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For immediate release
August 16, 1999

Columbia Receives Patent For Fast Method To Value Complex Securities

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 measured depths.

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 variables.

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 more accurate.

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 margins.

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 other methods.

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 to

To learn about obtaining rights to the patent or to license FINDER, contact Fred Kant, Columbia Innovation Enterprise, at This document is available at Working press may receive science and technology press releases via e-mail by sending a message to