Robust XVA. August 14, 2018

Transcription

1 Robus XVA Maxim Bichuch Agosino Capponi Sephan Surm Augus 14, 2018 Absrac We inroduce an arbirage-free framework for robus valuaion adjusmens. An invesor rades a credi defaul swap porfolio wih a defaulable counerpary, bu has incomplee informaion abou her credi qualiy. By consraining he acual defaul inensiy of he counerpary wihin an uncerainy inerval, we derive boh upper and lower bounds for he XVA process. We show ha hese bounds may be recovered as soluions of nonlinear ordinary differenial equaions. The presence of collaeralizaion and closeou payoffs inroduce fundamenal differences relaive o classical credi risk valuaion. The superhedging value canno be direcly obained by plugging one of he exremes of he uncerainy inerval in he valuaion equaion, bu raher depends on he relaion beween he XVA replicaing porfolio and he close-ou value hroughou he life of he ransacion. Keywords: robus XVA, counerpary credi risk, backward sochasic differenial equaion, arbirage-free valuaion. Mahemaics Subjec Classificaion 2010: 91G40, 91G20, 60H10 JEL classificaion: G13, C32 1 Inroducion Dealers need o accoun for marke inefficiencies relaed o funding and credi valuaion adjusmens when marking heir swap books. Those include he capial needed o suppor he rading posiion, he losses originaing in case of a premaure defaul by eiher of he rading paries, and he remuneraion of funding and collaeral accouns. I is common marke pracice o refer o hese coss as he XVA of he rade. Saring from 2011, major dealer banks have sared o mark hese valuaion adjusmens on heir balance shees; see, for insance, Cameron 2014 and Beker A large body of lieraure has sudied he implicaion of hese coss on he valuaion and hedging of derivaives posiions. Crépey 2015a and Crépey 2015b use backward sochasic differenial equaions o value he ransacion, accouning for funding consrains and separaing beween posiive and negaive cash flows which need o be remuneraed a differen ineres raes. Brigo and Pallavicini 2014 posulaes he exisence of a risk-neural pricing measure, and obain he valuaion mbichuch@jhu.edu, Deparmen of Applied Mahemaics and Saisics, Johns Hopkins Universiy ac3827@columbia.edu, Indusrial Engineering and Operaions Research Deparmen, Columbia Universiy ssurm@wpi.edu, Deparmen of Mahemaical Sciences, Worceser Polyechnic Insiue 1

2 equaion which accouns for counerpary credi risk, funding, and collaeral servicing coss. Bielecki and Rukowski 2014 consruc a semimaringale framework and provide he backward sochasic differenial equaion BSDE represenaion for he wealh process ha replicaes a defaul-free claim, assuming he rading paries o be defaul-free. Building on Bielecki and Rukowski 2014, Nie and Rukowski 2018 sudy he pricing of conracs boh from he perspecive of he invesor and her counerpary, and provide he range of fair bilaeral prices. The defaul risk of he rading paries involved in he ransacion is accouned for by Burgard and Kjaer 2011b, who derive he parial differenial equaion represenaions for he derivaive value, using replicaion argumens. Andersen e al view funding coss from a corporae finance perspecive. They develop a model ha is consisen wih asse pricing heories, and imporanly accoun for he impac of funding sraegies on he marke valuaion of he claim. We refer o Crépey e al for an overview of he lieraure on valuaion adjusmens. We consider a marke environmen, in which he invesor ransacs credi defaul swaps wih a counerpary and wans o compue he XVA of her rading posiion. The rading inefficiencies conribuing o he XVA include funding coss due o he difference beween reasury borrowing and lending raes, losses originaing if he invesor or her counerpary defaul premaurely, and coss of posing iniial and variaion margin collaeral. The disinguishing feaure of our framework, relaive o he lieraure surveyed above, is ha he invesor is uncerain abou he credi qualiy of her counerpary and hence unable o compue he acual XVA. As highlighed by European Banking Auhoriy 2015, over 75% of he counerparies do no have liquid CDS quoes see also Gregory An approach o deal wih missing quoes is o map a counerpary lacking liquidly quoed CDS spreads o he mos similar counerpary for which a liquid marke of such quoes exiss. The crieria used o define similariy beween counerpary credi qualiies is ypically based on regions, secor, and raings daa, bu ofen exhibis arbirariness and he resuling esimaes have significan errors Brummelhuis and Zhongmin We develop a framework of robus pricing for a credi defaul swap porfolio, ha we refer o as he uncerain inensiy model. A recen work by Fadina and Schmid 2018 develops a framework ha incorporaes model uncerainy ino defaulable erm srucure models. Similarly o us, hey assume prior informaion abou lower and upper bounds on he defaul inensiy, bu heir focus is on deriving inervals of defaulable bond prices, ignoring valuaion adjusmens. Our heory parallels ha for uncerain volailiy inroduced by Avellaneda e al Therein, he auhors consider a Black-Scholes ype model, in which he volailiy of he underlying asse is unknown and only a priori deerminisic bounds for is value are prescribed. They derive he Black-Scholes-Barenbla equaion characerizing he value of European opions in his model; see also Lyons 1995 for he case of one-dimensional barrier opions. Fouque and Ning 2017 generalize he analysis o he case ha he volailiy flucuaes beween wo sochasic bounds, arguing ha his beer capures he behavior of opions wih longer mauriy. Oher relaed works include Hobson 1998, El Karoui e al. 2009, and Denis and Marini 2006 who provide a probabilisic descripion using he heory of capaciies. Anoher line of subsequen work, building on he uncerain volailiy model, is he heory of G-Brownian moion and G-expecaion by Peng The mos criical source of uncerainy in counerpary risk environmens is he defaul risk of he invesor s counerpary. Hence, we focus on he impac ha he uncerainy in he counerpary defaul inensiy has on he valuaion of he rade, raher han analyzing he implicaions of uncerainy in he volailiy of he underlying. Using he inensiy uncerainy inerval, we compue he upper and lower bounds for he XVA. There are boh similariies and differences beween our seup and he 2

3 uncerain volailiy seup of Avellaneda e al On he one hand, he differenial equaions yielding he robus XVA price process are ordinary and of he firs order, as opposed o he case of uncerain volailiy in which he price bounds are obained by solving second-order parial differenial equaions. Addiional simplificaions arise because we do no need o deal wih he singulariy of probabiliy measures. On he oher hand, new echnical challenges are inroduced due o he complex relaionship beween he valuaion of he hedging porfolio, he deerminaion of collaeral levels, and he close-ou requiremens of he valuaion pary. In our framework, he invesor uses risky bonds underwrien by herself, her counerpary, and he reference eniies, o replicae he XVA process associaed wih he credi defaul swaps ransacion. There are several reasons behind our choices of using bonds as hedging insrumens. Firs, i is well known see, for insance, Gregory 2015 ha he vas majoriy of banks counerparies do no have liquidly raded CDSs. A clear example is he municipal bond marke ha, unlike he corporae bond marke, has no sufficienly fosered he developmen of he corresponding CDS marke Fabozzi and Feldsein The bankrupcy evens experienced by municipal eniies in recen years, such as he ciy of Deroi, increased pressure on municipal bond invesors and exchanged raded funds specializing in municipal bonds o heighen risk managemen. The mos direc insrumens o hedge agains his risk are he single name credi defaul swaps. However, he marke for credi defaul swaps is sill very hin compared o he primary marke of municipal bonds. According o Van Devener 2014, only 11 municipal CDSs have raded since he DTCC began reporing weekly rading aciviies in July Second, a problem ypically encounered when one considers CDS based replicaion sraegies is ha each CDS conrac will be on-he-run only for he firs hree monhs afer is issuance. Afer hese hree monhs, he CDS will change is saus o off-he-run and become very illiquid. We hus op for bonds as hedging insrumens because hey are ypically easier o rade a any ime prior o mauriy han he corresponding off-he-run CDSs. 2 We derive he nonlinear valuaion equaion ha uses corporae bonds o replicae he credi defaul swap posiion, and ake ino accoun counerpary credi risk and closeou payoffs exchanged a defaul. Our valuaion equaion is a special BSDE driven by Lévy processes, ha conains jump-o-defaul bu no diffusion erms. We characerize he super-hedging price of he ransacion as he soluion o a nonlinear sysem of ODEs, ha is obained from he nonlinear BSDE racking he XVA process, afer projecing such as BSDE ono he smaller filraion exclusive of credi evens informaion. The sysem consiss of an ODE, whose soluion is he value of he ransacion cash flows ignoring marke inefficiencies, and addiional ODEs ha yield he XVA of he porfolio. Inuiively, he superhedging price is he value aribued o he rade by an invesor who posiions herself in he wors possible economic scenario. We find ha he superhedging price and he corresponding superhedging sraegies may no be recovered by simply plugging one of he exremes of he uncerainy inerval ino he valuaion equaion. This consiues a crucial difference from he case of uncerain volailiy, where i is well known ha he price of a call or a pu opion under he wors-case scenario is obained by plugging he upper exreme of he volailiy uncerainy inerval ino he Black-Scholes formula; see, for example, Pham The difference in our seing is ha boh he collaeral and he close- 1 The Wall Sree Journal blog Traders Find Shor Bes on Puero Rico a Challenge claims ha defaul insurance on Puero Rico, sold in he form of derivaives called credi-defaul swaps, is available from few dealer banks. The conracs also have barely raded because here is no sufficienly available proecion for purchase and disclosures from he Commonwealh have been limied. 2 Bielecki e al inroduce rolling credi defaul swaps o deal wih he illiquidiy of off-he-run CDSs. These are non-radable insrumens, ha are economically equivalen o a self-financing rading sraegy ha a any given ime eners ino a CDS conrac of mauriy T and hen unwinds he conrac a ime + d. 3

4 ou value depend on exogenous, publicly observed, parameers, while he XVA hedging process is based on quaniies such as he invesor s borrowing and lending raes, ha are specific o he rader implemening he super-replicaing XVA hedging sraegy. Our analysis indicaes ha, depending on he relaion beween he curren value of he XVA hedge and he close-ou value of he ransacion, he lower or higher exreme of he inensiy uncerainy inerval should be used in he super-hedging sraegy. The rader always chooses he counerpary defaul inensiy ha maximizes her wealh process. For example, if he superhedge replicaing he XVA sraegy requires he invesor o be shor he counerpary s bond, i.e., a posiive jump would arise a he counerpary defaul his would be he case if he value of he XVA hedge process lies below he close-ou value, hen he rader would choose o use he larges possible value of he defaul inensiy, and vice-versa. The res of he paper is organized as follows. We develop he marke model in Secion 2. We inroduce he valuaion measure, collaeral process and close-ou valuaion in Secion 3. We inroduce he replicaing wealh process and he noion of arbirage in Secion 4. We develop a robus analysis of he XVA process in Secion 5. Secion 6 concludes. 2 Model Our framework builds on ha proposed by Bichuch e al in ha i uses a reduced form model of defauls and mainains he disincion beween universal and invesor specific insrumens. The model economy consiss of N firms, indexed by i = 1,..., N, whose defaul evens consiue he sources of risks in he porfolio. We use I and C o denoe, respecively, he rader also referred o as hedger or invesor hroughou he paper execuing he ransacion and her counerpary. Le Ω, F, P be a probabiliy space rich enough o suppor he following consrucions. We assume he exisence of N +2 independen and idenically disribued uni mean exponenial random variables E i, i = 1,..., N, I, C. The defaul ime of each firm i is defined o be he firs ime is cumulaed inensiy process exceeds he corresponding exponenially disribued random variable, i.e. τ i = sup { 0: 0 hp i sds > E i}. Accordingly, we use he defaul indicaor process H i = 1l {τi }, 0, o rack he occurrence of firm i s defaul. The background filraion F := F 0, where F := σ H j u; u : j {1,.., N}, conains informaion abou he risk of he porfolio, i.e., of he defaul of he N firms referencing he raded securiies, bu no abou he defauls of he invesor I and her counerpary C. The defaul inensiy processes h P i, i {1,..., N, I, C}, are consruced so ha hey are adaped o he 0 background filraion F, i.e., he defaul inensiy a a given ime depends on he firms defauls occurring before ime. To achieve his, we use he following sepwise procedure: Assume h P,0 i F 0 B[0, and define τi 0 := sup { 0: } 0 hp,0 i sds > E i. Then, we can define F 1 := σ H j u; u τ1 0 : j {1,.., N}, where τ1 0 is he ime of he firs defaul he firs order saisics. For k 1, P,k choose h i F k B[0, and define recursively h P,k i := h P,k 1 P,k i 1l [0,τ k 1 + h i 1l k [τ k 1,, k where we use he noaion τi k o denoe he i-h order saisics of he k-level sopping ime τ i k. Then we define τi k := sup { 0: } 0 hp,k i sds > E i as well as F k+1 := σ H j u; u τk+1 k : j {1,.., N} for k {1,..., N}. In his way, he inensiy h P,k i agrees wih h P,k 1 i up o he k-h defaul, bu accouns for informaion of he k-he defaul hereafer. Finally, we define he full filraion F as F = F N+1 0. We denoe he filraion conaining informaion abou he invesor and counerpary defauls by H = H 0, where H = σ H j u; u : j {I, C}. By consrucion, he defaul inensiies h P i, i {1,.., N, I, C}, are piecewise deerminisic funcions of ime we hus work in he framework 4

5 of piecewise-deerminisic Markov processes, see Davis We furhermore require ha hey are piecewise coninuous and uniformly bounded. The enlarged filraion, including boh porfolio risk defaul evens of he N firms referencing porfolio securiies and counerpary risk defaul evens of invesor and her counerpary, is denoed by G = G 0 = F H. We will consider he 0 augmened filraions, i.e., he smalles complee and righ-coninuous filraions, and denoe hem by F, H, G wih a sligh abuse of noaion. For fuure purposes, we define he maringale compensaor processes ϖ i,p of H i as ϖ i,p := H i 0 1 Hi s h P i s ds, i {1,..., N, I, C}. By consrucion, hese compensaor processes are F-maringales for i {1,..., N}, and G-maringales for i {1,..., N, I, C}. We ake he perspecive of a valuaion pary, responsible for deermining he price of he rade and is collaeral requiremens. The valuaion pary has full informaion on he defaul inensiies h P i, i {1,..., N}, of he firms referencing he raded porfolio. The hedger is fully informed on he porfolio marke risk defaul risk of he firms in he porfolio and she clearly knows her own defaul inensiy h P I. However, she only has limied informaion abou he acual defaul inensiy h P C = hp C 0 of her counerpary, and in paricular only knows is upper and lower bound, i.e., h C h P C h C for consans 0 < h C h C <. Hedging insrumens The goal is for he invesor o hedge a porfolio of credi defaul swaps CDS wrien on N differen reference eniies, denoed by 1, 2,..., N. All CDSs are assumed o maure a he same ime T. The credi risk exposure associaed wih his porfolio is hedged using boh universal and invesor specific insrumens. The universal insrumens are available o all marke paricipans, while he invesor specific insrumens are accessible solely o he hedger and no o oher marke paricipans. The universal insrumens include defaulable bonds underwrien by he reference eniies in he credi defaul swaps porfolio as well as by he rader and her counerpary. Under he physical measure P, for i = 1,..., N, I, C, and 0 T, he dynamics of he defaulable bond price processes wih zero recovery a defaul are given by dp i = µ i P i d P i dh i, P i 0 = e µ it, 1 where µ i, i {1,..., N, I, C}, are consan reurn raes. The invesor specific insrumens include he funding and collaeral accoun of he hedger. We assume ha he rader lends and borrows from her reasury desk a, possibly differen, raes r f + he lending rae and rf he borrowing rae. Denoe by Br± f he cash accouns corresponding o hese funding raes. An invesmen sraegy of ξ f := ξ f s; s 0 shares in he funding accoun yields an accoun value B r f := B r f s; s 0 given by B r f := B r f ξ f = e r 0 f ξ f sds, 2 where r f := r f y = r f 1l {y<0} + r + f 1l {y>0}. 3 5

6 Collaeral Hedger and counerpary use a collaeral accoun o miigae counerpary risk. Following he sandards se by he Basel Commiee on Banking Supervision BCBS and he Inernaional Organizaion of Securiies Commissions IOSCO see BIS Margin 2013, he collaeral consiss of variaion margins, racking he changes in marke value of he raded porfolio and denoed by V M, and iniial margins ha are used o miigae he gap risk a he close-ou of he ransacion and denoed by IM. 3. The European Marke Infrasrucure Regulaion EMIR posis a leas daily updaes for variaion margins and requires a revaluaion of iniial margins a leas every en days see EMIR OTC Regulaion In he Unied Saes, he Commodiy Fuures Trading Commission requires daily updaes on iniial margins CFTC Margin Requiremens Mahemaically, he collaeral process M := M ; 0, M = V M + IM, is an F adaped sochasic process which we assume o be posiive if he hedger poss collaeral is collaeral provider and negaive if he receives collaeral is he collaeral aker. Denoe by r m + he ineres rae on collaeral demanded by he hedger when he poss o her counerpary, and by rm he rae on collaeral demanded by he counerpary when he hedger is he collaeral aker. The value of he collaeral accoun a ime is hen given by where B rm = e rmmsds 0, r m := r m x = r m1l {x<0} + r + m1l {x>0}. Denoing by ψ m he number of shares of collaeral accoun B rm held by he rader a ime, we have he following relaion ψ m B rm = M. 4 The collaeral amoun M received or posed a ime will be deermined by a valuaion pary, as discussed in he nex secion. Figure 1 describes he mechanics of he enire flow of ransacions. 3 Valuaion Measure, Collaeralizaion and Close-ou We ake he perspecive of a rader who purchases a porfolio of credi defaul swaps, and deermine is value by consrucing a replicaing porfolio. Such a porfolio uses bonds of he underlying reference eniies o hedge away he marke risk of he ransacion, and bonds of he rading paries o hedge agains counerpary risk of he rading paries. Hence, from a corporae perspecive, we are ineresed in he enrance price of he ransacion. Because he rader does no know he exac defaul inensiy of her counerpary, such a replicaion argumen can only provide price bounds. In paricular, he lower bound provides a reliable benchmark o measure he poenial losses incurred if he porfolio is acquired a a higher price. Remark 3.1. The rader aims o compue he difference beween he enrance price, i.e. he price a which he ransacion is seled, and he marke value of he ransacion, so ha she can idenify he underlying risk facors and allocae hem o differen desks wihin he bank. This difference is referred o as XVA. 3 Noice ha iniial margins are updaed on a regular basis no jus posed once a he incepion of he rade as he name migh sugges, as i is he case for variaion margins. Variaion margins are usually updaed a a higher frequency inraday or a mos daily ha iniial margins, which are insead reseled daily or even a lower frequency 6

7 r + f Treasury Desk Cash Trader Bonds P 1,..., P N, P I, P C Bond Marke r f Variaion Margin r m + rm Margin Segregaed Accoun Counerpary Iniial r + m Figure 1: Trading: Solid lines are purchases/sales, dashed lines borrowing/lending, doed lines ineres due; blue lines are cash, and black lines are bond purchases for cash. Noe he difference beween he wo-sided variaion margin and he iniial margin ha is kep in a segregaed accoun. I is imporan o inroduce a finer disincion beween he differen sources of surcharges and unhedgable risk referred o, e.g., as CVA, FVA, KVA o correcly allocae hem o he managing desks. Hence, when calculaing he exi price, i.e. he price a which he porfolio can be liquidaed on he open marke his is relevan for ax and regulaory purposes, one needs o accoun for hese componens a a higher level of granulariy. One of hese componens is he KVA, defined as he amoun of capial a risk se aside by shareholders of he invesor s firm. KVA should be calculaed under he hisorical measure, which is ypically assumed o be he same as he risk neural measure o preserve analyical racabiliy. Our analysis deals wih enrance prices, i.e. prior o decomposing he rade ino risk sources and spliing i o he various desks. Neverheless, we compue he superhedging price of he ransacion, which is robus agains he specific choices of physical and pricing measure. Hence, our analysis migh also be used o provide bounds for KVA, which are robus o misspecificaion of he counerpary s defaul inensiy. Nex, we discuss public and privae valuaions. Privae valuaions are based on discoun raes, which depend on invesor specific characerisics, while public valuaions depend on publicly available discoun facors. Specifically, public valuaions are needed for he deerminaion of collaeral requiremens and he close-ou value of he ransacion. They are deermined by a valuaion agen who migh be eiher one of he paries involved in he ransacion or a hird pary, in accordance wih marke pracices reviewed by he Inernaional Swaps and Derivaives Associaion ISDA. The valuaion agen deermines he closeou value of he ransacion by calculaing he so-called clean price of he derivaive, using he discoun rae r D and he defaul inensiies of he firms in he porfolio, h P i, i {1,... N}, ha are known o he valuaion agen. Throughou he paper, we will use he superscrip when referring specifically o public valuaions. The hedging process will sop before mauriy if he hedger or her counerpary were o defaul premaurely. We hus define he erminal ime of he hedge i.e., he earlies beween he defaul ime of eiher pary or he mauriy T of he ransacion as τ := τ I τ C T. The valuaion done by he 7

8 agen is mahemaically represened as pricing he rade under he valuaion measure Q associaed wih he publicly available discoun rae r D chosen by he agen. The measure Q is equivalen o P and heir relaion is specified by he Radon-Nikodým densiy dq dp = Fτ τ1... τ N i {1,...,N,I,C} µi r D τ τ i τ τi 0 h P i sds Hi τ τ i e τ τi 0 r D µ i +h P i udu, 5 Remark 3.2. As he valuaion measure is used o deermine he clean price of he ransacion, i needs no depend on he defaul inensiies of he invesor I and her counerpary C. Neverheless, we have included boh of hese defaul inensiies in he definiion of Q because his will simplify he exposiion in laer secions of he paper. In paricular, we do no need o inroduce a differen measure for he hedger s valuaion. by The Q-dynamics of he defaulable bonds can be derived using Girsanov s heorem and are given dp i = r D P i d P i dϖ Q i, where ϖ Q i := ϖ Q i ; 0 τ τ N are F, Q-maringales. They can be represened explicily as ϖ Q i = ϖp i Hi u h P i u hq i udu where he processes hq i = µ i r D, i {1,..., N}, wih µ i, i = 1,..., N, I, C, being he rae of reurns of he bonds underwrien by he reference eniies, rader and her counerpary are he firms defaul inensiies under he valuaion measure and assumed o be posiive. 3.1 Collaeral The public valuaion process of he credi defaul swap porfolio, as deermined by he valuaion agen, is given by N ˆV = z i Ĉ i, i=1 where Ĉi is he ime value of he credi defaul swap referencing eniy i. The variables z i indicae if he rader sold he swap o her counerpary z i = 1 or purchased i from her counerpary z i = 1. In he case he swap is purchased, he rader pays he spread imes he noional o her counerpary, and receives he loss rae imes he noional a he defaul ime of he reference eniy, if i occurs before he mauriy T. This is he so-called clean price, and does no accoun for credi risk of he counerpary, collaeral or funding coss. Clearly, he public valuaion of he porfolio is jus he sum of he valuaion of he individual CDSs. Consisenly wih marke pracices, he collaeralizaion process consiss of wo pars, he iniial margin and he variaion margin. The variaion margin is jus se o be a fixed raio of he public valuaion of he porfolio, while he iniial margin is designed o miigae he gap risk and is calculaed using value a risk. Such a risk measure is se o cover a number of days of adverse price/credi spread movemens for he porfolio posiion wih a arge confidence level. 4 Noe ha here is an imporan difference beween iniial and variaion margins. Variaion margins are always direcional and can be rehypoecaed i.e., i flows from he paying pary o he receiving pary; he laer may use i for 4 Boh EU and US auhoriies require iniial margins o cover losses over a liquidaion period for en days in 99% of all realized scenarios EMIR OTC Regulaion 2016; CFTC Margin Requiremens

9 invesmen purposes, whereas iniial margins have o be posed by boh paries and need o be kep in a segregaed accoun, hus hey canno used for hedging or porfolio replicaion. Moreover, we assume ha collaeral is posed and received in he form of cash, which is pracically he mos common form of collaeral. 5 Thus, on he even ha neiher he rader nor her counerpary have defauled by ime, and he reference eniies in he porfolio have no all defauled, he collaeral process is defined as M := IM + V M = β V ar q ˆV +δ T ˆV + τ τ N > + α ˆV 1l {τ τn >}, 6 where 0 α 1 is he collaeralizaion level, δ > 0 is he delay in collaeral posing, q is he level of risk olerance and β is sress facor. The case α = 0 corresponds o zero collaeralizaion, while α = 1 means ha he ransacion is fully collaeralized. The posiive par of he value a risk quaniy capures he fac ha iniial margins canno be re-hypohecaed. Hence, he wealh process associaed wih he hedger s rading sraegy does no include received iniial margins. 3.2 Close-ou value of ransacion We follow he risk-free closeou convenion in he case of defaul by he hedger or her counerpary. According o his convenion, each pary liquidaes he posiion a he marke value when he oher rading counerpary defauls. Hence, he value of he replicaing porfolio will coincide wih he hird pary valuaion if he amoun of available collaeral is sufficien o absorb all occurred losses. If his is no he case, he rader will only receive a recovery fracion of her residual posiion, i.e., afer neing losses wih he available collaeral. Le us denoe by θ he value of he replicaing porfolio a τ < T. This is given by θ := ˆV τ + 1l {τc <τ I }L C Y 1l {τi <τ C }L I Y + where Y := ˆV τ M τ = 1 α ˆV τ β V ar q ˆV τ+δ T ˆV + τ is he value of he claim a defaul need of he posed collaeral, and 0 L I, L C 1 are he loss raes on he rader and counerpary claims, respecively. Alernaively, we can represen he value of he porfolio a defaul as where we define θ = θτ, ˆV, M = 1l {τi <τ C }θ I ˆV τ, M τ + 1l {τc <τ I }θ C ˆV τ, M τ, θ I ˆv, m := ˆv L I ˆv m + θ C ˆv, m := ˆv + L C ˆv m and recall M = α ˆV +. + β V ar q ˆV +δ T ˆV τ τ N > are also piecewise deerminisic and piecewise con- Noe ha θ I ˆV, M 0 and θ C ˆV, M 0 inuous F-adaped processes. 5 More precisely, cash is he predominan form of collaeral used for variaion margins, and i accouns for abou 80% of he oal posed variaion margin amoun. Iniial margins are usually delivered in he form of governmen securiies see, for insance, page 7 of ISDA Overall, he amoun of variaion margin posed for bilaerally cleared derivaives conracs was abou $ 173 billion in 2017, whereas he variaion margin accouned for $870 billions see page 1 herein. 9

10 4 Wealh process & Arbirage We consider a sylized model of single name credi defaul swaps, and view all exchanged cash flows from he poin of view of he proecion seller. If he rader purchases proecion from her counerpary agains he defaul of he i-h firm, hen he rader makes a sream of coninuous paymens a a rae S i of he noional o her counerpary, up unil conrac mauriy or he arrival of he credi even, whichever occurs earlier. Upon arrival of he defaul even of he i-h firm, and if his occurs before he mauriy T, he proecion seller pays o he proecion buyer he loss on he noional, obained by muliplying he loss rae L i by he noional. As he noional eners linearly in all calculaions, we fix i o be one. Recall ha ξ i denoes he number of shares of he bond underwrien by he reference eniy i, ξ f he number of shares in he funding accoun, and we use ξ I and ξ C o denoe he number of shares of rader and counerpary bonds, respecively. Using he ideniy 4, we may wrie he wealh process as a sum of conribuions from each individual accoun: N V := ξp i i + ξ I P I + ξ C P C i=1 + ξ f Br f ψ m B rm. 7 For he purpose of arbirage-free valuaion, i is imporan o consider no only he acual CDS porfolio, bu an arbirary muliple of i. Hence, we will consider a muliple γ of he acquired porfolio, and focus on self-financing sraegies. Definiion 4.1. A collaeralized rading sraegy ϕ := ξ 1,..., ξ N, ξ f, ξi, ξ C 0 associaed wih γ shares of a porfolio w = w 1, w 2,..., w N, where w i {0, 1} for i = 1,..., N, is self-financing if, for [0, τ τ N ], i holds ha N V γ := V 0 γ + ξu i dpu i + ξu I dpu I + ξu C dpu C + ξu f db r f u ψu m dbu rm i= N + γ w i S i τ i. 8 i=1 The above expression akes ino accoun he spread paymens received/paid by he invesor for all CDS conracs which she sold o resp. purchased from her counerpary. The se of admissible rading sraegies consiss of F-predicable processes ϕ such ha he porfolio process V γ is bounded from below cf. Delbaen and Schachermayer Remark 4.2. Noe ha he spread paymens received by he rader are coninuously reinvesed ino he hedging insrumens risky bonds or deposied in he funding accoun. This is a differen seup hen ha in Bielecki e al where he spread paymens are used o increase he posiions in he CDS insrumens. The difference sems from he fac ha, in heir model, he CDS conracs are liquidly raded hedging insrumens, whereas in our case hey are par of he porfolio o be replicaed. Before discussing he arbirage-free valuaion of he CDS porfolio, we have o clarify he assumpions under which he underlying marke is free of arbirage from he hedger s perspecive concepually, we follow Bielecki and Rukowski, 2014, Secion 3. Thus, o sar wih, we exclude he CDS insrumens from our consideraion, and consider a rader who is only allowed o buy or sell defaulable bonds wrien on he reference eniies, her counerpary or he invesor s firm iself and o borrow or lend money from he reasury desk. 10

11 Definiion 4.3. The marke P 1, P 2,..., P N, P I, P C admis hedger s arbirage if, given a nonnegaive iniial capial x 0, here exiss an admissible rading sraegy ϕ = ξ 1, ξ 2,..., ξ N, ξ f, ξ I, ξ C such ha P [ V τ e r+ f τ x ] = 1 and P [ V τ > e r+ f τ x ] > 0. If he marke does no admi hedger s arbirage for a given level x 0 of iniial capial, he marke is said o be arbirage free from he hedger s perspecive. We impose he following assumpion and argue ha i provides a necessary and sufficien condiion for he absence of arbirage. Assumpion 4.4. r D r + f < min i {1,...N,I,C} µ i. Remark 4.5. Necessiy: The condiion r D < min i {1,...N,I,C} µ i is needed for he exisence of he valuaion measure defined in Eq. 5 h Q i = µ i r D and risk-neural defaul inensiies mus be posiive. The condiion r f + < min i {1,...N,I,C} µ i has an even more pracical inerpreaion because i precludes he arbirage opporuniy of shor selling he risky bonds while invesing he proceeds in he funding accoun. Sricly speaking, he condiion r D < µ I µ C is no necessary from an arbirage poin of view, because i addresses only he soundness of he marke from he perspecive of he valuaion pary. While r D < min i {1,...N} µ i is necessary o conclude ha he valuaion pary s marke model is free of arbirage, one migh hypohesize a siuaion in which r D µ I µ C. From a pracical perspecive, his is however raher unlikely, as r D is ypically assumed o be an overnigh index swap OIS rae and as such lower han he reurn raes of he defaulable bonds. Having argued abou he necessiy in he above remark, we show ha Assumpion 4.4 is also sufficien o guaranee ha he underlying marke i.e., excluding he credi defaul swap securiies is free of arbirage. The proof proceeds along very similar lines as Proposiion 4.4 in Bichuch e al. 2017, and is delegaed o he appendix. Proposiion 4.6. Under Assumpion 4.4, he model does no admi arbirage opporuniies for he hedger for any x 0. As in Bichuch e al. 2017, we will define he noion of an arbirage free price of a derivaive securiy from he hedger s perspecive. We assume ha he hedger has zero iniial capial, or equivalenly, she does no have liquid iniial capial ha can be used for hedging he claim unil mauriy. The hedging porfolio will hus be enirely financed by purchases/sales of he risky bonds via he funding accoun. Definiion 4.7. The valuaion P R of a derivaive securiy wih erminal payoff ϑ F T is called hedger s arbirage-free if for all γ R, buying γ securiies for γp and hedging in he marke wih an admissible sraegy and zero iniial capial, does no creae hedger s arbirage. Le V represen he price process of he replicaing porfolio, and given by he supremum over all arbirage free prices. Then we define he oal valuaion adjusmen XVA as he difference beween his upper arbirage price and he clean price, i.e., XVA γ = V γ γ ˆV. 9 XVA hus quanifies he oal coss including collaeral, funding, and counerpary risk relaed coss incurred by he rader o hedge he sold CDS porfolio. Noice ha, a ime, he invesor does no know he exac defaul inensiy of he counerpary h P C for he ime inerval [, τ]. Hence, she is no able o execue he replicaion sraegy yielding he value process V, because all wha she knows 11

12 abou he defaul inensiy is ha h C h P C h C. Therefore, she will have o consider he wors case, accouning for all possible F-predicable dynamics of he defaul inensiy process in he inerval [ hc, h C ]. Denoe he valuaion of he replicaing porfolio when h P C = h by V h. The robus XVA is defined as rxva γ = ess sup V h γ1l {<τ} + V γ1l { τ} γ ˆV. 10 h [h C,h C ] Noice ha he supremum is aken over all admissible valuaions only prior o he occurrence of he hedger or her counerpary s defaul. In paricular, he defaul ime τ does no depend on he defaul inensiy inerval [ h C, h C ] capuring he uncerainy of he invesor. 5 Robus XVA for Credi Swaps In his secion, we derive explici represenaions for he robus XVA of a credi defaul swap porfolio. To highligh he main mahemaical argumens and economic implicaions of he resuls, we sar analyzing he case of a single credi defaul swap in Secion 5.1. We develop a comparison argumen o esablish he uniqueness of he robus XVA process and of he corresponding superhedging sraegies in Secion 5.2. We provide an explici compuaion of margins under he proposed framework in Secion 5.3. We generalize he analysis o a porfolio of credi defaul swaps in Secion BSDE represenaion of XVA This secion characerizes he XVA process given in Eq. 9 as he soluion o a BSDE. We sar analyzing he dynamics of he process V γ. Given a self financing sraegy, he invesor s wealh process in 8 under he risk neural measure Q follows he dynamics dv γ = r f ξ f Br f + r D ξ 1 P 1 + r D ξ I P I + r D ξ C P C r m ψ m B rm + γs 1 d ξ 1 P 1 dϖ 1,Q ξ I P I dϖ I,Q ξ C P C dϖ C,Q Seing = r f + ξ f Br f + r f ξ f Br f + r m + and using Eq. 7, we obain ha + rd ξ 1 P 1 + r D ξ I P I + r D ξ C P C M + r m M + γs1 d ξ 1 P 1 + dϖ 1,Q d ξ I P I dϖ I,Q ξ C P C dϖ C,Q. 11 Z 1,γ := ξ 1 P 1, Z I,γ := ξ I P I, Z C,γ := ξ C P C, 12 We may hen rewrie he wealh dynamics as dv γ = ξ f Br f = V γ ξ 1 P 1 ξ I P I ξ C P C M. 13 r f + V γ + Z 1,γ + Z I,γ + Z C,γ + γ M r f V γ + Z 1,γ + Z I,γ + Z C,γ γ M r D Z 1,γ r D Z I,γ r D Z C,γ + r + m γ M + r m γ M + γs 1 d + Z 1,γ dϖ 1,Q + Z I,γ dϖ I,Q + Z C,γ dϖ C,Q

13 To sudy he robus hedging sraegy, we use he above dynamics o formulae he BSDE associaed wih he porfolio replicaing he credi defaul swap. This is given by dv γ = f, V γ, Z 1,γ, Z I,γ, Z C,γ, γ; M d Z 1,γ dϖ 1,Q Z I,γ dϖ I,Q Z C,γ dϖ C,Q, V τ τ1 γ = γl 1 1l τ1 <τ + θ I γ ˆV τ, γ M τ 1l {τ<τ1 τ C T } + θ C γ ˆV τ, γ M τ 1l {τ<τ1 τ I T }, 15 where he driver f : Ω [0, T ] R 5, ω,, v, z, z I, z C, γ f, v, z, z I, z C, γ; M is given by f, v, z 1, z I, z C, γ; M := r f + v + z 1 + z I + z C + γ M r f v + z 1 + z I + z C γ M r D z 1 r D z I r D z C + r + m γ M + r m γ M + γs In he above expression, we highligh he dependence on he collaeral process M ha is used o miigae he defaul losses associaed wih he γ unis of he raded CDS conrac. In he case he reference eniy defauls before he invesor or her counerpary, τ 1 < τ, he erminal condiion is given by he loss erm γl 1. This is consisen wih he fac ha, a his ime, he value of he ransacion from he hedger s poin of view corresponds wih he hird pary valuaion γ ˆV τ1 = Ĉ1τ 1 = L 1 1l {τ1 T }. By posiive homogeneiy of he driver f wih respec o γ > 0, we will assume ha γ = 1 hroughou he paper and suppress i from he superscrip. The case γ = 1 follows from symmeric argumens. Nex, we sudy he dynamics of he credi defaul swap price process ˆV, viewed from he valuaion agen s perspecive. Such a process saisfies a BSDE ha can be derived similarly o Eq. 15 essenially ignoring he erms Z I, Z C as well as he collaeral erms, seing r f = r+ f = r D, and normalizing γ = 1. This is given by d ˆV = r D ˆV S 1 d Ẑ 1 dϖ 1,Q, ˆV τ1 T = L 1 1l τ1 <T. 17 This BSDE is well known o admi he unique soluion ˆV, Ẑ1, where ˆV can be represened explicily see he Appendix as ˆV = Ĉ1 = E Q [ T e u T hq 1 s+rdds S 1 du L 1 h Q u 1 ue We immediaely obain a BSDE for he XVA process given by where dxva = f, XVA, Z 1, Z I, Z C ; M d Z 1 dϖ 1,Q ] hq 1 s+rdds du F Z I dϖ I,Q Z C dϖ C,Q, 1l { τ1 }. 18 XVA τ τ1 = θ C ˆV τ, M τ 1l {τ<τ1 τ I T } + θ I ˆV τ, M τ 1l {τ<τ1 τ C T }, 19 and f, xva, z 1, z I, z C ; M := Z 1 := Z 1 Ẑ1, ZI := Z I, ZC := Z C, θ C ˆv, m := L C ˆv m, θi ˆv, m := L I ˆv m + 20 r + f xva + z 1 + z I + z C + L 1 M + r f xva + z 1 + z I + z C + L 1 M r D z 1 r D z I r D z C + r m + + M r m M rd L

14 Above, we have used he fac ha Ẑ1 = L 1 ˆV = L 1 ˆV by sochasic coninuiy and hus Z 1 = Z 1 + Ẑ1 = Z 1 + L 1 ˆV. We can now apply he reducion echnique developed by Crépey and Song 2015 o find a coninuous ordinary differenial equaion describing he XVA prior o he invesor and her counerpary s defaul. Proposiion 5.1. The BSDE dǔ = ǧ, Ǔ; ˆV, M d 22 Ǔ T = 0 in he rivial filraion F, wih driver ǧ, ǔ; ˆV, M = h Q I θi ˆV, M ǔ + h Q C θc ˆV, M ǔ h Q 1 ǔ + f, ǔ, ǔ, θ I ˆV, M ǔ, θ C ˆV, M ǔ; M 23 admis a unique soluion Ǔ, ha is relaed o he unique soluion XVA, Z 1, Z I, Z C of he BSDE in Eq. 19 as follows. On he one hand Ǔ := XVA τ τ1 24 is a soluion o he ODE reduced BSDE in Eq. 22, and on he oher hand a soluion o he full XVA BSDE 19 is given by XVA = Ǔ1l {<τ τ1 } + θc ˆV τc, M τc 1l {τc <τ 1 τ I T } + θ I ˆV τi, M τi 1l {τi <τ 1 τ C T } 1l { τ τ1 }, 25 Z 1 = Ǔ1l { τ τ1 }, ZI = θi ˆV Ǔ, M 1l { τ τ1 }, ZC = θc ˆV Ǔ, M 1l { τ τ1 }. The uniqueness of he soluion o he original BSDE for V as well as o heir projeced versions in he F-filraion follows from he definiion of XVA. Corollary 5.2. The BSDE 15 admis a unique soluion. soluion Ū of he ODE dū = g, Ū; ˆV, M d This soluion is relaed o he unique Ū T = 0 26 in he filraion F wih g, ū; ˆV, M = h Q I θi ˆV, M ū + h Q C θc ˆV, M ū h Q 1 ū + f, ū, L 1 ū, θ I ˆV, M ū, θ C ˆV, M ū; M via he following relaions. On he one hand Ū := V τ τ1 is a soluion o he reduced BSDE 26, while on he oher hand a soluion o he full BSDE 15 is given by V := Ū1l {<τ τ1 } + L 1 1l τ1 <τ + θ C ˆV τc, M τc 1l {τc <τ 1 τ I T } + θ I ˆV τi, M τi 1l {τi <τ 1 τ C T } 1l { τ τ1 }, Z 1 := L 1 Ū1l {<τ τ1 }, Z I := θ I ˆV, M Ū 1l { τ τ1 }, Z C := θ C ˆV, M Ū 1l { τ τ1 }. 14

15 Using he above represenaion, we can provide explici represenaions for he replicaion sraegies of he XVA. We will use he ilde symbol o denoe hese replicaing sraegies e.g., ξ 1, ξ I, ξ C denoe, respecively, he number of shares of he bond underwrien by he reference eniy, rader and her counerpary so o disinguish hem from he sraegies used o replicae he CDS price process. Using he maringale represenaion heorem for he probabiliy space Ω, F, F, Q and bond price dynamics, we obain ha ξ 1 = Z 1 P 1 1l {<τ τ1 } = Invoking Theorem 5.1 along wih equaions 12 and 20 we conclude ha Ǔ P 1 1l {<τ τ1 }. 27 ξ I = Z I P I 1l { τ τ1 } = L I ˆV M + + Ǔ P I 1l { τ τ1 }, 28 ξ C = Z C P C 1l { τ τ1 } = L C ˆV M + Ǔ P C 1l { τ τ1 }, 29 and from equaions 4 and 6 i follows ha ψ m = M 1l {τ τ1 >}. 30 B rm Finally, using Eq. 13 and he ideniy V = XV A + ˆV, we obain ξ f = V ˆV ξ 1 P 1 ξ I P I ξ C P C M B r 1l f {τ τ1 >} = 2Ǔ + L C ˆV M L I ˆV M + M B r 1l f {τ τ1 >}, 31 where in he las equaliy above, we have used he definiion of XVA given in Eq. 9 ogeher wih he ideniy XVA = Ǔ on {τ τ 1 > }. Noe ha he hedging sraegies are specified only in erms of bond prices and are hus known o he hedger a ime. However, hey neiher give informaion on he value of he XVA process nor on he evoluion of he hedging sraegy, because he defaul inensiy process h Q C is unknown o he hedger. 5.2 Comparison and Superhedging This secion develops a comparison principle for he reduced BSDE 22 solves by he XVA process. We subsequenly use his resul o consruc a superhedging sraegy for he XVA. The BSDE given in Eq. 22 is effecively an ODE. To mainain consisency wih he heory of ODEs, we swich he direcion of ime by defining ˆv := ˆV T and m := M T. I follows from Eq. 18 ha ˆv is bounded, i.e., ˆv M 0 for some consan M 0. Similarly, se ǔ = ǓT. Applying he reducion echnique of Crépey and Song 2015 o Eq. 17, similarly o how i was done above in Proposiion 5.1, we ge ˆv = S 1 h Q 1 L 1 h Q 1 + r Dˆv, 32 ˆv0 = 0. 15

16 We may hen rewrie Eq. 22 as ǔ = ǧ, ǔ; ˆv, m, 33 ǔ0 = 0. Taken ogeher, he wo ODES 32 and 33 consiue a sysem of ODEs. The funcions h Q 1, hq I, hq C, [0, T ], are all piecewise deerminisic coninuous. The following heorem, whose proof is repored in he appendix, provides an exisence and uniqueness resul. Proposiion 5.3. There exiss a unique piecewise classical soluions o he sysem of ODEs Recall ha he rader does no know he acual defaul inensiy h P C and hus hq C of her counerpary. She only knows ha his is lower han h Q C and higher han h Q C. The following comparison principle heorem, whose proof is repored in he appendix, will be used o find he superhedging price of he XVA. Theorem 5.4 Comparison Theorem. Assume ha here exiss h C h C > 0 such ha h C h Q C h C and le ǔ be he soluion of ODE 33. Le h Q C ˆv, m, ǔ = h C 1l { θc ˆv,m ǔ 0} + h C1l { θc ˆv,m ǔ 0}, h Q C ˆv, m, ǔ = h C 1l { θc ˆv,m ǔ 0} + h C1l { θc ˆv,m ǔ 0}, and define he drivers g and g by plugging he defaul inensiies h Q C and h Q C ino he expression of ǧ given by 23, i.e., g, ǔ; ˆv, m = h Q I θi ˆv, m ǔ + h Q C ˆv, m, ǔ θc ˆv, m ǔ h Q 1 ǔ + f, ǔ, ǔ, θ I ˆv, m ǔ, θ C ˆv, m ǔ ; ˆv, m, g, ǔ; ˆv, m = h Q I θi ˆv, m ǔ + h Q C ˆv, m, ǔ θc ˆv, m ǔ h Q 1 ǔ + f, ǔ, ǔ, θ I ˆv, m ǔ, θ C ˆv, m ǔ ; ˆv, m. Le ǔ and ǔ be he soluions o ODE 33 where ǧ is replaced by g and g respecively, i.e., Then ǔ ǔ ǔ. ǔ = g, ǔ ; ˆv, m, ǔ 0 = 0 ǔ = g, ǔ ; ˆv, m, ǔ 0 = The ODEs 34 may be undersood as he credi risk counerpars of he Black-Scholes-Barenbla PDEs for he uncerain volailiy model; see Avellaneda e al The main difference beween our sudy and heirs is ha, in heir paper, he uncerainy comes from he volailiy which appears as a second order erm in he differenial operaor. Hence, he indicaor funcion specifying he value of volailiy o use in he pricing formula depends on he second order derivaive of he opion price wih respec o he underlying, i.e., he Gamma of he opion. In our seing, he indicaor funcion specifying he value of counerpary s defaul inensiy o use depends on he relaion beween he curren value of he XVA hedge and he close-ou value. The value of he hedge jumps o he close-ou value when he counerpary defauls. If he size of his jump is posiive, i.e., he close-ou value of 16

17 he ransacion is higher, hen he rader needs o be shor he counerpary s bond o hedge his jump-o-defaul risk. As he rader wans o consider he wors possible scenario for her hedge, she would choose he larges value of he counerpary s defaul inensiy h C because his yields he larges rae of reurn on he counerpary bond ha she holds. Vice-versa, if he jump is negaive, he rader needs o be long he counerpary bond. Consequenly, he rader would use he smaller counerpary s defaul inensiy h C o accoun for he wors possible hedge scenario. Our objecive is o provide a igh upper bound for he XVA price process, because his would imply a igh superhedging price. We achieve his by connecing such a superhedging price o he rxva defined in Eq. 10. Define he process Ǔ := ǔ T. The following heorem shows ha he rxva coincides wih he superhedging price, and addiionally specifies he superhedging sraegy. The laer is obained by aking he sraegy given in and using he superhedging price Ǔ in place of Ǔ. Theorem 5.5. The robus XVA admis he explici represenaion given by rxva = Ǔ 1l {<τ τ1 } + θc ˆV τc, M τc 1l {τc <τ 1 τ I T } + θ I ˆV τi, M τi 1l {τi <τ 1 τ C T } 1l { τ τ1 }, 35 and he corresponding superhedging sraegies for rxva are given by ξ 1, = Ǔ P 1 1l {<τ τ1 }, ξ I, = L I ˆV M + + Ǔ P I 1l { τ τ1 }, ξ C, ψ m, ξ f, = L C ˆV M + Ǔ P C = M 1l {<τ τ1 }, B rm 1l { τ τ1 }, = 2Û + L C ˆV M L I ˆV M + M B r 1l f {<τ τ1 }. 36 Proof. The proof consiss of wo pars. In he firs par, we verify ha he expression of rxva given in Eq. 35 is he smalles superhedging price. In he second par, we show ha he sraegy given in 36 is a superhedging sraegy. This requires showing ha he implemenaion of his sraegy does no require any cash infusion, and ha he wealh process conrolled by his sraegy is exacly he rxva process. V h Define XVA h := V h ˆV for h F, h C h h C and XVA := V hq C ˆV, where we recall ha is he valuaion process of he replicaing porfolio obained by seing he counerpary defaul inensiy equal o h; see also he discussion before Eq. 10. Firs, noe ha XVA XVA h. This follows direcly from Theorem 5.4, which provides a comparison resul for he Ǔ erm on he righ hand side of he XVA expression in 25, noing ha he wo closeou erms are jus independen of h. Hence, he righ hand side of Eq. 35 is smaller han he lef hand side: he laer represens a specific F- predicable inensiy process saisfying he boundary condiions, while he former is he supremum over all such inensiy processes. This shows ha he lef side of Eq. 35 is less or equal han he righ side. To show he reverse inequaliy, i.e., ha he lef side of Eq. 35 is greaer or equal han he righ side, we noe ha he family XVA h h F,h [hc,h C ] is direced upwards, i.e., for h, h F, h C h, h h C, 17

18 here exiss a process h F, h C h h C, such ha XVA h XVA h XVA h. Indeed, seing A := {ω Ω : XV A h > XV A h } we can define h direcly by seing h s := h s1l A + h s1l A c for s and h s = 0, for 0 s <. Such a process is clearly F s -measurable because A is F - measurable. As he essenial supremum of an upward direced se can be wrien as monoone limi see Föllmer e al., 2004, Theorem A.32, lim n XVA hn = rxva. Thus, as he counable union of nullses is sill a nullse we have ha, for all, rxva is smaller or equal han he righ side of Eq. 35. Nex, we provide he expressions for he superhedging sraegies. These are derived by replacing Ǔ wih Ǔ ino equaions Using he hedging sraegies defined in 36, we obain ha, on he se { < τ}, he value of he hedging porfolio a ime is ξ 1, u Pu 1 + ξ I, P I + ξ C, P C + ξ f, On he se { < τ}, he change in value of he porfolio is ξu 1, = dpu 1 + ξ I, dp I + ξ C, dp C + ξ f, db r f r D + h Q 1 Ǔ + L I ˆV M + + Ǔ B r f ψ m, B rm = Ǔ. 37 ψ m, db rm 38 r D + h Q I + Ǔ L C ˆV M r D + h Q C + r + mm + + r mm + r + f 2Û + L C ˆV M L I ˆV M + M + + r f 2Û + L C ˆV M L I ˆV M + M d. Addiionally, for he hedging sraegy o be self-financing, we need o include he cash flow r f ξ f, r D L 1 d. 39 The presence of his cash flow is due o he fac ha he clean valuaion ˆV is compued using he publicly available discoun rae r D, while he privae valuaion V is obained using he funding rae r f. Such a cash flow needs o be accouned for in he implemenaion of he superhedging sraegy. Taken ogeher, equaions 38 and 39 describe he change in value of he superhedging porfolio. Nex, we compare i wih he change in value of he robus XVA process given by dǔ = r D + h Q I L I ˆV M + + Ǔ rd + h Q C ˆV, M, Ǔ L C ˆV M Ǔ + r D + h Q 1 Ǔ + r + mm + + r mm + r + f 2Û + L C ˆV M L I ˆV M + + L 1 M + + r f 2Û + L C ˆV M L I ˆV M + + L 1 M r D L 1 d. 40 Using he fac ha h Q C ˆV, M, Ǔ hc Q θc ˆV, M Ǔ 0, i follows ha 38 ogeher wih 39 dominae 40 from above, i.e., ξ 1, u dpu 1 + ξ I, dp I + ξ C, dp C + ξ f, db r f ψ m, db rm + r f ξ f, r D L 1 d dǔ. 41 The above compuaions were done on he se { < τ}. A he sopping ime τ i can be easily checked ha boh Ǔ and he superhedging porfolio are zero. Togeher wih 41 and Theorem 5.4, i follows ha he superhedging porfolio dominaes Ǔ for all imes. 18

19 We noice ha if we use he robus superhedging sraegies given in 36 and sar wih an iniial capial rxva 0, hen here will be no racking error in he sense of El Karoui e al In oher words, he error commied for implemening he robus sraegy ξ 1,, ξi,, ξc,, ξ f,, ψm, in he real marke where he defaul inensiy of he counerpary is h Q C insead of he robus marke model where defaul inensiy is h Q C is zero. This may be undersood as follows: Eq. 37 shows ha he value of he superhedging porfolio is always rxva. However, unil he earlies among he defaul ime of he counerpary, hedger, or mauriy of he CDS conrac, whichever comes firs, he superhedging porfolio keeps generaing profis because he change in he value of he superhedging porfolio is greaer han he change in he value of he Ǔ, as shown in 41. In oher words, during a ime inerval d, he hedger pockes an exra cash h Q C ˆV, M, Ǔ hc Q θc ˆV, M Ǔ d a any ime prior o he end of he replicaion sraegy. The robus sraegies depend only on he XVA price process and he bond prices, and are independen of he defaul inensiy h Q C, he value of which is unknown o he hedger Figure 2: We use he following benchmark parameers: r ± f = r D = 0.001, α = β = 0, T = 3, L I = L C = 0.5, S 1 = 2, h 1 = 0.11l {0 <1} l {1 <T }, L 1 = 10, h Q I = 0.2, h C = 0.25, h C = 0.15, h Q C = 0.2. Lef panel: Plo of ǔ solid, ǔ dashed and ǔ doed as a funcion of ime. Righ panel: Plo of θ C ˆv, 0 dash-doed, θ I ˆv, 0 doed and ǔ solid as a funcion of ime. In he lef panel, he defaul inensiy a which we swich beween he sub-and super-soluions is he crossing poin of he dashed and doed lines wih he x-axis, ha occurs a approximaely = In he righ panel, he hird pary valuaion ˆv becomes posiive a approximaely = In he case of zero margins, i follows direcly from Eq. 20 ha he hird pary valuaion ˆv = ˆv + ˆv may be expressed in erms of he closeou value, and given by θ C ˆv,0 L C θ I ˆv,0 L I. Hence, we deduce from he righ panel of Figure 2 ha he hird pary valuaion is negaive prior o = 1.83, and posiive for > Figure 2 also shows ha he superhedging sraegy is non-rivial in he sense ha i is no monoone in he defaul inensiy. As i can be seen from he righ panel of Figure 2, he quaniy θ C ˆv, 0 ǔ is zero a , non-negaive for < 0, and sricly negaive for > 0. This implies ha h Q C = h C1l {0 0 } + h C 1l {0 < 3}. In oher words, prior o 0 he rader will use he larges value of he inensiy h C for her superhedging porfolio because he jump of he superhedging porfolio o he close-ou value when he counerpary defauls, given by θ C ˆv, 0 ǔ, is posiive. Afer ime 0, he rader will choose he smalles value h C of he defaul inensiy because his jump would be negaive. This is direcly visibile from he righ panel of Figure 2, because he dashdoed line dominaes he solid one. This analysis highlighs a fundamenal difference wih respec 19