Quasi Monte Carlo in FinanceMay 13, 2013 · 1 minute read · Comments
I have been wondering if there was any better alternative than the standard Sobol (+ Brownian Bridge) quasi random sequence generator for the Monte Carlo simulations of finance derivatives.
Here is what I found:
- Scrambled Sobol. The idea is to rerandomize the quasi random numbers slightly. It can provide better uniformity properties and allows for a real estimate of the standard error. There are many ways to do that. The simple Cranley Patterson rotation consisting in adding a pseudo random number modulo 1, Owen scrambling (permutations of the digits) and simplifications of it to achieve a reasonable speed. This is all very well described in Owen Quasi Monte Carlo document
- Lattice rules. It is another form of quasi random sequences, which so far was not very well adapted to finance problems. A presentation from Giles & Kuo look like it’s changing.
- Fast PCA. An alternative to Brownian Bridge is the standard PCA. The problem with PCA is the performance in O(n^2). A possible speedup is possible in the case of a equidistant time steps. This paper shows it can be generalized. But the data in it shows it is only advantageous for more than 1024 steps - not so interesting in Finance.