Three New York inventors have shown that complex securities
can be valued much faster and more accurately than by the
method widely used by financial institutions.
Joseph F. Traub, the Edwin Howard Armstrong Professor of
Computer Science at Columbia University; Spassimir Paskov, a
former Ph.D. student of Professor Traub's, now associate
director in risk management at the New York office of
Barclays Capital, the investment arm of Barclays Bank; and
Irwin Vanderhoof, professor of finance at New York
University, have received a patent to be issued Aug. 17.
(Patent # 5,940,819) The patent has been assigned to
Columbia University, where the work was conducted.
A very important instance of a complex security is a
financial derivative, that is, an instrument whose value is
derived from an underlying asset. The significance of this
innovation is illustrated by the size of the derivatives
market which, according to Alan Greenspan, the chairman of
the Federal Reserve, had an estimated value of $70 trillion
in 1998 and as much as $80 trillion in 1999 (New York Times,
March 20, 1999).
Financial derivatives can be extremely complicated and
require large amounts of computer time to value. An example
of a financial derivative is a collateralized mortgage
obligation (CMO), which is constructed from a mortgage pool.
Assuming that the pool consists of 30-year mortgages and
that the interest rate and prepayments can change monthly,
the expected cash flows from the pool is a problem that must
be solved in 360 variables.
The usual method employed by financial institutions to value
financial derivatives is called Monte Carlo. To understand
the idea behind Monte Carlo, consider the problem of
estimating the average depth of a pond with an uneven
bottom. Points at which the depth is measured are chosen at
random. The average depth is estimated by the average of the
In quasi-Monte Carlo, the points are chosen uniformly with
as few points as possible such that the average of the
measured points is close to the true average depth. These
are known as low discrepancy points and are given by a
formula rather than chosen at random. This is a problem in
two variables, the coordinates of the pool's surface, while
the CMO problem is in 360 variables. Measuring the "depths"
in the CMO problem is extremely expensive; it can take a
million computer operations to value a single complex
security by sampling points. It is therefore advantageous to
choose as few points as possible.
The conventional wisdom in the early 1990s, shared by the
world's leading experts, was that quasi-Monte Carlo would
not be effective for problems with more than a dozen or so
In 1992, Professor Vanderhoof saw comment in a technical
publication on the work of Henryk Wozniakowski, professor of
computer science at Columbia, and realized that Professor
Wozniakowski's results on quasi-Monte Carlo could be applied
to complex securities. He arranged to have the New York
investment house, Goldman Sachs and Co., give a complicated
CMO to Professor Traub, who asked one of his Ph.D. students,
Dr. Paskov, to try valuing this CMO using both Monte Carlo
and quasi-Monte Carlo methods.
To their amazement, quasi-Monte Carlo was faster than Monte
Carlo by factors ranging from ten to a thousand. It was also
Dr. Paskov built a software system called FINDER (FINancial
DERivatives) that uses quasi-Monte Carlo to value financial
derivatives and other complex securities. He made major
improvements in known quasi-Monte Carlo methods and
incorporated them into FINDER. After Dr. Paskov received his
Ph.D. and left Columbia, Anargyros Papageorgiou, a former
Ph.D. student of Professor Traub's and now a research
scientist at Columbia, made further improvements to FINDER.
He tested a wide variety of path-dependent financial
derivatives, such as Asian, lookback, and barrier options,
with the same conclusion, that quasi-Monte Carlo is
consistently superior to Monte Carlo, and by wide
Although quasi-Monte Carlo beats Monte Carlo, it is well
known among mathematicians that by analyzing a particular
problem, Monte Carlo can be improved by a number of
techniques, such as variance reduction. Quasi-Monte Carlo is
fast without such an analysis. This is advantageous when a
financial institution wishes to value an investment with
hundreds or thousands of instruments, to quickly value a new
instrument, or to provide a standard against which to test
Why did all the leading experts believe that quasi-Monte
Carlo was not good for problems with many variables, and yet
computer experimentation showed that it was vastly superior
to Monte Carlo for many problems of finance?
For problems such as CMOs, as well as other securities with
multiple future cash flows, a possible explanation was
provided in a recent paper written by Ian H. Sloan of the
University of New South Wales and Professor Wozniakowski.
They use the fact that because of the discount factor in
computing the present value of cash flows, the variables
representing cash flows in the distant future are less
important than short-term variables. They formalize this
fact to achieve their results. For some problems, such as
the CMO, Professors Sloan and Wozniakowski have developed an
explanatory theory. For other types of securities, where
quasi-Monte Carlo has been successful, research continues
toward obtaining a theoretical understanding.
FINDER is available on various platforms, including Windows
95/NT and UNIX. It integrates easily with custom and
off-the-shelf applications. A version of FINDER is available
as a Microsoft Excel add-in. To learn more about FINDER and
to see the results of using quasi-Monte Carlo and Monte
Carlo on a variety of instruments, go tohttp://www.cs.columbia.edu/~traub.