Design of Financial CDO Squared Transactions Using

DesignofFinancialCDOSquaredTransactionsUsingConstraintProgrammingPierreFlener1,JustinPearson1,LuisG.Reyna2,andOlofSivertsson11DepartmentofInformationTechnologyUppsalaUniversity,Box337,SE–75105Uppsala,[e-mail],[e-mail],[e-mail]2SwissReFinancialProductsSwissRe,ParkAvenuePlaza,NewYork,NY10055,[e-mail]tenextremelyfastmethodofbuildingaparticularkindofportfolioinfinance,herecalledaportfoliodesign(PD),withapplicationsinthecreditderivativesmarket,forexam-plewhendesigningcollateraliseddebtobligationssquared(CDO2)trans-actions.APDgeneralisesabalancedincompleteblockdesign(BIBD)andisusuallyhardertobuild.Worse,typicalfinancialPDsareanorderofmagnitudelargerthanthelargestBIBDsbuiltsofarbyconstraintpro-grams,andinpracticeanoptimisationversionoftheproblemofbuildingPDshastobesolved.Ourmethodisbasedonembeddingsmalldesigns,whosedeterminationisitselfaconstraintsatisfactionproblem,intotheoriginallargedesign.TogetherwiththedetectionofwhenaPDmightbeaBIBD,symmetrybreaking,extendedreuseofpreviouslybuiltPDs,andadmissibilitycheckingduringsearch,theperformanceofthemethodbecomesgoodenoughfordesigning(near-)optimalCDO2transactions,withsizescommoninthecreditderivativesmarket,withinminutes.Forexample,weoptimallybuildatypicalfinancialPD,whichhasover10746symmetries,injustafewminutes.Thehighqualityofourapproximatedesignscanbeassessedbycomparisonwithalowerboundontheop-timum.OurdesignssufficientlyimprovethecurrentlybestonessoasoftentomakethedifferencebetweenhavingornothavingafeasibleCDO2transactionduetoinvestorandrating-agencyconstraints.1IntroductionThestructuredcreditmarkethasseentwoimportantnewproductsoverthelastdecade:creditderivativesandcollateraliseddebtobligations(CDOs).Thesenewproductshavecreatedtheabilitytoleverageandtransformcreditriskinwaysnotpossiblethroughthetraditionalbondandloanmarkets.CDOstypicallyconsistofaspecial-purposevehiclethathascreditexposuretoaroundonehundreddifferentissuers.Suchvehiclespurchasebondsandloansandotherfinancialassetsthroughtheissuanceofnotesorobligationswithvary-inglevelsofrisk.Inatypicalstructure,creditlossesintheunderlyingpoolareManuscriptClickheretodownloadManuscript:opd.psallocatedtothemostsubordinatedobligationsornotesfirst.Anaturalprogres-sionofthemarkethasbeentousenotesfromexistingCDOsasassetsintoanewgenerationofCDOs,calledCDOSquared(CDO2)orCDOofCDO[13].ThecreditderivativesmarkethasallowedamoreefficientmechanismforcreatingCDO2.Theideaistousetranchesofcreditdefaultswapsinsteadofnotes.Thetranchesarechosenfromacollectionofcreditswiththelevelofliquidityandriskadequatetothepotentialinvestors.ThesetransactionsaresometimeslabelledsyntheticCDO2.InthecreationofasyntheticCDO2,thenaturalquestionarisesonhowtomaximisethediversificationofthetranchesgivenalimiteduniverseofprevi-ouslychosencredits.InatypicalCDO2,thenumberofavailablecreditsrangesfrom250to500andthenumberoftranchesfrom4toasmanyas25.TheinvestmentbankerarrangingforaCDO2usuallyseekstomaximisethereturnofthesubordinatednotesundertheconstraintsimposedbytheratingagenciesandtheinvestors.Thisisachallengethattypicallyisonlypartiallyaddressed,inpartduetothedifficultyofpricingtheunderlyingassets[6].1Inthispaper,weanalysethefinanciallyrelevantabstractedproblemofse-lectingthecreditscomprisingeachofthetrancheswithaminimaloverlap,ormaximumdiversification.Theminimisationoftheoverlapusuallyresultsinbet-terratingsforthenotes,typicallyresultinginmoreefficientstructures.Thecontributionsandsignificanceofthispaperareasfollows:–Weintroduceanewmethodofbuildingportfoliodesignstothefinanceworld,withpracticalapplicationsinthecreditderivativesmarket,suchasthede-signofCDO2transactions.Themethodisfullyautomated,oftenextremelyfast,andbuildsdesignsthatareasclosetotheoptimumasoneiswillingtowaitfor.Itimprovesontheusualmethodofadhocmanualpermutations.–Weintroduceportfoliodesigns(PDs)asanewbenchmarkproblemandanewsuccessfultechnologytransfertotheconstraintprogrammingcommunity.–Wepresentthenewconceptofdesignembeddingsbygeneralisingthewell-knownnotionofdesignmultiples,andsuccessfullyapplyittosolvelargePDinstances(near-)optimally.–Wesignificantlyimprovetherun-timesandqualityofourpreviousresultsin[11],wherethediscussedproblemwasactuallyoriginallyintroduced,usingthenewtheoreticalandmodellingresultsof[17].Theremainderofthispaperisorganisedasfollows.InSection2,wepresentPDsfirstintheirfinancialdomainandtheninanabstractedway,whileintroducingnecessarytheoreticalbackgroundandresults.Next,inSection3,wepresentasophisticatedmethod,implementedasaconstraintprogram,forexactlybuildingaPDbyglobalsearch.SincethismethoddoesnotscaleforthesolutionoftypicalfinancialinstancesoftheoptimisationversionoftheproblemofbuildingPDs,weintroduceinSection4amethodofapproximatelybuildingsuchlargerdesigns,usinganotionofembeddingoccurrencesofsmallerdesignsinalargerone.The1Thereareveryfewpubliclyaccessiblepaperswecanciteinthisintroduction,asmostareconfidentialduetothepotentialfinancialvalueoftheirresults.2determinationofthesmalldesignsisitselfaconstraintsatisfactionprobl