3. 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 √det (2) (2)³ LLL 1 exp(-x²-¹x)dx (3) where a = (a1, a2, a3), −∞ < ai < +∞ ( i = 1, 2, 3), x Є R³, dx =dx₁dx2dxn, Σ= (σij)3×3 is a positive definite correlation matrix. It can be proved that can be expressed as (you do not need to show this) Σ = CCT, where C = (Cij)3×3 is a lower triangular matrix and CT is the transpose of C. Since is positive definite, so Cii > 0 for 1 ≤ i ≤ 3; C is strictly diagonally dominant in this sense: i Cii > Cij, 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) 3) x2 = x2(Y1, 92, 93, x3 = x3 (Y1, Y2,Y3) " (4) 1 then, √, f(x1, x2, x3)dxdxdx3 = √(Y1, 42, 43) · det (J) - dy dyzdys (5) D1 1 D2 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 Əx1 მx1 მე1 მყ1 მყ2 მყ3 Jx2 8x2 Əx2 მყვ მყ1 მყ2 8x3 Jx3 მ3 მყ1 მყ2 მყ3 and det(J) is the determinant of J. (a) Let y =C-¹x. N(s, a, Σ) = (2π)¯¯ Show that (3) becomes using the transformation (7): rb₁ b2 .ხვ е 1* e dy √ e dy • e-³ dy ∞ ∞ where i-1 b₁ = a1/c11, b = (ai Cijyj)/Cii, for 2≤ i ≤ 3. j=1 (6) (7) (8)