Hi,My question is related to the technicalities of arriving at the weights w1, w2 and w3 when applying the Vanna-Volga method. Castanga & Mercurio (2005) say that the solution for these weights may be found using the Cramer's Rule implying that the determinant of the "vega-vanna-volga"-matrix exists. The determinant is then defined as a sum of multiples of d1's and d2's. I believe that this is really the case (for at least reasonable strikes), but how one goes about proving it?Additionally, I am unable to collapse the sum to the multiple of ln-strikes. So do you guys have any intuition to guide me into the right direction. Full proof is not necessary, just a hint.
! My hypothesis is that since we are dealing with d1s and d2s there is a simple way to use the normal-density identities to write the sum of multiples in a different way to reduce the summation into more intuitive form. However, I have a high tendency to over-complicate things and I am assuming the solution to this problem is far more simple.I took the equation (23) in their paper and used identity d2=d1-sigma*sqrt(T) and substituted into the sum in 23 which yields:(d1(K2))^2*[d1(K3)-d1(K1)]+(d1(K3))^2*[d1(K1)-d2(K2)]+(d1(K1))^2*[d1(K2)-d1(K3)]A=(d1(K2))^2 >0B=(d1(K3))^2 >0C=(d1(K1))^2 >0A*[d1(K3)-d1(K1)]+B*[d1(K1)-d2(K2)]+C*[d1(K2)-d1(K3)]A*[ln(K1/K3)/sigma*sqrt(T)]+B*[ln(K2/K1)/sigma*sqrt(T)]+C*[ln(K3/K2)/sigma*sqrt(T)]Now, it seems that the first term with A is negative and rest is positive. But this approach seems quite complicated..