interacts with the integrand.

Tags

StatisticsDerivatives pricingBacktesting
Full text
# What are the merits of pseudo random numbers over quasi random numbers in monte-carlo simulation?


# What are the merits of pseudo random numbers over quasi random numbers in monte-carlo simulation?












I understand that quasi-random numbers have much better convergence, but are there any reasons for me to use pseudo-random numbers instead?

## Answer by Quartz (score 7, accepted)

https://quant.stackexchange.com/a/10783

Quasi Random Numbers are more tricky than it might seem, using them as a black box like with PRNGs is risky. E.g. an unscrambled Sobol' sequence is uniform only asymptotically, while for realistic sample sizes there are subvolumes with significantly different densities. You often do not realize that because the convergence graph looks good anyway, it gives no clue of the bias, and worse the low dimensional marginals $are$ indeed uniform, thus masking the problem. The same holds for lattice rules and other sequences, where empty spaces might interact with important features of the integrand.

On the other hand a good PRNG gives an unbiased result by default (see also here).

A useful compromise is to use randomized QMC, that is multiple QMC point sets shifted by a random offset. That way you get an unbiased estimator and also an error estimate; the downside is of course that convergence will not be as fast as for QMC.

Only use QMC if you know well what you're doing.

## Answer by Probilitator (score 3)

https://quant.stackexchange.com/a/10791

I would argue (this is also what Quartz already hinted at) that PRNGs are far easier to set up than a well functioning QMC and are thus generally user-friendlier

Excel and R both offer a PRNG. (but not a QMC) Thus someone working with these software will be more likely to use a PRNG than to painstakingly implement a QMC. Also as Quartz explained one needs a certain level of Know-How to understand the intricacies of Quasi Monte Carlo. Not many people have that or a willing to invest the time and effort if a PRNG-approach also works (albeit not so fast)

## Answer by SCallan (score 1)

https://quant.stackexchange.com/a/14552

Giuseppe Bruno, Bank of Italy, did some interesting work in R showing that the use of Quasi Random Numbers in Monte Carlo simulations was superior to Pseudo-Random. Here is an abstract of what he presented at useR! 2014: Pricing Credit Risk Derivative with R

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.