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?

• could you provide a link for "quasi-random numbers have much better convergence" ? Apr 1, 2014 at 9:49
• Hi, here's the link. It converges faster because of the use of low discrepancy series. Apr 1, 2014 at 10:02
• This should probably be on Cross Validated. Apr 1, 2014 at 14:28
• @Raphael Why should it go to Cross Validated? Apr 3, 2014 at 8:34

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.

• Thank you very much, Quartz. For anyone interested in reading more, I've found this Apr 1, 2014 at 15:13
• @Quartz - you are the boss ;) Apr 1, 2014 at 17:57
• I have been through the pain of getting a QMC to work, the literature is indeed very sparse and it's non trivial! I would say that the major eye opener for me was checking the statistical properties of each dimension of uncorrelated numbers generated - your samples need to have the correct moments (when converted to normal). Those need to be fixed in order to remove the biases they create (i.e. the sobol numbers i was using had a variance consistently slightly below one). Either you can adjust the generated numbers such that the moments are closer to what you want, or you can rank the dimen...
– will
Jul 5, 2022 at 1:56
• ...sions and then generate new random numbers using a brownian bridge, and then attribute the "best" dimensions of your QMC to the dimensions in the MC that contribute the most variance to the payoff (for this you need to do some guesswork. In my code i simply assumed that the variance contributed was proportional to the total variance of the timesteps for each asset (with a bias to easlier timesteps if both were the same). This works if your number of paths is not so large that you're able to generate them all at once and then adjust them prior to the uniform->normal->correlated transforms.
– will
Jul 5, 2022 at 2:02

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)

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