Valuation Overview

Valuing financial instruments frequently requires the computation of the expected value of a derivative product. Typically, the underlying asset, e.g., a stock, follows a stochastic model and the value of the derivative depends on the value of the underlying asset at various moments in time. In most cases the expectations cannot be computed analytically. In this respect, one has to approximate the value of an integral, which is often high dimensional.

Monte Carlo methods have been traditionally used for the approximation of such derivative products. In fact, in 1992, the conventional wisdom was that although the theory suggested that low discrepancy methods were superior to Monte Carlo, this theoretical advantage was not seen for high dimensional problems.

Joseph Traub and Spassimir Paskov, a Ph.D. student at that time and currently an associate director in risk management for the investment arm of Barclays Bank, decided to compare the efficacy of low discrepancy sequences and Monte Carlo methods in valuing financial derivatives. They used a model problem provided by Goldman Sachs. Their results showed that low discrepancy methods were significantly superior to Monte Carlo.

Software construction and testing of low discrepancy methods for valuing financial derivatives began at Columbia University in autumn of 1994. At that time, the first version of the software system FinDer was built. It included the a standard implementation of the Halton sequence and an improved version of the Sobol sequence.

FinDer undergoes continuous development. The generalized Faure sequence was added to FinDer in the Spring of 1996. In the Fall of 2000, the generalized Faure was expanded to include two new constructions.The software has been upgraded, expanded and refined to its present form. Our tests indicate that generalized Faure sequences are the best among the methods that we considered.

We have used the low discrepancy sequences in FinDer to price financial derivatives and to calulate Value at Risk (VaR). We have found that the Sobol and the generalized Faure sequences:

  1. beat Monte Carlo by a wide margin
  2. achieve a small error with a small number of sample points (e.g. for a 360-dimensional Collateralized Mortgage Obligation 170 generalized Faure points yield error 10^{-2}),
  3. can be as much as 1000 times faster than Monte Carlo when the accuracy demand is high.