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 2₁ = +(yi) = Show that (8) is changed to: edt, for, 1≤i≤3. 0 dz₂ rea Jo £1 N(s, a, Σ) = √ dzi Г by using (9), where dz3 (10) (c) Let i-1 €₁ = $(b₁) = $(α1/€11), e₁ = (b;) = [(a; -Σci-¹(z;))/c], for 2≤i≤3. Cij j=1 Ziew; for 1≤i≤3. (11) Show that (10) is converted to: N(s, a, Σ) = f(w, w2, wx)dw₁dwɔdw's (12) by using (11), where f(w1, W2, W3) Ie = 3 i=1 3 e₁ = II i=1 a₁- Σ(e,w) Φ Cii