Chapter 2: FAQs 225
What are Low Discrepancy Numbers?
Short Answer
Low-discrepancy sequences are sequences of num-
bers that cover a space without clustering and without
gaps, in such a way that adding another number to the
sequence also avoids clustering and gaps. They give the
appearance of randomness yet are deterministic. They
are used for numerically estimating integrals, often in
high dimensions. The best known sequences are due to
Faure, Halton, Hammersley, Niederreiter and Sobol’.
Example
You have an option that pays off the maximum of 20
exchange rates on a specified date. You know all the
volatilities and correlations. How can you find the value
of this contract? If we assume that each exchange rate
follows a lognormal random walk, then this problem
can be solved as a 20-dimensional integral. Such a high-
dimensional integral must be evaluated by numerical
quadrature, and an efficient way to do this is to use
low-discrepancy sequences.
Long Answer
Some financial problems can be recast as integrations,
sometimes in a large number of dimensions. For ex-
ample, the value of a European option on lognormal
random variables can be written as the present value
of the risk-neutral expected payoff. The expected payoff
is an integral of the product of the payoff function and
the probability density function for the underlying(s) at
expiration. If there arenunderlyings then there is typ-
ically ann-dimensional integral to be calculated. If the
number of dimensions is small then there are simple
efficient algorithms for performing this calculation. In
one dimension, for example, divide the domain of inte-
gration up into uniform intervals and use the trapezium
