In cooperation with Luis G. Reyna (then at Merrill Lynch, NY, USA), we designed an approximate and often extremely fast method of optimally designing synthetic squared collateralised debt obligations (CDO²) transactions, with v baskets drawn from a universe of b credits, subject to (at present) the following constraints:
- Each basket contains r credits.
- Any two baskets share at most L credits.
The objective of this constrained optimisation problem is to minimise L [FPRS07] [Slides]. Typically, there are v=4...25 baskets of about r=100 credits to be drawn among b=250...500 credits. We assume not only the baskets but also the credits are fully interchangeable, which is not devoid of financial justification, due to the difficulty of risk analysis.
For more realism, side constraints on diversification, ratings, regulation, spread, etc, can readily be added, possibly making the credits only partially interchangeable: contact us if you are interested in doing this extension.
Our method relies on a tight lower bound on L [Siv05, SFP08]. It also includes a new technique for symmetry breaking, namely the embedding of small designs, whose determination is itself a constraint satisfaction problem, into the given large design. For example, we optimally build the typical <v=10,b=350,r=100> CDO² design, which has over 10^746 symmetries, in just a few minutes, for L=22, which is the predicted lower bound:
Optimal <10,350,100> CDO² design, with an overlap of maximum 22 credits between any two baskets, embedding 11 copies of the optimal <10,30,9> CDO² design (on the left, with an overlap of maximum 2 credits between any two baskets), and 1 copy of the optimal <10,20,1> CDO² design (on the right, without any overlaps between any two baskets)
The method is implemented as a constraint program (performing global search), and our most recent results show that (constraint-based) local search gives even better results [ÅFP05b], without even needing the embeddings, but still relying on the lower bound on L.