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Hull and White: CDO without Monte Carlo
Posted: April 21st, 2008, 2:07 pm
by alyukov
I have a question about this paper
http://www.rotman.utoronto.ca/~hull/dow ... aper.pdfIn the equation 1, when authors choose Gaussian distribution for M and Z, the resulting distribution for x would also be normal, so we can calculate inverse for F in the equation 2. Easy to implement and quite fast.But later in the paper they picked t-distribution for M and Z. If I am not mistaken, sum of two t-distributed variables is not t-distributed, so how did they find the distribution F and its inverse for the equation 2?
Hull and White: CDO without Monte Carlo
Posted: April 22nd, 2008, 6:32 am
by katastrofa
Assuming that the comulative and partial distribution function of both M and Z variables are denoted by Phi and phi, respectively, and thatX = aM + bZ,than P(X < x|M=m) = P(Z < (x - am)/b) = Phi((x - am)/b) .Hence,P(X < x) = \int_{-\infty}^\infty Phi((x - am)/b) phi(m) dmThis integral is usually evaluated numerically. Inverting this function may or may not be a PITA, depending on your time constraints. For a typical CDO, you'll need to do it a few thousands of times (125 underlyings x 10-60 of time points, assuming quarterly premium payments).
Hull and White: CDO without Monte Carlo
Posted: April 23rd, 2008, 5:47 pm
by alyukov
numerical integration...not so good and fast and accurate...ok, thanks!
Hull and White: CDO without Monte Carlo
Posted: April 23rd, 2008, 6:30 pm
by katastrofa
For a smooth enough function, numerical quadrature can be pretty accurate.