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Finance, Economics, and Mathematics by Robert C. Merton, Oldrich A. Vasicek

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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 c33-math-0001, 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

equation

with

equation

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 equation

where the bivariate normal density is given by

2 equation

This need arises in at least the following areas:

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

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