By Richard Martin (main author)
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Extra info for Credit Suisse's Credit Portfolio Modeling Handbook
Example 1. Take X1 = g1(Z) = Z, and X2 = g2(Z) = –Z. Then X1 and X2 have the same distribution (because the distribution of Z is symmetrical about the origin), and so we cannot tell the difference between a long and a short position in an asset. e. long or short, but the next example shows that the problem is much worse. Example 2. Take X1 = g1(Z) = Z 2 –1 X2 = g2(Z) = Ga (½;2;Φ(Z)) –1 X3 = g3(Z) = Ga (½;2;Φ(–Z)). 13 Either by Jacobians or by the following argument. Denote by F the cumulative distribution function of X.
1 Source: Credit Suisse First Boston In the top right-hand figure the numbers have been constructed so as to make X and Y independent, and are obtained by multiplying the relevant marginal probabilities. One can recognize independence very easily: each row is just a multiple of the top (marginal) row, and each column is a multiple of the left (marginal) column. So the distribution of X, conditionally on Y, does not depend on what value Y happens to take—and vice versa. There is only one way to make random variables independent.
3). Dots represent samples from an ‘imaginary Monte Carlo simulation’. (Bottom) Typical form of codependence representable by one of the ‘standard’ copulas, with positive and negative correlation. Neither of the bottom two graphs looks like the top one, so a standard copula is a poor representation of the codependence. The transfer function approach, on the other hand, captures it in a natural way. X2 X1 Variable 2 Variable 2 Variable 1 Variable 1 We are not, however, dismissing the copulas as entirely irrelevant.
Credit Suisse's Credit Portfolio Modeling Handbook by Richard Martin (main author)