Vasicek a Series Expansion for the Bivariate Normal Integral

A Series Expansion forthe Bivariate Normal Integral

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Page ii Release Date: 1996 Revision: 01-April-1998

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A Series Expansion for the Bivariate Normal Integral

Page iii Release Date: 1996 Revision: 01-April-1998

Abstract

An infinite series expansion is given for the bivariate normal cumulative

distribution function. This expansion converges as a series of powers of 1 2− ρd i ,

where ρ is the correlation coefficient, and thus represents a good alternative to

the tetrachoric series when ρ is large in absolute value.


Page 1 Release Date: 1996 Revision: 01-April-1998

Introduction

The cumulative normal distribution function

N n dx u ux

a f a f=−∞z

with

n u ua f a f d i= −−2 2π ½ exp ½

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

N n d d2 2x y u v u vyx

, , , ,ρ ρb g b g=−∞−∞zz (1)

where the bivariate normal density is given by

n21 2

2 2

22 121

u vu uv v

, , exp ½½

ρ π ρ ρρ

b g a f d i= − − − +−

FHG

IKJ

− −(2)

This need arises in at least the following areas:

1. Pricing exotic options. Option with payout depending on the prices of two lognormallydistributed assets, or two normally distributed factors, involve the bivariate normaldistribution function in the pricing formula. Examples include the so-called rainbowoptions (such as calls on maximum or minimum of two assets), extendible options,spread and cross-country swaps, etc.

2. Correlation of derivatives. While the instantaneous correlation of two derivatives is thesame as the correlation of the underlying assets, calculation of the correlation over non-infinitesimal intervals often requires the bivariate normal function.

3. Loan loss correlation. If a loan default occurs when the borrower's assets fall below acertain point, the covariance of defaults on two loans is given by a bivariate normalformula. This covariance is needed when evaluating the variance of loan portfolio losses.

A standard procedure for calculating the bivariate normal distribution function is thetetrachoric series,

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N N N n n He He21

0

11

x y x y x yk

x yk kk

k

, ,!

ρ ρb g a f b g a f b g a f a f b g= ++

+

=

∞

∑ (3)

where

Heki i k i

i

k

x =k

i k ixa f a f a f

!! !−

− − −

=∑ 2

1 2 2

0

2

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, wegive an alternative series that converges approximately as a geometric series with quotient1 2− ρd i .

The Expansion

The starting point of this note is the formula

dd

N nρ

ρ ρ2 2x y x y, , , ,b g b g= (4)

proven in the appendix.

Because of the identity

N N sign N signifif2 2 2 2 2 2 2

02

02

0 00

x y xx y

x xy yx y

y x

x xy yy

xyxy

, , , , , ,½

ρρ

ρ

ρ

ρb g = −

− +

FHGG

IKJJ +

−

− +

FHGG

IKJJ −

><

LNM

OQP

for xy ≠ 0 , we can limit ourselves to calculation of N2 0x, ,ρb g . Suppose first that ρ > 0 . Then byintegrating equation (4) with y = 0 from ρ to 1 we get

N N2 0x x Q, , min ,½ρb g a fb g= − (5)

where

Q x r r

rx

rr

=

= − −−

FHG

IKJ

zz− −

n d

d

2

1

1 21 2

2

0

2 11

, ,

exp ½½

a f

a f d iρ

ρ

π



To evaluate the integral, substitute

r s= −1

to obtain

Q s sxs

s= − −FHGIKJ

− − −−

z½ exp ½½ ½2 11

0

1 22

πρ

a f a f d (6)

Using the expansion

12

222

0

− =− −

=

∞

∑sk

ksk k

k

a f a fa f

½ !

!

we get

Qk

ks

xs

sk k

k

= −FHGIKJ

− − −

=

∞−

∑z½!

!exp ½½2

221

22

00

1 22

πρ

a f a fa f d (7)

Because

22 12

22

2k

ks

xs

sk k k ka fa f d i!

!exp ½½ ½ ½− − − −

−FHGIKJ ≤ ≤ − ρ (8)

for k s> ≤ ≤ −0 0 1 2, ρ , the series in equation (7) converges uniformly in the interval 0 1 2, − ρand can be integrated term by term. It can be easily established that

sxs

sk

kx

xt

ii

x tx

t

k k k kt

i i i i

i

k

− + +

− − − +

=

−FHGIKJ =

+−

−FHGIKJ − − −

FHGIKJ

LNMM

OQPP

z∑

½

½ ½

exp ½!

!

. exp ½!

!

21 2 1

0

22 1

0

2 11 2

21 2 2

d

N

a f a f

a f a f a fπ

for k ≥ 0 . Substitution into (7) then gives

Q Akk

==

∞

∑0

(9)

where

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Ak k

x

x ii

xx

kk k k

i i i i

i

k

=+

−

−−

FHG

IKJ − −∑ − −

−

FHGG

IKJJ

LNMM

OQPP

− +

− − − − +

=

−

1 12 1

1 2

21

21 2 1 2

1

2 1

12

22 1 2

0 2

!

exp ½!

!½ ½

a f

a f a f a f d i a fπρ

ρ πρ

N (10)

Equations (5), (9) and (10) give an infinite series expansion for N2 0 0x, , ,ρ ρb g > . When ρ < 0 ,integration of equation (4) from -1 to ρ yields

N N2 0 0x x Q, , max ½,ρa f a fb g= − +(11)

with Q still given by (9) and (10).

A convenient procedure for computing the terms in the expansion (9) is using the recursiverelationships

Ak

k kx A B

Bk

k kB

k k k

k k

= − −+

+

=−

+−

−

−

2 12 2 1

2 12 2 1

1

21

22

1

a fa fa f d iρ

with

Bx

A xx

B

01 2

2

2

0 2 0

2 11

21

= − −−

FHG

IKJ

= − −−

FHGG

IKJJ

+

−

−

π ρρ

πρ

a f d i

a f

½

½

exp ½

N

To determine the speed of convergence of (9), integrate the first half of inequality (8) from 0 to1 2− ρ . This results in the bound

0 21

11 2< ≤+

−− +A

kk

k½

½½

π ρa f d i

for k > 0 , and therefore the series (9) converges approximately as 1 2−∑ ρd ik k . As the

tetrachoric series for N2 0x, ,ρb g

N N n He2 22 1

0

0 21 1

2 11 2x x x

k kxk k

kk

k

, , ½!

½ρ π ρb g a f a f a f a f a f= ++

−− − +

=

∞

∑ (12)



converges approximately as ρ2k k∑ , a reasonable method for calculating N2 0x, ,ρb g is to use

the tetrachoric series (12) when ρ2 ≤ ½ and the expressions (5) and (11) with the series (9) whenρ2 > ½ .

The error in the calculation of N2 0x, ,ρb g resulting from using m terms in the expansion (9) isbound in absolute value by

Amk

k m

m

=

∞− +∑ <

+−½

½½

21 1

112

2πρ

ρa f d i

Numerical Results

A comparison of the convergence of the tetrachoric series (12) and the alternative calculation (5)or (11) with the series (9) in calculating N2 0x, ,ρb g for high values of the correlation coefficient isgiven in the following tables:

Table 1 Partial sums of the tetrachoric and alternative series

x = − =1 95, .ρ x = − =1 99, .ρ

Number of terms Tetrachoric Alternative Tetrachoric Alternative

1

2

3

5

10

20

30

50

100

200

300

.171033 .158632

.171033 .158631

.167298 .158631

.161764 .158631

.157961 .158631

.158466 .158631

.158660 .158631

.158632 .158631

.158631 .158631

.158631 .158631

.158631 .158631

.174894 .158655

.174894 .158655

.170304 .158655

.162651 .158655

.156068 .158655

.158068 .158655

.159374 .158655

.158599 .158655

.158711 .158655

.158657 .158655

.158654 .158655

Exact .158631 .158655

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Table 2 Number of terms necessary for precision 10-4

ρ = .8 .9 .95 .99

Tetrachoric series

x=0 8 16 30 121

x=±1 7 14 22 75

x=±2 6 11 18 42

Alternative series

x=0 4 3 2 1

x=±1 3 1 1 1

x=±2 1 1 1 1



Appendix

We prove equation (4) by stating a slightly more general result. Let

n ppξξξξ ξξξξ ξξξξ, exp ½ '½S S Sb g a f d i= −− − −2 2 1π

N n dp pξξξξ υυυυ υυυυξξξξ

, ,S Sb g a f=−∞z

be the p-variate normal density function and cumulative distribution function, respectively,

where ξξξξ = x x xp1 2, , ,!d i ′ is a p × 1 vector and S = σ σ σ11 12, , ,! ppn s is a p p× symmetric

positive definite matrix. We prove the following lemma:

Lemma. Let p ≥ 2 . Then for i j≠ ,

∂∂

= ∂∂ ∂

= − −−− −

σ ijp

i jp

p

x xN N

n N

ξξξξ ξξξξ

ξξξξ ξξξξ ξξξξ

, ,

, ,

S S

S S S S S S S

b g b g

e j e ja f a f a f a f a f a f a f a f a f a f

2

2 1 11 2 2 21 111

1 22 21 111

12

Here ξξξξ 1a f d i= x xi j, ’ is the 2 1× vector of x xi j, , ξξξξ 2a f is the p − ×2 1b g vector of the remaining

components of ξξξξ , and S 11a f , S 12a f , S 21a f , S 22a f is the 2 2× , 2 2× −pb g , p − ×2 2b g , and p p− × −2 2b g b gdecomposition of S into the i -th and j -th row and column and the remaining rows and columns.

Proof. We have

∂∂

= − ∂∂

FHG

IKJ +

∂∂

FHG

IKJ

− − −

σ σ σijp

ij ijpn tr nξξξξ ξξξξ ξξξξ ξξξξ, ½ ½ ,S S

SS

SS Sb g b g1 1 1 .

Define ςςςς = v v vpp11 12, , ,!n s by ςςςς = −S 1 and put ψψψψ = y y yp1 2, , ,!d i ’=Vx. Since for i j≠

∂∂

= +Sσ ij

ij jiΕΕΕΕ ΕΕΕΕ

where ΕΕΕΕ ij is the matrix having unity for the ij -th element and zeros elsewhere, we get on

substitution

∂∂

= − +σ ij

p ij i j pv y yn nξξξξ ξξξξ, ,S Sb g d i b g

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On the other hand,

∂∂

= −x

yj

p j pn nξξξξ ξξξξ, ,S Sb g b g

∂∂ ∂

= − +2

x xv y y

i jp ij i j pn nξξξξ ξξξξ, ,S Sb g d i b g

and therefore

∂∂

= ∂∂ ∂σ ij

pi j

px xn nξξξξ ξξξξ, ,S Sb g b g

2

Integrating with respect to ξξξξ and exchanging the order of integration and differentiation yieldsthe first equality of the lemma. The second equality follows from the factorization

n n np p xξξξξ ξξξξ ξξξξ, , ,S S S S S S S Sb g e j e ja f a f a f a f a f a f a f a f a f a f= − −−− −

2 1 11 2 2 21 111

1 22 21 111

12



References

Gupta, Shanti S., 1963, Probability integrals of multivariate normal and multivariate t, Annals ofMathematical Statistics 34, 792-828.

Johnson, N.L. and S. Kotz, 1972, Distributions in Statistics, Continuous Multivariate Distributions,Wiley, New York.

Vasicek, Oldrich A., 1997, The loan loss distribution, working paper, KMV Corporation.

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Vasicek a Series Expansion for the Bivariate Normal Integral