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A Series Expansion forthe Bivariate Normal Integral
KMV Corporation
Page ii Release Date: 1996 Revision: 01-April-1998
COPYRIGHT 1996, 1998, KMV CORPORATION, SAN FRANCISCO, CALIFORNIA, USA. Allrights reserved. Document Number: 999-0000-043. Revision 2.0.1.
KMV Corporation or Kealhofer, McQuown, Vasicek Development, L.P. retain all trade secret,copyright and other proprietary rights in this document. Except for individual use, this documentshould not be copied without the express written permission of the owner.
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All other trademarks are the property of their respective owners.
Published by: Authors:
KMV Corporation Oldrich Alfons Vasicek1620 Montgomery Street, Suite 140San Francisco, CA 94111 U.S.A.Phone: +1 415-296-9669FAX: +1 415-296-9458email: [email protected]: http: // www.kmv.com
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.
A Series Expansion for the Bivariate Normal Integral
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,
KMV Corporation
Page 2 Release Date: 1996 Revision: 01-April-1998
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
A Series Expansion for the Bivariate Normal Integral
Page 3 Release Date: 1996 Revision: 01-April-1998
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
KMV Corporation
Page 4 Release Date: 1996 Revision: 01-April-1998
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)
A Series Expansion for the Bivariate Normal Integral
Page 5 Release Date: 1996 Revision: 01-April-1998
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
KMV Corporation
Page 6 Release Date: 1996 Revision: 01-April-1998
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
A Series Expansion for the Bivariate Normal Integral
Page 7 Release Date: 1996 Revision: 01-April-1998
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
KMV Corporation
Page 8 Release Date: 1996 Revision: 01-April-1998
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
A Series Expansion for the Bivariate Normal Integral
Page 9 Release Date: 1996 Revision: 01-April-1998
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.