# How to apply quasi-Monte Carlo to path-dependent options?

Following up on my recent question on variance reduction in a Cox-Ingersoll-Ross Monte Carlo simulation, I would like to learn more about using a quasi-random sequence, such as Sobol or Niederreiter, to generate quasi-Monte Carlo interest rate paths.

As before, the objective is to estimate the value of a path-dependent interest rate option. I am evaluating the integral defined by my valuation equation independently for $N$ paths, each of which are $T$ periods long. $T$ may be as high as 40 (quarters, or 10 years) or more in some cases. Should I use the first $N$ $T$-dimensional sequences? The particular Niederreiter implementation I found only allows up to 20 dimensions, which is too small for most securities in my sample. Is this a limitation of the particular implementation or a general limitation of the Niederreiter algorithm? Is it valid to append successive 20-dimensional sequences (with a different random seed each time)?

I've also seen some applications where the first 1000 or so numbers in the sequence are discarded, or where successive innovations to the time-series skip as many as 100 numbers in the sequence between time steps. What is the basis for these modifications to the basic sequences, and does it improve convergence in practical applications?

Finally, is there any research on the pros and cons of the various low-discrepancy sequence algorithms for financial applications? Why does Brian B favor Niederreiter, for example?

For such high-dimensional path problems you will want to use the Morokov technique (you can find the paper online), which takes QR samples for the "important" dimensions and then reverts to pseudorandom for the less important dimensions in an interest rate problem remarkably similar to yours. (Similar principles apply to using QR sequences in factor model based simulations). Effectively, each sample in $T$-dimensional space has $P$ of its dimensions from the QR sequence and $T-P$ dimensions from a pseudorandom sequence.

Skipping techniques are generally considered a good idea, though the need for them is mitigated by the Morokov ordering. The general principle is to start-skip about as many sample points (in $T$ dimensional space) as you plan to take overall. Intermediate skips are, in my opinion, pointless when Morokov ordering is used.

Do not forget to form any normal samples from the uniform QR samples using Moro inversion rather than Box-Mueller.

As for sequence preferences, my firm did a study on a sample set of equity derivatives and found Niederreiter sequences to perform better than Halton or Sobol, by a slight margin.

• Reducing the dimensionality by using a Brownian bridge to fill in gaps over a relatively low-dimension quasi-random sequence seems a bit complicated, given the post-processing which must be applied to prevent interest rates from going negative in the bridge. I wonder if Morokoff's second idea from the paper, sorting by last observation and applying the next $N$ QR innovations in order, is useful by itself. I'll try it out. Thanks again for the suggestions. – Tal Fishman Dec 6 '11 at 21:29
• Is the post-processing necessary? Granted I am only guessing the CIR math is tractable enough to form bridge distributions analytically but if it is then you are guaranteed positive rates when using it. How were you thinking to do the bridge where you would get negative rates? – Brian B Dec 6 '11 at 22:22
• I was reading MathWorks' article on the topic. – Tal Fishman Dec 6 '11 at 22:45
• Ah. They are not being very smart and are simulating the CIR integrated square root process SDE by Euler approximation rather than using the actual density. Also note they were not doing any kind of bridge for CIR there, though now that I think of it one could Euler approximate a bridge in principle. – Brian B Dec 6 '11 at 23:02
• How do you generate a process that produces the desired density? The only method I've seen to go beyond 1st order Euler is Milstein's 2nd order scheme, but I've also seen writeups that suggest that it doesn't work in more than 1 dimension. – Foster Boondoggle Dec 20 '11 at 4:04