An Efficient, Distributable, Risk Neutral Framework for CVA Calculation

Transcription

1 An Effcent, Dstrbutable, Rsk Neutral Framework for CVA Calculaton Dongsheng Lu and Frank Juan September 2010 Abstract The mportance of counterparty credt rsk to the dervatve contracts was demonstrated consstently throughout the fnancal crss of Accurate valuaton of Credt value adjustment (CVA) s essental to reflect the economc values of these rsks. In the present artcle, we revewed several dfferent approaches for calculatng CVA, and compared the advantage and dsadvantage for each method. We also ntroduced an more effcent and scalable computatonal framework for ths calculaton. 1. Introducton The mportance of counterparty credt rsk to the dervatve contracts was demonstrated consstently throughout the fnancal crss of Credt value adjustment (CVA), whch s the accountng treatment to reflect the far value of counterparty credt rsk, frequently caused large tradng proft and loss (P&L) swng for some dervatve houses durng the crss. Whle t may be temptng to consder CVA as loan loss reserve equvalent for dervatves, there s a fundamental dfference; namely, CVA s ntended to reflect the economc value for carryng counterparty credt rsk based on the market at the partcular pont n tme, the loan loss reserve, on the other hand, s smply the presumed objectve estmate of expected credt loss. In other words, n addton to the expected loss CVA ncludes the current market prce of rsk or market s sentment to rsk. As the market for credt dervatves matures over the past decade, more and more tools are now avalable to the traders for managng the counter party credt rsk assocated wth dervatves tradng. Managng counterparty credt exposure no longer depends purely on settng credt rsk lmts accordng to potental future exposure. Rather, credt rsk can now be hedged wth varous credt dervatves readly avalable such as CDS 1. Managng counterparty credt rsk n fact has become an ntegrated part of many dervatve tradng desks' day to day actvtes 2. The cost for managng ths credt rsk (CVA) therefore s routnely ncluded n the quoted prces for dervatves provded by dealers nowadays. Utlzng the nformaton from CDS market s therefore essental for reflectng the economc value of credt cost n any CVA calculaton. The accuracy and sophstcaton of CVA calculaton s also gettng ever more demandng. For Example, varous detaled features of CSA (Credt Support Annex), such as mutual put breaks and automatc trgger event wll need to be addressed n a CVA calculaton. The economc values of these credt mtgants have to be properly accounted for so that there s proper ncentve to 1 One ssue s how to hedge one s own credt rsk or the correspondng economc value, DVA. The authors beleve that ths should be accounted for naturally n the fundng cost of the tradng actvty n a normal market envronment. There s currently however an asymmetry between CDS spread and fundng spread due to the fact certan fnancal nsttutons have no need for fundng from market n an envronment the central bank s runnng a quanttatve easng polcy. 2 In many fnancal nsttutons, a dedcated CVA desk was establshed to warehouse all [counterparty] credt rsk centrally so that ths rsk can be managed more effcently.

2 utlze these rsk mtgaton technques and the busness can compete farly n a very compettve market. There have been several studes over the years on ssues related to the calculaton for counterparty credt rsk exposure and CVA. For example, Canabarro and Duffe [CD03] provded an excellent ntroducton on measurng and valung counterparty credt rsk. "Counterparty Credt Rsk Modelng" [PYK05] edted by Pykhtn provded more detaled revews on varous facets n the CVA calculatons. Pykhkn and Zhu [PZ07] also gave an excellent overvew on a comprehensve framework for prcng CVA. There are two general frameworks for CVA calculaton, forward smulaton and backward nductons. The framework dscussed n detals by Pykhkn and Zhu (PZ) s what we wll call forward smulaton framework, where computatons are done along all smulated paths gong forward n tme. In ths framework, a large number of Monte Carlo scenaros are generated for market rsk factors that mpact the valuaton of the dervatve portfolo at dscrete tme steps from current tme to the tme when all nstruments wthn the portfolo expre. Each ndvdual poston wthn the portfolo s valued at every tme step along each smulated path, from whch the exposure profles of portfolo are calculated at every tme step. Nettng agreement, collateral thresholds and other credt mtgants are appled durng the calculaton of exposure profle. Assumng defaults are ndependent of market rsk factors, the CVA can be calculated as: + CVA = PV (( 1 Rc ) E ( t ) dpd c ( t ) (1 Ro ) E ( t ) dpd, o ( t )) (1), + Where PV ndcates present value operaton, E t ) and E ( t ) are the expected postve and ( negatve exposures respectvely, dp s the dscrete default probablty, and R s the recovery rate. We use the subscrpt o, and c to dfferentate between the counterparty and one s own credt rsk. One trcky ssue wthn ths approach s the consstency between the rsk factors used for dervatve valuaton and the rsk factors used for CVA calculaton. We wll dscuss further about ths pont at later sectons. The second approach s to calculate CVA through a backward prcng framework. Wthn ths framework, valuatons of all nstruments are performed twce, once n a default free settng and once n the presence of default rsk. In ths framework, CVA s smply the dfference between these two valuatons: CVA = V d V nd (2) Durng both the default free and defaultable settngs, all nstruments wll be valued usng the same rsk factors under the same prcng measure. The prcng settng could be trees, fnte dfference or Monte Carlo smulatons. The remanng of ths artcle s organzed as follows: frst, we wll dscuss a lttle more n detals about the forward smulaton and backward prcng framework. Pros and cons wll be consdered as well as mplementaton detals. Credt smulatons wth correlatons to market rsk factors are dscussed wthn the framework of structural model. Dfferent approaches to obtan a rsk

3 neutralzed transton matrx and technques to the mgraton n trees/lattce/monte Carlo smulatons are dscussed. Afterwards, we wll present an effcent CVA calculaton methodology, whch can be related to both frameworks. Ths methodology s scalable, dstrbutable and easy to mplement. Credt mgraton, ratng based threshold and ATEs are also consdered wthn credt smulatons. We wll address the senstvty calculatons for CVA, ncludng market rsk senstvty and CDS senstvty as well as wrong way exposure. Fnally we wll dscuss the mpact of exercse boundary decson n the presence of default and some practcal ssues. 2 Forward Smulaton Framework Forward smulaton framework nvolves the valuatons of all nstruments at every pont of tme and every scenaro. The loss profles L(t) are obtaned at every tme pont t, whch gves the dstrbuton of portfolo valuatons at dfferent market rsk scenaros. Collateral thresholds can be appled wth netted portfolo value. The maxmum exposure at a gven tme pont and scenaro would be mn[nettngvalue, Collateral Threshold]. The computaton requrement nvolved n forward smulaton framework s very ntensve: N M calculatons for all K nstruments. Whle ths mght stll be manageable for lnear nstruments wth the powerful computng machnes routnely employed by the top fnancal nsttutons, the brute force approach quckly becomes not economcal for more complcated dervatves and exotc nstruments. In order to reduce the requred computaton tme, prcng accuracy has to be compromsed by makng approxmatons n prcng functon. To mplement the forward smulaton, the fnancal nstruments wll need to be aged as they are smulated through tme, and the prcng functons have to be adapted to handle such agng. For example, fxngs and resets of swaps and exercses of optons wll need to be tracked at every tme step. Whle numerc trcks lke Brownan brdge can be used to handle past fxngs by nterpolatng rates between tme ponts, complcaton arses wth Bermudan exercse and/or trgger events due to ther path-dependent characterstc. At each tme step durng the smulaton a decson has to be made regardng whether the deal has been exercsed or trggered prevously. Addtonal approxmatons based on algorthms lke condtonal valuaton are therefore necessary to solve ths ssue. PZ's framework does not perform credt smulaton, therefore t would have dffculty handlng ratng based collateral thresholds. For example, assumng the followng CSA agreements wth the followng collateral thresholds aganst a counterparty: AA Threshold: A Threshold: BBB Threshold: 50M 10M 0 M Ths means that f the counterparty ratng drops to A from AA, the collateral threshold wll decrease from 50M to 10M, or the maxmum exposure to the counterparty wll be 10M; f the ratng drops to BBB or below, the threshold wll decrease to 0, meanng counterparty wll have to post suffcent collateral to cover the current exposure (any postve MTM values). Obvously the

4 ratng based thresholds are valuable credt mtgants, and ts economc value should be reflected n CVA calculatons. In a forward smulaton framework, ths would mply we know the ratng of counterparty at each scenaro or the dstrbuton of ratng across scenaros, whch would requre the smulaton of credt mgratons. We defer the dscusson on extenson of credt smulatons n ths framework to a later secton. 3 Backward Prcng Framework Backward prcng based on default and no-default valuatons provdes an effcent alternatve to the forward smulaton framework. In ths secton we dscuss n detals how t can be mplemented. In general, the portfolo of nstruments are prced wthn the same backward prcng mechansm, regardless t s a tree, a lattce or a Monte Carlo smulaton. Knowng the value of each nstrument at every node/path, the nettng across the portfolo can be performed easly. Defaults and mgraton of credt can be naturally accounted for by steppng backwards n tme through dscountng. We wll explan the mplementaton n detals below. In the frst step, we wll generate a generc tme grd for prcng all nstruments. The last tme grd pont s selected based on the longest maturty of these nstruments. The tme grd s set up to ncludes as many cashflow, exercse/trgger schedules as possble. The grds are dstrbuted wth more tme ponts n the front and less ponts n the back. Gven that Cashflow at 10 years 1 week s barely dstngushable from cashflow at 10 years, fewer tme grd ponts are selected as tme goes further nto the future. Ths should not pose any problem wth the much hgher tolerance n accuracy for CVA. Based on the generc tme grd establshed, a generate rsk neutral tree or scenaros s produced as n regular dervatve prcng. Ether short rate tree or Lbor Market model should serve the present purpose of CVA calculaton for vanlla nterest rate dervatves. However, for structure desk wth exotc nstruments prced wth many dfferent prcng models, a consstent model should be selected for the CVA calculatons. For nterest rate dervatves, we found that a globally calbrated cross currency LMM seems to be a good choce as most fxed ncome dervatves can be prced reasonably well wth ths model. Before gettng nto detals about nstrument prcng, t s worthwhle to dscuss the default and ratng based threshold mechansm. There are two dfferent approaches to handle the credt settng: one s to use the specfc default/credt nformaton to dentfy the exposure at default from scenaros, whch we wll dscuss later. The other s to embed the credt settng n the credt spread enabled dscountng. We wll concentrate on the second approach for the tme beng. Frst, we assume that all credt dscountng spread from market mpled CDS spreads. Gven default probablty P d from market CDS spreads, the dscountng spread s gven by: [ 1 (1 R P ] ( r sp t rt e + ) = e ) (3) d where the sp s the addtonal spread needs to be added to the rskfree rate for dscountng, and P d s the cumulatve default probablty. If only default s concerned, the step by step

5 dscountng mechansm s qute smple: f the nettng value of the portfolo s postve, whch means the counterparty owes money to the concerned fnancal nsttuton, counterparty s credt spread wll be used for addtonal dscountng n that partcular scenaro and tme step; on the other hand, f the nettng value s negatve, whch means the fnancal nsttuton owes money to the counterparty, the nsttuton s own credt spread should be used for the addtonal dscountng. Smlar concept can be extended to account for credt mgraton. Assume a reasonable transton matrx s avalable 3. By propagatng the transton matrx through tme, we wll have the probablty of any tradng party at a specfc ratng at a gven tme. In another words, the probablty of a partcular tradng party A at t, scenaro j and ratng k: P 4 jk. Ths probablty wll allow us to be able to dscount cashflows under dfferent ratngs: where CF = ( Max[ CF, j, Thrk ] Dk + ( CF, j Max[ CF, j, Thrk ]) 0 ) (4) 1, j Pjk D k D k s the dscountng correspondng to ratng k, CF, j s the cashflow(or MTM) at tme t and scenaro j, and Thr s the ratng based threshold. Here we account for the ratng based threshold by applyng two dscount factors, rsk free and rsky, to the correspondng exposures separately. Ths of course mples that the dscountng wll have to be appled after all exposures wth a counterparty has been netted. On the tree, the smplest approach would be to assume that there s no correlaton between credt mgraton and the market rsk factors. Therefore, the ratng dstrbuton at a tree node only depends on the tme. P jk s reduced to P k, and at any gven tme pont can be obtaned by propagatng the transton matrx. There are now two credt ratng dstrbutons cpty P jk and self Pjk at every tme pont, one for the counterparty and one for the consdered fnancal nsttuton. At any gven pont n tme durng the backward nducton prcng, the MTM values of all fnancal nstruments n the portfolo wll be netted frst, and a decson wll be made n terms of the spreads to be used for credt dscountng based on the sgn of netted MTM. If MTM s postve, dscountng, otherwse would acheve the above objectve. cpty P jk wll be used for self P jk wll be used. In later sectons, we wll dscuss credt smulatons that Now we are ready to prce all nstruments through the prcng framework, whether t s HW trees/lattce or LMM scenaros. There are generally four logc steps at any gven tme: - calculate MTM for all fnancal nstruments - make exercse/trgger decson and calculate opton value - net all MTM values and calculate exposure by applyng contractual agreements 3 There are n fact many practcal ssues n terms of obtanng an accurate credt transton matrx, the dscusson of whch wll deferred to later sectons. 4 Of course, ths s based on the assumpton that transton matrx s tme nvarant.

6 - Applyng dscountng backward to next tme step At maturty, all the cashflows for all nstruments are known wth certanty wth certanty and there s no exercse decson to make. Once all the cashflows are netted, t wll be obvous whch spread need to be appled for dscountng the cashflows: f the MTM of the portfolo s postve, counterparty's dscount spread wth dfferent credt ratngs wll be appled; f the MTM of the portfolo s negatve, the concerned fnancal nsttuton s dscount spread wth dfferent credt ratngs wll be appled. At each tme step, exercse/trgger decson wll also be made for ndvdual trades. Followng that, nettng wll be performed for the portfolo of nstruments. Followng the above steps backward to tme zero, the prcng of a portfolo of nstruments n the presence of default rsk wth mportant features n CSA accounted for s obtaned. 4 Smulatng Credt We now dscuss the frst approach: credt smulaton. Credt smulaton not only gves us a way to examne default and ratngs mgratons drectly n scenaros, t also provdes us a tool to reflect the correlaton between market rsk factors and credt defaults/mgratons. Gven the smulated credt n scenaros, one would be able to account for all credt related events. Agan, t s assumed a reasonable transton matrx has been obtaned, whch provdes the rght default and mgraton nformaton over tme. One way to perform credt smulaton s based on the structural model, whch was frst proposed by Merton [MER74]. In a structural model, default occurs when the underlyng asset value decreases below than the lablty threshold, smlar to corporate default behavor. The frst step of credt smulaton process would be calbraton of the default threshold to CDS market. For each ndvdual counterparty, we need to determne the threshold for default at each pont n tme. Smlarly, ratng mgraton can be acheved by ntroducng ratng thresholds for the underlyng asset value. Wth the probablty of transtng to dfferent ratng levels readly avalable from transton matrx, these ratng thresholds can also be determned easly 5. For calbraton, one can generate a large number of scenaros through each tme steps followng smple Geometrc Brownan Moton (GBM), da = σadw (5) Where A stands for asset value and σ s the volatlty of the asset returns. At a specfc tme, the probablty of the reference entty at a partcular ratng level (ncludng default) corresponds to a range of returns. By examnng the returns across scenaros, one can fnd the relevant ratng thresholds. The thresholds are determned so that f the asset return crosses these partcular levels, t should result n result n a correspondng ratng change. These thresholds wll be calbrated for 5 Techncally, the approach we are takng s not a true structural model, as the asset values were never really determned. Rather, we are followng the conceptual framework usng a dummy varable representng asset value. Dfferent thresholds on asset returns for defaultng and transton are determned based observed market data.

7 each credt entty j and at all reference tmes t, denoted as H. Correlatons between the credt and the market rsk factors can be acheved through the followng structural model: s s m m d M = c dw + c dw (6a) s s A A d A = b dw + c dw (6b) j j s s j ρ ( M, A ) = c b, j s s j ρ ( A, A ) = b b, j s s j ρ ( M, M ) = c c (6c) Here A agan s the asset values and M s the market rsk factors that we want to have correlaton wth credt rsk factors, c and b matrx are the coeffcents or factor loadng on the random factors. s m A dw, dw, and dw are for systematc, dosyncratc market rsk and dosyncratc credt rsk factors. ρ are the correlatons among market and credt rsk factors. Gven the already calbrated sngle name thresholds, the credt states for all scenaros can be known easly. The above formulatons are not lmted to the normal dstrbuton framework. One can extend the normalzed random numbers to fat taled dstrbutons, whch would allow better modelng of jump events. The sngle name calbraton under normal dstrbutons can be translated nto fat taled dstrbutons by converson of cumulatve dstrbutons: H 1 = F ( N( )) (7) FT H N Where N() s the cumulatve normal dstrbuton functon, H N s the threshold under 1 normal dstrbuton and F() s the cumulatve fat tal dstrbuton, so F () s the nverson of t. The above formulaton wll also allow changes of CDS spreads durng smulatons, ndcatng the change of credt envronment. The easest way s to make the default thresholds stochastc, for example a mean reverted process: dh = k( a H ) dt + σdw (8) where a s the mean threshold and k s the mean reverson speed. Wth H beng stochastc, the asset wll move closer or further from the default boundary, showng worse or better credt stuatons. As was dscussed earler, credt exposure from CVA can be hedged by tradng CDS dynamcally. To properly reflect the economc values of CDS hedge and CVA, we would need to consstently account for the default rsk n both CDS valuaton and CVA calculaton. Ths can be acheved wth a consstent credt smulatons framework.

8 One complcaton s the CVA calculaton for CDS contract. We know that the CVA for CDS n a smulaton framework comes from the jont default of counterparty and reference entty. Therefore we need to smulate jont defaults n order to have accurate CDS CVA calculatons. The same s true for CDOs and related credt products, where jont defaults are mportant. One key nput for smulatng jont defaults s the default correlaton. However, there s no product traded n the market drectly lnked to jont default, therefore there s no real market data to calbrate the jont default dstrbuton. One lkely possblty s to calbrate to market traded ndex (CDX, ITRAXX) tranches usng structural models (not reduced form copula). Ths topc deserves wll be left for future nvestgaton.. In addton to smulate credt events n Monte Carlo scenaros, t s also possble to generate credt states wthn prcng trees/lattces. Wth credt states and probabltes generated, CVAs can be calculated n exactly the same way wthn trees/lattces as n Monte Carlo smulatons. We wll use tree as an example. For a tree, we have the probabltes for each node j at every pont n tme : P j, and we have P j = 1. If we assume there s no correlaton between market rsk factors and the credt default, j t would be trval to determne the jont credt and market moves, as t s smply the cross product of credt mgraton and the market rsk trees. It s however a lttle bt trcker when there s correlaton. Wth correlatons between market rsk factors and credt events known, the credt nformaton at every node can be generated usng a three-step process. In the frst step, we wll look at the dstrbuton of market rsk factors, and the calbrated credt rsk factors, so we have all the necessary dstrbuton nformaton, such as return percentles. For example, at a gven tme t, we have S S, S,..., 0, 1 2 S N for spot values, and the correspondng probablty at each node: P P, P,..., 0, 1 2 P N That gves a dscrete dstrbuton P(S) for the market rsk factors. Ths would be the same dstrbuton f we have used Monte Carlo smulaton and have followed the same rsk neutral process. In the second step, we would generate the correlated random numbers for all tme steps gven the correlatons between market rsk factors and credt rsk factors. Ths can be acheved ether wth factor models or takng the square root of the correlaton matrx 6. The thrd step would nvolve the mappng of correlated random numbers nto ndvdual credt states: The 6 Ether Cholesky decomposton or sngular value decomposton can be used for ths purpose.

9 market rsk random numbers wll be mapped to the tree nodes gven the percentle dstrbutons, and the credt states can be generated for each node by lookng at the correspondng credt rsk random numbers. Gven the credt states for each node, one can evaluate the exposure for relevant credt states. Another possble approach to generate credt nformaton usng tree s to create correlated tree wth both assets and market rsk factors, whch lkely wll be a qute nvolved process. Wth all asset nformaton known for every tree node, the credt states can be calculated from calbrated threshold nformaton. Wth smulated credt scenaros, forward smulaton framework can be adjusted to manage ratng based collateral thresholds and ATEs. MTM wll only need to be calculated for defaulted scenaros, wth exposures calculated based on ratng based threshold. 5 Rsk Neutralzed Transton Matrx Hstorcal transton matrx, propagatng n tme, can provde tme dependent ratng mgratons nformaton. However, the defaults generated from the propagaton n general do not match the rsk neutral default probabltes derved from CDS market. The reason for ths dscrepancy s obvous: the hstorcal default behavor s dfferent from what s predcted n the market place for future, and does not reflect current credt envronment. In addton, market mpled default rates also nclude some rsk premum n t. To consstently prce credt rsk, we need rsk neutralzed transton matrx for our credt smulatons, just as market traded nstruments are prced n a rsk neutral measure. McNulty and Levn [NV00] proposed a way to translate hstorcal transton matrx to rsk neutral transton matrx based on CAPM model. In ther methodology, they assumed that asset returns n real/hstorcal world can be translated nto rsk neutral world by ntroducng a rsk premum term ρθ, where ρ s the correlaton between asset and market. Assumng the asset returns are normally dstrbuted, the rsk neutral probablty can be therefore derved wth a smple rsk premum term: P RN ( R < b) = N( b + ρθ ) (9) Where N() s the cumulatve normal dstrbuton functon. Usng ths translaton, they were able to lnk the hstorcal transton matrx to the market traded defaults, whch s mpled n CDS tradng. McNulty s approach s smple and gves approxmate defaults n the rsk neutral world, however, may not be consstent wth all default nformaton from market. For our CVA calculatons, we need a more accurate transton matrx, whch s consstent wth avalable CDS term structures, so that the scenaro defaults are relable. In developng the term structure model of credt spreads, Jarrow, Lando and Turnbull [JLT97] used a transton matrx approach wth matrx parameters calbrated to the market traded spreads. The calbraton s done by applyng rsk premum to each ratng and tme perods so to match the

10 market. Ths process s done year over year wth dfferent rsk premum at dfferent tme, mplyng the transton matrx wll not beng Markovan. The followng shows the transton matrx from t to t+1: q~ q ~ = 11 ( t, t + 1) ( t, t + 1) ( t, t + 1) ( t, t + 1) ~ n t, t+ 1 Q... 0 q~ q~ q~ q~ 1n ( t, t + 1) ( t, t + 1)... 1 where tld ndcates n the rsk neutral world. The lnkng between hstorcal world and rsk neutral world s through: q ~ ( t, t + 1) = ( t) q ( t, t + 1) j π j wth π (t) beng the rsk premum appled to the hstorcal transtons. Thus JLT s dea s to start wth a hstorcal transton matrx and apply rsk premum as necessary to make the matrx rsk neutral over tme. Even though the hstorcal transton matrx s arguably dffcult to obtan n the frst place for dfferent tradng enttes, such as corporates vs. fnancals vs. nsurance companes etc, we feel t s prudent to make no assumpton, such as non-markovan property n JLT, about what we start wth and derve a transton matrx. It s also nterestng to see to what extent market traded CDS spreads can be accounted for wth a Markovan transton matrx. To ths end, we decde to derve the transton matrx by optmzng (constraned) the matrx varables, so that the propagated default probabltes are consstent wth the default probabltes mpled from market. Below s a smple example to llustrate the dervaton of the rsk neutralzed transton matrx (RNTM). Assume you have a transton matrx of 4x4 wth A,B,C,D ratng and the market traded default probablty (based on recovery assumpton) at dfferent pont of tme s P(Default) A B C % 1.72% 6.28% % 4.27% 11.80% % 13.00% 25.60% % 30.00% 48.00% Table 1 Where the default probabltes for ratngs A, B, C at lsted for terms 1, 2, 5, and 10 yrs. Then the 3x3 matrx excludng the default row and column can be used as parameters n fttng to the default matrx. The optmzed transton matrx Transton Matrx A B C D A 96.00% 2.50% 1.19% 0.31% B 0.40% 83.00% 14.87% 1.73% C 0.41% 1.00% 92.30% 6.29% D 0.00% 0.00% 0.00% %

11 Table 2 should recover all the market traded default probabltes wthn reasonable range, and therefore are deemed to be rsk neutral. In ths case, we say the transton process s almost perfectly Markovan. However, teratvely propagatng transton matrx forward assumes that every year credt mgraton and default would stay the same. The default/transton ntensty s therefore not gong to change for a specfc ratng. On the other hand, market mpled default probablty reflect dfferent vews about default ntenstes for dfferent tme perods (terms), whch could cause problem n fttng the CDS mpled defaults probabltes wth the rsk neutralzed transton matrx. For a more realstc transton matrx wth many ratngs, the optmzaton of rsk neutralzed transton matrx s a even more complcated problem. In practcal mplementatons, we use Levenberg-Maquart algorthm to optmze the transton matrx parameters n a multdmensonal space. In general we can consder the market traded defaults as: Market Impled Defaults = Transton Matrx Mgratons + Perturbatons from Average Long Term Vew The perturbatons from long term vew ncludes the specfc rsk of the partcular company from the average ratng vew and noses n the market placec due to lqudty. They could be real, or smply nose, whch may be used n makng tradng decsons. In the followng we gve an example of our rsk neutralzaton process for a more realstc matrx and the devatons of mgraton vs market traded CDS. The rsk neutralzaton of transton matrx s done through an optmzaton process n a multdmensonal space. We acheve the optmzaton through the Levenberg-Marquardt algorthm. As an example, we have the probabltes of default term structure estmated as the followng: P(Default) AAA AA A BAA BA B C % 0.81% 0.92% 1.25% 2.87% 6.67% 12.50% % 1.93% 2.24% 3.06% 7.12% 13.85% 24.52% % 3.36% 3.94% 5.38% 12.36% 24.19% 37.30% % 6.75% 7.89% 10.74% 21.94% 41.23% 56.80% % 9.97% 11.69% 15.80% 31.00% 55.46% 70.68% % 14.96% 17.83% 23.79% 44.54% 72.65% 87.80% % 23.74% 27.89% 36.21% 58.11% 83.77% 97.28% % 32.38% 37.46% 48.07% 72.34% 97.30% 99.00% % 39.91% 46.12% 58.85% 84.17% 99.00% 99.00% % 46.17% 53.70% 68.20% 93.18% 99.00% 99.00% Table 3 In our optmzaton, we also use a weghted scheme, where the weghtngs of short/long term defaults are dfferent: there s vrtually no long term default tradng n the market, therefore we put n much less weghts on the long term ponts. The optmzaton parameters are the transton

12 matrx parameters from AAA to C, whch totals 49 parameters. The followng gves one optmzed transton matrx, whch does a good job n replcatng most of the default nformaton: Year 1 AAA AA A BAA BA B C D AAA 89.13% 9.76% 0.46% 0.08% 0.03% 0.07% 0.10% 0.37% AA 0.97% 87.58% 9.61% 0.51% 0.33% 0.18% 0.21% 0.61% A 0.22% 2.35% 86.92% 6.98% 1.54% 0.78% 0.51% 0.69% BAA 0.37% 0.66% 4.00% 81.64% 8.20% 3.30% 0.78% 1.05% BA 0.10% 0.30% 0.50% 2.00% 80.97% 11.00% 3.07% 2.05% B 0.06% 0.07% 0.07% 0.07% 0.50% 75.00% 19.70% 4.53% C 0.10% 0.10% 0.10% 0.10% 0.10% 0.10% 93.00% 6.40% D 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% % Table 4 Ths matrx gves the followng probablty of default: Ftted P(D) AAA AA A BAA BA B C % 1.07% 1.20% 1.51% 3.51% 7.56% 14.65% % 2.20% 2.51% 3.30% 7.46% 15.96% 27.15% % 3.39% 3.92% 5.35% 11.80% 24.54% 37.82% % 5.96% 7.12% 10.21% 21.23% 40.70% 54.67% % 8.80% 10.79% 15.81% 30.91% 54.44% 66.94% % 13.55% 17.04% 24.90% 44.41% 70.05% 79.37% % 22.55% 28.49% 39.62% 61.94% 85.52% 90.55% % 32.24% 39.81% 52.00% 73.39% 93.00% 95.62% % 41.82% 49.98% 61.67% 80.60% 96.53% 97.93% % 50.71% 58.65% 69.09% 85.24% 98.19% 98.98% Table 5 There are two aspects when we lnk ths matrx and the specfc names. Frst n practce, the specfc name CDS wll be dfferent from the average ratng default. Two companes wth the same ratng could have qute dfferent traded CDS spreads. The second s the devatons of ftted transton matrx default from the market traded defaults, whch could be regarded as nose from market lqudty as well as assumpton breakdown of transton matrx model: The default and transtonal ntensty may not be regarded as constant through tme, and there are resdue non- Markovan characterstcs n the real transton process. In mplementng the rsk neutralzed transton matrx n the CVA calculatons for specfc names, the generated ratngs dstrbuton can be adjusted to accormodate the market traded default probabltes. At a specfc pont n tme, the default probabltes from transton matrx propagaton wll be set equal to the market traded default, whereas the rest of the ratngs populatons wll be scaled accordngly. Ths can be mplemented n the sngle name calbraton process, where the default threshold wll be adjusted to match the market traded defaults. Ths adjustment would ensure the ndvdual defaults are matched exactly and CDS prces wll be recoverd n the valuatons. Rsk neutralzed transton matrx wth ratngs mgraton could also provde some nsght nto the underlyng dynamcs mpled by CDS market. Whle most of defaults are through transtons,

13 RNTM provdes a nce way to examne the economcs mpled from the CDS spreads. One can decompose the CDS spread several components. From transton matrx computatons, one can derve, for example out of AA 10yr total default 15%, how much t s comng from transtonal default through a specfc ratng and how much s comng from jump to default. Ths lne of thnkng could provde a useful vew and another layer of hedgng strategy to hedge generc market credt moves. 6 A More Effcent Framework In ths secton, we wll present a new CVA calculaton methodology based on backward prcng framework, but from a dfferent perspectve: [ 1 R) D mn( Threshold, V ) ] Q CVA = E t t A< H d ( (10) where A s asset value, H s the default threshold, R s the recovery assumpton, D s the dscountng and Q ndcates t s rsk neutral measure. In ths defnton, CVA s the ntegraton of all dscounted rsk neutral default exposures from current tme to maturty. What s dfferent from the backward prcng n prevous sectons s that mpled n the above formula we are no longer valuatng the portfolo twce anymore, once s suffcent: we are only concerned about the defaulted scenaros, and CVA s calculated by collectng the defaulted values through tme. We wll use Monte Carlo smulaton as an example. Tree/lattce mplementaton would be trcker f ratng mgraton s consdered. In general the process would nvolve three dfferent steps: Scenaro generaton (market and credt) Valuatons under the market scenaros, and save all values Collect CVAs by aggregatng defaulted exposures (market by credt) The frst step has been descrbed n detals n prevous sectons. In the second step, deal s valued exactly the same way as what one would usually do n regular valuatons, except that t s done wth the same valuaton framework: generc tme grd, generc term structure and market rates, whch are all embedded n the market scenaros. As n the backward prcng framework, valuatons of all nstruments are done wth the same prcng trees/lattces/mc scenaros. The valuatons of each deal can be separate or aggregated. Once the valuatons are done, ther values at the generc tme grd and each scenaro wll be saved n a database. For credt nstruments lke CDS, one would also need the credt scenaros n valuatons. When the valuatons are done separately for each deal and each market scenaro set, the computatons can be dstrbuted on a computer grd easly as the prcng of every deal per scenaro set s a separate computer job. Whle the memory and computaton requrement could be ntensve for large portfolos, we can resort to some smple trcks. One way to save tme and

14 memory s to work wth cashflow nstruments more effcently. For smple lnear nstruments wth only cashflows, all cashflows could be aggregated together as one cashflow nstrument wth many cashflows along tme. For example, f there 1000 swaps wth one counterparty, all these swaps can be collapsed nto one sngle cashflow nstrument wth cashflows at many tme ponts. Ths would reduce the requrement for memory and computatons: only one value, nstead of 1000 values, for each scenaro and tme step needs to be stored n memory. There certanly could be some approxmatons requred for dstrbutng remote cashflows. For non-lnear nstruments, t s also possble to collapse multple nstruments nto one generc nstrument. For example, the caps can be collapsed nto one cap as caplets are separable. One can also convert daly dgtal caps nto weekly or even monthly ones, whch would save computatons sgnfcantly. The thrd step s the aggregaton of CVAs. Ths step s the overlap of credt and market scenaros. Based on the credt scenaros we generated n the prevous secton, we would know the credt state of a tradng party at tme t and scenaro j. In each of the scenaros, we would frst apply the trgger events: f termnaton event s trggered wthout default, the aggregaton would dscontnue. Snce we are only collectng defaulted exposures n ths framework, we would look for all defaulted scenaros for both tradng partes, n whch we net all the trade values and apply ratng based thresholds to calculate the defaulted exposure. The exposure at a gven tme would be: max( 0, mn( H, NetValue* defaultflag)) (11) where the default flag s 1 f counterparty defaults and -1 othewse. So when counterparty defaults and netted value s negatve, there s no credt exposure. In applyng the threshold, we also need to go backward one step to fnd the ratng of the tradng party rght before default, whch s used n lookng up the ratng based threshold table. The most sgnfcant advantage of ths new framework s the effcency and computatonal flexblty. Every deal valuaton for a specfc scenaro set s one separate computer job, whch can be dstrbuted on a computatonal grd. For each CVA calculaton, every deal has to be valued only once wthn the generc prcng model setup. The calculatons of ncremental CVA, market/credt rsk greeks and evaluatons of wrong way exposure can be acheved wth great effcency. Another mportant advantage of ths new framework over the regular backward prcng framework s the accuracy ganed when calculatng CVA for exotc structures, n whch sgnfcant amount of regressons have to be performed n determnng the exercse boundary f Longstaff Schwartz algorthm s used [LS98]. Whle the regressons under rsky and rskless settngs can be sgnfcantly dfferent, the nose from the regresson n practce can be qute sgnfcant as we are takng the dfference of two bg numbers. Ths s especally the case when we are lmted to small number of smulaton paths for the reason of practcal cost. Wth the new framework, the regresson nose ssue s avoded by aggregaton, where only defaulted exposures are collected. To acheve better convergence, for example n dealng wth jont defaults, one may need a lot of

15 credt scenaros. Ths does not cause problem n the current framework. In the thrd aggregaton step, one can aggregate for example 100,000 credt scenaros wth 10,000 saved market scenaro values. So every market scenaro wll have 10 credt scenaros correspond to t, as f every market scenaro has a credt dstrbuton. Assumng there s no CVA for CSA wth zero collateral threshold, the ATE can also be treated as beng zero-threshold. 7 Incremental CVA, Greeks Computatons and Wrong Way Exposure Estmaton It s straghtforward to calculate ncremental CVA effcently, gven the saved portfolo values at every market scenaro and the generated market/credt scenaros. Wth a new trade comng n, the trade wll be prced wthn the same market scenaros, and the result wll be aggregated along wth the saved scenaro values. The dfference n new total CVA and old CVA would be the ncremental CVA. For the hedgng of CVA, greeks are needed n terms of market rsk factor senstvtes lke delta, gamma, vega etc, as well as the credt rsk senstvtes lke the CDS delta. Wth the flexble dstrbuton of computaton tasks, the computaton of market rsk senstvtes are smply new market scenaro generaton, whereas the credt rsk senstvtes means new credt scenaros based on changes n specfc CDS spread. In calculatng the CDS senstvty, the CDS spread shft can be acheved by re-calbratng the default thresholds for the specfc name and re-generate the credt scenaro set. Wth all valuatons per deal and per market scenaro set calculated separately on the dstrbuted computatonal grd, senstvty calculatons can be acheved wth great effcency. As credt scenaros are generated wth dfferent correlaton assumpton between the credt and the market rsk factors, wrong way exposure can also be evaluated by changng the correlaton assumpton, whch would gve dfferent CVAs showng the lkely co-jump n default and market rsk factors. Ths can be accomplshed wth the same market scenaros: the random numbers that are used n generatng the credt scenaros can be re-mapped to reflect the new correlaton. Ths would, by desgn, save addtonal computatons from market scenaro changes. 8 Exercse Boundary of Optons n the Presence of CVA The above approach makes assumpton that the exercse decson n the valuaton of optons wll not be affected by the credt exposure or CVAs. Ths s mostly due to the fact that we want to be able to compute the CVA more effcently through dstrbuted computng, where deal by deal valuatons and credt aggregatons are done at dfferent stages, and are dfferent jobs n mplementatons. As such, the opton exercse boundary s not regressed based on future potental defaults. So how sgnfcant s ths approxmaton? For a typcal Bermudan swapton, whch physcally settles, meanng the exercse would lead to an underlyng swap. The decson of whether exercse the opton s based the comparson of two quanttes: the value of future optons and the current value of underlyng swap. If current value of underlyng swap (meanng exercsng) s greater than the future opton value (meanng not exercsng), then you would exercse the opton, and vce versa. The counterparty credt

16 mplcaton on ths exercse decson les n the dfference n credt value adjustment for underlyng swap and future opton. Now let us explore t wth a specfc example: a 20y Bermudan 4.5% fxed coupon vs. lbor flat rght before an exercse date. Based on a low nterest rate envronment (we used 2010/09/10 market data for ths exercse) wthout any credt consderatons, the optmal decson s to exercse the opton. The followng table shows the change of exercse boundary when counterparty credt rsk s consdered. We compared three dfferent possble default rsk levels (0%, 1%, and 2% annual default rate). The exercse boundary changes by roughly 20bps for counterparty wth 1% annual default, and dropped another 20bps f the counterparty has 2% annual default rate. Scenaro Exercse Boundary No Credt 4.5% 1% annual Default 4.3% 2% annual Default 4.1% Table 6 The reason for the change of exercse boundary s qute obvous: the opton value s one sded wth values localzed at low lbor rates and also tends to be longer term than the underlyng swap, where the fx/float coupons cashflow exchanges are two sded and are more front loaded. Therefore we would expect the counterparty default effect s less for the underlyng swaps than the future optons. Hence more CVA effect from the future opton value, meanng future opton value drops more than the underlyng swap when default s consdered. And the exercse boundary drops as a result. Intutvely, the exercse boundary behavor would be dfferent f the curve shape changes from steep upward slopng to flat and to downward slopng shape. For a downward slopng curve, the opton values wll be even more backend loaded and the dfference between the future opton and underlyng swap CVAs could be even greater. The next queston naturally s what knd effect ths change of exercse boundary would have on the CVA calculaton of ths Bermudan swapton structure. For a 10M 20yr no call 6 month Bermudan swapton wth 5% fxed coupon vs lbor flat, the CVA s around $305K for a default probablty of 2% per year. The change of exercse boundary effect due to the presence of CVA s around $1800. For a no call 2y Bermudan swapton, the CVA exercse boundary effect s $6000 out of $330K CVA. So ths s rather a small effect. Ths s expected snce only the states around the exercse boundary would be mpacted, whch generally have rather small or close to zero values. The rest of the exercse decsons for a majorty of the dstrbuton would reman the same. The exercse boundary would also change when the nettng of portfolo s consdered and when addtonal credt provsons of CSA are ncluded n the calculaton. For example, consderng one partcular opton by tself may ndcate that economcally t should be exercsed; ncludng the nettng wth other postons wth the same counter party mght ndcate the overall poston s more an asset rather than a lablty, whch would then change the exercse boundary and exercse decson.

17 The correlaton between market rsk factor and asset default would obvously have sgnfcant effect on the exercse boundary as well. It s expected the exercse boundary regresson wll be skewed by defaults, meanng the opton exercse decson would be affected by the changes of asset values (a proxy for credt state) as well. Therefore, If Longstaff Schwartz regresson s utlzed to determne the exercse boundary, one would nclude the structural asset varables among the explanatory varables. In such cases, the CVA would be affected by the above assumpton. Wthn the current framework, we are gnorng the effect of CVA on exercse boundary, whch are small most of the tme. However, t s always prudent to understand the characterstcs of the products, portfolos and market envronment, as well as the effects of such assumptons. Ths can be acheved by comparng results from rgorous backward prcng framework wth and wthout credt ncluded decson makng. In the case of szable mpact, one may take addtonal correctons n exercsng the opton. We wll leave ths for future exploratons. 9 Practcal Issues There are some practcal ssues that are relevant to the realzaton of all economc values from the CVA calculatons. For example, n order to realze the values from automatc trgger event, one would need to know as soon as the downgrade event happens, and then make decson about exercsng the trgger opton. The same s true for mutual put breaks. Mutual puts are the same as a seres of Bermudan optons, whch should be executed as f they are the same as normal optons wth exercse decsons. Mssng the exercse decson of such opton when counterparty owes a lot of money could mean sgnfcant loss economcally. In practce ATE and Mutual put breaks may not get executed economcally because of practcal reasons. For example, relatonshp between the two tradng party, or the economc value s small and there s not enough ncentve to call off the deal. However, the economc value of these optons s clearly there and the executon ssue s more of practcal decson, not economc ssue. In CVA calculatons, one could put n a barrer, below whch ATEs and MPs wll not be executed. Thus smulate the real stuaton. 10 Concluson We have revewed several approaches for CVA calculaton, and compared the advantages and dsadvantages of these methods. We proposed an effcent computaton framework, whch can be easly mplemented for dstrbuton computng. We also touch on potental mpact of counterparty rsk on opton exercsng decson. 11 Reference [CD03] Canabarro, E. and D. Duffe, Measurng and Markng Counterparty Rsk. In Asset/Lablty Management for Fnancal Insttutons,edted by L. Tlman. Insttutonal Investor Books, 2003.

18 [JLT97] Jarrow, R., D. Lando and S. Turnbull (1997). A Markov Model for the Term Structure of Credt Spreads, Revew of Fnancal Studes, 10: [LS98] Longstaff, F and E. Schwartz, Valung Amercan optons by smulaton: a least squares approach, Revew of Fnancal Studes, 14: , 1998 [MER74] Merton, Robert C. On the prcng of corporate debt: the rsk structure of nterest Rates, Journal of Fnance, 29: , [NV00] McNulty, C and R. Levn, Modelng Credt Mgraton, Rsk, 10(2): , [PYK05] Pykhtn, M, Counterparty Credt Rsk Modelng, Rsk Books, London, [PZ07] Pykhtn, M and S. Zhu, A Gude to Modellng Counterparty Credt Rsk, ssrn.com/abstract=