Multi Currency Credit Default Swaps

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1 Muli Currency Credi Defaul Swaps Quano effecs and FX devaluaion jumps Damiano Brigo Nicola Pede Andrea Perelli arxiv: v2 [q-fin.pr] 21 Jan 2018 Firs posed on SSRN and arxiv on December 2015 Second version posed on SSRN on February 2017 This version: January 23, 2018 Credi Defaul Swaps (CDS) on a reference eniy may be raded in muliple currencies, in ha proecion upon defaul may be offered eiher in he currency where he eniy resides, or in a more liquid and global foreign currency. In his siuaion currency flucuaions clearly inroduce a source of risk on CDS spreads. For emerging markes, bu in some cases even in well developed markes, he risk of dramaic Foreign Exchange (FX) rae devaluaion in conjuncion wih defaul evens is relevan. We address his issue by proposing and implemening a model ha considers he risk of foreign currency devaluaion ha is synchronous wih defaul of he reference eniy. As a fundamenal case we consider he sovereign CDSs on Ialy, quoed boh in EUR and USD. Preliminary resuls indicae ha perceived risks of devaluaion can induce a significan basis across domesic and foreign CDS quoes. For he Republic of Ialy, a USD CDS spread quoe of 440 bps can ranslae ino a EUR quoe of 350 bps in he middle of he Euro deb crisis in he firs week of May More recenly, from June 2013, he basis spreads beween he EUR quoes and he USD quoes are in he range around 40 bps. We explain in deail he sources for such discrepancies. Our modeling approach is based on he reduced form framework for credi risk, where he defaul ime is modeled in a Cox process seing wih explici diffusion dynamics for defaul inensiy/hazard rae and exponenial jump o defaul. For he FX par, we include an explici defaul driven jump in he FX dynamics. As Imperial College, London, U.K. (damiano.brigo@imperial.ac.uk) Imperial College, London, U.K. (n.pede13@imperial.ac.uk) Credi Suisse, London, U.K. (andrea.perelli@credi-suisse.com). 1

2 our resuls show, such a mechanism provides a furher and more effecive way o model credi / FX dependency han he insananeous correlaion ha can be imposed among he driving Brownian moions of defaul inensiy and FX raes, as i is no possible o explain he observed basis spreads during he Euro deb crisis by using he laer mechanism alone. AMS Classificaion Codes : 60H10, 60J60, 91B70; JEL Classificaion Codes : C51, G12, G13 Keywords: Credi Defaul Swaps, Liquidiy spread, Liquidiy pricing, Inensiy models, Reduced Form Models, Capial Asse Pricing Model, Credi Crisis, Liquidiy Crisis, Devaluaion jump, FX devaluaion, Quano Credi effecs, Quano CDS, Muli currency CDS. Conens 1. Inroducion Overview of he Modelling Problem Previous Lieraure Quano CDS Main Conribuion Model Descripion The Roles of he Currencies Two Markes Measures Modeling Framework for he Quano CDS Correcion A diffusive correlaion model: exponenial OU / GBM A Jump o Defaul Framework Resuls Numerical Mehods Quano CDS Par Spreads Parameers Dependence Tes on he Impac of Tenor and Credi Worhiness on he Quano Correcion Correlaion Impac on he Shor Term Versus Long Term Model Calibraion o Marke Daa for Conclusions and Furher Work 36 Appendices 36 Appendix A. Proof of Proposiion

3 Appendix B. Proof of Proposiion

4 1. Inroducion 1.1. Overview of he Modelling Problem The need for quano defaul modeling arises naurally when pricing credi derivaives offering proecion in muliple currencies. Reasons for enering ino Credi Defaul Swaps (CDS) in differen currencies can come from financial, economic, or even legislaive consideraions: hey range from he composiion of he porfolio ha has o be hedged o he accouning rules in force in he counry where he invesor is based. In case he reference eniy is sovereign, economic reasons play a major role since for an invesor i migh be more appealing o buy proecion on, for example, Republic of Ialy s defaul in USD raher han in EUR. Indeed, in he laer case he currency value iself is srongly relaed wih he reference eniy s defaul. Figure 1 shows he ime series of par spreads for USD denominaed and EUR denominaed CDSs on Republic of Ialy from he beginning of 2011 unil he end of The ime range has been chosen so o include he 2011 Euro deb crisis. Spreads (bps) S 5Y EUR S 5Y USD 0 May 2011 Sep 2011 Jan 2012 May 2012 Sep 2012 Jan 2013 May 2013 Sep Spreads (bps) May 2011 Sep 2011 Jan 2012 May 2012 Sep 2012 Jan 2013 May 2013 Sep 2013 Figure 1: In he op char, 5Y par spread ime series for USD denominaed CDSs, SUSD 5Y, and EUR denominaed CDSs, SEUR 5Y, on Ialy. The difference beween he wo par spreads is showed in he boom char. The difference beween he par spreads for USD denominaed and EUR denominaed CDSs is shown in he boom char. In order o build a model which accouns for he defaul informaion and generae he spreads in he wo currencies, he join evoluion of he obligors hazard rae and of he FX rae beween he wo currencies mus be modelled. In he presen paper we show wo ways o model he join dynamics of credi and FX raes. In he firs approach he ineracion beween he credi and he FX componen is enirely explained by an insananeous correlaion beween he Brownian moions driving he sochasic hazard rae and he FX rae. In a second, more sophisicaed modelling approach, a furher source of dependence beween he wo componens is inroduced in 4

5 he form of a condiional devaluaion jump of he FX rae upon defaul of he reference obligor. The diffusive approach emphasizes he limiaions of confining he credi/marke ineracion o insananeous correlaion beween hazard rae and marke risk facors. As shown by comparing he model implied quano spreads in Figure 5 wih he observed quano spreads in Figure 1, insananeous correlaion alone is no able o explain he observed quano spread. This phenomenon is akin o he pricing of credi correlaion insrumens where i has been observed ha insananeous correlaion beween hazard raes is unable o generae he sufficien level of dependence o hi he marke spreads of index ranches (see, for example, Brigo e al. [2013], Brigo and Mercurio [2006], Cherubini e al. [2004]). Using he laer modelling approach we will show how he inroducion of jump o defaul effecs achieves a much sronger FX/Credi dependence han correlaed Brownian moions. In paricular, he addiion of FX jumps allows o recover boh he EUR and he USD spreads (see he resuls presened in Secion 3.5). Furhermore, we show a powerful, ye simple, way of exracing he magniude of currency devaluaion upon defaul from he CDS marke daa (see Secion 3.5.2). In addiion o muli currency CDSs, he quano effec in credi modelling finds a naural applicaion in he credi valuaion adjusmen (CVA) space. CVA is an adjusmen o he fair value of a derivaive conrac ha accouns for he expeced loss due o he counerpary s defaul. We refer he ineresed reader o Brigo e al. [2013] for a comprehensive overview of CVA modelling and o Cherubini [2005] for specific discussions abou collaeral modelling. Modelling he dependence beween credi and marke risk facors is crucial o accuraely calculae he CVA charge. One of he main challenges in calculaing CVA is he lack of liquid CDS marke daa o calibrae model parameers. The calibraion and approximaion echniques showed in his paper o connec currency devaluaion wih muli currency CDS par spreads can as well be applied o CVA modelling for example, o beer reflec righ way or wrong way risk. The resuling FX/Credi cross modelling improvemen is crucial, especially in hose cases where he ineracion beween he counerpary credi and he FX componen is srong, i.e. wih emerging marke credis and sysemically relevan counerparies. In Secion 2.5, we show how he inroducion of defaul driven FX jumps changes he dynamics of he sochasic hazard rae afer a measure change. This happens because, from a mahemaical perspecive, he FX rae is a componen of he Radon Nikodym derivaive ha links he risk neural probabiliy measures associaed o wo differen currencies. As saed by Girsanov Theorem (see, for example, Jeanblanc e al. [2009]), he dynamics of he compensaed defaul process under differen risk neural measures differ in heir drif componen. Such drif depends on he quadraic covariaion beween he FX rae and he 5

6 defaul process (and i is zero when such covariaion is null) and can be inerpreed as he sochasic hazard rae of he reference eniy. The above resul is srongly linked o anoher aspec of FX rae modelling, which we will refer o as FX symmery hroughou his documen (see he discussion in secion 2.2). Consisency beween an FX rae process and is reciprocal is no guaraneed under every possible disribuional assumpions made on is dynamics. For example, in case of sochasic volailiy FX modelling, he reciprocal FX rae would no necessarily have he same dynamics ha one would expec given ha he reciprocal FX rae is also a Radon Nikodym derivaive. For geomeric Brownian moions, however, his consisency is guaraneed. Due o he change in he hazard rae in he second pricing measure induced by he jump o defaul feaure of he FX rae/radon Nikodym derivaive process, we prove in secion ha he symmery is preserved also for our specific FX model Previous Lieraure We refer o Bielecki e al. [2005] for an overview of he general problem of deducing a PDE o price defaulable claims and o Bielecki e al. [2008] for he specific problem of CDS hedging in a reduced form framework. For an inroducion o he join modelling of credi and FX in a reduced form framework wih applicaion o Quano CDS pricing, we refer o Ehlers and Schönbucher [2006], EL- Mohammadi [2009]. Ehlers and Schönbucher [2006] propose he idea o link FX and hazard rae by considering a jump diffusion model for he FX rae process where he jump happens a he defaul ime. Differenly from he presen work, no explici derivaion of he PDE is presened, as he focus is on affine processes modelling. The same idea is presened and developed in EL-Mohammadi [2009]. In ha work i is shown how o calculae quano correced survival probabiliies using a PDE based approach. In order o do ha, he auhor deduces a Fokker Planck equaion for he join disribuion of FX and hazard rae. The approach we presen in Secion 2 below is based on he same Jump o Defaul framework as he one used in he references above. In our case, however, he calculaion of he quano correced survival probabiliies depends on solving a Feynman Kac equaion, he soluion of which is a price, while in EL-Mohammadi [2009] a probabiliy densiy disribuion was calculaed. A implemenaion level, he difference beween he wo approaches lies in he fac ha in he laer case an addiional inegraion sep would be required o calculae a price. Addiionaly, he way we work ou our main pricing equaion makes clear wha insrumens and in wha amouns one would need o effecively implemen a dela hedging sraegy. An algorihm using a fixed poin approach has been recenly proposed o calculae CVA in Kim and Leung [2016]. 6

7 The echniques showed in his paper seem paricularly relevan for long mauriy rades, where he effecs of idiosyncraic jump o-defaul componens on counerpary risk can be more pronounced and where, herefore, hey can have a big impac on wrong ray risk esimaion. For a relevan example of CVA calculaions relaed o long mauriy rades, we refer o Biffis e al. [2016], where he cos of CVA and collaeralizaion are calculaed for longeiviy swaps. The use of Lévy processes wih local volailiy o price opions on defaulable asses has been recenly explored in Lorig e al. [2015], where a family of asympoic expansions for he ransiion densiy of he underlying is derived. Differenly from he approach presened in his paper, in ha case a single sochasic process drives boh he defaul inensiy and he opion s underlying. On he oher hand, being able o accoun for he implied volailiy skew is feaure currenly missing from he framework presened in Secion 2 and ha will be explored in fuure works. Wih respec o he Republic of Ialy s es case ha is presened in he resuls secion 3, we noe ha he Euro area siuaion presens ineresing problems ha go beyond he mere credi FX ineracion which is he focus of he presen work. An addiional layer of complexiy is provided in his case by he inerconnecedness beween he credi risk of he differen currencies. Empirically, Germany quano CDS basis ends o be more pronounced han he Greece one (see Pykhin and Sokol [2013]), reflecing higher correlaion beween EUR/USD and Germany hazard rae of defaul and higher EUR/USD devaluaion upon Germany defaul Quano CDS Quano CDS are designed o provide proecion upon defaul of a cerain eniy in a given currency. There are cases, like for sovereign eniies or for sysemically imporan companies, when an invesor migh prefer o buy proecion on a currency oher han he one in which he asses of he reference eniy are denominaed. A ypical reason for enering his ype of rades would be o avoid he FX risk linked o he devaluaion effec associaed o he reference eniy s defaul. Alernaively, proecion migh be needed in a differen currency from he one in which he asses of he reference eniy are denominaed because i serves as a hedge on a securiy denominaed in ha specific currency. The discouned cashflows of he premium leg, Π Premium, are given (as seen from he proecion seller s perspecive) by Π Premium = S c N i=0 1 τ>ti D ccy 0 (T i) (1) 7

8 S ccy1 (ccy1) S ccy2 (ccy2) A B A B LGD(ccy1) LGD(ccy2) Figure 2: Proecion on a given reference eniy can be bough by A from B in differen currencies. The sream of paymens in Eq (1) is indicaed by he solid arrow, while he dashed arrow is used for he coningen paymen in Eq (2). The LGD paymen, albei seled in differen currencies, is he same percenage of he noional in he wo conracs. where (T 0,..., T N ) is he se of quarerly spaced paymen imes; D ccy (T ) is he sochasic discoun facor for currency ccy a ime for mauriy T ; S c is he conracual spread; τ is he defaul ime of he reference eniy. The proecion leg is made of a single cash flow, Π Proecion, paid upon defaul of he reference eniy on a reference obligaion: where Π Proecion = LGD 1 τ TN D ccy 0 (τ), (2) LGD is he loss given defaul relaed o he conrac. The spread S c ha makes he expeced value of he cash flows in Eq (1) equal o he expeced value of he cash flow in Eq (2) is referred o as par spread and we will usually use S o denoe i. The exisence of CDSs on he same reference eniy whose premium and proecion cashflows are paid in differen currencies creaes a basis spread beween he par spreads of hese conracs. Figure 2 provides a schemaic represenaion of wo possible conracs seled in wo differen currencies. We refer o Elizalde e al. [2010] and references herein for an overview on quano CDS markes and for a horough exposiion of he rules governing hese conracs. We noe here ha he sandard conracs for sovereign CDS are denominaed in USD. This means in paricular ha for counries of he EUR zone, like Ialy, Greece or Germany, he modeling se up o use when including a devaluaion approach is he one deailed in Secion 2.5.5; 8

9 upon defaul of he reference eniy, a common aucion ses he loss given defaul (LGD). The LGD so defined is valid for all he CDSs, irrespecively of he currency hey are denominaed in Main Conribuion In his paper, we derive he pricing equaions for quano CDS in differen models wihin he reduced form framework. In doing so, we show wo of he main mechanisms o model dependence beween he credi and he FX rae componen. We will refer o he currency in which he CDSs wrien on he reference eniy are more liquid as o he liquid currency, ha will also define he risk neural measure used for pricing. We will assume ha CDSs in a differen currency from he liquid one exis and we will refer o his second currency as he conracual currency. In paricular, we discuss he mahemaical implicaions of he inroducion of a devaluaion jump on he spo FX rae beween he conracual currency and he liquid corrency, boh on he pricing equaions and on he main risk facors. More in deail: 1. in Proposiion 1 we show ha, if we assume for he FX rae defining he value of one uni of conracual currency in he liquid currency a dynamics dz = µ Z Z d + σz dw + γ Z Z dd, Z 0 = z, (3) where D = 1 τ< is he defaul process, hen he hazard raes in he wo currencies are linked by ˆλ = (1 + γ Z )λ ; (4) where ˆλ is he hazard rae in he measure linked o he conracual currency and λ is he hazard rae in he currency linked o he liquid currency. An imporan corollary of his resul is ha, in cases where CDS par spreads can be approximaed hrough he relaion S = (1 R)λ, a similar resul holds for par spreads, oo: Ŝ = (1 + γ Z )S. (5) We show in Secion 3 how such an approximaion is applicable o Republic of Ialy s par spreads in he ime period ranging from 2011 o 2013; 2. in Secion we show ha if we assume for he FX rae he dynamics given in Eq (3), hen i) by no arbirage consideraions, he drif of (Z, 0) is given by µ Z = r ˆr γ Z λ (1 D ); 9

10 where r is he risk free rae in he pricing measure linked o he liquid currency and ˆr is he risk free rae in he conracual measure. Alernaively, by symmery consideraions, we could model he reciprocal FX rae X = 1 /Z using he same ype of jump diffusion process dx = µ X X d σx dw + γ X X dd, X 0 = 1 z, and in his second case we would obain a drif given by µ X = ˆr r γ X ˆλ (1 D ), where γz γ X = 1 + γ Z ; ii) in Proposiion 2 we show ha he no arbirage dynamics implied for (X, 0) is of he same ype as he no arbirage dynamics of (Z, 0). This is a resul ha migh no hold in general, for example when sochasic volailiy is also included, or wih a price inhomogeneous local volailiy model like CEV; 3. in Proposiion 3 we show an approximaed formula, valid for shor mauriy CDSs, o esimae he devaluaion rae paramener γ and we presen numerical resuls corroboraing i in Secion 3. We sudy in deail he case of he currency basis spread for CDSs wrien on Ialy in he period providing, for each day in ha ime range, he resuls of he calibraion of a model ha includes a jump o defaul effec on he FX rae. We show he calibraed parameers and how he calibraed model parameers produce esimaes which are consisen wih he approximaed formula in Eq (5). 2. Model Descripion Our modelling framework for credi risk falls ino he reduced form approach and, as such, describes no only he evoluion of survival probabiliies, bu also he defaul even. In Secion 2.1 we inroduce some definiions concerning he role of differen currencies involved in pricing a quano CDS. In Secion 2.2 we inroduce he general framework ha we will refer o o work wih wo financial markes. In Secion 2.3 we inroduce some useful formulae and definiions o price muli currency credi defaul swaps. In Secion 2.4 we will model a sochasic hazard rae as a exponenial Ornsein Uhlenbeck process and he FX rae as a Geomeric Brownian Moion (GBM) and we will consider 10

11 he wo driving diffusions o be correlaed. In Secion 2.5 we presen our proposal o embed a facor devaluaion approach ono he FX rae dynamics. This provides a way o exend he model shown in Secion 2.4. We begin by considering a probabiliy space (Ω, F, Q, (F, 0)) saisfying he usual hypoheses. In paricular (F, 0) is a filraion under which he dynamics of he risk facors are adaped and under which he defaul ime of he reference eniy is a sopping ime. Depending on he specific examples, we will also consider spaces wih a differen equivalen measure, for example he risk neural measure associaed o he liquid money marke or he risk neural measure associaed o he conracual currency money marke. We will be using a Cox process model for he defaul componen and we will refer o he sochasic inensiy of he defaul even simply as hazard rae or inensiy, using he wo erms inerchangeably. Unlike he usual approach followed in he so called reduced form framework for credi risk modelling (see Lando [2004], Brigo and Mercurio [2006]), we do no inroduce a second filraion wih respec o which only he sochasic processes driving he marke risk facors are measurable The Roles of he Currencies In his secion we se up some definiions concerning he role of he currencies ha will be used in our modelling approach. For of any quano CDS pricing, we will be considering he following wo relevan currencies: conracual currency This currency is a conrac s aribue: i is he currency in which boh premium leg and proecion leg paymens are seled. When considering applicaions o quano CDS, for a given reference eniy, CDSs are available in a leas wo differen conracual currencies; liquid currency This is he conracual currency of he mos liquidly raded CDS on a given eniy. calibrae he model. I is used o define a risk neural measure used o price and We lis here wo examples o illusrae he use of he conracual and liquid currencies. 1. he pricing in USD measure of a CDS on Republic of Ialy seled in EUR; 1 The oal filraion (F, 0), inclusive of marke and defaul risk, is he only filraion we will consider (ha is called (G ) in Brigo and Mercurio [2006]). We noe ha he pracical reason for considering his second filraion is because ha allows o apply heoreical resuls developed o price ineres raes derivaive o credi risk derivaives pricing. Due o he specific model choices we make in he following, however, his would no presen any real advanage, while, as shown in secions and 2.5.5, working wih a single filraion gives us he possibiliy o calculae he quano adjusmen using a PDE approach. 11

12 Tes case 1 Tes case 2 Conracual currency EUR USD Liquid currency USD USD Table 1: Currencies involved in he priicing of he es cases deailed in Secion he pricing in USD measure of a CDS on Republic of Ialy seled in USD. We specified he values of he wo currencies for each of hese es cases in Table 1. We chose he es cases so ha for all of hem USD is he he liquid currency, bu his is no necessarily rue for all CDS available in muliple currencies. I is worh noing ha he es case 2 can be priced using a usual single currency approach. Tes cases 1 and 2 will be used in Secion 3.5 o illusrae he capabiliy of he model specified in Secion o explain he currency basis observed in he marke Two Markes Measures In his secion we summarize known resuls abou change of measure in presence of FX effecs. This is mosly done o esablish noaion and se he scene for he following original developmens. Le us consider he wo economies linked o he liquid currency and o he conracual currency, respecively. Le us also consider he corresponding money marke accouns as he numeraires for boh he economies. We will use a ha ˆ, o denoe variables in he conracual currency economy, so ha, for example, he wo numeraires are (B, 0) for he liquid currency economy and ( ˆB, 0) for he conracual currency economy. The money marke accouns dynamics are given by db = r B d, B 0 = 1, (6) d ˆB = ˆr ˆB d, ˆB0 = 1, (7) where (r, 0) and (ˆr, 0) are he sochasic processes describing he shor raes in he wo economies. Le us also consider an exchange rae (X, 0) beween he currencies of he wo economies. X is defined as he price of one uni of he liquid currency expressed as unis of he foreign currency in a spo exchange a ime. We are ineresed in finding an expression for he Radon Nikodym derivaive ha changes he probabiliy measure from ˆQ o Q. This can be worked ou by using he Change of Numeraire echnique and a generic payoff denominaed in he conracual currency, represened by he funcion ˆφ T. To do so, we consider, as said above, he conracual currency money marke accoun, ( ˆB, 0), as a numeraire for he measure ˆQ, while for he mea- 12

13 sure Q we sill use he liquid currency money marke accoun, bu wih value denominaed in he conracual currency, ((XB), 0). The price of he conracual currency payoff ˆφ can be expressed in he wo measures as: [ ] [ ] ˆB B X Ê ˆφT = E ˆφT. (8) ˆB T B T X T The Ê [ ] expecaion on he lef hand side, on he oher hand, can be wrien as [ ] [ ˆB ˆB B T X T Ê ˆφT = ˆB Ê T ˆB T B X ] B X ˆφT B T X T (9) and he wo expressions above can be used o obain he Radon Nikodym derivaive ha defines he change of measure from ˆQ o Q: L T := dq d ˆQ F T = B T X T B X ˆB ˆB T (10) F. In deducing he form of (L, 0) we sared from expeced values condiioned on Throughou his work, however, we will mainly be ineresed in expeced values condiioned a F 0 so ha for all he applicaions in he following secions we will be using he formula above wih = 0 and T =, namely L = B ˆB X X 0, L 0 = 1. (11) Assumpion 1. In he following we will be considering deerminisic ineres raes boh for he liquid currency and for he conracual currency economy. This means ha he money marke accouns will be described by db = r()b d, B 0 = 1, (12) d ˆB = ˆr() ˆB d, ˆB0 = 1, (13) in place of (6) and (7). To lighen he noaion, in mos cases we will drop he dependency for r() and ˆr() in he following equaions. The process defined in Eq (11) has o be a maringale in he foreign measure. This condiion can be used o deermine, ogeher wih Assumpion 1, he drif of (X, 0). By Io s formula, he dynamics of (L, 0) can be wrien as dl = d ) X ˆB X 0 ( B = B ˆB X 0 (dx + rx d ˆrX d), L 0 = 1. (14) 13

14 If for example we assume a lognormal dynamics for he FX rae dx = µ X X d + σx dŵ, X 0 = x 0, (15) hen asking ha (L, 0) in Eq 14 is a maringale brings o he familiar condiion µ X = ˆr r. (16) Remark 1. More generally, he same resul holds rue in case of a (X, 0) of he ype dx = µ X X d + ν dî, X 0 = x, (17) where (Î, 0) is a generic ˆQ maringale. An equivalen argumen would lead, saring from he conracual currency measure and going o he liquid currency one, o se a drif condiion for he process (Z, 0) defined as Z = 1 X. We can define i along he same lines of wha was done wih (X, 0), as a geomeric Brownian moion wih a drif o be deermined hrough arbirage consideraions dz = µ Z Z d + σ Z Z dw, Z 0 = z. The Radon Nikodym measure in his case would be given by L l c = Z ˆB Z 0 B, L l c 0 = 1. (18) Requiring ha (L l c, 0) has o be a maringale under he liquid currency measure, would se he drif erm as µ Z = r ˆr. (19) Remark 2 (Symmery). Alernaively, one could deduce he dynamics for (Z, 0) in Q saring from (X, 0), whose dynamics is known in ˆQ. By applying Io s formula o he process given by Z = f(x ) where f(x) = 1 /x, i would be possible o deduce he dynamics of (Z, 0) in ˆQ. Once is dynamics is known, he form of he driving maringales under Q can be worked ou using Girsanov Theorem. Under he log normal dynamics chosen for he FX raes, his laer approach and he one saring from he Radon Nikodym derivaive in Eq (18) lead o he same resul. A deailed calculaion in case he dynamics of he FX rae is subjec also o jump o defaul effec, is presened in Secion below. There are cases, for example sochasic volailiy FX rae models, where saring from a differen specificaion of he FX rae can make a difference, because he consisency beween he arbirage free dynamics obained under he wo differen specificaions is no 14

15 guaraneed. In hese models, if one sars from X as a primiive modelling quaniy, and hen implies he disribuion of Z a some ime from he law of X, wha will be obained can be a differen disribuion from he one ha one would have had by saring from Z as a primiive modelling quaniy based on he same dynamical properies as X. In applicaions o quano CDS pricing, where he FX rae is used in Eq (8),and where, depending on he circumsances, we migh be ineresed in pricing or calibraing eiher under he liquid currency measure or under he conracual currency measure, here is a degree of arbirariness in using one specificaion or he oher. Having consisency beween he wo specificaions is a desirable propery o avoid resuls ha depend on he aforemenioned choice Modeling Framework for he Quano CDS Correcion In his secion we derive model independen formulas o price coningen claims where conracual currency is differen from he liquid currency used o define he pricing measure. In he nex secions we will show he applicaion of hese formulas under differen dynamics assumpions for he main risk facors. Le us sar by calculaing he value of a defaulable zero coupon bond; i will be hen used as a building block o calculae CDS values. To do so, we choose a payoff funcion ˆφ T = 1 τ>t in Eq (9) and wrie ˆV (T ) = Ê [ ˆB ˆB T ] 1 τ>t = E ˆB d 1 ˆQ τ>t. (20) ˆB T dq Using he Radon Nikodym derivaive in (10), he price of he coningen claim in he conracual currency economy can be calculaed by aking he expecaion in he liquid currency economy: ˆV (T ) = B [ ] ZT E 1 τ>t. Z B T Under Assumpion 1 he above can be rewrien as ˆV (T ) = B(, T ) Z E [Z T 1 τ>t ], (21) where B(, T ) = B /B T is he discoun facor from ime T o ime T. I migh be useful 2 o define he foreign currency survival probabiliies as ˆp (T ) := ˆV (T ) ˆB(, T ). (22) 2 Mosly for compuaional reasons because such definiion would easily allow CDS pricers defined for single currency calculaions o be re used for quano CDS pricing. 15

16 Le us now consider he price, expressed in liquid currency, of he defaulable zero coupon bond seled in he conracual currency, U. This is given by: U (T ) = ˆV (T )Z = B(, T )E [Z T 1 τ>t ]. (23) Being he Q price of a radable asse, he drif of he process (U, 0) has o be given by r()u d. Therefore, we can wrie a Feynman Kac equaion o calculae U (T ). Once U (T ) is known, ˆp (T ) can be calculaed as ˆp (T ) = U (T ) Z ˆB(, T ). (24) 2.4. A diffusive correlaion model: exponenial OU / GBM In his secion we presen a specific model o calculae U. We will be working wih a hazard rae process and a FX rae process which are defined and calibraed in he liquid measure. Le us denoe by (λ, 0) a sochasic process given by λ = e Y an Ornsein Uhlenbeck process defined as he soluion of where (Y, 0) is dy = a(b Y ) d + σ Y dw (1), Y 0 = y, (25) where he parameers (a, b, σ Y, y) R + R + R + R +. Le us also consider a GBM process for he FX rae dz = µ Z Z d + σ Z Z dw (2) Z 0 = z, (26) where µ Z is se by no arbirage consideraions and i is given in his case by Eq (19), and where (σ Z, z) R R +. The dependence beween FX and credi can be specified in his model h rough he insananeous correlaion (quadraic covariaion) beween he wo driving Brownian moions, ρ [ 1, 1], d W (1), W (2) = ρ d. From he resuls in Secion 2.2, he FX rae in he opposie direcion o Z, ha is X = 1 /Z follows a dynamics given by wih µ X given by Eq (16) and σ X = σ Z. dx = µ X X d + σ X X dŵ (2), X 0 = x, (27) Le finally (D, 0) be he defaul process D = 1 τ<. 16

17 Remark 3. Due o he symmery relaion holding for FX raes ha are modeled as geomeric Brownian moions ha was saed in Remark 2, i does no maer if we choose o model (Z, 0), or (X, 0), as he wo dynamics are consisen. Remark 4. The choice of he (exponenial OU and GBM) dynamics has been mainly driven by he need for he hazard rae process o say non negaive. However, differen hazard raes dynamics, possibly wih local volailiies, can easily be accouned for using he same framework presened below as far as hey only driven by Wiener processes and no jump processes are involved. Exensive lieraure has been produced on he use of square roo processes for defaul inensiy, mosly due o heir racabiliy in obaining closed form soluions for Bonds, CDS and CDS opions, see for example Brigo and El-Bachir [2010] and Brigo and Alfonsi [2005], where exac and closed form calibraion o CDS curves is also discussed. For he FX rae dynamics, insead, here is no such freedom of choice as he drif is given by no arbirage condiions, and inroducing local or sochasic volailiies migh break he symmery relaion beween he FX rae and is reciprocal Hazard Rae s Dynamics in he ˆQ Measure We are assuming ha he hazard rae process dynamics is known in Q. Knowing he Radon Nikodym derivaive beween measure Q and measure ˆQ would allow us o wrie he dynamics of he hazard rae in ˆQ. Tha can be obained by using Girsanov s Theorem, from which so ha Pricing Equaion dŵ (1) = dw (1) d W (1), Z = dw (1) ρσ Z d (28) Z dy = a(b Y ) d σ Y ρσ Z d + σ Y dŵ (1). (29) In his secion we deduce a pricing equaion o calculae he value of U. We follow he approach used in Bielecki e al. [2005]. Given he srong Markov propery of all he processes defined so far, U (T ) can be expressed as a funcion of, Z, Y and D. Le us denoe is value a for Z = z, Y = y and D = d by f(, z, y, d). f is a funcion depending on boh coninuous and jump processes, and is Io differenial can be wrien as (see, for example, Jeanblanc e al. [2009]) ( df = rf d + f d + z f µ Z z d + σ Z z dw (2) ) ( + y f a(b Y ) d + σ Y dw (1) ( ) 2 σ Z z zz f d + 1 ( ) σ Y 2 yy f d + ρσ Z σ Y z zy f d + f dd, (30) 2 ) 17

18 where, wih some abuse of noaion, we have defined he jump o defaul erm as f := f(, Z + Z, Y, D + D ) f(, Z, Y, D ). (31) A compensaor for (D, 0) in he measure Q is defined as he process (A, 0) such ha D A is a Q maringale wih respec o (F, 0). The compensaor for (D, 0) is given by (see Lemma in Jeanblanc e al. [2009]) da = 1 τ> λ d. (32) We define he resuling maringale as (M, 0). I is given by M = D A. (33) Consequenly, he compensaor of he las erm in Eq (30) can be wrien as 1 τ> e Y f, (34) which, condiional on F, D = d, Z = z, and Y = y, is equal o ( ) (1 d)e y f(, z(1 + γ Z ), y, 1) f(, z, y, 0) d. (35) I is possible o wrie a Feynman Kac ype PDE o compue he value of U (T ). Indeed (U, 0) is a Q price and, as such, i mus locally grow a he rae r. Therefore, is drif mus saisfy he following equaion f + µ Z z z f + a(b Y ) y f + 1 ( 2 σ z) Z zz f ( ) σ Y 2 yy f + ρσ Z σ Y z zy f + e y (1 d) f = 0, 2 where he explici dependence of f on he sae variables (x, y,, d) has been omied for clariy of reading. If i wasn for he las erm, his would be he ypical PDE for defaul free payoffs. Incidenally, his jump o defaul erm is also he only erm of he equaion where he values f(, z, y, 0) and f(, z, y, 1) appear ogeher. In fac, by condiioning firs on d = 1 and hen on d = 0 we can decouple he wo funcions u(, z, y) := f(, (1 + γ Z )z, y, 1), (36) v(, z, y) := f(, z, y, 0) (37) and calculae hem by solving ieraively wo separae PDE problems. We firs solve for u, as for d = 1 he las erm does no appear in he equaion, and, once u has been 18

19 calculaed, we use i o solve for v. Final condiions for he wo funcions are respecively given by v(t, z, y) = f(t, z, y, 0) = z; (38) u(t, z, y) = f(t, z, y, 1) = 0. (39) The PDE problem ha mus be solved o obain u is hen given by u = µ Z z z u a(b y) y u 1 ( 2 σ x) Z zz u 2 1 ( ) σ Y 2 yy u ρσ Z σ Y z zy u (40) 2 u(t, z, y) = 0. (41) The soluion o his problem is u 0, herefore in his case one can solve direcly he PDE for v, which is hen given by v = µ Z z z v a(b y) y v 1 ( 2 σ x) Z zz v 2 1 ( ) σ Y 2 yy v ρσ Z σ Y z zy v + e y v 2 (42a) v(t, z, y) = z. (42b) Remark 5 (Inerpreaion of u and v). The funcions u and v accoun for he pre defaul and pos defaul value of a derivaive wih payoff φ(x, y, d). The price of his derivaive can be wrien as [ V = 1 τ> E φ(xt, Y T, D T ) X = x, Y = y, D = d ], (43) where, due o he srong Markov propery of he processes (X, 0), (Y, 0), and (D, 0), he expeced value on he righ hand side can be wrien as [ f(, x, y, d) = E φ(xt, Y T, D T ) X = x, Y = y, D = d ]. (44) This can be decomposed as f(, x, y, d) = 1 d=1 u(, x, y) + 1 d=0 v(, x, y) where [ v(, x, y) := E φ(xt, Y T, D T ) X = x, Y = y, D = 0 ], (45) [ u(, x, y) := E φ(xt, Y T, D T ) X = x, Y = y, D = 1 ], (46) 19

20 in fac [ f(, x, y, d) = E φ(xt, Y T, D T ) X = x, Y = y, D = d ] [ = 1 τ> E φ(xt, Y T, D T ) X = x, Y = y, D = 0 ] [ + 1 τ E φ(xt, Y T, D T ) X = x, Y = y, D = 1 ] = 1 τ> v(, x, y) + 1 τ u(, x, y) (47) as boh 1 τ> and 1 τ are measurable in he F filraion. The derivaive price can hen be wrien as V = 1 τ> v(, X, Y ) + D u(, X, Y ), (48) where we defined D := 1 τ> 1 τ>. (49) 2.5. A Jump o Defaul Framework The exponenial OU based model described in Secion 2.4 can be exended by incorporaing a devaluaion mechanism in he FX rae dynamics. By linking he devaluaion o he defaul even, i is possible o inroduce a furher source of dependence beween (λ, 0) and (X, 0). In Secion 3 i will be shown ha his will prove o be a more suiable mechanism o model he basis spread for quano CDS. This secion is organised as follows: in he firs subsecions, from Secion o Secion 2.5.4, we will discuss in general how he dynamics of he risk facors are affeced by he inroducion of a jump o defaul effec on he FX componen. Given ha he Radon Nikodym derivaive depends on he FX rae, his change is expeced o have an impac on all he risk facors whose dynamics has o be wrien in a measure differen from he one in which hey have been originally calibraed and, poenially, on he FX symmery discussed is in Remark 2. This is proven o hold rue also in his new, more general, framework (see Proposiion 2). In Secion we will apply he general resuls from he firs subsecions o he pricing of quano CDS Risk Facors Dynamics Le us hen consider a jump diffusion process for he FX rae in place of (26), while we will be keeping he same model choice for he hazard rae λ = e Y : dy = a(b Y ) d + σ Y dw (1), Y 0 = y, (50) dz = µz d + σ Z Z dw (2) + γ Z Z dd, Z 0 = z, (51) d W (1), W (2) = ρ d (52) 20

21 where, as before, he parameers (a, b, σ Y, y) R + R + R + R +, (σ Z, z) R R +, ρ [ 1, 1], and where γ Z [ 1, ) is he devaluaion/revaluaion rae of he FX process. The ypical case in which his devaluaion facor is used is for reference eniies whose defaul can negaively impac he value of heir local currency. As an example, we expec he value of EUR expressed in USD o fall in case of Ialy s defaul. We leave unspecified he drif erm of (Z, 0) and we simply use µ for i in order o disinguish i from µ Z. I will be shown in Secion ha he inroducion of he jump erm will lead o a resul differen from Eq (19) if we wan he process defined in Eq (18) o sill be a maringale. Remark 6 (Jumps). The jump erm in SDE for jump diffusion processes can be described equivalenly using (D, 0) or he compensaed process (M, 0), he effec of using one erm or he oher being jus a change in he drif erm. We prefer using he non compensaed erm when inroducing he FX process in order o highligh he jump srucure and hence he addiional source of dependence beween he FX and he credi componen. On he oher hand, he descripion in erms of he compensaed maringale (M, 0) will arise naurally every ime he Fundamenal Theorem of Asse Pricing will be used o derive no arbirage drif condiions, e.g. when Eq (14) is used o deduce Eq (61) below and, as i will be shown in Secion 2.5.5, o deduce he main pricing equaion Hazard Rae s and FX Rae s Dynamics in ˆQ Given he dependence of (L l c, 0) on (D, 0) via (Z, 0), in his case he change of measure modifies no only he expeced value of (W, 0), bu also he expeced value of (M, 0) which was originally given by dm = dd (1 D )λ d in Q. However, Girsanov s Theorem provides he adjusmens for each of hese processes needed o obain a maringale in he new measure. dŵ = dw d W, Z Z = dw σ Z d, (53a) d ˆM = dm (1 D )γ Z λ d. (53b) The Wiener process decomposiion in ˆQ is given by he same formula used in Secion 2.4, while we derive he maringale decomposiion for (D, 0) as a resul of he following Proposiion 1. Le (M, 0) be he maringale associaed o he defaul process (D, 0) in he domesic currency measure dm = dd (1 D )λ d, 21

22 hen an applicaion of he Girsanov Theorem allows o wrie he corresponden maringale in he foreign measure ( ˆM, 0) as d ˆM = dm d M, Ll c L l c = dm d D, γ Z D = dm (1 D )γ Z λ d (54) = dd (1 D )(1 + γ Z )λ d (55) where he dynamics of (L l c, 0) is defined by Eq (18) and Eq (51). Eq (55) saes ha he inensiy of he Poisson process driving he defaul even in he foreign currency is given by Proof. Inegraion by pars gives d ( ˆM L l c ) = L l c ˆλ := (1 + γ Z )λ (56) d ˆM + ˆM dl l c = L l c d ˆM + ˆM dl l c = L l c + d [ ˆM, L l c ] + γ Z L l c dd (dm (1 D )γ Z λ d) + ˆM dˆl + γ Z L l c dd = L l c dm + ˆM dl l c + γ Z L l c dm so he process ((L l c ˆM), 0) is a maringale in he domesic measure as i can be wrien as a sum of sochasic inegrals on local maringales. process ( ˆM, 0) is a local maringale in he foreign measure. As a consequence, he Remark 7 (CDS par spreads approximaion). In all he cases where he well known approximaion λ S 1 R beween hazard raes, CDS par spreads, S, and recovery raes, R, holds, he relaion in Eq (56) can be wrien in erms of CDS par spreads raher han hazard raes as (57) Ŝ = (1 + γ Z )S. (58) This happens, for example, where he hazard rae is consan in ime and when he premium leg s cash-flows can be approximaed by a sream of coninuously compounded paymens (see Brigo and Mercurio [2006]) Hazard Raes Dynamics in he Two Measures As shown by Proposiion 1, he hazard rae s magniude changes depending on wheher we are pricing a coningen claim in ˆQ or Q. 22

23 If we sill consider an exponenial OU model for he evoluion of he hazard rae, he relaion obained in Proposiion 1, ˆλ = (1 + γ Z )λ can be ranslaed in erms of he driving processes (Y, 0) and (Ŷ, 0) as eŷ Y = log 1 + γ Z from which dy = dŷ. (59) This resul could be useful when wriing he pricing PDE, because he price could be calculaed as an expecaion in he domesic measure, while he se of sochasic processes migh be defined in he foreign measure FX Raes Dynamics in he Two Measures and Symmery The FX rae in his model is a jump diffusion process, whose jumps are given by (see Eq (51)) Z = γ Z Z D. (60) Noice ha also his specificaion of he FX rae is subjec o arbirage consrains such ha he Radon-Nikodym derivaive defined by Eq (18) be a maringale. The condiion equivalen o Eq (19) in he case where he FX dynamics is given by Eq (51) is provided by µ = µ Z λ γ Z 1 τ> = r ˆr λ γ Z 1 τ>. (61) Despie he inroducion of he jump in he FX rae dynamics, he consisency highlighed in Remark 2 beween (X, 0) and (Z, 0) is mainained. From a pracical poin of view his means ha we do no need o worry abou which FX rae we use, as one can be obained as a ransformaion of he firs one and i is guaraneed o saisfy he no-arbirage relaions for he associaed Radon Nikodym derivaive. This is proved in he nex Proposiion 2 (FX raes symmery under devaluaion jump o defaul). Le us consider an FX rae process whose dynamics in he domesic measure Q is specified by Eq (51) and whose drif is given by Eq (61). Then he dynamics of he process (X, 0) where X = 1 /Z in he foregin measure ˆQ is given by dx = (ˆr r)x d σ Z X dŵ (2) + X γ X d ˆM, X 0 = 1 z, (62) 23

24 where he devaluaion rae for (X, 0) is given by γz γ X = 1 + γ Z. (63) In paricular, (62) is such ha he Radon Nikodym derivaive defined by Eq (11) is a ˆQ-maringale. Proof. See Appendix A Alernaively, a represenaion where he jumps are highlighed can be used for he ˆQ-dynamics of (X, 0) dx = (ˆr r (1 D )γ X λ ) X d σ Z X dw (2) + X γ X dd, X 0 = 1 z. (64) Pricing Equaion In his secion, we consider he case where liquid currency and pricing currency coincide and are differen from he conracual currency. As discussed in Secion 1.3, his is he ypical seup arising o price in he USD marke measure CDSs wrien on European Moneary Union counries, as he sandard currency for hem is USD. If one wans o price a EUR denominaed conrac for such reference eniies in he USD measure, one has firs o calibrae he hazard rae o USD denominaed conracs and hen he pricing can be carried ou using he equaions derived in his secion. This is also he procedure followed o produce he resuls showed in Secion 3.5 below. Wihou loss of generaliy, we will sudy he case of liquid currency and pricing currency associaed o he domesic measure Q. dy = a(b Y ) d + σ Y dw (1), (65) dz = µ Z Z d + σ Z Z dw (2) + γ Z Z dd (66) d W (1), W (2) = ρ d (67) wih dm = dd (1 D )λ d (68) so ha he no-arbirage drif is given by (see Eq (61)) µ Z = r ˆr γ Z (1 D )λ. (69) An applicaion of he generalized Io formula (see, for example, Jeanblanc e al. [2009]) 24

25 allows us o wrie he Q dynamics of (U, 0). Using U = f(, Z, Y, D ): df = rf d+ f d+ z f ( µ Z z d + σ Z z dw (2) + γ Z z dd )+ y f ( a(b Y ) d + σ Y dw (1) ( ) 2 σ Z z zz f d + 1 ( ) σ Y 2 yy f d + ρσ Z σ Y z zy f d + f dd z f Z. 2 The pricing equaion could be deduced by he f dynamics in he same way discussed in Secion 2.4.2: v = (r ˆr)z z v a(b y) y v 1 ( 2 σ z) Z zz v 2 1 ( ) σ Y 2 yy v ρσ Z σ Y z zy v + e y (v γ Z z z v) 2 (70a) ) v(t, z, y) = z. (70b) Inferring Defaul Probabiliy Devaluaion Facor from he FX Rae Devaluaion Facor I is possible o link he FX rae devaluaion facor inroduced in (51) wih a probabiliy rescaling facor. This is done in he following Proposiion 3 (Defaul probabiliies devaluaion). Under he hypoheses of i) small enors: T 0, (71) ii) independence beween he Brownian moions driving he FX and hazard rae processes: ρ = 0, (72) he raio of he quano-correced and single-currency defaul probabiliies can be approximaed hrough Proof. See Appendix B. 1 ˆp 0 (T ) 1 p 0 (T ) 1 + γz. (73) 3. Resuls 3.1. Numerical Mehods In order o produce he resuls presened in his secion, he PDE sysem(70) has been solved numerically, boh for direc calculaions of quano adjused survival probabiliies and for he calibraion problems described laer in he paper in Secion

26 z µ σ Z a b y σ Y T Table 2: Parameers used o produce he par spreads impac in Figure 3 For his purpose, we implemened a finie difference mehod belonging o he family of alernaing direcion implici (ADI) schemes. The descripion of he scheme ha has been used can be found in During e al. [2013]. I mus be noed ha he PDE sysem (70) consiss of a pricing PDE and of a erminal condiion. In order o apply he chosen scheme o such PDE sysems, we also have o specify boundary condiions. For his purpose, we chose o use neiher Neumann nor Dirichle condiions raher, he second derivaive of he soluion was se o zero on he boundaries Quano CDS Par Spreads Parameers Dependence In his secion we show how he quano-correced CDS par spreads are affeced by changing he value of some of he parameers. Specifically we show in Figure 3 he dependence of CDS par spread on he values of ρ and γ Z. we show in Figure 4 he dependence of CDS par spreads on he value of σ Z for differen values of σ Y. For he ranges of values chosen, a sronger dependence is showed on σ Y han on σ F X ; we show in Figure 5 he dependence of CDS par spreads on he value of ρ for differen values of σ Y. In paricular, we show how he impac of correlaion increases wih σ Y. The parameer which affeced he mos he value of he spreads in his analysis is, as one expecs, he devaluaion rae, γ Z (see Figure 3). For he chosen value of he parameers, a change in he insananeous correlaion from is exreme values, 1 and 1, can usually move he par spread of less han 10bps, while moving he devaluaion rae o is exreme value, 1, can bring o zero he level of he par spread. Figure 4 shows ha par spreads sensiiviy o he volailiy of he FX rae process is slighly weaker han he one o he log-hazard rae s volailiy for he chosen ranges of parameers values. In our example, a 5Y par spread can change of around 10 bps wih σ Z ranging from 1% o 20%, while i can range up o 30 bps wih σ Y going from 20% o 70% and wih σ Z fixed a 20%. In Figure 5 we show he sensiiviy of par spreads o he value of diffusive correlaion ρ. The dependence of par spreads on he correlaion is exremely weak for values of σ Y in he range of 20%. Around his level of log-hazard rae volailiy, he maximum 26

27 γ = γ =0.6 S (bps) γ = γ =1 S (bps) ρ ρ Figure 3: 5Y CDS par spread impac vs ρ and γ. The reference value for he par spread is calculaed using he parameers values in Table σ Y = σ Y =0.4 S (bps) σ Y = σ Y =0.6 S (bps) σ Z σ Z Figure 4: 5Y CDS par spread impac vs σ Z and σ Y. The reference value is produced using he parameers values in Table 3. z µ ρ a b y T Table 3: Parameers used o produce he resuls shown in Figure 4. 27

28 σ Y = σ Y =0.4 S (bps) σ Y = σ Y =0.6 S (bps) ρ ρ Figure 5: 5 years par spread impac vs ρ and σ Y. parameers values in Table 4. The reference value is produced using he z µ σ Z a b y T Table 4: Parameers used o produce he resuls shown in Figure 5. change ha correlaion can produce on he quano-par spreads is 10 bps. From Figure 5, a more realisic value of σ Y of 60% is required o observe an impac of around 30 bps on he 5Y par spread when changing he correlaion from 1 o 1, showing he limis of a purely diffusive correlaion model in explaining large differences beween domesic and quano-correced CDS par-spreads. There are circumsances where he basis beween par spreads of CDSs in differen currencies can be sensibly higher han hese values. In hose cases, a purely diffusive model for he hazard rae is no sufficien o explain he observed basis and an approach where dependence is induced by devaluaion jumps is required. As an example of an hisorical occurrence of such a wide basis, we show in Secion 3.5 resuls of model calibraions o he ime series of par spreads for EUR-denominaed and USD-denominaed 5Y CDSs on he Ialian Republic. In he differen conex of impac of dependence on CDS credi valuaion adjusmens, even under collaeralizaion, Brigo e al Brigo and Chourdakis [2009], Brigo e al. [2014, 2013] show ha a copula funcion on he jump o defaul exponenial hresholds may be necessary o obain sizable effecs when looking a credi credi dependence, pure diffusive correlaion no being enough. 28