余弦密度近似：在掉期定价中的应用

MILLIMAN WHITE PAPER Cosine densities approximations:Applicationsto swaptionspricing Approximating densitiesto speeduppricing Sophian MehallaLucien Morice Economic Scenario Generators (ESGs) areessential toolsfor insurancecompanies. The production of market-consistent scenariosrequires the models tobe calibratedwiththe current market information.Within ESGs, interest ratemodels focus the attention of practitioners. Their complexity has significantlyincreased over the last decade,andso hasthe needforfast and accuratepricingmethodsfor derivatives.This paperdescribesan efficient swaptionspricingmethod based on density approximation with Fourier seriesunder theLIBOR1Market Model with DisplacedDiffusion and Stochastic Volatility(DD-SV-LMM)framework.A comparison to standard methodsis made. TheLIBOR Market Model (LMM)and its different versions, on theother hand,modelquantities that are directly observablein themarket. In their simplest specification,theyprovide a theoreticalframework consistent with the use of theBlack or Bachelierformulas.Adding a shift coefficient to “displace” the simulateddistribution into the non-positiveregion (i.e.,allowing it to generatenegative rateenvironments)and stochastic volatility(to bettermatch theobservedskewin the market)yields the DisplacedDiffusion with Stochastic Volatility LMM (DD-SV-LMM). Motivations ESGs are the cornerstone of many processes in insurancecompanies such asthe computation of risk managementindicators, orthe valuation of long-term commitments withoptional warranties depending on the economic situation. AnESG is defined as a set ofmodels used to project the jointbehaviour ofrelevant economic or financial risk factors overmultiple scenarios.In order to be consistent with currenteconomic conditions,simulations should be generated bymodels calibratedtocurrent market data:this is the so-calledmarket-consistency.TheMilliman Economic ScenarioGenerator2, that has beenemployedto lead the present work,isan ESGused worldwide. In its original specification, the DD-SV-LMM contains too muchrandomness—roughly speaking—to be analytically tractable.Togetan exploitable version of the model, somestochasticquantitiesare therefore“frozen” to their initial valuestoremovesome hazard and thussimplify the model.Thisis the so-calledfreezing technique. This assumptionyields a Heston-like modelunder whichthe moment-generating function can benumericallycomputed,andtheEuropean options can then beevaluatedthrough moment-generating function integration.This method requires asignificant computation timebecausethe moment-generating functionmustbe computed multipletimes for eachoption to be priced. In the case of insurance companies,modellinginterestraterisk isa prioritybecauselife insurance policies consist inlong-termcommitments embeddingoptional warrantiesthatmay be activated depending on policyholder behaviourinvarious economic environments.Moreover,bonds andinterest rate derivativesconstituteamajor part ofthe assetallocation of insurance companies.This is why,for the sakeof consistency, interest rate models areusuallycalibrated toEuropeanswaptions. In the DD-SV-LMM,one has access to the(approximate)moment-generating functionof the swap rate process,whichallowsus, by applying Fourier transformation,torecover thedensity function of the process.However, this transformation iscostly fromanumerical point of viewas itrequires numericalapproximationof integrals based on Gaussian quadrature.Other methods have been proposedto efficiently approximate Mostinterest-ratemodelsfallintotwo categories:“short rate”modelor“market”model.Theformerfocusses onmodellingtheshort rate,an unobservablefactorcorresponding to thereturn of an investment over an infinitesimal period, as intheHull&WhiteandG2++ models. the density function based on polynomial expansions(seenotably[MEH21]). Those methods are competitive (both interms of computational timeand data replication accuracy) buttheirvalidity domain can berestrictedin some cases.Theso-called“cosine expansion” methodintroduced in [FAN10] andpresentedin this document still takes advantage of the linkbetweenmoment-generatingand density functionsbut allowsus, by applying a number of approximations,to performcomputations without requiringcumbersome quadratures.Inthe end, we would be able to compute swaption pricesbasedon this competitive approach.This method is aimed atreducingthecomputational timededicatedtoeachoption price withoutexcessive accuracy losses. Weignore minor valuation adjustments needed forswapswhichincorporateoneortwobusinessday paymentdelaystoaccommodate for the final rate fixingbeing unknown until themorning after for reformed benchmark rates. Both forward and swap rates, as functions of ZC bonds, arequantities directly observable on financial markets. In itsprimary version, the LMM assumes a lognormal type dynamicfor those rates,allowing to use the Black formula forthepricingof derivatives on forwardrates (floorlet, caplet). In presentDD