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