Journal of Computational Finance, 1 (4) (1998), 5–10.
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.
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
where the bivariate normal density is given by
This need arises in at least the following areas:
A standard procedure for calculating the bivariate normal distribution function is the tetrachoric series,
where
are the Hermite polynomials. For a comprehensive review of the literature, see Gupta (1963).
The tetrachoric series (3) converges only slightly faster than a geometric series with quotient ρ, and it is therefore not very practical to use when ρ is large in absolute value. In this note, we give an alternative series that converges approximately as a geometric series with quotient .
The starting point of this chapter is the formula
which is proven in the Appendix.
Because of the identity
for , we can limit ourselves to calculation of . Suppose first that . Then by integrating Equation (4) with from ρ to 1 we get
where
To evaluate the integral, substitute
to obtain
Using the expansion
we get
Because
for and , the series in Equation (7) converges uniformly in the interval and can be integrated term by term. It can be easily established that
for . Substitution into (7) then gives
where
Equations (5), (9), and (10) give an infinite series expansion for with . When , integration of Equation (4) from –1 to ρ yields
with Q still given by (9) and (10).
A convenient procedure for computing the terms in the expansion (9) is using the recursive relationships
with
To determine the speed of convergence of (9), integrate the first half of inequality (8) from 0 to . This results in the bound
for , and therefore the series (9) converges approximately as . As the tetrachoric series for ,
converges approximately as , a reasonable method for calculating is to use the tetrachoric series (12) when and expressions (5) and (11) with the series (9) when .
The error in the calculation of resulting from using m terms in the expansion (9) is bound in absolute value by
A comparison of the convergence of the tetrachoric series (12) and the alternative calculations (5) or (11) with the series (9) in calculating for high values of the correlation coefficient is given in Tables 33.1 and 2.
Table 33.1 Partial Sums of the Tetrachoric and Alternative Series
Number of Terms | Tetrachoric | Alternative | Tetrachoric | Alternative |
1 | .171033 | .158632 | .174894 | .158655 |
2 | .171033 | .158631 | .174894 | .158655 |
3 | .167298 | .158631 | .170304 | .158655 |
5 | .161764 | .158631 | .162651 | .158655 |
10 | .157961 | .158631 | .156068 | .158655 |
20 | .158466 | .158631 | .158068 | .158655 |
30 | .158660 | .158631 | .159374 | .158655 |
50 | .158632 | .158631 | . 158599 | .158655 |
100 | .158631 | .158631 | .158711 | .158655 |
200 | .158631 | .158631 | .158657 | .158655 |
300 | .158631 | .158631 | .158654 | .158655 |
Exact | .158631 | .158655 |
Table 33.2 Number of Terms Necessary for Precision 10–4
ρ = .8 | ρ = .9 | ρ = .95 | ρ = .99 | |
Tetrachoric series | ||||
8 | 16 | 30 | 121 | |
7 | 14 | 22 | 75 | |
6 | 11 | 18 | 42 | |
Alternative series | ||||
4 | 3 | 2 | 1 | |
3 | 1 | 1 | 1 | |
1 | 1 | 1 | 1 |
We prove equation (4) by stating a slightly more general result. Let
and
be the p-variate normal density function and cumulative distribution function, respectively, where is a vector and is a symmetric positive-definite matrix. We now prove the following lemma:
Lemma. Let . Then for
Here is the vector of is the vector of the remaining components of x, and are the , and decompositions of Σ into the i-th and j-th row and column and the remaining rows and columns.
Proof. We have
Define by and put . Since for
where Eij is the matrix having unity for the (i,j)-th element and zeros elsewhere, we get on substitution
On the other hand,
and
and therefore
Integrating with respect to and exchanging the order of integration and differentiation yields the first equality of the lemma. The second equality follows from the factorization
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