Arbitrage-Free XVA. August 9, 2016

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1 Arbirage-Free XVA Maxim Bichuch Agosino Capponi Sephan Surm Augus 9, 216 Absrac We develop a ramework or compuing he oal valuaion adjusmen XVA o a European claim accouning or unding coss, counerpary credi risk, and collaeralizaion. Based on noarbirage argumens, we derive backward sochasic dierenial equaions BSDEs associaed wih he replicaing porolios o long and shor posiions in he claim. This leads o he deiniion o buyer s and seller s XVA, which in urn ideniy a no-arbirage inerval. In he case ha borrowing and lending raes coincide, we provide a ully explici expression or he unique XVA, expressed as a percenage o he price o he raded claim, and or he corresponding replicaion sraegies. In he general case o asymmeric unding, repo and collaeral raes, we sudy he semilinear parial dierenial equaions PDE characerizing buyer s and seller s XVA and show he exisence o a unique classical soluion o i. To illusrae our resuls, we conduc a numerical sudy demonsraing how unding coss, repo raes, and counerpary risk conribue o deermine he oal valuaion adjusmen. 1 Keywords: XVA, counerpary credi risk, unding spreads, backward sochasic dierenial equaions, arbirage-ree valuaion. Mahemaics Subjec Classiicaion 21: 91G4, 91G2, 6H1 JEL classiicaion: G13, C32 1 Inroducion When managing a porolio, a rader needs o raise cash in order o inance a number o operaions. Those include mainaining he hedge o he posiion, posing collaeral resources, and paying ineres on collaeral received. Moreover, he rader needs o accoun or he possibiliy ha he posiion may be liquidaed premaurely due o his own or his counerpary s deaul, hence enailing addiional coss due o he closeou procedure. Cash resources are provided o he rader by his reasury desk, and mus be remuneraed. I he is borrowing, he will be charged an ineres rae depending on curren mbichuch@jhu.edu, Deparmen o Applied Mahemaics and Saisics, Johns Hopkins Universiy ac3827@columbia.edu, Indusrial Engineering and Operaions Research Deparmen, Columbia Universiy ssurm@wpi.edu, Deparmen o Mahemaical Sciences, Worceser Polyechnic Insiue 1 This aricle subsumes he wo permanen working papers by he same auhors: Arbirage-Free Pricing o XVA - Par I: Framework and Explici Examples, and Arbirage-Free Pricing o XVA - Par II: PDE Represenaion and Numerical Analysis. These papers are accessible a hp://arxiv.org/abs/ and hp://arxiv.org/abs/ , respecively. 1

2 marke condiions as well as on his own credi qualiy. Such a rae is usually higher han he rae a which he rader lends excess cash proceeds rom his invesmen sraegy o his reasury. The dierence beween borrowing and lending rae is also reerred o as unding spread. Even hough pricing by replicaion can sill be pu o work under his rae asymmery, he classical Black-Scholes ormula no longer yields he price o he claim. In he absence o deaul risk, ew sudies have been devoed o pricing and hedging claims in markes wih dierenial raes. Korn 1995 considers opion pricing in a marke wih a higher borrowing han lending rae, and derives an inerval o accepable prices or boh he buyer and he seller. Cvianić and Karazas 1993 consider he problem o hedging coningen claims under porolio consrains allowing or a higher borrowing han lending rae. El Karoui e al sudy he super-hedging price o a coningen claim under rae asymmery via nonlinear backward sochasic dierenial equaions BSDEs. The above sudies do no consider he impac o counerpary credi risk on valuaion and hedging o he derivaive securiy. The new se o rules mandaed by he Basel Commiee Basel III 21 o govern bilaeral rading in OTC markes requires o ake ino accoun deaul and unding coss when marking o marke derivaives posiions. This has originaed a growing sream o lieraure, some o which is surveyed nex. Crépey 215a and Crépey 215b inroduce a BSDE approach or he valuaion o counerpary credi risk aking unding consrains ino accoun. He decomposes he value o he ransacion ino hree separae componens, he conrac porolio o over-he-couner derivaives, he hedging asses used o hedge marke risk o he porolio as well as counerpary credi risk, and he unding asses needed o inance he hedging sraegy. Brigo and Pallavicini 214 and Brigo e al. 212 derive a risk-neural pricing ormula by aking ino accoun counerpary credi risk, unding, and collaeral servicing coss, and provide he corresponding BSDE represenaion. Pierbarg 21 derives a closed orm soluion or he price o a derivaive conrac, which disinguishes beween unding, repo and collaeral raes, bu ignores he possibiliy o counerpary s deaul. Moreover, he assumes ha borrowing and lending raes are equal, an assumpion which has been laer relaxed by Mercurio 214a. Burgard and Kjaer 211a and Burgard and Kjaer 211b generalize Pierbarg 21 s model o include deaul risk o he rader and o his counerpary. They derive PDE represenaions or he price o he derivaive via a replicaion approach, assuming he absence o arbirage and suicien smoohness o he derivaive price. Bielecki and Rukowski 214 develop a general semimaringale marke ramework and derive he BSDE represenaion o he wealh process associaed wih a sel-inancing rading sraegy ha replicaes a deaul-ree claim. As in Pierbarg 21, hey do no ake counerpary credi risk ino accoun. Nie and Rukowski 213, sudy he exisence o air bilaeral prices. A good overview o he curren lieraure is given in Crépey e al In he presen aricle we inroduce a valuaion ramework which allows us o quaniy he oal valuaion adjusmen, abbreviaed as XVA, o a European ype claim. We consider an underlying porolio consising o a deaul-ree sock and wo risky bonds underwrien by he rader s irm and his counerpary. Sock purchases and sales are inanced hrough he securiy lending marke. We allow or asymmery beween reasury borrowing and lending raes, repo lending and borrowing raes, as well as beween ineres raes paid by he collaeral aker and received by he collaeral provider. We derive he nonlinear BSDEs associaed wih he porolios replicaing long and shor posiions in he raded claim, aking ino accoun counerpary credi risk and closeou payos exchanged a deaul. Due o rae asymmeries, he BSDE which represens he valuaion process o he porolio replicaing a long posiion in he claim canno be direcly obained via a sign change rom he one replicaing a shor posiion. More speciically, here is a no-arbirage inerval which can be deined in erms o he buyer s and he seller s XVA. 2

3 We show ha our ramework recovers he model proposed by Pierbarg 21, as well is exension o he case in which he hedger and his counerpary can deaul. In boh cases, we can express he oal valuaion adjusmen in closed orm, as a percenage o he publicly available price o he claim. This gives an inerpreaion o he XVA in erms o unding coss o a rade and counerpary risk, and has risk managemen implicaions because i pushes banks o compress rades so as o reduce heir borrowing coss and counerpary credi exposures see ISDA 215. One o he crucial assumpions o he Pierbarg s seup is ha raes are symmeric. In he case o asymmeric raes, closed orm expressions are unavailable, bu we can sill exploi he connecion beween he BSDEs and he corresponding nonlinear PDEs o sudy numerically how unding spreads, collaeral and counerpary risk aec he oal valuaion adjusmen. In his regard, our sudy exends he previous lieraure in wo direcions. Firs, we develop a rigorous sudy o he semilinear PDEs associaed wih he nonlinear BSDEs yielding he XVA. Relaed sudies o he PDE represenaions o XVA include Burgard and Kjaer 211a and Burgard and Kjaer 211b, who consider an exended Black-Scholes ramework in which wo corporae bonds are inroduced in order o hedge he deaul risk o he rader and o his counerpary. They generalize heir ramework in Burgard and Kjaer 213 o include collaeral miigaion and evaluae he impac o dieren unding sraegies. Second, we provide a comprehensive numerical analysis which explois he previously esablished exisence and uniqueness resul. We ind srong sensiiviy o XVA o unding coss, repo raes, and counerpary risk. Viewing boh buyer s and seller s XVA as uncions o collaeralizaion levels deines a no-arbirage band whose widh increases wih he unding spread and he dierence beween borrowing and lending repo raes. As he posiion becomes more collaeralized, he rader needs o inance a larger posiion and he XVA increases. Boh buyer s and seller s XVA may decrease i he raes o reurn o rader and counerpary bonds are higher han he unding coss incurred or replicaing he closeou posiion. The paper is organized as ollows. We develop he model in Secion 2 and inroduce he replicaed claim and collaeral process in Secion 3. We analyze arbirage-ree valuaion and XVA in he presence o unding coss and counerpary risk, reerred o as XVA, in Secion 4. Secion 5 provides an explici expression or he XVA under equal borrowing and lending raes. Secion 6 develops a numerical analysis when borrowing and lending raes are asymmeric. Secion 7 concludes he paper. Some proos o echnical resuls are delegaed o an Appendix. 2 The model We consider a probabiliy space Ω, G, P rich enough o suppor all subsequen consrucions. Here, P denoes he physical probabiliy measure. Throughou he paper, we reer o I as he invesor, rader or hedger ineresed in compuing he oal valuaion adjusmen, and o C as he counerpary o he invesor in he ransacion. The background or reerence ilraion ha includes all marke inormaion excep or deaul evens and augmened by all G, P-nullses, is denoed by F := F. The ilraion conaining deaul even inormaion is denoed by H := H. Boh ilraions will be speciied in he sequel o he paper. We denoe by G := G he enlarged ilraion given by G := F H, augmened by G, P-nullses. Noe ha because o he augmenaion o F by nullses, he ilraion G saisies he usual condiions o G, P-compleeness and righ coninuiy; see Secion 2.4 o Bélanger e al. 24. We disinguish beween universal insrumens, and invesor speciic insrumens, depending on wheher heir valuaion is public or privae. Privae valuaions are based on discoun raes, which depend on invesor speciic characerisics, while public valuaions depend on publicly available discoun 3

4 acors. Throughou he paper, we will use he superscrip when reerring speciically o public valuaions. Secion 2.1 inroduces he universal securiies. Invesor speciic securiies are inroduced in Secion Universal insrumens This class includes he deaul-ree sock on which he inancial claim is wrien, and he securiy accoun used o suppor purchases or sales o he sock securiy. Moreover, i includes he risky bond issued by he rader as well as he one issued by his counerpary. The sock securiy. We le F := F be he G, P-augmenaion o he ilraion generaed by a sandard Brownian moion W P under he measure P. Under he physical measure, he dynamics o he sock price is given by ds = µs d + σs dw P, where µ and σ are consans denoing, respecively, he appreciaion rae and he volailiy o he sock. The securiy accoun. Borrowing and lending aciviies relaed o he sock securiy happen hrough he securiy lending or repo marke. We do no disinguish beween securiy lending and repo, bu reer o all o hem as repo ransacions. We consider wo ypes o repo ransacions: securiy driven and cash driven, see also Adrian e al The securiy driven ransacion is used o overcome he prohibiion on naked shor sales o socks, ha is he prohibiion o he rader o selling a sock which he does no hold and hence canno deliver. The repo marke helps o overcome his by allowing he rader o lend cash o he paricipans in he repo marke who would pos he sock as a collaeral o he rader. The rader would laer reurn he sock collaeral in exchange o a pre-speciied amoun, usually slighly higher han he original loan amoun. Hence, eecively his collaeralized loan has a rae, reerred o as he repo rae. The cash lender can sell he sock on an exchange, and laer, a he mauriy o he repo conrac, buy i back and reurn i o he cash borrower. We illusrae he mechanics o he securiy driven ransacion in Figure 1. The oher ype o ransacion is cash driven. This is essenially he oher side o he rade, and is implemened when he rader wans a long posiion in he sock securiy. In his case, he borrows cash rom he repo marke, uses i o purchase he sock securiy posed as collaeral o he loan, and agrees o repurchase he collaeral laer a a slighly higher price. The dierence beween he original price o he collaeral and he repurchase price deines he repo rae. As he loan is collaeralized, he repo rae will be lower han he rae o an uncollaeralized loan. A mauriy o he repo conrac, when he rader has repurchased he sock collaeral rom he repo marke, he can sell i on he exchange. The deails o he cash driven ransacion are summarized in Figure 2. We use r r + o denoe he rae charged by he hedger when he lends money o he repo marke and implemens his shor-selling posiion. We use rr o denoe he rae ha he is charged when he borrows money rom he repo marke and implemens a long posiion. We denoe by B r+ r and B r r he repo accouns whose dris are given, respecively, by r r + and rr. Their dynamics are given by For uure purposes, deine B rr db r± r = r ± r B r± r d. := B rr ψ r = e rrψr s ds, 1 4

5 Sock Marke Treasury Desk Trader Repo Marke 6 3 r + r Figure 1: Securiy driven repo aciviy: Solid lines are purchases/sales, dashed lines borrowing/lending, doed lines ineres due; blue lines are cash, red lines are sock. The reasury desk lends money o he rader 1 who uses i o lend o he repo marke 2 receiving in urn collaeral 3. He sells he sock on he marke o ge eecively ino a shor posiion 4 earning cash rom he deal 5 which he uses o repay his deb o he unding desk 6. As a cash lender, he receives ineres a he rae r r + rom he repo marke. There are no ineres paymens beween rader and reasury desk as he paymens 1 and 6 cancel each oher ou. where r r x = r r 1l {x<} + r + r 1l {x>}. 2 Here, ψ r denoes he number o shares o he repo accoun held a ime. Equaions 1-2 indicae ha he rader earns he rae r r + when lending ψ r > shares o he repo accoun o implemen he shor-selling o ξ shares o he sock securiy, i.e., ξ <. Similarly, he has o pay ineres rae rr on he ψ r ψ r < shares o he repo accoun ha he has borrowed by posing ξ > shares o he sock securiy as collaeral. Because borrowing and lending ransacions are ully collaeralized, i always holds ha ψ r B rr = ξ S. 3 The risky bond securiies. Le τ i, i {I, C}, be he deaul imes o rader and counerpary. These deaul imes are exponenially disribued random variables wih consan inensiies h P i, i {I, C}, and are independen o he ilraion F. We use H i = 1l {τi },, o denoe he deaul indicaor process o i. The deaul even ilraion is given by H = H, H = σhu, I Hu C ; u. Such a deaul model is a special case o he bivariae Cox process ramework, or which he H- hypohesis see Ellio e al. 2 is well known o hold. In paricular, his implies ha he F-Brownian moion W P is also a G-Brownian moion. We inroduce wo risky bond securiies wih zero recovery underwrien by he rader I and by his counerpary C, and mauring a he same ime T. We denoe heir price processes by P I and P C, 5

6 Sock Marke Treasury Desk Trader Repo Marke 6 5 r r Figure 2: Cash driven repo aciviy: Solid lines are purchases/sales, dashed lines borrowing/lending, doed lines ineres due; blue lines are cash, red lines are sock. The reasury desk lends money o he rader 1 who uses i o purchase sock 2 rom he sock marke 3. He uses he sock as collaeral 4 o borrow money rom he repo marke 5 and uses i o repay his deb o he unding desk 6. The rader has hus o pay ineres a he rae rr o he repo marke. There are no ineres paymens beween rader and reasury desk as he paymens 1 and 6 cancel each oher ou. respecively. For T, i {I, C}, he dynamics o heir price processes are given by dp i = µ i P i d P i dh i, P i = e µ it, 4 wih reurn raes µ i. We do no allow bonds o be raded in he repo marke. Our assumpion is driven by he consideraion ha he repurchase agreemen marke or risky bonds is oen illiquid. We also reer o he inroducory discussion in Blanco e al. 25 saing ha even i a bond can be shored on he repo marke, he enor o he agreemen is usually very shor. Throughou he paper, we use τ := τ I τ C T o denoe he earlies o he ransacion mauriy T, rader and counerpary deaul ime. 2.2 Hedger speciic insrumens This class includes he unding accoun and he collaeral accoun o he hedger. Funding accoun. We assume ha he rader lends and borrows moneys rom his reasury a possibly dieren raes. Denoe by r + he rae a which he hedger lends o he reasury, and by r he rae a which he borrows rom i. We denoe by B r± unding raes, whose dynamics are given by he cash accouns corresponding o hese db r± = r ± Br± d. 6

7 Le ξ he number o shares o he unding accoun a ime. Deine B r := B r ξ = e r ξs ds, 5 where r := r y = r 1l {y<} + r + 1l {y>}. 6 Equaions 5-6 indicae ha i he hedger s posiion a ime, ξ, is negaive, hen he needs o inance his posiion. He will do so by borrowing rom he reasury a he rae r. Similarly, i he hedger s posiion is posiive, he will lend he cash amoun o he reasury a he rae r +. Collaeral process and collaeral accoun. The role o he collaeral is o miigae counerpary exposure o he wo paries, i.e he poenial loss on he ransaced claim incurred by one pary i he oher deauls. The collaeral process C := C ; is an F adaped process. We use he ollowing sign convenions. I C >, he hedger is said o be he collaeral provider. In his case he counerpary measures a posiive exposure o he hedger, hence asking him o pos collaeral so as o absorb poenial losses arising i he hedger deauls. Vice versa, i C <, he hedger is said o be he collaeral aker, i.e., he measures a posiive exposure o he counerpary and hence asks her o pos collaeral. Collaeral is posed and received in he orm o cash in line wih daa repored by ISDA 214, according o which cash collaeral is he mos popular orm o collaeral. 2 We denoe by r c + he rae on he collaeral amoun received by he hedger i he has posed he collaeral, i.e., i he is he collaeral provider, while rc is he rae paid by he hedger i he has received he collaeral, i.e., i he is he collaeral aker. The raes r c ± ypically correspond o Fed Funds or EONIA raes, i.e., o he conracual raes earned by cash collaeral in he US and EURO markes, respecively. We denoe by B r± c he cash accouns corresponding o hese collaeral raes, whose dynamics are given by db r± c = r c ± B r± c d. Moreover, le us deine where B rc := B rc C = e rccsds, r c x = r + c 1l {x>} + r c 1l {x<}. Le ψ c be he number o shares o he collaeral accoun B rc mus hold ha held by he rader a ime. Then i ψ c B rc = C. 7 The laer relaion means ha i he rader is he collaeral aker a, i.e., C <, hen he has purchased shares o he collaeral accoun i.e., ψ c >. Vice versa, i he rader is he collaeral provider a ime, i.e., C >, hen he has sold shares o he collaeral accoun o her counerpary. Beore proceeding urher, we visualize in Figure 3 he mechanics governing he enire low o ransacions aking place. 2 According o ISDA 214 see Table 3 herein, cash represens slighly more han 78% o he oal collaeral delivered and hese igures are broadly consisen across years. Governmen securiies insead only consiue 18% o oal collaeral delivered and oher orms o collaeral consising o riskier asses, such as municipal bonds, corporae bonds, equiy or commodiies only represen a racion slighly higher han 3%. 7

8 Sock & Repo Marke r r Sock r + r r + Treasury Desk Cash Trader Bonds P I, P C Bond Marke r Collaeral r + c r c Counerpary Figure 3: Trading: Solid lines are purchases/sales, dashed lines borrowing/lending, doed lines ineres due; blue lines are cash, red lines are sock purchases or cash and black lines are bond purchases or cash. 3 Replicaed claim, close-ou value and wealh process We ake he viewpoin o a rader who wans o replicae a European ype claim on he sock securiy. Such a claim is purchased or sold by he rader rom/o his counerpary over-he-couner and hence subjec o counerpary credi risk. The closeou value o he claim is decided by a valuaion agen who migh eiher be one o he paries or a hird pary, in accordance wih marke pracices as reviewed by he Inernaional Swaps and Derivaives Associaion ISDA. The valuaion agen deermines he closeou value o he ransacion by calculaing he Black Scholes price o he derivaive using he discoun rae r D. Such a publicly available discoun rae enables he hedger o inroduce a valuaion measure Q deined by he propery ha all securiies have insananeous growh rae r D under his measure. The res o he secion is organized as ollows. We give he deails o he valuaion measure in Secion 3.1. We inroduce he valuaion process o he claim o be replicaed and o he collaeral process in Secion 3.2. We deine he class o admissible sraegies in Secion 3.3 and speciy he closeou procedure in Secion The valuaion measure We irs inroduce he deaul inensiy model. Under he physical measure P, deaul imes o rader and counerpary are assumed o be independen exponenially disribued random variables wih 8

9 consan inensiies h P i, i {I, C}. I hen holds ha or each i {I, C} ϖ i,p := H i 1 H i u h P i du is a G, P-maringale. The valuaion measure Q associaed wih he publicly available discoun rae r D chosen by he valuaion agen is equivalen o P and given by he Radon-Nikodým densiy dq dp Gτ = e rdµ σ W P τ r D µ2 2σ 2 τ µ I r D h P I H I τ e r Dµ I +h P I τ µ C r D H C τ h P e r Dµ C +h P C τ. C We also recall ha µ I and µ C denoe he rae o reurns o he bonds underwrien by he rader and counerpary respecively. Under Q, he dynamics o he risky asses are given by ds = r D S d + σs dw Q, 8 dp I = r D P I d P I dϖ I,Q, dp C = r D P C d P C dϖ C,Q, where W Q := W Q ; τ is a G, Q-Brownian moion and ϖi,q := ϖ I,Q ; τ as well as ϖ C,Q := ϖ C,Q ; τ are G, Q-maringales. The above dynamics o P I and P C under he valuaion measure Q can be deduced rom heir respecive price processes given in 4 via a sraighorward applicaion o he Iô s ormula. By applicaion o Girsanov s heorem, we have he ollowing relaions: W Q = W P + µr D σ, ϖ i,q = ϖ i,p + 1 H i u h P i h Q i du. The quaniy, hq i = µ i r D, i {I, C}, is he deaul inensiy o name i under he valuaion measure and is assumed o be posiive. 3.2 Replicaed claim and collaeral speciicaion The price process o he coningen claim ϑ L 2 Ω, F T, Q o be replicaed is, according o he valuaion agen, given by ˆV ϑ := e r DT E Q[ ϑ F ]. We will drop he superscrip ϑ and jus wrie ˆV when i is undersood rom he conex. In he case o a European opion we have ha ϑ = ΦS T, where Φ : R > R is a real valued uncion represening he erminal payo o he claim and hus ˆV = ˆV, S. Addiionally, he hedger has o pos collaeral or he claim. As opposed o he collaeral used in he repo agreemen, which is always he sock, he collaeral miigaing counerpary credi risk o he claim is always cash. The collaeral is chosen o be a racion o he curren exposure process o one pary o he oher. I he hedger sells a European call or pu opion on he securiy o his counerpary he would hen need o replicae he payo ΦS T, Φ, and deliver i o he counerpary a T, he counerpary always measures a posiive exposure o he hedger, while he hedger has zero exposure o he counerpary. As a resul, he rader will always be he collaeral provider, while he counerpary he collaeral aker. By a symmeric reasoning, i he hedger buys a European call or pu opion rom his counerpary he would hen replicae he payo ΦS T, Φ, o hedge his posiion, hen he will always be he collaeral aker. On he even ha neiher he rader nor he counerpary have deauled by ime, he collaeral process is deined by C := α ˆV 1l {τ>} = α ˆV, S 1l {τ>}, 9 9

10 where α 1 is he collaeralizaion level. The case when α = corresponds o zero collaeralizaion, α = 1 gives ull collaeralizaion. Collaeralizaion levels are indusry speciic and are repored on a quarerly basis by ISDA, see or insance ISDA 211, Table 3.3. herein. 3 Our collaeral rule diers rom Pierbarg 21, where he collaeral is assumed o mach he value o he conrac inclusive o unding coss, repo spreads and collaeralizaion. Using such an approach, boh he hedger and his counerpary generally disagree on he level o posed collaeral i heir unding, repo or collaeral raes dier or i hey use dieren models o measure credi risk. Hence, he wo counerparies would need o ener ino negoiaions o agree on a collaeral level. In our model, his is avoided because he valuaion agen deermines he collaeral requiremens based on he Black-Scholes price ˆV o he claim, exclusive o unding and counerpary risk relaed coss. 3.3 The wealh process We allow or collaeral o be ully rehypohecaed. This means ha he collaeral aker is graned an unresriced righ o use he collaeral amoun, i.e., he can use i o purchase invesmen securiies. This is in agreemen wih mos ISDA annexes, including he New York Annex, English Annex, and Japanese Annex. We noice ha or he case o cash collaeral, he percenage o rehypohecaed collaeral amouns o abou 9% see Table 8 in ISDA 214 hence largely supporing our assumpion o ull collaeral re-hypohecaion. As in Bielecki and Rukowski 214, he collaeral received can be seen as an ordinary componen o a hedger s rading sraegy, alhough his applies only prior o he counerpary s deaul. We denoe by V ϕ he wealh process o he hedger and emphasize ha he collaeral can acually be used by him via rehypohecaion i he is he collaeral aker. Le ϕ := ξ, ξ, ξi, ξ C ;, where we recall ha ξ denoes he number o shares o he securiy, ξ he number o shares in he unding accoun, and we use ξi and ξ C o denoe he number o shares o rader and counerpary bonds, respecively, a ime. Recalling Eq. 7, and expressing all posiions in erms o number o shares muliplied by he price o he corresponding securiy, he wealh process V ϕ is given by he ollowing expression V ϕ := ξ S + ξ I P I + ξ C P C + ξ Br + ψ r B rr ψ c B rc, 1 where we noice ha he number o shares ψ r o he repo accoun and he number o shares ψ c held in he collaeral accoun are uniquely deermined by equaions 3 and 7, respecively. Deiniion 3.1. A collaeralized rading sraegy ϕ is sel-inancing i, or [, T ], i holds ha V ϕ := V ϕ + ξ u ds u + ξu I dpu I + ξu C dpu C + ξu db r u + ψu r dbu rr ψu c dbu rc, where V ϕ = V is he iniial endowmen. Moreover, we deine he class o admissible sraegies as ollows: Deiniion 3.2. The admissible se o rading sraegies is given as class o F-predicable processes such ha he porolio process is bounded rom below, see also Delbaen and Schachermayer The average collaeralizaion level in 21 across all OTC derivaives was 73.1%. Posiions wih banks and broker dealers are he mos highly collaeralized among he dieren counerpary ypes wih levels around 88.6%. Exposures o non-inancial corporaions and sovereign governmens and supra-naional insiuions end o have he lowes collaeralizaion levels, amouning o 13.9%. 1

11 3.4 Close-ou value o ransacion The ISDA marke review o OTC derivaive collaeralizaion pracices, see ISDA 21, secion 2.1.5, saes ha he surviving pary should evaluae he ransacions ha have been erminaed due o he deaul even, and claim or a reimbursemen only aer miigaing losses wih he available collaeral. In our sudy, we ollow he risk-ree closeou convenion meaning ha he rader liquidaes his posiion a he marke value when his counerpary deauls. Nex, we describe how his is modeled in our ramework. Denoe by θ he value o he replicaing porolio a τ, where we recall ha τ has been deined in Secion 2. This value represens he amoun o wealh ha he rader mus hold in order o replicae he closeou payos when he ransacion erminaes. I is given by θ = θτ, ˆV := ˆV τ + 1l {τc <τ I }L C Y 1l {τi <τ C }L I Y + = 1l {τi <τ C }θ I ˆV τ + 1l {τc <τ I }θ C ˆV τ, 11 where Y := ˆV τ C τ = 1 α ˆV τ is he value o he claim a deaul, need o he posed collaeral and we deine θ I v := v L I 1 αv +, θ C v := v + L C 1 αv, 12 where or a real number x we are using he noaions x + := maxx,, and x := max, x. The erm 1l {τc <τ I }L C Y originaes he residual CVA erm aer collaeral miigaion, while 1l {τi <τ C }L I Y + originaes he DVA erm, see also Brigo e al. 214 and Capponi 213 or addiional deails. The quaniies L I 1 and L C 1 are he loss raes agains he rader and counerpary claims, respecively. Remark 3.3. We elaborae on why θ is he amoun which needs o be replicaed by he rader when he ransacion erminaes. Suppose ha he rader has sold a call opion o he counerpary hence ˆV, S > or all. This means ha he rader is always he collaeral provider, C = α ˆV, S > or all, given ha he counerpary always measures a posiive exposure o he rader. I he counerpary deauls irs and beore he mauriy o he claim, he rader will ne he amoun ˆV τ, S τ owed o he counerpary wih his collaeral posed o he counerpary, and only reurn o her he residual amoun Y. As a resul, his rading sraegy mus yield an acual wealh equal o his amoun. The above expression o closeou given by θ = ˆV τ, S τ indicaes ha his is indeed he case. Because he counerpary already holds he collaeral, he rader only needs o reurn o her he amoun Y, which is precisely he wealh process o he rader a τ. 4 Arbirage-ree valuaion and XVA The goal o his secion is o ind a valuaion or he derivaive securiy wih payo ΦS T ha is ree rom arbirage in a cerain sense. Beore discussing arbirage-ree valuaions, we have o make sure ha he underlying marke does no admi arbirage rom he hedger s perspecive as discussed in Bielecki and Rukowski, 214, Secion 3. In he underlying marke, he rader is only allowed o borrow/lend sock, buy/sell risky bonds and borrow/lend rom/o he unding desk. In paricular, neiher he derivaive securiy, nor he collaeral process is involved. Deiniion 4.1. We say ha he marke S, P I, P C admis hedger s arbirage i we can ind a rading sraegy ϕ = ξ, ξ, ξi, ξ C ; such ha, given a non-negaive iniial capial x o he hedger and denoing he wealh process corresponding o i V ϕ, x, we have ha P[ V τ ϕ, x e r+ τ x ] = 1 11

12 and P [ V τ ϕ, x > e r+ τ x ] >. I he marke does no admi hedger s arbirage or he hedger s iniial capial x, we say ha he marke is arbirage ree rom he hedger s perspecive. We will omi he argumens x, ϕ or boh in he wealh process V ϕ, x, whenever undersood rom he conex. In he sequel, we make he ollowing assumpion: Assumpion 4.2. The ollowing relaions hold beween he dieren raes: r + r r, r+ r, and r + r D < µ I µ C. Remark 4.3. The above assumpion is necessary o preclude arbirage. The condiion r D < µ I µ C is needed or he exisence o he valuaion measure as discussed a he end o secion 3.1 h Q i = µ i r D and risk-neural deaul inensiies mus be posiive. I, by conradicion, r r + > r, he rader can borrow cash rom he unding desk a he rae r and lend i o he repo marke a he rae r+ r, while holding he sock as a collaeral. This resuls in a sure win or he rader. Similarly, i he rader could und his sraegy rom he reasury a a rae r < r+, i would clearly resul in an arbirage. The condiion r + < µ I and muais muandis r + < µ C has a more pracical inerpreaion: i precludes he arbirage opporuniy o shor selling he bond underwrien by he rader s irm and invesing he proceeds in he unding accoun. We nex provide a suicien condiion guaraneeing ha he underlying marke is ree o arbirage. Proposiion 4.4. Suppose ha in addiion o Assumpion 4.2, r + r r + r r. Then he model does no admi arbirage opporuniies or he hedger or any x. We remark ha in a marke model wihou deaulable securiies, similar inequaliies beween borrowing and lending raes have been derived by Bielecki and Rukowski 214 Proposiion 3.3, and by Nie and Rukowski 213 Proposiion 3.1. We impose addiional relaions beween lending raes and reurn raes o he risky bonds given ha our model also allows or counerpary risk. Proo. Firs, observe ha under he condiions given above we have r r ψ r = r + r ψ r 1l {ψ r >} + r r ψ r 1l {ψ r <} r + ψr 1l {ψ r >} + r + ψr 1l {ψ r <} = r + ψr r ξ = r+ ξ 1l {ξ >} + r ξ 1l {ξ <} r+ ξ 1l {ξ >} + r+ ξ 1l {ξ <} = r+ ξ Nex, i is convenien o wrie he wealh process under a suiable measure P speciied via he sochasic exponenial d P dp Gτ r + µ = e W σ τ P r + µ2 2σ 2 τ µ I r + h P I H I τ e r+ µ I+h P I τ µ C r + h P C H C τ e r+ µ C+h P C τ By Girsanov s heorem, P is an equivalen measure o P such ha he dynamics o he risky asses are given by ds = r + S d + σs dw P, dp I = r + P I d P I dϖ I, P, dp C = r + P C d P C dϖ C, P 12

13 where W P := W P ; τ is a G, P-Brownian moion and ϖ I, P := ϖ I, P ; τ as well as ϖ C, P := ϖ C, P ; τ are G, P-maringales. The r + discouned asses S := e r+ S, P I := e r+ P I and P C := e r+ P C are hus G, P-maringales. In paricular, W P = W P + µr+ σ and he deaul inensiy o he hedger and o his counerpary under P are given by h P i = µ i r +, i {I, C}, which is posiive in ligh o he assumpions o he proposiion. Denoe he wealh process associaed wih S, P I, P C in he underlying marke by ˇV. Using he sel-inancing condiion, is dynamics are given by Then we have ha ˇV τ ϕ, x ˇV ϕ, x = Thereore, i ollows ha d ˇV = r ξ Br + r + ξ S + r r ψ r B rr + r + ξi P I + r + ξc P C + ξ σs dw P ξ I P I dϖ I, P ξ C P C dϖ C, P = r ξ Br + r r ψ r B rr d + ξ ds + ξ I dp I + ξ C dp C. = τ + τ τ + τ τ r + ξ S + r ξ Br ξ σs dw P τ r + ξ S + r + ξ Br ξ σs dw P r + ˇV x d + e r+ τ ˇVτ ϕ, x ˇV ϕ, x τ τ τ + r r ψ r B rr ξ I P I dϖ I, P + r + ψr B rr d + r + ξi P I + r + ξc P C ξ I P I dϖ I, P ξ σs dw P τ ξ C P C dϖ C, P + r + ξi P I + r + ξc P C τ τ ξ C P C dϖ C, P ξ I P I dϖ I, P ξ d S τ ξ I d P τ I ξ C τ d P C. d d ξ C P C dϖ C, P. Noe ha he righ hand side o he above inequaliy is a local maringale bounded rom below as he value process is bounded rom below by he admissibiliy condiion, and hereore is a supermaringale. Taking expecaions, we conclude ha E P [ e r+ τ ˇVτ ϕ, x ˇV ϕ, x ]. Thus eiher P [ ˇVτ ϕ, x = e r+ τ x ] = 1 or P [ ˇVτ ϕ, x < e r+ τ x ] >. As P is equivalen o P, his shows ha arbirage opporuniies or he hedger are precluded in his model he would receive e r+ τ x by lending he posiive cash amoun x o he reasury desk a he rae r +. Nex we wan o deine he noion o an arbirage ree price o a derivaive securiy rom he hedger s perspecive. We will assume ha he hedger has zero iniial capial, or equivalenly, he does no have liquid iniial capial which can be used or hedging he claim unil mauriy. The hedging porolio will hus be enirely inanced by purchase/sale o he sock via he repo marke and purchase/sale o bonds via he unding accoun. While our enire analysis migh be exended o he case o nonzero iniial capial, such an assumpion will simpliy noaion and allow us o highligh he key aspecs o he sudy. 13

14 Deiniion 4.5. The valuaion P R o a derivaive securiy wih erminal payo ϑ F T is called hedger s arbirage-ree i or all γ R, buying γ securiies or γp and hedging in he marke wih an admissible sraegy and zero iniial capial, does no creae hedger s arbirage. Beore giving a characerizaion o hedger s arbirage-ree valuaions, we analyze he dynamics o he wealh process. We will rewrie i under he valuaion measure Q or noaional simpliciy. Using he condiion 3, we obain rom Eq. 1 ha dv = r ξ Br + r D r r ξ S + r D ξ I P I + r D ξ C P C r c ψ c B rc d + ξ σs dw Q ξ I P I dϖ I,Q ξ C P C dϖ C,Q Seing = r + ξ rc = r + ξ + r c + Br + r ξ Br ψ c B rc Br + r + c ψ c B rc + r ξ Br + rd r r ξ S + rd r + r ξ S + rd ξ I P I + r D ξ C P C d + ξ σs dw Q ξ I P I dϖ I,Q ξ C P C dϖ C,Q + rd rr + ξ S rd r r + ξ S + rd ξ I P I + r D ξ C P C C + r c C d + ξ σs dw Q ξ I P I dϖ I,Q d d ξ C P C dϖ C,Q. 13 Z = ξ σs, Z I = ξ I P I, Z C = ξ C P C, 14 and using again he condiion 3 and Eq. 1, we obain ξ Br = V ξ I P I ξ C P C + ψ c B rc Then he dynamics in Eq. 13 reads as dv = r + V + Z I + Z C + C r V + Z I + Z C C = V ξ I P I ξ C P C C r D rr 1 + Z rd r r + 1 Z rd Z I r D Z C + r c + σ σ + Z dw Q + Z I dϖ I,Q + Z C dϖ C,Q. We nex deine he drivers +, v, z, z I, z C ; ˆV := r + C + r c C d v + z I + z C α ˆV + r v + z I + z C α ˆV + r D r r 1 σ z+ r D r + r 1 σ z r D z I r D z C + r c + + α ˆV r c α ˆV 16, v, z, z I, z C ; ˆV := +, v, z, z I, z C ; ˆV, 17 which depend on he marke valuaion process ˆV via he collaeral C and we omi indicaors as we are ineresed in hedging only up o deaul. In paricular ± : Ω [, T ] R 4, ω,, v, z, z I, z C ±, v, z, z I, z C ; ˆV are drivers o BSDEs admiing unique soluions as implied by Corollary 4.8. Moreover, deine V +,γ and V,γ as soluions o he BSDEs dv +,γ V +,γ τ = +, V +,γ, Z +,γ, Z I,+,γ, Z C,+,γ ; ˆV d Z +,γ dw Q Z I,+,γ dϖ I,Q Z C,+,γ dϖ C,Q, = γ θ I ˆV τ 1l {τi <τ C T } + θ C ˆV τ 1l {τc <τ I T } + ϑ1l {τ=t }

15 and dv,γ V,γ τ =, V γ, Z,γ, Z I,,γ, Z C,,γ ; ˆV d Z,γ dw Q Z I,,γ dϖ I,Q Z C,,γ dϖ C,Q, = γ θ I ˆV τ 1l {τi <τ C T } + θ C ˆV τ 1l {τc <τ I T } + ϑ1l {τ=t }. 19 We noe ha V +,γ describes he wealh process when replicaing he claim γϑ or γ > hence hedging he posiion aer selling γ securiies wih erminal payo ϑ wih zero iniial capial. On he oher hand, V,γ describes he wealh process when replicaing he claim γϑ, γ > hence hedging he posiion aer buying γ securiies wih erminal payo ϑ wih zero iniial capial. Noice ha by posiive homogeneiy o he drivers + and o he BSDEs 18 and 19, i is enough o consider he cases when γ = 1. To ease he noaion, we se V +,1 = V + and V,1 = V. We also noe ha he wo BSDEs are inrinsically relaed: V, Z, Z I,, Z C, is a soluion o he daa, θ I ˆV τ 1l {τi <τ C T } + θ C ˆV τ 1l {τc <τ I T }, ϑ i and only i V, Z, Z I,, Z C, is a soluion o he daa +, θ I ˆV τ 1l {τi <τ C T } + θ C ˆV τ 1l {τc <τ I T }, ϑ. Our goal is o compue he oal valuaion adjusmen XVA ha needs o be added o he Black- Scholes price o he claim o ge he acual valuaion. As we have seen, he siuaion is asymmeric or sell- and buy-valuaions, so we will have o deine i boh rom he seller s and buyer s viewpoin. Deiniion 4.6. The seller s XVA is he G-adaped sochasic process XVA deined as while he buyer s XVA is deined as XVA + := V + ˆV 2 XVA := V ˆV. 21 XVA + quaniies he oal coss including collaeral, unding, and counerpary risk relaed coss incurred by he rader o hedge a long posiion in he opion, whereas XVA quaniies he oal coss incurred when hedging a shor posiion. As we will see in Secion 5.1, hese wo XVAs agree only i he drivers o he BSDEs are linear. Noing ha he agen s valuaion process ˆV saisies he Black-Scholes BSDE d ˆV = r D ˆV d Ẑ dw Q, ˆV T = ϑ, which is well known o admi a unique soluion, we can immediaely obain BSDEs or he XV A ± : wih dxva ± = ±, XVA ±, Z ± Z ± dw Q I,±, Z, I,± Z dϖ I,Q C,± Z ; ˆV d C,± Z dϖ C,Q, XVA ± τ = θ C ˆV τ 1l {τc <τ I T } + θ I ˆV τ 1l {τi <τ C T }, 22 Z ± := Z ± Ẑ, ZI,± = Z I,±, ZC,± = Z C,±, θ C ˆv := L C 1 αˆv, θi ˆv := L I 1 αˆv +, 23 15

16 and +, xva, z, z I, z C ; ˆV : = r + xva + z I + z C + 1 α ˆV + r xva + z I + z C + 1 α ˆV + r D r r 1 σ z+ r D r + r 1 σ z r D z I r D z C + r + c α ˆV + r c α ˆV + r D ˆV, 24, xva, z, z I, z C ; ˆV : = +, xva, z, z I, z C ; ˆV. 25 Noe ha comparing 16 wih 24 and 17 wih 25 we see ha ±, v, z, z I, z C ; ˆv = ±, v + ˆv, z, z I, z C ; ˆv + r Dˆv. 26 Nex, we can apply he reducion echnique developed by Crépey and Song 215 o ind a coninuous BSDE describing he XVA prior o deaul. Theorem 4.7. The BSDEs dǔ ± = ǧ ±, Ǔ ±, Ž± ; ˆV d Ž± dw Q Ǔ ± T = 27 in he ilraion F wih ǧ +, ǔ, ž; ˆV := h Q I θi ˆV ǔ + h Q C θc ˆV ǔ + +, ǔ, ž, θ I ˆV ǔ, θ C ˆV ǔ; ˆV 28 ǧ, ǔ, ž; ˆV := ǧ +, ǔ, ž; ˆV, 29 admi unique soluions Ǔ ±, Ž±, which are relaed o he unique soluions XVA ±, Z ±, Z I,±, o he BSDEs 22 via he ollowing relaions. On he one hand Ǔ ± := XVA ± τ, Ž ± := Z ± 1l {<τ}, Z C,± are soluions o he reduced BSDE 27, and on he oher hand he soluions o he ull XVA BSDEs given by Eq. 22 are given by XVA ± := Ǔ ± 1l {<τ} + θc ˆV τc 1l {τc <τ I T } + θ I ˆV τi 1l {τi <τ C T } 1l { τ}, Z ± := Ž± 1l {<τ}, Z I,± := θi ˆV Ǔ ± 1l { τ}, Z C,± := θc ˆV Ǔ ± 1l { τ}. Proo. As he equaions 27 are coninuous BSDEs wih Lipschiz driver, exisence and uniqueness are classical c., e.g., El Karoui e al., 1997, Theorem The equivalence o he ull G-BSDEs and he reduced F-BSDEs ollows rom Crépey and Song, 215, Theorem 4.3: In our case we do no change he probabiliy measure and hus condiion A is saisied as he H-hypohesis holds noe ha condiion B also holds in our conex. Condiion J is rivially saisied as he erminal condiion does no depend on he auxiliary processes Z, ZC and Z I. Finally, by he maringale represenaion heorems wih respec o F and G see Bielecki and Rukowski, 21, Secion 5.2, he maringale problems considered in Crépey and Song 215 and he acual BSDEs considered in he presen aricle have he same unique soluions. 16

17 The uniqueness o he soluions o he original BSDEs or V ± as well as o heir projeced versions in he F-ilraion ollows rom he deiniion o XVA: Corollary 4.8. Boh he BSDE 18 and 19 admi a unique soluion. These soluions are relaed o he unique soluions Ū ±, Z ± o he BSDEs in he ilraion F wih dū ± = g ±, Ū ±, Z ± ; ˆV d Z ± dw Q Ū ± T = ϑ 33 g +, ū, z; ˆV := h Q I θi ˆV ū + h Q C θc ˆV ū + +, ū, z, θ I ˆV ū, θ C ˆV ū; ˆV g, ū, z; ˆV := g +, ū, z; ˆV, via he ollowing relaions. On he one hand Ū ± := V τ ± Z ± := Z ± 1l {<τ}, are soluions o he reduced BSDEs 33, while on he oher hand he soluions o he ull BSDEs given by equaions 18 and 19 are given by V ± := Ū ± 1l {<τ} + θ C ˆV τc 1l {τc <τ I T } + θ I ˆV τi 1l {τi <τ C T } 1l { τ}, Z ± := Z ± 1l {<τ}, Z I,± := θ I ˆV Ū ± Z C,± := θ C ˆV Ū ± 1l { τ}, 1l { τ}. Remark 4.9. We discuss how he replicaing sraegies o he XVA process can be obained rom he above given resuls. We use he ilde symbol o denoe he sraegies replicaing he XVA process e.g., ξ, ξ I, ξ C denoe, respecively, he number o sock, rader and counerpary bond shares used o replicae XVA. This enables us o disinguish hem rom he sraegies used o replicae he price process V o he claim. From he maringale represenaion heorem and he dynamics o he sock price see also he discussion a he end o he proo o Proposiion 5.3, i ollows ha ξ ± = Z ± σs 1l {<τ}, ψr,± = ξ ± S 1l B rr {<τ}. From Theorem 4.7 along wih equaions 14 and 23, i ollows ha ξ I,± ξ C,± = = I,± Z P I C,± Z P C = L I1 α ˆV + Ǔ ± P I = L C1 α ˆV Ǔ ± P C 1l { τ}, 1l { τ}. 17

18 From equaions 9 and 7 i ollows ha ψ c = α ˆV 1l {τ>}. Finally, rom Eq. 15, replacing V ± wih XV A ± + ˆV we obain ξ,± = V ± B rc ˆV ξ I P I ξ C Pα C ˆV B r 1l {τ>} = L C1 α ˆV L I 1 α ˆV + + Ǔ ± α ˆV B r 1l {τ>}, where in he las equaion we have used he deiniions o XVA ± wih he ac ha XVA ± = Ǔ ± on he se {τ > }. given in equaions 21 and 2 along Nex, we analyze under which condiions he valuaions obained by solving he BSDEs are arbirage ree. Theorem 4.1. Le Φ be a uncion o polynomial growh. Assume ha and r + r r + r r, r + r, r+ r D < µ I µ C, 34 r + c r c r µ I µ C. 35 I V V +, where V + and V are he irs componens o he soluions o he BSDEs 18 and 19, hen here exis valuaions π sup and π in, π in π sup, or he opion ϑ = ΦS T such ha all values in he closed inerval [π in, π sup ] are ree o hedger s arbirage. All valuaions sricly bigger han π sup and sricly smaller han π in provide hen arbirage opporuniies or he hedger. In paricular, we have ha π sup = V + and π in = V. Proo. Firs, noice ha by virue o he condiions in 34, he underlying marke model is ree o hedger s arbirage. Nex, we noe ha he rader can perecly hedge a shor call posiion wih erminal payo ϑ = ΦS T using he iniial capial V + as he polynomial growh o Φ implies ϑ L2 Ω, F T, P. Thus i is clear ha any value P > V + is no arbirage ree, as we could jus sell he opion or ha value, use V + o hedge he claim and deposi P V + in he unding accoun. Using he same argumen, we can conclude ha buying an opion or any value P < V will lead o arbirage. Second, assume by conradicion ha a valuaion P V + would lead o an arbirage when selling he opion. This means ha saring wih iniial capial P, he rader can perecly hedge a claim wih erminal payo ϑ F T, where ϑ ϑ = ΦS T a.s. and P[ϑ > ϑ] >. As h Q j = µ j r D, j {I, C}, 18

19 we have g + ϑ, ū, z; ˆV g +, ū, z; ˆV ϑ = h Q ϑ I θi ˆV θ I ˆV ϑ + h Q C θc + r r+ r θ I + r D θ I + αrc ϑ ˆV µ I r θi ˆV ϑ θ C ˆV ϑ ϑ ϑ θi ˆV + θ C ˆV ū α ϑ ˆV θ I ˆV ϑ + θ C ϑ ˆV θ I ˆV ϑ + θ C ˆV ϑ αr + c ϑ ˆV θ I ˆV ϑ + α r r+ c r c ˆV ϑ ˆV ϑ ϑ ˆV θ C ˆV ϑ α ϑ ˆV ϑ ˆV θ C ˆV ϑ ˆV ϑ + ˆV ϑ + + µ C r θi ˆV ϑ. + θi ˆV ϑ + θ C ˆV ϑ ˆV ϑ ϑ ˆV θ I ˆV ϑ ū α ˆV ϑ + To deduce he irs inequaliy, we have used ha he erm muliplying r r+ can be direcly seen rom he direc compuaion below is posiive. This ϑ ϑ ϑ θ I ˆV + θ C ˆV α ˆV = 1 α1 L I ϑ ˆV + 1 α1 + LC ϑ ˆV 1 α1 L I ˆV ϑ + 1 α1 + LC ˆV ϑ = θi ˆV ϑ + θ C ˆV ϑ α ˆV ϑ. To deduce he las inequaliy, we have used 35. Thus, we can apply he comparison principle or F-BSDEs El Karoui e al., 1997, Theorem 2.2 o Ū and hen noice ha ϑ ϑ ϑ implies ha ˆV ˆV ϑ, ϑ which in urn leads o he ollowing inequaliy beween he closeou erms: θ C ˆV θ C ˆV ϑ and ϑ θ I ˆV θ I ˆV ϑ see also Eq. 12 or heir deiniions. I hus ollows ha V ϑ V ϑ and in paricular P > V + using sric comparison, i.e., P = V implies ϑ = ϑ a.s., conradicing our assumpion. Using a symmeric argumen, i ollows ha P V. Thus, i V V +, we can conclude ha all valuaions in he inerval [π in = V, V + = π sup ] are arbirage-ree, whereas no arbirage ree valuaion exiss i V > V +. We noe ha he widh o he no-arbirage inerval can be described boh in erms o he valuaions o he claim and o he XVA as XVA + XVA = V + V = πsup π in. The reduced BSDE also enables us o provide a PDE represenaion or he case ϑ = ΦS T using jus classical argumens based on he nonlinear Feynman-Kac ormula. We provide such a represenaion in he ollowing proposiion whose proo is repored in he appendix. Proposiion The wo-dimensional semilinear Cauchy problem u ± + Lu ± = g ±, u ±, σu ± x + û x ; û, u ± T, x =, ŵ + Lŵ = r D ŵ, ŵt, x = Φe x, 36 where he dierenial operaor L is deined by L := r D σ2 x σ2 2 2 xx, 19

20 admis he unique viscosiy soluions u ± and i holds ha u ±, log S = XVA ± 1l {<τ}. Moreover, i Φ is piecewise coninuously diereniable wih Φ and Φ where deined having a mos polynomial growh, hen he Cauchy problems 36 has classical soluions. Remark The ransormaion rom he BSDE 18 or 19 o he PDE 36 was derived by projecing a G-BSDE o a F-BSDE and hen applying he Feynman-Kac ormula o i. One can ollow an alernaive roue, deriving irs a PDE in our variables relaed o he BSDE wih jumps via a nonlinear Feynman-Kac ormula, and hen reducing he dimension. This is shown in deail in Bichuch e al. II 215, and we provide here he gis o he argumen. On { < τ}, deine he pre-deaul soluions v ±, S, ϖ I,Q, ϖ C,Q = V ± 1l {τ>} o he BSDEs 18 and 19. The exisence o hese measurable uncions v ±, i.e., he ac ha V ± are Markovian, ollows rom Proposiion in Delong 213. Speciically, Theorem 3.2 in Bichuch e al. II 215 shows ha v ± saisy v ± j {I,C} h Q j θ j ˆv, s v ±, s, w I, w C v j ± r D sv s ± 1 2 σ2 s 2 v ± ss ±, v ±, σsv s ±, s, w I, w C, θ I ˆv, s v ±, s, w I, w C, θ C ˆv, s v ±, s, w I, w C ; ˆv, s =, v ± T, s,, = Φs in he viscosiy sense. In he above expression, we have used he noaion v i ± = v±, i {I, C}. w i Addiionally, Theorem 3.2 in Bichuch e al. II 215 shows ha v ± are he unique viscosiy soluions o he PDEs 37 saisying lim v ±, e x,, e c log2 x =, c >. x The above sep ransers he BSDEs 18 and 19 o he PDEs 37. The key sep in he PDE domain ha parallels he ransormaion rom he original BSDEs o he reduced BSDEs is given in Remark 3.3 o Bichuch e al. II 215. Essenially, we are only concerned wih V ± beore any deaul occurs, hence we do no need o keep rack o he maringale erms ϖ j,q s. These are only needed o rack he occurrence o a deaul. I no deaul has occurred, ϖ j,q = h Q j. I hen ollows ha v± becomes a uncion o only wo variables, i.e. v ±, s := v ±, s, h Q I, hq C. The PDEs 37 become v ± + h Q I + hq C v±, s r D s v ± s 1 2 σ2 s 2 v ± ss ±, v ±, σs v ± s, s, θ I ˆv, s v ±, sθ C ˆv, s v ±, s; ˆv, s = v ± T, s = Φs. j {I,C} h Q j θ jˆv, s, From here, applying he sandard change o variables x = log s, w ±, x = v ±, e x and ŵ, x = ˆv, e x, we obain he PDEs w ± r D σ2 w x ± σ2 w xx ± + h Q I + hq C w ± ±, w ±, x, σ w x ± ŵ,, x, θ I x w ± ŵ,, x, θ C x w ±, x; ŵ, x = h Q j θ jŵ, x, j {I,C} w ± T, x = Φe x. 37 2

21 The PDEs 36 ollow by deriving he PDEs or w ± = w ± ŵ, which are associaed wih he BSDEs represenaion o he XVA given by equaions 2 and 21. Hence, we can express XVA as a soluion o a Cauchy problem or a wo-dimensional sysem o semilinear PDEs, w ± + Lw ± = ±, w ± + ŵ, σw x ± + ŵ x ±, ˆθ I ŵ w ±, ˆθ C ŵ w ± ; ŵ + ˆθj ŵ w ± + r D ŵ, w ± T, x =, j {I,C} h Q j ŵ + Lŵ + r D ŵ =, ŵt, x = Φe x. 38 Recalling he deiniions o ±, ±, and g ± and heir relaions given in equaions 16, 17, 24, 25, 26 and 28, 29 we obain ha equaions 38 and 36 coincide. A his poin, we can conclude ha he unique soluions w ± o he PDEs 38 are only in he viscosiy sense. I Φ is piecewise coninuously diereniable and also Φ where deined has a mos polynomial growh, hen i ollows rom Theorem 4.11 ha he uniqueness o he soluions also holds in he classical sense. Moreover, we can apply Theorem A.1 o Bichuch e al. II 215 and obain ha, on he se { < τ}, we have ha 5 Explici examples Z ± = σs w x ±, logs, Z i,± = θ i ˆV, S w ±, logs, i {I, C}. We specialize our ramework o deal wih a concree example or which we can provide ully explici expressions or he oal valuaion adjusmen. More speciically, we consider Pierbarg 21 s model as well as an exension o i, accouning or counerpary credi risk and closeou coss. This means ha deaulable bonds o rader and counerpary become an inegral par o he hedging sraegy. Throughou he secion, we make he ollowing assumpions on he ineres raes, as in Pierbarg s seup: r + = r = r, r + c = r c = r c, r D = r + r = r r = r r. We also assume ha r > r r > r c, he case o be expeced in pracice according o Pierbarg 21. Under he above assumpions, he securiy, unding and collaeral accouns do no depend on wheher he posiion in he securiy is long or shor, wheher he amoun is borrowed rom or len o he reasury, and wheher he collaeral is posed or received. Due o he symmery beween raes, he buyer s and seller s XVA coincide, and hence we can drop boh he plus and minus superscrips in he BSDEs. The dierence beween he discoun rae r D chosen by he valuaion agen and he repo rae may also be inerpreed as a proxy or illiquidiy o he repo marke. Under his inerpreaion, r D = r r corresponds o a regime o ull liquidiy. The BSDEs become linear and lead o explici addiive decomposiions o XVA in erms o dieren adjusmens as deailed in he sequel. We also remark ha similar decomposiions have been obained by Brigo e al. 212, see Theorem 4.3 and subsequen remarks herein. 5.1 Pierbarg s model This secion provides an explici represenaion o XVA and he associaed hedging sraegy in he ramework proposed by Pierbarg 21. Besides symmery beween raes, Pierbarg 21 precludes 21