From CVA to EPE - Chirikhin

1 BUILDING TOMORROW CVA to EPEC omparisons and contrastsAndrey Chirikhin , Han LeeRoyal Bank of Scotland3 February 2012 OutlineCVA and EPE IntroductionComparisons and contrastsFuture workConclusionOutlineCVA and EPE IntroductionComparisons and contrastsFuture workConclusionCVA and EPE IntroductionFundamental questionsIAre CVA and EPE scopes overlapping or complimentary?IShould modelling approaches be similar or different?ICan CVA infrastructure be reused for EPE?ICan EPE infrastructure be reused for CVA?IWhat are the implications forIactive counterparty risk management,Iregulatory capital management,Ioptimization of funding?CVA and EPE IntroductionComparison and scopeDespite aiming at seemingly same goals, the scopes of EPE and CVAsystems are quite FMTM yesyesMain outputEEPE, CVA VAR (EE)CVA, DVA, FVA, ..Applicationwhole bookactively hedgedMeasurehistorical or risk neutralrisk neutralCollateral logicyes, approximateyes, part of payoffScenarios/deltasstress test, ad-hocdeltas, PNL predictPre-dealyesyes, including deltasBacktestregular, statisticalad-hoc, PNL explainCalibration/estimationat least quarterlydriven by hedgingCVA and EPE IntroductionCVA DefinitionICVA is the floating leg of an FTD with stochastic notionalCVA=CcyRZT0EQ MtM+t(1 Rct) cte Rt0( cu+ su)duDt dt CcyRN i=1EQ MtM+ti(1 Rcti) Dti pcFTD(ti)=CcyRN i=1 ELtiDti pcFTD(ti).

2 IThus, the essential component in pricing is the strip of terminalexpected losses (ELti), which is to be computed using the samemega-model that will also diffuse all other factors showing in thepricing and EPE IntroductionCVA VAR: spread risk component (Basel III)ISince ultimately CVA VAR is computed by applying VAR model toregulatory CVA, its own definition depends on the type of the VARmodel used by the banks with full IMM approval for bonds and advance methodfor collateral, pretty much the intuitive definition of CVA:CVA=LGD i EEi 1Di 1+EEiDi2 e si 1ti 1/LGD e siti/LGD +IBanks with different scope of IMM model are provided with otherformulas, CS01 CVA beyond 1 year are relevant under Basel and EPE IntroductionEEPE: counterparty risk component (Basel II+)IDenoteMtMtthe netted set s mark-to-market at timet,computedby in-the future by the EPE evolution the regulation definesExpected Exposure:EEt=E MtM+t Expected Positive Exposure:EPEt=1tRt^10 EEuduEffective EEEEEt=max(EEt,EEEt )Effective EPEEEPEt=1tRt^10 EEEuduI"Effective" exposures are defined for the shortest of the longestmaturity in the netting set and one defined as EAD for deals in trading risk: exposure type examplesCVA and EPE IntroductionHistorical EPE vs risk neutral CVA?

3 IRegulators do not require CCR model to be historical. This, inprinciple, opens the door for the banks with strong risk neutralCVA infrastructure to use it also for issue is that it is not easy to make a risk neutral model a"sufficiently" risk premia to the drifts will not make dynamics of themarket data similar to the one observed in the risk factors will be missing:ITo price CVA on an equity option, one can use Black-Scholes (BS)economy. Only stock needs to be risk neutrally CCR in BS economy, one wants to make not only stockhistorically stochastic, but also implied vol and evolve both stockand vol in a correlated and EPE IntroductionEPE: historical vs risk neutralIConsider BS economy with a single stock as neutral with historical drift:St=S0exp t+ pt Xt ,Xt N(0,1).Risk neutral drift would be =r 2 :St=S0exp t+ 0exp t+ pt Xt+q1 2Yt ,Xt,Yt N(0,1).OutlineCVA and EPE IntroductionComparisons and contrastsFuture workConclusionComparisons and contrastsRates, Inflation and FXIAs we will show CVA and EPE analysis can be complementary, inparticularIto benchmark one model against the other,Ito identify the risk-neutral model s consider a realistic setup of running CVA and EPE kind ofcalculations forI10 year IRS,I10 year RPI swap,I10 year FX outputs to EPE, ENE produced by CVA and EPE data for IR and Inflation models as of January 24, and contrastsCVA Model for IR/InflationIHybrid rates-inflation risk-neutral model (HW/HW/BS):dr=(fr a r(t))dt+ r(t)dWrt,di=(fi a i(t))dt+ i(t)dWit,dY/Y=i(t)dt+ to ATM IR caplets and YOY Inflation capletsIProjection curve = discount curve (LIBOR)IIR/Break even correlation markedIOther correlations set to zero (for this exercise)Comparisons and contrastsEPE Modelling.

4 Arbitrage-free curvesIWe work with "flat instantaneous forward curves" (easier to ruleout arbitrage).ISelect an array of "break" timesft0,t1,..,tngdefines(t) =btc 1 k=0e hk(tk+1 tk)e hbtc(t tbtc),wherebtc=min(i:t tk 0),is segment selection functionandjfh0, ..,hngis the array of "forward rates":ddt[ lns(t)]= and contrastsEPE Modelling: stochastic Nelson-Siegel modelIThe vector offh0, ..,hngis driven by a stochastized version ofNelson-Siegel (NS) model:hn,t= 1,tf1(tn) + 2,tf2(tn) + 3,tf3(tn)f1( ) =1,f2( ) =1 exp( ) ,f3( ) =1 exp( ) exp( ), i,t+1= i+ i i,t+1+ iei,t,ei,t N(0,1).Comparisons and contrastsEPE Modelling: stochastic Nelson-Siegel modelNS postulates the "smoothed" versions of the typical principalcomponents, which makes itmuch handierfor (stressed) factors represent thecomponents building up second and the thirdfactors are related to theLaguerre shapes are typicallyextracted by PCA analysis ofthe individual tenor and contrastsEPE Modelling: NS versus "market standard"IA more standard approach is to model eachhidirectly.

5 Forexample, if non-negativity is essential then i,t+1= i+ i i,t+1+ iei,t,ei,t N(0,1)hi,t=exp( i,t)ei,tdependentITypically this requires 10-20 tenors to be matrix ofei,tis usually PCA ed to produce 3-5principal key advantage is that this can fit the initial data better at theexpense much heavier and contrastsEPE Modelling: equivalenceNote however that if i= in NS, then hn,t=3 i=1 i,tfi(tn) =3 i=1( i+( i 1) i,t+ iei,t)fi(tn)=3 i=1( i+( 1) i,t+ iei,t)fi(tn)=3 i=1 ifi(tn) +( 1)3 i=1 i,tfi(tn) +3 i=1fi(tn) iei,t=An+( 1)hn,t+ n,t,which is equivalent to "standard" approach with constant .ComparisonsEPE Model: specificationIRates and break-even curves: 3 factor index:dIt=( + X(t,1))dt+ dWt,whereX(t,1)is the one-year tenor point of the inflation break evencurveIAll estimated from history:I5 year history,Idaily returns where available,Imonthly data for inflation marginal model is estimated first.

6 Correlations between thediffusion terms are then and contrastsInflation models: calibration/EstimationCVA ModelTenor31/12/2012 01/01/2013 31/12/2013 01/01/2014 31/12/2014 01/01/2015 31/12/2015 01/01/2016 +Rate vol, bp568484109109119119116BE vol, bp179179179179179179202202 Index ModelRateBreak evenIndexParameterLevelCurvatureTiltLeve lCurvatureTiltLT level, bp460-11040332-65-35MR (ann), + and contrasts10 year IRS: broadly sameComparisons and contrasts10 year RPI: CVA riskier?Comparisons and contrastsRPI swap history: curve titling in 2008 ComparisonsRPI modelsINS-based EPE model correctly picks up 2008 curve tilting byassigning higher vol to the second factor and making itslong-term level model we chose cannot reproduce this dynamics, as1-factor HW essentially produces parallel future evolution of CVA model will move the whole curve inparallel, resulting in higher model will mostly tilt the curve, by moving the short s why CCR EPE graph is so flat: shortening maturity isoffset by higher short term volatility induced by NS vol of the first NS factor to the level of second NS factor,makes NS model much more similar to and contrasts10 year RPI: CVA vol brought to EPE levelComparisons and contrasts10 year RPI.

7 NS first factor vol = NS second factor volComparisons and contrastsFX option modelsICVAIB lack 76 setup, rates set to zeroF=F0exp 2Ft2+ FWF , BS= addition to the above BS= Fexp 2 t2+ WF +W q1 2 .ICorrelation kept constant through EPE evolutionI3 cases of correlation: 2f ,0, and contrastsFX option: WWR in EPEMore variance in the model implies higher F= = and contrastsSummaryIDifference may be to be more pronounced for non-linear products,where WWR can be present because of correlations not presentin the CVA model can be a useful tool to identify pitfalls of risk neutralmodels, trying to just fit the and EPE IntroductionComparisons and contrastsFuture workConclusionFuture workOpen topicsIGoing CVA way: dimensionality reduction; analytical approachIGoing EPE way: capital optimization problem; numericalapproachIWe have seen in some explicit model examples the effect onCVA/EPE based upon on choice of risk factors anddimensionality.

8 How can these be looked at from the sameperspective especially from a practical implementation point ofview ?Future workLocal measure changeICVA modelling is based on American Monte Carlo (AMC). Thechoice of a Risk-Neutral measure in CVA has an implication forfeasibility of this extended use of AMC within connection between the two approaches is to use a localmeasure change (historical to risk-neutral), equivalent to theDelta-Gamma approximation (Duffie and Pan, Finance andStochastics, 5, 155-180; 2001) and then numerically making useof a bundling of paths to sample efficiently the additional dynamicparameters ( volatility)IFor accuracy of callability, can make use of similar result thatdimensionality of exercise boundary can be dimensionality ofunderlying risk factors (Hunt and Kennedy 2005; SSRN: )Future workAMC for CVA motivationIWe specifically refer to adaptation of LS rollback to it implements MC within MCIRisk neutral assumption is essential to justify the the future MtM (FMTM) distribution attn+1,AMC rollbackautomatically (in the limit) generates the continuation values,which can be used to values are provided in terms of projection on somebasis functions of the timet measurable variables, typically,easily huge performance gains not only for exotics, but also forthe aggregated portfolios of vanillas (especially if long/shortpositions are present, which allows to offset AMC rollback bias).

9 Future workAMC for EPE challengesIRisk-neutral AMC does not automatically extend to EPE, becauseEPE MC will evolve not only observables in historical measure,but also model CVA (pricing) rollback is the one with the parametersfixed once, during the model CVA projections (strictly speaking) cannot be used forEPE, as they are conditioned on the fixed model AMC, adding exotics to EPE methodology impliesconsiderably higher hardware requirements, mostly driven byvaluation AMC would thus be "MC within MC within MC".IGiven regulators requirements, EPE pricers need to besufficiently accurate for the trades to receive IMM workAMC for EPE: setupIGiven time slicestnandtn+1, in CVA AMC we would discount androll back the vectors of continuation valuesV(tn+1,Xtn+1j RN),Ithe functions of the projection basisXtn+1,Iconditioned on the fixed risk-neutral model parameters EPE MC, assume we knowV(tn+1,Xtn+1, RNtn+1j EPE), (per path) realizations of the values continuation valuesV,Ias functions of the "market" variablesXtn+1and time-tn+1values ofthe valuation model parameters RNtn+1on the same EPE path,Iconditioned on the fixed risk-neutral model parameters need to projectV(tn+1,Xtn+1, RNtn+1j EPE)risk neutrally ontoXtn+1,conditioned on both RNtnand workAMC for EPE.

10 ExtensionA possible an interpolator onV(tn+1,Xtn+1, RNtn+1j EPE)in terms ofXtn+1, RNtn+ such interpolator can be a low dimensionalcorrection to some analytical "Black" formula, which willeffectively serve as a "control variate". a single risk-neutral sub-sampling step fromtninto thisinterpolator, using per-path values of the worst case one would sample from each path attn, but withsmaller number of path per "sub" paths with close RNtnallows doing traditional AMCprojection within each bundle, thus avoiding direct and EPE IntroductionComparisons and contrastsFuture workConclusionConclusionIWe need both CVA and EPE methodologies. They want EPE methodology to be richer in terms of the risk factorcoverage (or at least to be easily extendable to include new riskfactors).IA uniform architecture based on AMC may be possible,containing CVA AMC as a special may yield a holistic solution to bothIlocal "risk neutral" hedging,Icapital/term funding optimization communication has been prepared by The Royal Bank of Scotland , The Royal Bank of Scotland plc or an affiliatedentity ( RBS ).