(b) Let j= Zi 1 Yi z; = (y) = √ √ edt, for,1≤i≤3. Show that (8) is changed to: √2πT Lo (9) rei N(s, a, Σ) = 0 by using (9), where dz1 e2 So dz2 i-1 S rez dz3 (10) €1 = Þ(b1) = $(a1/C11), Ci = $(bi) = ¢[(ai-Σcij§¯¹(zj))/cii], for 2 ≤ i ≤ 3. j=1 (c) Let Zieiwi for 1 ≤ i ≤ 3. Show that (10) is converted to: (11) N(s, a, Σ) = [0,1] 3 01 F(1, 2, 3) dwdw2dw (12) by using (11), where 3 3 ƒ (w1, W2, W3) = П ei = II i=1 i=1 Φ - 2-1 'ai – Σ cijÞ¯¹(e;w;)` Cii 2 This question is about to transform the following multivariate normal CDF to an integral over the unit cube [0, 1]3 by a series of variable transforms. How to estimate such integrals by MC/QMC simulation is crucial in option pricing and sensitivites computation. Denote H(a, Σ) = 1 pai 2 ՐԱՑ (3) where a = (a1, a2, as), 0 < a <+oo (i = 1, 2, 3), x = R³, dx =dx₁d2dxn, Σ (0)3x3 is a positive definite correlation matrix. It can be proved that Σ can be expressed as (you do not need to show this) Σ = CCT, where C = (C)3x3 is a lower triangular matrix and CT is the transpose of C. Since is positive definite, so ci > 0 for 1 ≤ i ≤ 3; C is strictly diagonally dominant in this sense: Cii>Σcij, 1≤i≤3; j=0 and C¹ exists and is also lower triangular. Note: The following variable change for multiple variables integration is needed for this question. Recall from calculus about integration with variable changes: if x1 = x1 (y1, Y2,Y3) = 12 2(Y1, 92, 93) x3 = 13 (Y1, Y2,Y3) (4) 1 then, f(x1, x2, 3) 2x3 = √(V1, 92, 93) - det (J) - dy dyzdys (5) where D2 is the image of the integration domain D₁ under the variable change (4), J is the Jacobian matrix of the variable change (4) and is given by θει θει θει Dys да By: Dyz Dys aza aza aza дуг Byz дуз and det(J) is the determinant of J. (a) Let y=C¹x. Show that (3) becomes using the transformation (7): N(s, a, Σ) = (2) where i-1 (6) (7) b₂ bs dy1 e dyi e (8) = - b₁a₁/c11, b (aΣCity;)/C, for 2≤i≤3. j=1
(b) Let j= Zi 1 Yi z; = (y) = √ √ edt, for,1≤i≤3. Show that (8) is changed to: √2πT Lo (9) rei N(s, a, Σ) = 0 by using (9), where dz1 e2 So dz2 i-1 S rez dz3 (10) €1 = Þ(b1) = $(a1/C11), Ci = $(bi) = ¢[(ai-Σcij§¯¹(zj))/cii], for 2 ≤ i ≤ 3. j=1 (c) Let Zieiwi for 1 ≤ i ≤ 3. Show that (10) is converted to: (11) N(s, a, Σ) = [0,1] 3 01 F(1, 2, 3) dwdw2dw (12) by using (11), where 3 3 ƒ (w1, W2, W3) = П ei = II i=1 i=1 Φ - 2-1 'ai – Σ cijÞ¯¹(e;w;)` Cii 2 This question is about to transform the following multivariate normal CDF to an integral over the unit cube [0, 1]3 by a series of variable transforms. How to estimate such integrals by MC/QMC simulation is crucial in option pricing and sensitivites computation. Denote H(a, Σ) = 1 pai 2 ՐԱՑ (3) where a = (a1, a2, as), 0 < a <+oo (i = 1, 2, 3), x = R³, dx =dx₁d2dxn, Σ (0)3x3 is a positive definite correlation matrix. It can be proved that Σ can be expressed as (you do not need to show this) Σ = CCT, where C = (C)3x3 is a lower triangular matrix and CT is the transpose of C. Since is positive definite, so ci > 0 for 1 ≤ i ≤ 3; C is strictly diagonally dominant in this sense: Cii>Σcij, 1≤i≤3; j=0 and C¹ exists and is also lower triangular. Note: The following variable change for multiple variables integration is needed for this question. Recall from calculus about integration with variable changes: if x1 = x1 (y1, Y2,Y3) = 12 2(Y1, 92, 93) x3 = 13 (Y1, Y2,Y3) (4) 1 then, f(x1, x2, 3) 2x3 = √(V1, 92, 93) - det (J) - dy dyzdys (5) where D2 is the image of the integration domain D₁ under the variable change (4), J is the Jacobian matrix of the variable change (4) and is given by θει θει θει Dys да By: Dyz Dys aza aza aza дуг Byz дуз and det(J) is the determinant of J. (a) Let y=C¹x. Show that (3) becomes using the transformation (7): N(s, a, Σ) = (2) where i-1 (6) (7) b₂ bs dy1 e dyi e (8) = - b₁a₁/c11, b (aΣCity;)/C, for 2≤i≤3. j=1
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
please do part b
![(b) Let
j=
Zi
1
Yi
z; = (y) = √ √ edt, for,1≤i≤3.
Show that (8) is changed to:
√2πT
Lo
(9)
rei
N(s, a, Σ) =
0
by using (9), where
dz1
e2
So
dz2
i-1
S
rez
dz3
(10)
€1 = Þ(b1) = $(a1/C11), Ci = $(bi) = ¢[(ai-Σcij§¯¹(zj))/cii], for 2 ≤ i ≤ 3.
j=1
(c) Let
Zieiwi for 1 ≤ i ≤ 3.
Show that (10) is converted to:
(11)
N(s, a, Σ) =
[0,1] 3
01 F(1, 2, 3) dwdw2dw
(12)
by using (11), where
3
3
ƒ (w1, W2, W3) = П ei = II
i=1
i=1
Φ
-
2-1
'ai – Σ cijÞ¯¹(e;w;)`
Cii
2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F643c9073-bfc9-41d3-b50c-3b14187fea75%2Fd3359b3d-f7a1-4e2b-9b16-9a3dc6cd79d3%2F93bftjl_processed.png&w=3840&q=75)
Transcribed Image Text:(b) Let
j=
Zi
1
Yi
z; = (y) = √ √ edt, for,1≤i≤3.
Show that (8) is changed to:
√2πT
Lo
(9)
rei
N(s, a, Σ) =
0
by using (9), where
dz1
e2
So
dz2
i-1
S
rez
dz3
(10)
€1 = Þ(b1) = $(a1/C11), Ci = $(bi) = ¢[(ai-Σcij§¯¹(zj))/cii], for 2 ≤ i ≤ 3.
j=1
(c) Let
Zieiwi for 1 ≤ i ≤ 3.
Show that (10) is converted to:
(11)
N(s, a, Σ) =
[0,1] 3
01 F(1, 2, 3) dwdw2dw
(12)
by using (11), where
3
3
ƒ (w1, W2, W3) = П ei = II
i=1
i=1
Φ
-
2-1
'ai – Σ cijÞ¯¹(e;w;)`
Cii
2
![This question is about to transform the following multivariate
normal CDF to an integral over the unit cube [0, 1]3 by a series of variable
transforms. How to estimate such integrals by MC/QMC simulation is crucial
in option pricing and sensitivites computation. Denote
H(a, Σ)
=
1
pai
2
ՐԱՑ
(3)
where a = (a1, a2, as), 0 < a <+oo (i = 1, 2, 3), x = R³, dx =dx₁d2dxn,
Σ (0)3x3 is a positive definite correlation matrix. It can be proved that Σ
can be expressed as (you do not need to show this)
Σ = CCT,
where C = (C)3x3 is a lower triangular matrix and CT is the transpose of C.
Since is positive definite, so ci > 0 for 1 ≤ i ≤ 3; C is strictly diagonally
dominant in this sense:
Cii>Σcij, 1≤i≤3;
j=0
and C¹ exists and is also lower triangular.
Note:
The following variable change for multiple variables integration is
needed for this question.
Recall from calculus about integration with variable changes: if
x1 = x1 (y1, Y2,Y3)
=
12 2(Y1, 92, 93)
x3 = 13 (Y1, Y2,Y3)
(4)
1
then,
f(x1, x2, 3) 2x3 = √(V1, 92, 93) - det (J) - dy dyzdys (5)
where D2 is the image of the integration domain D₁ under the variable change
(4), J is the Jacobian matrix of the variable change (4) and is given by
θει θει θει
Dys
да
By: Dyz Dys
aza
aza aza
дуг
Byz дуз
and det(J) is the determinant of J.
(a) Let
y=C¹x.
Show that (3) becomes using the transformation (7):
N(s, a, Σ) = (2)
where
i-1
(6)
(7)
b₂
bs
dy1 e dyi
e
(8)
=
-
b₁a₁/c11, b (aΣCity;)/C, for 2≤i≤3.
j=1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F643c9073-bfc9-41d3-b50c-3b14187fea75%2Fd3359b3d-f7a1-4e2b-9b16-9a3dc6cd79d3%2Fbks97c_processed.jpeg&w=3840&q=75)
Transcribed Image Text:This question is about to transform the following multivariate
normal CDF to an integral over the unit cube [0, 1]3 by a series of variable
transforms. How to estimate such integrals by MC/QMC simulation is crucial
in option pricing and sensitivites computation. Denote
H(a, Σ)
=
1
pai
2
ՐԱՑ
(3)
where a = (a1, a2, as), 0 < a <+oo (i = 1, 2, 3), x = R³, dx =dx₁d2dxn,
Σ (0)3x3 is a positive definite correlation matrix. It can be proved that Σ
can be expressed as (you do not need to show this)
Σ = CCT,
where C = (C)3x3 is a lower triangular matrix and CT is the transpose of C.
Since is positive definite, so ci > 0 for 1 ≤ i ≤ 3; C is strictly diagonally
dominant in this sense:
Cii>Σcij, 1≤i≤3;
j=0
and C¹ exists and is also lower triangular.
Note:
The following variable change for multiple variables integration is
needed for this question.
Recall from calculus about integration with variable changes: if
x1 = x1 (y1, Y2,Y3)
=
12 2(Y1, 92, 93)
x3 = 13 (Y1, Y2,Y3)
(4)
1
then,
f(x1, x2, 3) 2x3 = √(V1, 92, 93) - det (J) - dy dyzdys (5)
where D2 is the image of the integration domain D₁ under the variable change
(4), J is the Jacobian matrix of the variable change (4) and is given by
θει θει θει
Dys
да
By: Dyz Dys
aza
aza aza
дуг
Byz дуз
and det(J) is the determinant of J.
(a) Let
y=C¹x.
Show that (3) becomes using the transformation (7):
N(s, a, Σ) = (2)
where
i-1
(6)
(7)
b₂
bs
dy1 e dyi
e
(8)
=
-
b₁a₁/c11, b (aΣCity;)/C, for 2≤i≤3.
j=1
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