The numerical computation of a multivariate normal probability is often a difficult problem. This article describes a transformation that simplifies the problem and places it into a form that allows efficient calculation using standard numerical multiple integration algorithms. Test results are presented that compare implementations of two algorithms that use the transformation with currently available software.

Numerical Computation of Multivariate Normal Probabilities

Algorithms for the computation of bivariate and trivariate normal and t probabilities for rectangles are reviewed. The algorithms use numerical integration to approximate transformed probability distribution integrals. A generalization of Plackett's formula is derived for bivariate and trivariate t probabilities. New methods are described for the numerical computation of bivariate and trivariate t probabilities. Test results are provided, along with recommendations for the most efficient algorithms for single and double precision computations.

Numerical computation of rectangular bivariate and trivariate normal and t probabilities

https://rss.onlinelibrary.wiley.com/doi/pdf/10.2307/2347897

Multivariate normal probabilities with error bound

We propose a simple and efficient way to calculate bivariate normal probabilities. The algorithm is based on a formula for the partial derivative of the bivariate probability with respect to the correlation coefficient.

On the computation of the bivariate normal integral

R. Geske and H. E. Johnson (1984) develop an equation for the American put price and obtain accurate prices using a method requiring quadrivariate normal integrals evaluated over an interval containing four equally spaced exercise points. The authors show that a modification of their method, which uses optimal placement of exercise points, yields, in most cases, accurate values using nothing more than bivariate normals. In the more difficult (deep-in-the-money) cases, trivariate normals suffice. Copyright 1992 by American Finance Association.

A Simple and Numerically Efficient Valuation Method for American Puts Using a Modified Geske‐Johnson Approach

Algorithm 462: bivariate normal distribution

An alogorithm is described by which uniform pseudorandom integers may be used to construct binary “numbers” in which the probability that each bit in the word is a 1-bit and can assume any desired parameter value.Techniques for making use of such “numbers” in simulation programming are described.

Thomas G. Donnelly

Papers

Algorithm 462: bivariate normal distribution

Some techniques for using pseudorandom numbers in computer simulation