# Chapter 33A Series Expansion for the Bivariate Normal Integral

*Journal of Computational Finance*, 1 (4) (1998), 5–10.

## Abstract

An infinite series expansion is given for the bivariate normal cumulative distribution function. This expansion converges as a series of powers of , where ρ is the correlation coefficient, and thus represents a good alternative to the tetrachoric series when ρ is large in absolute value.

## Introduction

The cumulative normal distribution function

with

appears frequently in modern finance: Essentially all explicit equations of options pricing, starting with the Black-Scholes formula, involve this function in one form or another. Increasingly, however, there is also a need for the bivariate cumulative normal distribution function

1
where the bivariate normal density is given by

2
This need arises in at least the following areas:

*Pricing exotic options*. Options with payout depending on the prices of two lognormally distributed assets, or two normally distributed ...