Prime Numbers

8.3 Quasi-Monte Carlo (qMC) methods 413
N/10 6
πN
1 3.14158
2 3.14154
3 3.14157
4 3.14157
5 3.14158
6 3.14158
7 3.14158
8 3.141590
9 3.14158
10 3.1415929
Table 8.1 Approximations to π via prime-based qMC (Halton) sequence, using
primes p =2, 3, 5, the volume of the unit 3-ball is assessed for various cumulative
numbers of qMC points, N =10 6 through N =10 7 . We have displayed decimal
digits only through the first incorrect digit.
a similar Monte Carlo √table to be in the third or so digit to the right of the
decimal (because log10 N is about 3.5 in this case). This superiority of qMC
to direct methods—which is an advantage of several orders of magnitude—is
typical for “millions” of points and moderate dimensions.
Now to the matter of Wall Street, meaning the phenomenon of computational
finance. If the notion of very large dimensions D for integration has
seemed fanciful, one need only cure that skepticism by observing the kind of
calculation that has been attempted in connection with risk management theory
and other aspects of computational finance. For example, 25-dimensional
integrals relevant to financial computation, of the form
I = ··· cos |x| e −x·x d D x,
x∈R
were analyzed in [Papageorgiu and Traub 1997], with the conclusion that,
surprisingly enough, qMC methods (in their case, using the Faure sequences)
would outperform direct Monte Carlo methods, in spite of the asymptotic
estimate O((ln D N)/N ), which does not fare too well in practice against
O(1/ √ N)whenD = 25. In other treatments, for example [Paskov and Traub
1995], integrals with dimension as high as D = 360 are tested. As those
authors astutely point out, their integrals (involving collateralized mortgage
obligation, or CMO in the financial language) are good test cases because the
integrand has a certain computational complexity and so—in their words—
“it is crucial to sample the integrand as few times as possible.” As intimated
in [Boyle et al. 1995] and by various other researchers, whether or not a
qMC is superior to a direct Monte Carlo in some high dimension D depends
very much on the actual calculation being performed. The general sentiment
is that numerical analysts not from the financial world per se tend to use