Arbitrage-Free Pricing with Funding Costs and Collateralization

Transcription

1 Arbirage-Free Pricing wih Funding Coss and Collaeralizaion Andrea Pallavicini Dep. of Mahemaics, Imperial College London Financial Engineering, Banca IMI Seminar on Credi Risk Pisa, 29 May 2015 A. Pallavicini Funding Coss 29 May / 86

2 Talk Ouline Talk Ouline 1 Securiies, Derivaives and Trading Sraegies 2 Arbirage-Free Pricing 3 Funding Coss A. Pallavicini Funding Coss 29 May / 86

3 Disclaimer Disclaimer The opinions expressed in his work are solely hose of he auhors and do no represen in any way hose of heir curren and pas employers. A. Pallavicini Funding Coss 29 May / 86

4 Reference Books and Papers Reference Books Duffie, D. (1992,2001) Dynamic Asse Pricing Theory. Princeon Universiy Press. Björk, T. (1999,2009) Arbirage Theory in Coninuous Time. Oxford Universiy Press. Bielecki, T., Rukowski, M. (2002,2004) Credi Risk: modeling, valuaion and hedging. Springer Finance. Brigo, D., Morini, M., Pallavicini, A. (2013) Counerpary Credi Risk, Collaeral and Funding wih Pricing Cases for All Asse Classes. Wiley. Crépey, S., Bielecki, T., Brigo, D. (2014) Counerpary Risk and Funding: A Tale of Two Puzzles. Chapman and Hall. A. Pallavicini Funding Coss 29 May / 86

5 Reference Books and Papers Reference Papers Bergman, Y.Z. (1995) Opion pricing wih differenial ineres raes. Review of Financial Sudies 8 (2) Burgard C., Kjaer, M. (2010,2011) Parial Differenial Equaion Represenaions of Derivaives wih Bilaeral Counerpary Risk and Funding Coss. Journal of Credi Risk 7 (3) Crépey, S. (2011,2013) Bilaeral Counerpary Risk under Funding Consrains. Forhcoming in Mahemaical Finanace. Pallavicini, A., Perini, D., Brigo, D. (2011,2012) Funding Valuaion Adjusmen. and Funding, Collaeral and Hedging. Bielecki, T., Rukowski, M. (2013,2014) Valuaion and Hedging of Conracs wih Funding Coss and Collaeralizaion. A. Pallavicini Funding Coss 29 May / 86

6 Securiies, Derivaives and Trading Sraegies Talk Ouline 1 Securiies, Derivaives and Trading Sraegies Marke Securiies Self-Financing Trading Sraegies Funding and Discouning 2 Arbirage-Free Pricing 3 Funding Coss A. Pallavicini Funding Coss 29 May / 86

7 Securiies, Derivaives and Trading Sraegies Marke Securiies Marke Securiies I We sar wih he simple seing of a marke wih defaul-free securiies. We laer add counerpary credi risk, funding coss and collaeralizaion. We assume ha he marke quoes he prices of some securiies we name {S 1,..., S n }. When holding a securiy we may face he possibiliy o receive or pay a quaniy of cash. The owner of a bond receives coupons on a regular basis. Share holders receive dividends over ime. Many bilaeral conracs consiss in a srip of random cash flows. A. Pallavicini Funding Coss 29 May / 86

8 Securiies, Derivaives and Trading Sraegies Marke Securiies Marke Securiies II We name {γ T1,..., γ TN } he coupons, dividends or cash flows received or paid while holding a securiy We define he cumulaive dividend process as D := N γ Ti 1 {Ti } i=1 The profis and losses achieved holding a securiy are described by he gain process, which is defined as G := S + D A. Pallavicini Funding Coss 29 May / 86

9 Securiies, Derivaives and Trading Sraegies Marke Securiies Trading Porfolios and Toal Wealh I A rading sraegy in he marke securiies consiss in holding a porfolio of securiies. We name {q 1,..., q n } he quaniies of each securiy held in he porfolio. A each ime he rader may change he composiion of he porfolio. The quaniies q i may be eiher posiive or negaive. The oal wealh realized by he sraegy can be compued by aking ino accoun he profi and losses along ime. If we can rade only on imes { 0 = 0, 1,..., m = }, we can wrie he oal wealh as W := n m q i 0 S i 0 + i=1 k=1 i=1 n q i k 1 (G i k G i k 1 ) A. Pallavicini Funding Coss 29 May / 86

10 Securiies, Derivaives and Trading Sraegies Marke Securiies Trading Porfolios and Toal Wealh II We can subsiue he definiion of gain process in he oal wealh formula o highligh how he dividends conribue o i. W = n m q i 0 S i 0 + i=1 k=1 i=1 n q i k 1 (S i k S i k 1 + ) N γ Ti 1 {k 1 <T i k } A simple example of rading sraegy is enering a posiion and never changing i, namely q does no depend on ime. In his case we should obain ha he oal wealh is simply he sum of he gain processes of each securiy imes heir quaniies. i=1 A. Pallavicini Funding Coss 29 May / 86

11 Securiies, Derivaives and Trading Sraegies Marke Securiies Trading Porfolios and Toal Wealh III The wealh of a consan-quaniy rading sraegy. n W = q i S i 0 + = = = i=1 n q i S i 0 + i=1 n i=1 q i n q i ( S i ) m + D m i=1 n q i G i i=1 m k=1 ( S i k S i k 1 + n q (S i i m S i 0 + i=1 i=1 ) N γ Ti 1 {k 1 <T i k } i=1 ) N γ Ti 1 {Ti m} A. Pallavicini Funding Coss 29 May / 86

12 Securiies, Derivaives and Trading Sraegies Self-Financing Trading Sraegies Self-Financing Trading Sraegies I An ineresing class of rading sraegies is given by he self-financing sraegies. The wealh process of a self financing sraegy is always equal o he liquidaion value of he porfolio.. n W = qs i i Which is he consequence of such consrain on he quaniies q? i=1 A. Pallavicini Funding Coss 29 May / 86

13 Securiies, Derivaives and Trading Sraegies Self-Financing Trading Sraegies Self-Financing Trading Sraegies II We focus on he incremen in he wealh process over ime. W k W k 1 = n q i k 1 (S i k S i k 1 + D i k D i k 1 ) i=1 If we require ha he sraegy is self-financing, we ge W k W k 1 = n (q i k S i k q i k 1 S i k 1 ) i=1 If we equae he wo expressions, we obain n q i k S i k = i=1 n i=1 q i k 1 ( S i k + D i k D i k 1 ) A. Pallavicini Funding Coss 29 May / 86

14 Securiies, Derivaives and Trading Sraegies Self-Financing Trading Sraegies Self-Financing Trading Sraegies III Thus, he quaniies are seleced so ha dividends are re-invesed in he sraegy; furher cash is no required and no cash ouflow is generaed. In his sense he sraegy is self-financing. Some examples are: A sraegy in shares of a company. Every ime a dividend is paid he rader mus buy more shares. This sraegy is self-financing. A sraegy in a zero-coupon bond. A mauriy he zero-coupon bond pays he noional, bu we canno re-inves in i since he conrac is erminaed. We need a second securiy o build a self-financing sraegy. A. Pallavicini Funding Coss 29 May / 86

15 Securiies, Derivaives and Trading Sraegies Self-Financing Trading Sraegies Trading Sraegies in Coninuous Time In he following we use a coninuous-ime noaion, and we express he cumulaive dividend process as D := D 0 + dπ u, π := 0 N γ Ti 1 {Ti } while he wealh process for he rading sraegy q is given by W := q 0 S 0 + q u dg u 0 where he inernal producs is in securiy space. If he sraegy is self-financing we wrie. W = q S i=1 A. Pallavicini Funding Coss 29 May / 86

16 Securiies, Derivaives and Trading Sraegies Funding and Discouning Treasury Bank Accoun I Implemen a rading sraegies we need o access some cash-paying (and cash-receiving) securiies o fund (and o inves) dividends. For insance, if we have o pay a a fuure ime T a uni of cash, we can buy a zero-coupon bond paying such cash a T. Since rading sraegies have heir own rading horizons, we wish o access cash-paying (and cash-receiving) securiies wihou a mauriy ime. In pracice we need a bank accoun. We can ener ino a bank accoun by paying one uni of cash a incepion, and receiving i back a any laer ime along wih a compensaion. On he oher hand, we can also ge one uni of cash a incepion o pay i back a a laer ime along wih a fee. Do bank accouns exis in he marke? A. Pallavicini Funding Coss 29 May / 86

17 Securiies, Derivaives and Trading Sraegies Funding and Discouning Treasury Bank Accoun II On he marke we have saving accouns, bu heir are inended for reail operaions. Traders may access a special bank accoun, named he Treasury Bank Accoun (TBA), which is managed by he bank reasury deparmen. The TBA is no a real securiy raded on he marke, bu i behaves as a securiy from he poin of view of raders. The TBA is implemened by he reasury by issuing bonds, using collaeral porfolios, accessing saving accouns, ec... The compensaion rae, received when borrowing cash, and he fees, required when lending cash, are decided by he reasury. A. Pallavicini Funding Coss 29 May / 86

18 Securiies, Derivaives and Trading Sraegies Funding and Discouning Treasury Bank Accoun III If we assume ha he lending and borrowing raes are he same, name hem r, we can calculae he price process B of he TBA as he soluion of db = r B d, B 0 = 1 namely { } B = exp du r u 0 In he following we assume ha he TBA is one of he securiy used o implemen rading sraegies. We discuss again his assumpion when funding coss are inroduced. A. Pallavicini Funding Coss 29 May / 86

19 Securiies, Derivaives and Trading Sraegies Funding and Discouning Price Deflaors When we say ha he price process of a securiy is given by S we are hinking of liquidaing he securiy o obain an amoun of cash equal o S. Cash behaves as a uni of measure for prices. Ye, we canno access cash wihou paying fees or receiving compensaions, since we lend and borrow cash by means of he TBA. Thus, o ake ino accoun he cos of money, we need o express he wealh processes in erm of he TBA, namely where W is he deflaed wealh. W := W B How can we define deflaed price and cumulaive dividend processes? A. Pallavicini Funding Coss 29 May / 86

20 Securiies, Derivaives and Trading Sraegies Funding and Discouning Invariance of Self-Financing Trading Sraegies I We require ha he propery of a rading sraegy of being self-financing is invarian under deflaion. We define he deflaed price and cumulaive dividend processes o ensure his propery.. If q is a self-financing sraegy (W = q S ) we can wrie W = W B = q S where we define he deflaed price process as S := S B The definiion of he deflaed cumulaive dividend process is less obvious, since we mus consider ha dividends are paid over ime, and he TBA value depends on ime oo. A. Pallavicini Funding Coss 29 May / 86

21 Securiies, Derivaives and Trading Sraegies Funding and Discouning Invariance of Self-Financing Trading Sraegies II Saring from he definiion of deflaed wealh, we can wrie ( ) W = W dwu 0 + W u r u B u du 0 B u ( ) dgu = q 0 S 0 + q u S u r u B u du 0 B u ( dsu = q 0 S 0 + q u S u r u B u du + dd ) u B u B u = q 0 S q u dḡu where we define he deflaed cumulaive dividend and gain processes dd u D := D 0 +, Ḡ := 0 B S + D u A. Pallavicini Funding Coss 29 May / 86

22 Securiies, Derivaives and Trading Sraegies Funding and Discouning Invariance of Self-Financing Trading Sraegies III If he bank accoun is risky, as in a foreign-currency accoun, he definiion of he deflaed processes mus ake ino accoun he covariaion of he dividend process wih he deflaor. For a generic posiive process Y (deflaor) we can follow Duffie (2001) o wrie: W Y = q 0 S0 Y + 0 q u dgu Y, G Y := S Y + D Y where we define he deflaed price and cumulaive dividend processes S Y := Y S, D Y := Y 0 D 0 + (Y u dd u + d Y, D u ) 0 A. Pallavicini Funding Coss 29 May / 86

23 Arbirage-Free Pricing Talk Ouline 1 Securiies, Derivaives and Trading Sraegies 2 Arbirage-Free Pricing Pricing Formuale and Derivaive Replicaion Counerpary Credi Risk Margining Procedures 3 Funding Coss A. Pallavicini Funding Coss 29 May / 86

24 Arbirage-Free Pricing Pricing Formuale and Derivaive Replicaion Arbirages I In efficien markes securiies are always raded a heir fair value. Invesors can possibly obain higher reurns only by purchasing riskier invesmens. The possibiliy o make money from nohing wihou risks should be excluded from he se of possible rading sraegies. We name arbirages such sraegies. A more formal definiion of arbirage is needed o going on. We refer again o Duffie (2001) for he huge lieraure on arbirages and heir relaionship wih maringale pricing. A. Pallavicini Funding Coss 29 May / 86

25 Arbirage-Free Pricing Pricing Formuale and Derivaive Replicaion Arbirages II We inroduce a probabiliy space (Ω, F, P) endowed wih he sandard filraion F := (F ) 0 generaed by he securiy price processes, and he physical probabiliy measure P represening he acual disribuion of supply-and-demand shocks on securiy prices. We can define arbirages as a self-financing rading sraegy q whose wealh a incepion ime is non-posiive, namely W 0 while a mauriy T i is never negaive, and i is sricly posiive in some sae, so ha we can wrie W T 0, P { W T > 0 } > 0 To avoid arbirages we can impose some condiions on he wealh process W. A. Pallavicini Funding Coss 29 May / 86

26 Arbirage-Free Pricing Pricing Formuale and Derivaive Replicaion Equivalen Maringale Pricing I Given he TBA as price deflaor, we can ensure he absence of arbirages, if we can find a measure Q, equivalen o he physical measure P, such ha he deflaed gain process Ḡ is a maringale under such measure. The measure Q is known as risk-neural measure. Arbirages are forbidden even if we use a generic deflaor Y. In his case he measure Q Y depends on he choice of he deflaor, and i is known as equivalen maringale measure. The reverse is no rue in general. A. Pallavicini Funding Coss 29 May / 86

27 Arbirage-Free Pricing Pricing Formuale and Derivaive Replicaion Equivalen Maringale Pricing II Under suiable echnical condiions on he rading sraegy q, he maringale condiion allows us o wrie E [ ] T W T F = W + E [ ] q u dḡ u F = W where he expecaions are aken under he risk-neural measure. If q is an arbirage, we have W 0 and W T 0 = W T 0 = W = E [ WT F ] 0 on he oher hand, he equivalence beween he measures implies P { W T > 0 } > 0 = Q { W T > 0 } > 0 = Q { WT > 0 } > 0 leading o W > 0 which conradics he hypohesis. A. Pallavicini Funding Coss 29 May / 86

28 Arbirage-Free Pricing Pricing Formuale and Derivaive Replicaion Equivalen Maringale Pricing III If we assume he exisence of a risk-neural measure, we can price marke securiies wih mauriy dae T by exploiing he maringale condiion of deflaed gain processes. Ḡ = E [ Ḡ T F ] Then, we can expand he gain process o obain he arbirage-free pricing formula under Q-expecaion [ ] S T T dd u S = B E + F B T B u or for a generic deflaor Y under Q Y -expecaion [ S = 1 ] T E Y Y T S T + (Y u dd u + d Y, D u ) F Y A. Pallavicini Funding Coss 29 May / 86

29 Arbirage-Free Pricing Pricing Formuale and Derivaive Replicaion Replicaion of Derivaive Conracs I We can exend pricing formulae o derivaive securiies no raded on he marke. We consider a derivaive wih price process V and cumulaive dividend process Q. In order o replicae he derivaive in erms of marke securiies, we can implemen a sraegy q o inves (or o fund) he dividend s received (or paid) by he derivaive, namely Q. = W q S Furhermore, we require ha a mauriy he price of he consiuens of he sraegy is equal o he price of he derivaive. V T. = qt S T A. Pallavicini Funding Coss 29 May / 86

30 Arbirage-Free Pricing Pricing Formuale and Derivaive Replicaion Replicaion of Derivaive Conracs II The derivaive price can be calculaed a any ime from he marke securiy prices. We consider a rading sraegy q which invess in he marke securiies as he sraegy q and shors one uni of he derivaive, namely q := (q, 1) The wealh generaed by such sraegy is given by W = q 0 S 0 V 0 + (q u dg u dv u dq u ) = q S V 0 so ha we can conclude ha he sraegy q is self-financing wih null final wealh, W T = 0. A. Pallavicini Funding Coss 29 May / 86

31 Arbirage-Free Pricing Pricing Formuale and Derivaive Replicaion Replicaion of Derivaive Conracs III If we require absence of arbirages, we obain ha a any ime < T we mus have W T 0 = W 0 = q S V On he oher hand, we can consider he sraegy ( q, 1) leading o q S V Thus, we have a any ime up o mauiry T ha V = q S We can wrie ha he derivaive gain process is equal o he wealh generaed by he replicaing sraegy q. W = V + Q A. Pallavicini Funding Coss 29 May / 86

32 Arbirage-Free Pricing Pricing Formuale and Derivaive Replicaion Replicaion of Derivaive Conracs IV If we assume he exisence of a risk-neural measure for he marke securiies, we have ha he deflaed gain process of he derivaive is a maringale oo, leading o he pricing equaion V = B E [ V T T dq u + B T B u F or for a generic deflaor Y under Q Y -expecaion [ V = 1 ] T E Y Y T V T + (Y u dq u + d Y, Q u ) F Y which can be solved once a erminal condiion for V T is seleced. ] A. Pallavicini Funding Coss 29 May / 86

33 Arbirage-Free Pricing Counerpary Credi Risk Marke and Enlarged Filraions I The nex elemen we add o he pricing framework is he possibiliy of defaul of one of he counerparies of he conrac. How can we deal wih he defaul even under he risk-neural measure? We need o describe he filraion o adop o calculae he risk-neural expecaions. Marke risks for conracs wih defaulable counerparies arise from he uncerainy boh in defaul probabiliies and in he defaul imes. We could add risks specific of he underlying asse and recoveries as well. As a firs sep we inroduce he marke filraion F represening all he observable marke quaniies bu he defaul evens. A. Pallavicini Funding Coss 29 May / 86

34 Arbirage-Free Pricing Counerpary Credi Risk Marke and Enlarged Filraions II Then, we define he defaul evens of he counerpary τ C and of he invesor τ I along wih he firs defaul ime τ := τ C τ I We define he enlarged filraion G conaining also he defaul monioring. See Bielecki and Rukowski (2001) for deails. G := F H C H I F H k := σ({τ k u} : u ), k {C, I} A. Pallavicini Funding Coss 29 May / 86

35 Arbirage-Free Pricing Counerpary Credi Risk Marke and Enlarged Filraions III From he definiion of G, we can wrie g G f F : g {τ C > } {τ I > } = f {τ C > } {τ I > } or simply g G f F : g {τ > } = f {τ > } Thus, for any G-adaped process x we can inroduce he pre-defaul F-adaped process x such ha 1 {τ>} x = 1 {τ>} x We use his propery for numerical implemenaions o express expecaions under he enlarged G filraion as expecaions under he marke F filraion. A. Pallavicini Funding Coss 29 May / 86

36 Arbirage-Free Pricing Counerpary Credi Risk Trading Sraegies wih Defaulable Counerparies I The counerpary credi risk is defined as he risk ha he counerpary o a ransacion could defaul before he final selemen of he ransacion cash flows. When one of he counerpary defauls he rade is erminaed. An economic loss would occur if he ransacion wih he counerpary has a posiive economic value a he ime of defaul. We can accommodae counerpary risk by erminaing he dividend process a he firs defaul even, and seing he erminal condiion for he securiy price accordingly. S T τ := 1 {τ T } θ τ, D := D {τ>u} dπ u where θ τ is he cash flow paid if he defaul occurs, and wihou loss of generaliy we se 1 {τ>t } S T. = 0. A. Pallavicini Funding Coss 29 May / 86

37 Arbirage-Free Pricing Counerpary Credi Risk Trading Sraegies wih Defaulable Counerparies II To avoid arbirages we require ha he deflaed gain processes are maringale under he G filraion. The pricing equaion becomes [ ] S = B E 1 {τ T } θ τ B τ + T 1 {τ>u} dd u B u A similar expression holds for generic deflaors Y. Since credi defaul risk inroduces an elemen of non-predicabiliy, we canno implemen a replicaion sraegy o price derivaive securiies, bu in simple cases. However, we can price hem as any oher marke securiy. G A. Pallavicini Funding Coss 29 May / 86

38 Arbirage-Free Pricing Counerpary Credi Risk Close-Ou Neing Rules I In case of defaul of one pary, he surviving pary should evaluae he ransacions jus erminaed, due o he defaul even occurrence, o claim for a reimbursemen afer he applicaion of neing rules o consolidae he ransacions. The amoun of he cash flow θ τ resuls from such analysis. The cash flow θ τ is described by he ISDA documenaion as given by ( θ τ := 1 {τc <τ I } RC ε + τ + ε ) ( τ + 1{τI <τ C } ε + τ + R I ε ) τ = ε τ 1 {τc <τ I }(1 R C )ε + τ + 1 {τi <τ C }(1 R I )ε τ where R C and R I are he recovery raes, and ε τ is he close-ou amoun represening he exposure measured by he surviving pary on he defaul even. A. Pallavicini Funding Coss 29 May / 86

39 Arbirage-Free Pricing Counerpary Credi Risk Close-Ou Neing Rules II I is difficul o define he close-ou amoun, and also ISDA is no very asserive on he opic. See Brigo, Morini and Pallavicini (2013) for a review. You may have a risk free close-ou, where he residual deal is priced a mid marke wihou any residual counerpary risk. ε τ. = Bτ T τ T [ dπu E B u You may have a replacemen close-ou, where he remaining deal is priced by aking ino accoun he credi qualiy of he surviving pary and of he pary ha replaces he defauled one. A possible guess for he pre-defaul close-ou is given by G τ ] ε τ. = Sτ A. Pallavicini Funding Coss 29 May / 86

40 Arbirage-Free Pricing Counerpary Credi Risk Close-Ou Neing Rules III The replacing pre-defaul close-ou is he firs example of non-lineariies in he pricing equaion. Indeed, if we wrie he pre-defaul price we ge 1 {τ>} S = 1 {τ>} B E [ 1 {τ T } θ τ ( S τ ) B τ + T 1 {τ>u} dd u B u G ] The above expression is an implici equaion for he he pre-defaul price of he securiy, which could be wihou soluions. In he following, when we inroduce collaeralizaion and funding coss, we discuss again such problem. A. Pallavicini Funding Coss 29 May / 86

41 Arbirage-Free Pricing Margining Procedures Collaeralizaion and Counerpary Credi Risk The growing aenion on counerpary credi risk is ransforming OTC derivaives money markes: an increasing number of derivaive conracs is cleared by CCPs, while mos of he remaining conracs are raded under collaeralizaion. Boh cleared and bilaeral deals require collaeral posing, along wih is remuneraion. Collaeralized bilaeral rades are regulaed by ISDA documenaion, known as Credi Suppor Annex (CSA). Cenralized clearing is regulaed by he conracual rules described by each CCP documenaion. See Brigo e al. (2012) and Brigo and Pallavicini (2014) for a descripion of bilaeral-raded and cenrally-cleared conracs. A. Pallavicini Funding Coss 29 May / 86

42 Arbirage-Free Pricing Margining Procedures Trading Sraegies wih Margining Procedures I We can include he margining procedure wihin arbirage-free pricing by exending he definiion of he gain and he cumulaive dividend process. In general, a margining pracice consiss in a pre-fixed se of daes during he life of a deal when boh paries pos or wihdraw collaerals, according o heir curren exposure, o or from an accoun held by he Collaeral Taker. We consider ha a posiive collaeral accoun C is held by he invesor, oherwise by he counerpary. Moreover, as we se a null erminal condiion for he securiy price, we se C T. = 0. The Collaeral Taker remuneraes he accoun a rae c fixed by he collaeralizaion agreemen. The collaeral rae may depend on he sign of he collaeral accoun. A. Pallavicini Funding Coss 29 May / 86

43 Arbirage-Free Pricing Margining Procedures Trading Sraegies wih Margining Procedures II Thus, he cumulaive dividend process can be exended in he following way D := D {τ>u} (dπ u + dc u c u C u du) Noice ha including he collaeral accoun in he cumulaive dividend process means ha we can re-hypohecae is conen. Moreover, a rade erminaion we have o wihdraw collaeral asses kep in our accouns, so ha he gain process can be re-defined as G := S + D C A. Pallavicini Funding Coss 29 May / 86

44 Arbirage-Free Pricing Margining Procedures Trading Sraegies wih Margining Procedures III To avoid arbirages we require ha he deflaed gain processes are maringale under he G filraion. Thus, we ge Ḡ = E [ [ ] ] T Ḡ T τ G = S = C + E S T τ C T τ + 1 {τ>u} d D u G The inegral over deflaed dividends can be wrien as T 1 {τ>u} d D u = T 1 {τ>u} ( dπu B u = C T τ B T τ C τ B τ + + dc u B u T c ) uc u du B u 1 {τ>u} ( dπu B u + (r ) u c u )C u du B u A. Pallavicini Funding Coss 29 May / 86

45 Arbirage-Free Pricing Margining Procedures Trading Sraegies wih Margining Procedures IV If we subsiue he expression for he dividend inegral, we ge he pricing equaion [ θ T ( τ dπu 1 {τ>} S = 1 {τ>} B E 1 {τ T } + 1 {τ>u} + (r ) ] u c u )C u du G B τ B u B u According o ISDA he definiion of he on-defaul cash flow in presence of collaeralizaion and re-hypohecaion is given by θ τ := ε τ 1 {τc <τ I }(1 R C )(ε τ C τ ) + 1 {τi <τ C }(1 R I )(ε τ C τ ) Anoher source of non-lineariies occurs if he collaeral accoun is proporional o he pre-defaul price of he derivaive. C. = α S where α is a F-adaped process. A. Pallavicini Funding Coss 29 May / 86

46 Funding Coss Talk Ouline 1 Securiies, Derivaives and Trading Sraegies 2 Arbirage-Free Pricing 3 Funding Coss Defaulable Bank Accouns Funding Policies and Neing Ses Addiive Price Adjusmens A. Pallavicini Funding Coss 29 May / 86

47 Funding Coss Defaulable Bank Accouns How o Consruc a Bank Accoun When we derive he pricing equaions, we assume he availabiliy of a reasury bank accoun. Now, we analyse how i is implemened by he reasury, and if counerpary risk may change his consrucion. Bank accouns are used by raders boh for cash lending and borrowing. Trading sraegies o borrow and o lend cash are differenly implemened, leading o differen bank accouns. See Bergman (1995), Crépey (2011), Pallavicini, Perini and Brigo (2011). We consider he following sylized procedure up o ime. Lending: a rading desk has a surplus of cash o be invesed a ime 0, a ime he desk ges he cash back wih a premium. Borrowing: a rading desk needs cash a ime 0, a ime he desk gives he cash back wih a fee. A. Pallavicini Funding Coss 29 May / 86

48 Funding Coss Defaulable Bank Accouns Lending Bank Accoun I We sar by discussing he lending case. In paricular, we assume ha a bank I invess cash in zero-coupon bonds of a counerpary C. Along wih he posiion in bonds he bank shall buy proecion for losses due o he defaul of he counerpary. The bank can buy a Credi Defaul Swap (CDS) for each bond in he sraegy. A CDS conrac proecs he bond owner from losses occurring on defaul ime by paying a fee s l. If he counerpary defauls he CDS covers all losses, and he bank may open a new posiion wih anoher counerpary. The sraegy can be implemened up o ime or up o he defaul of he bank. In paricular, we assume o roll he posiions on a ime grid { 0 = 0, 1,..., m = } A. Pallavicini Funding Coss 29 May / 86

49 Funding Coss Defaulable Bank Accouns Lending Bank Accoun II A ime 0 he bank buys a zero-coupon bond of he counerpary wih mauriy 1 and noional q l 0 := 1 P l 0 ( 1 ) where P l 0 ( 1 ) is he bond marke price, so ha we have a cash flow of 1 {τ>0}γ buy 0 := 1 {τ>0}q l 0 P l 0 ( 1 ) A he same ime he bank eners a par ino a CDS conrac wih mauriy 1 on he same bond. A. Pallavicini Funding Coss 29 May / 86

50 Funding Coss Defaulable Bank Accouns Lending Bank Accoun III A ime 1 he noional of he bond is reurned o he bank and he CDS fee is paid, if neiher he bank nor he counerpary has defauled beween 0 and 1. 1 {τ>1}γ receive 1 := 1 {τ>1}q l 0, 1 {τ>1}γ fee 1 := 1 {τ>1}q l 0 s l ( 1 0 ) If a defaul happens, and he defauling pary is he counerpary, he CDS covers all losses, and on he nex ime-sep he posiion is opened wih anoher counerpary. If he bank survives, all conracs are opened again wih noional q l 1 := ql 0 (1 s l ( 1 0 )) P l 1 ( 2 ) so o build a self-financing sraegy, namely γ receive 1 + γ fee 1 + γ buy 1 = 0 A. Pallavicini Funding Coss 29 May / 86

51 Funding Coss Defaulable Bank Accouns Lending Bank Accoun IV Thus, we can sum all he conribuions up o ime, or up o he defaul of he bank, o define he wealh generaed by he invesing sraegy. m 1 W τ l I := {τi > k }γ buy k + = m k=1 k=0 m k=1 1 s l 1 k ( k k 1 ) {τi > k } P l k 1 ( k ) ( 1 {τi > k } γ receive k + γ fee ) k We can wrie he wealh of he sraegy in coninuous ime as { τi W τ l I = exp du ( yu l su) } l, y l := T log P(T l ) T = 0 where y l is he marke yield of he bond issued by he counerpary. A. Pallavicini Funding Coss 29 May / 86

52 Funding Coss Defaulable Bank Accouns Lending Bank Accoun V Up o he defaul of he bank (included) he wealh process is a locally risk-free bank accoun, independenly of he counerpary issuing he bonds. All hese accouns are derived securiies, so ha, o avoid arbirages, all he bond/cds bases mus be equal o he same rae r. r := y l s l In he pracice many facors, like bond and CDS marke liquidiy, CDS collaeralizaion and gap risk, defaul even specificaion, ec..., preven o exrac r from bond and CDS quoes. For laer convenience, we cas he bond/cds basis as a spread l l over he overnigh rae e, and we wrie l l := y l s l e A. Pallavicini Funding Coss 29 May / 86

53 Funding Coss Defaulable Bank Accouns Borrowing Bank Accoun I We coninue he discussion wih he borrowing case. In paricular, we assume ha a bank I obains cash by issuing zero-coupon bonds. Noice ha he bank canno buy proecion on herself o hedge is own defaul even. A ime 0 he bank issues a zero-coupon bond wih mauriy 1 and noional q b 1 0 := P b 0 ( 1 ) where P b 0 ( 1 ) is he bond marke price, so ha we have a cash flow of 1 {τi > 0}γ issue 0 := 1 {τi > 0}q b 0 P b 0 ( 1 ) A. Pallavicini Funding Coss 29 May / 86

54 Funding Coss Defaulable Bank Accouns Borrowing Bank Accoun II If he bank defauls, he sraegy is erminaed and he bond owner recovers only a fracion R I of he noional. 1 {0<τ I 1}γ recovery τ I := 1 {0<τ I 1}R I q b 0 If he bank survives, a ime 1 he noional of he bond is reurned o he counerpary. 1 {τi > 1}γ pay 1 := 1 {τi > 1}q b 0 and all conracs are opened again wih noional q b 1 := qb 0 P b 1 ( 2 ) so o build a self-financing sraegy (bu on bank defaul even), namely + γ issue 1 = 0 γ pay 1 A. Pallavicini Funding Coss 29 May / 86

55 Funding Coss Defaulable Bank Accouns Borrowing Bank Accoun III Thus, we can sum all he conribuions up o ime, or up o he defaul of he bank, o define he wealh generaed by he funding sraegy. m 1 W τ b I := 1 + = m k=1 k=0 1 {τi > k } ( 1 {τi > k } γ issue k m 1 1 P b k 1 ( k ) R I + 1 {τi k+1 }γ recovery τ I 1 {k <τ I k+1 } k=0 j=1 ) + m k k=1 1 {τi > k }γ pay k 1 P b j 1 ( j ) We can wrie he wealh of he sraegy in coninuous ime as { } W τ b I = W τ b I 1 {=τi }(1 R I ) W τ b b I, W := exp du yu b 0 where y b is he marke yield of he bond issued by he bank. A. Pallavicini Funding Coss 29 May / 86

56 Funding Coss Defaulable Bank Accouns Borrowing Bank Accoun IV Only up o he defaul of he bank (excluded) he wealh process is a locally risk-free bank accoun. On bank defaul he posiion is erminaed wih an addiional cash flow, given by { τi } (1 R I ) exp du yu b 0 We noice ha, in case of defaul of he bank, we have a funding benefi since only a par of he reimbursemen will be fulfilled. See Crépey (2011) and Pallavicini, Perini and Brigo (2011). For laer convenience, we express he yield of bank bonds as a spread l b over he overnigh rae e, and we wrie l b := y b s b e where s b is he CDS spread of he bank. A. Pallavicini Funding Coss 29 May / 86

57 Funding Coss Defaulable Bank Accouns Trading Sraegies wih Funding Coss I Lending and borrowing sraegies are used by he reasury o assis rading aciviies. We can focus on a paricular rading sraegy in marke or derived securiies which is funded by he reasury on a neing base (funding neing se). The assignmen of a securiy o a paricular neing se is decided by he reasury. A possible choice is a neing se including all he rades of he bank. We assume ha conracs of he same counerpary are no spli among differen neing ses. See Pallavicini, Perini and Brigo (2011), Albanese and Andersen (2015). Now, we ry o esablish a pricing formula for he whole neing se. This exercise requires o re-define he TBA o ake ino accoun he possibiliy of defaul of he bank. A. Pallavicini Funding Coss 29 May / 86

58 Funding Coss Defaulable Bank Accouns Trading Sraegies wih Funding Coss II We can accommodae he funding benefi by seing he erminal condiion on bank defaul as given by S T τ := 1 {τ T } θ τ + 1 {τ=τi <T }(q f τ I ) + (1 R I )B f τ I where q f is he quaniy of cash allocaed by he reasury o fund he securiy S wihin he neing se, and he TBA is defined as B f := 1 {q f >0}B b + 1 {q f 0}B l where he lending and borrowing bank accouns are defined as { B l := W l = exp du ( yu l su l ) } { }, B b := W b = exp du yu b 0 We define also a TBA rae as given by f := 1 {q f >0}yu b ( + 1 {q f 0} y l u su l ) 0 A. Pallavicini Funding Coss 29 May / 86

59 Funding Coss Defaulable Bank Accouns Trading Sraegies wih Funding Coss III To avoid arbirages we require ha he gain processes, deflaed by he TBA, are maringales under he G filraion. The equivalen maringale measure depends on he neing se, so ha we are removing only he arbirages wihin he neing se. See Bielecki and Rukowski (2014) for a discussion of arbirages in non-linear pricing. The pricing equaion becomes ] + 1 {<τ=τi T }(qτ f I ) + (1 R I ) G 1 {τ>} S = B f E f [ 1 {<τ T } θ τ B f τ [ T ( + B f E f dπu 1 {τ>u} B f u + (f ) ] u c u )C u du Bu f G where hw f over he expecaion symbols is reminder of he dependency of he measure on he funding sraegy. A. Pallavicini Funding Coss 29 May / 86

60 Funding Coss Funding Policies and Neing Ses Funding Policies I A firs possibiliy o fix he unknown value of (q f τ I ) + is pricing he whole neing se by replicaing i wih he conained securiies. This price-and-hedge problem requires o simulaneously solve hree equaions: he definiion of he neing se, he self-financing condiion, and he erminal condiion inclusive of he funding benefi. The soluion consiss boh in he value of he neing se and in he sraegy in cash and securiies used o hedge i. See Crépey (2011), Bielecki and Rukowski (2014). The exisence of a soluion in a general seing is difficul o prove, since he erminal condiion is no predicable (gap risk). Conagion effecs, or a delay in he defaul procedure, canno be hedged in he pracice. The marke securiies may jump if sensiive o credi risk. A. Pallavicini Funding Coss 29 May / 86

61 Funding Coss Funding Policies and Neing Ses Funding Policies II Alernaively, we could implemen a parial hedging, and we could size he cash amoun by some opimal argumen. See Crépey (2011), Burgard and Kjaer (2011). Here, we assume a diffusive seing for underlying risk facors and predicable erminal condiions for price processes. Under hese assumpions, we know ha, in case of a complee marke, he hedging sraegy is given by dela hedging. See Crépey (2011), Pallavicini, Perini and Brigo (2011). Thus, if we can consider a neing se formed by one derivaive securiy V along wih is dela-hedging asses S, we ge ha he quaniy of cash needed o implemen he hedging sraegy is given by F := V C V (S C ) S V, q f. = F + ɛ B f where C and C V are he collaeral accouns of marke and derived securiies, and ɛ is he hedging error. A. Pallavicini Funding Coss 29 May / 86

62 Funding Coss Funding Policies and Neing Ses Funding Policies III In he same way we can consider neing ses formed by many derived securiies. In his case, we can ne all he funding requiremens. F i := V i C V (S C ) S V i, q f. = 1 B f n (F i + ɛ i ) i=1 Noice ha his choice effecively reduces funding requiremens, since we have ( n + n (F i + ɛ)) i ( F i + ɛ i + ) i=1 In he following we focus on he case of a single derived securiy. In he las secion we discuss again his problem. i=1 A. Pallavicini Funding Coss 29 May / 86

63 Funding Coss Funding Policies and Neing Ses Pricing Formulae wih Funding Coss I In he case of a neing se formed by a single derived securiy, and under he assumpion of a diffusive seing wihou gap risk, we can wrie he price equaion as given by [ ] T 1 {τ>} Ṽ = B f E f + + (f u c u )Cu V du G 1 {<τ T } θ V τ B f τ 1 {τ>u} dπ V u B f u + B f E f [ 1 {<τ=τi T }(1 R I ) (F τ I + ɛ τi ) + I is useful o make explici he dependency of he TBA process and rae on he cash used o implemen he hedging sraegy. B f τ I G ] B f u A. Pallavicini Funding Coss 29 May / 86

64 Funding Coss Funding Policies and Neing Ses Pricing Formulae wih Funding Coss II We can apply he Feynman-Kac heorem o wrie flows deflaed w.r.. a bank accoun B e accruing a he overnigh rae e. 1 {τ>} Ṽ = B e + B e T T B e T [ E e dπu V + (e u c u )Cu V du θu V 1 {τ>u} Bu e + 1 {τ du} Bu e ] G E e [ 1 {τ>u} (f u e u ) F u B e u E e [ 1 {τ=τi du}(1 R I ) (F u + ɛ u ) + B e u G ] where he pricing measure is such ha he marke securiies can be priced as 1 {τ>} S = B e T [ E e dπ u + (e u c u )C u du θ u 1 {τ>u} Bu e + 1 {τ du} Bu e G ] G ] A. Pallavicini Funding Coss 29 May / 86

65 Funding Coss Funding Policies and Neing Ses Does Funding Coss Exiss? I We can check ha in a complee marke funding coss does no exis. See Pallavicini, Perini and Brigo (2011), Burgard and Kjaer (2011), Hull and Whie (2012), Albanese and Andersen (2015).. We swich o marke filraion, remove collaerals, and we pu ɛ = 0. T [ B 1 {τ>} Ṽ = 1 {τ>} E e e+λ ( dπ V Bu e+λ u + θu V λ u du ) ] F 1 {τ>} T 1 {τ>} T [ B du E e e+λ B e+λ u [ B du E e e+λ B e+λ u l l u (V u S u S V u ) F ] (l b u + s b u (1 R I )λ I u) (V u S u S V u ) + F ] where we have subsiued he TBA rae in erm of CDS spreads and liquidiy bases, while he defaul inensiies are defined as λ d := E [ 1 {τ } G ], λ I d := E [ 1 {τi } G ] A. Pallavicini Funding Coss 29 May / 86

66 Funding Coss Funding Policies and Neing Ses Does Funding Coss Exiss? II Then, we apply again he Feynman-Kac heorem o group all he adjusmens ino an effecive discoun rae. [ T 1 {τ>} Ṽ = 1 {τ>} E ζ B ζ+λ ( dπ V Bu ζ+λ u + θu V λ u du ) ] F where he effecive rae ζ is given by ζ := e + (l b + s b (1 R I )λ I )1 {V>S S V } + l l 1 {V<S S V } If he bank has he possibiliy o rade her own CDS, we have ha s b. = (1 R I )λ I leading o ζ = e + l b 1 {V>S S V } + l l 1 {V<S S V }. = r where he las sep holds assuming no CDS/bond basis. A. Pallavicini Funding Coss 29 May / 86

67 Funding Coss Addiive Price Adjusmens Fair Value Policies I Some cash flows in he pricing equaion happen afer he defaul of he invesor. These flows are erms in θ V and he funding benefis. The invesor can ignore such flows while rading wih oher counerparies, since hey maer only when he defaul procedure is in place afer he invesor defaul. We can spli accordingly he derivaive price as V := V 1 + V 2 where V 1 is he rading par and V 2 he reasury par of he derivaive price. Furhermore, we spli he on-defaul cash flow o isolae he par occurring on invesor s defaul. θ V τ := 1 {τc <τ I }θ V,C τ C + 1 {τi <τ C }θ V,I τ I A. Pallavicini Funding Coss 29 May / 86

68 Funding Coss Addiive Price Adjusmens Fair Value Policies II We sar wih an approximaion. We consider he funding adjusmens only for conracual cash flows, and we discard he hedging error, so ha we ge V + T T [ E e B e ] ( 1 {τ>u} dπ V Bu e u + (e u c u )Cu V du + 1 {τ du} θu V ) G [ E e B e ( 1 {τ>u} Bu e 1{τI du}(1 R I )(Fu 0 ) + (fu 0 e u )Fu 0 du ) ] G T [ ] B F 0 := V 0 C V (S C ) S V 0, V 0 := E e e Bu e dπ u f 0 := e + 1 {F 0 >0} ( s b + l b ) + 1{F 0 0} l l Noice ha if he marke is complee his approximaion is exac. A. Pallavicini Funding Coss 29 May / 86

69 Funding Coss Addiive Price Adjusmens Fair Value Policies III MM V 1 := V 0 CVA LVA FCA FBA + DVA V 2 := FDA + T T T T T T [ E e B e ] 1 {τ=τc du} Bu e (Vu 0 θu C ) G [ E e B e ] 1 {τ>u} Bu e (c u e u )Cu V du G [ E e B e ] ( 1 {τ>u} s b Bu e u + l b ) u (F 0 u ) + du G [ E e B e ] 1 {τ>u} Bu e l l ( Fu 0 ) + du G [ E e B e ] 1 {τ=τi du} Bu e (θu I Vu 0 ) G [ E e B e ] 1 {τ=τi du} Bu e (1 R I )(Fu 0 ) + G A. Pallavicini Funding Coss 29 May / 86

70 Funding Coss Addiive Price Adjusmens Fair Value Policies on Neing Ses I If we look a he whole neing se, we can apply he previous decomposiion o each conrac of he se, bu wih a cash amoun F 0 := n i=1 F 0,i := n i=1 V 0,i C V,i (S C ) S V 0,i Since he adjusmens are non-linear funcions of he cash amoun, we need a recipe o decompose he adjusmens. We can define he cash amoun o compue funding coss F b,0,i := ( n j=1 F 0,j ) + ( n j=1,j i F 0,j ) + and he cash amoun o compue funding benefis F l,0,i := ( n j=1 F 0,j ) ( n j=1,j i F 0,j ) A. Pallavicini Funding Coss 29 May / 86

71 Funding Coss Addiive Price Adjusmens Fair Value Policies on Neing Ses II Thanks o he definiions of he cash amouns o compue funding coss and benefis, we can add a new conrac ino he neing se wihou re-compuing he funding adjusmens of he oher ones. FCA i FBA i + FDA i + T T T [ E e B e ] ( 1 {τ>u} s b Bu e u + l b u) F b,0,i u du G [ E e B e ] 1 {τ>u} Bu e l l ( Fu l,0,i ) du G [ E e B e ] 1 {τ=τi du} Bu e (1 R I )Fu l,0,i G A. Pallavicini Funding Coss 29 May / 86

72 Appendix: Calculaion Tools Feynman-KAc Theorem Probabilisic Inerpreaion of Pricing Equaions I The Feynman-Kac Theorem Consider a vecor of Markov risk facors S wih infiniesimal generaor L µ := (µ S ) S Tr S, S 2 S and assume ha he derivaive price V solves he PDE ( + L µ ν ) V + π = 0, V T = 0 Hence, he soluion of he PDE is given by V = T E µ [ B ν B ν u dπ u ] where under he pricing measure Q µ he risk facors grow a rae µ. A. Pallavicini Funding Coss 29 May / 86

73 Appendix: Calculaion Tools Feynman-KAc Theorem Probabilisic Inerpreaion of Pricing Equaions II A useful applicaion of he heorem is changing he discoun facor by adding a sream of coupons. V = = = T E µ T E µ T E ρ [ B ν B ν u [ B ρ B ρ u [ B ρ B ρ u dπ u ] ] dπ u + (µ u ρ u )V u du ] dπ u + (µ u ρ u )V u du (ν u ρ u )S u S V u du where under he pricing measure Q ρ he risk facors grow a rae ρ. A. Pallavicini Funding Coss 29 May / 86

74 Appendix: Calculaion Tools Filraion Swiching Tools Pricing Cash Flows Occurring before he Defaul Even I For any G-adaped process φ, we can consider he G-adaped process x. = E [ 1{τ>T } φ T G ] If we observe x only before he defaul even, and we ake he expecaions of boh side under F filraion, we ge x E [ 1 {τ>} F ] = E [ 1{τ>} E [ 1 {τ>t } φ T G ] F ] = E [ 1{τ>T } φ T F ] On he oher hand, we have from he definiion of pre-defaul process 1 {τ>} x = 1 {τ>} E [ 1 {τ>t } φ T G ] leading o 1 {τ>} E [ 1 {τ>t } φ T G ] = 1{τ>} E [ 1 {τ>t } φ T F ] Q{ τ > F } A. Pallavicini Funding Coss 29 May / 86

75 Appendix: Calculaion Tools Filraion Swiching Tools Pricing Cash Flows Occurring before he Defaul Even II Firs Filraion Swiching Lemma In a marke wih defaulable names, where τ is he firs defaul even, we can price cash flows occurring before he firs defaul even by swiching o he marke filraion F. 1 {τ>} E [ 1 {τ>t } φ T G ] = 1{τ>} E [ Q{ τ > T F T } φ T F ] Q{ τ > F } where φ is a G-adaped process, and φx is he corresponding pre-defaul process. In paricular, we have also 1 {τ>} Q{ τ > T G } = 1 {τ>} Q{ τ > T F } Q{ τ > F } A. Pallavicini Funding Coss 29 May / 86

76 Appendix: Calculaion Tools Filraion Swiching Tools Pricing Cash Flows Occurring on he Defaul Even I A second useful lemma can be derived for cash flows paid only if a defaul occurs. For any G-adaped process φ we can proceed as before, bu, now, we consider he G-adaped process x. = E [ 1{τ<T } φ τ G ] leading o 1 {τ>} E [ 1 {τ<t } φ τ G ] = 1{τ>} E [ 1 {<τ<t } φ τ F ] Q{ τ > F } As before we wish o remove he explici dependency on he defaul even on he righ-hand side. A. Pallavicini Funding Coss 29 May / 86

77 Appendix: Calculaion Tools Filraion Swiching Tools Pricing Cash Flows Occurring on he Defaul Even II We go on by localizing he defaul even, and we ge 1 {τ>} E [ ] T 1 {<τ<t } φ τ F = 1{τ>} E [ ] 1 {τ du} φ u F To proceed furher we require ha φ is also predicable. We obain 1 {τ>} E [ ] T 1 {<τ<t } φ τ F = 1{τ>} du E [ ] 1 {τ>u} λ u φ u F where we define he firs-defaul inensiy as he densiy of he compensaor of 1 {τ<}, namely λ d := E [ 1 {τ d} G ] A. Pallavicini Funding Coss 29 May / 86

78 Appendix: Calculaion Tools Filraion Swiching Tools Pricing Cash Flows Occurring on he Defaul Even III Second Filraion Swiching Lemma In a marke wih defaulable names, where τ is he firs defaul even, we can price cash flows occurring on he firs defaul even by swiching o he marke filraion F. 1 {τ>} E [ ] T 1 {τ<t } φ T G = 1{τ>} du E[ Q{ τ > u F u } λ ] u φu F Q{ τ > F } where λ is he firs-defaul inensiy and φ is a G-predicable process, while λ and φ are he corresponding pre-defaul processes. A. Pallavicini Funding Coss 29 May / 86

79 Seleced References Seleced references I Amraoui, S., Hiier, S. (2008). Opimal Sochasic Recovery for Base Correlaion. Basel Commiee on Banking Supervision Inernaional Convergence of Capial Measuremen and Capial Sandards A Revised Framework Comprehensive Version" (2006), Srenghening he Resilience of he Banking Secor" (2009), Inernaional Framework for Liquidiy Risk Measuremen, Sandards and Monioring (2009), Basel III: a Global Regulaory Framework for More Resilien Banks and Banking Sysems (2010) and (2011), Liquidiy Transfer Pricing: a guide o beer pracice (2011) Available a Bank for Inernaional Selemens and Inernaional Organizaion of Securiies Commissions Second Consulaive Documen on Margin requiremens for non-cenrally cleared derivaives (2013). Bergman, Y.Z. (1995) Opion pricing wih differenial ineres raes Review of Financial Sudies 8 (2) Beumee, J., Brigo, D., Schiemer, D., Soyle, G. Charing a Course hrough he CDS Big Bang Available a ssrn.com. Bielecki, T. R, Cialenco, I., Iyigunler I. (2011) Counerpary Risk and he Impac of Collaeralizaion in CDS Conracs Available a Bielecki, T.R. and Rukowski, M. (2001) Credi risk: modeling, valuaion and hedging. Springer Finance. Bielecki, T.R. and Rukowski, M. (2013) Valuaion and hedging of OTC conracs wih funding coss, collaeralizaion and counerpary credi risky Available a arxiv.org. Brigo D. (2011) Counerpary Risk FAQ: Credi VaR, CVA, DVA, Closeou, Neing, Collaeral, Re-hypohecaion, Wrong Way Risk, Basel, Funding, and Margin Lending. Available a ssrn.com. A. Pallavicini Funding Coss 29 May / 86

80 Seleced References Seleced references II Brigo, D., Alfonsi, A. (2005) Credi Defaul Swaps Calibraion and Derivaives Pricing wih he SSRD Sochasic Inensiy Model, Finance and Sochasic, 9 (1). Brigo D., Buescu, C., Pallavicini, A., Qing L. (2012) Illusraing a Problem in he Self-Financing Condiion in Two Papers on Funding, Collaeral and Discouning Available a ssrn.com. Brigo, D. Capponi, A. (2008) Bilaeral counerpary risk wih applicaion o CDS Risk Magazine (2010) Exended version available a arxiv.org. Brigo, D., Capponi, A., Pallavicini, A., Papaheodorou, V. (2011) Collaeral Margining in Arbirage-Free Counerpary Valuaion Adjusmen including Re-Hypohecaion and Neing. IJTAF (2013) 16, 2. Exended version available a ssrn.com. Brigo, D., Mercurio, F. (2006) Ineres Rae Models: Theory and Pracice wih Smile, Inflaion and Credi. Second ediion. Springer Verlag Financial Series. Brigo, D., Morini, M., Pallavicini A. (2013) Counerpary Credi Risk, Collaeral and Funding wih Pricing Cases for All Asse Classes. Wiley. Brigo, D., Pallavicini, A. (2006) Counerpary Risk and Coningen CDS Valuaion under Correlaion beween Ineres-Raes and Defaul. Available a ssrn.com. Brigo, D., Pallavicini, A. (2007) Counerpary Risk under Correlaion beween Defaul and Ineres-Raes. In Numerical Mehods for Finance" ed. Miller, J., Edelman, D., Appleby, J., Chapman & Hall/Crc Financial Mahemaics Series. Brigo, D., Pallavicini, A. (2008) Counerpary Risk and Coningen CDS under correlaion. Risk Magazine 2. A. Pallavicini Funding Coss 29 May / 86

81 Seleced References Seleced references III Brigo, D., Pallavicini, A. (2013) CCPs, Cenral Clearing, CSA, Credi Collaeral and Funding Coss Valuaion FAQ: Re-Hypohecaion, CVA, Closeou, Neing, WWR, Gap-Risk, Iniial and Variaion Margins, Muliple Discoun Curves, FVA? Available a ssrn.com. Brigo, D., Pallavicini, A. (2014) CCP Cleared or Bilaeral CSA Trades wih Iniial/Variaion Margins Under Credi, Funding and Wrong-Way Risks: A Unified Valuaion Approach. Available a ssrn.com. Brigo, D., Pallavicini, A., Papaheodorou, V. (2009) Bilaeral Counerpary Risk Valuaion for Ineres-Rae Producs: Impac of Volailiies and Correlaions. Available a ssrn.com. Shor updaed version IJTAF (2011) 14, 6. Brigo, D., Pallavicini, A., Torresei, R. (2006) The Dynamical Generalized-Poisson loss model. Risk Magazine, (2007) 6. Exended version available a ssrn.com. Brigo, D., Pallavicini, A., Torresei, R. (2007) Cluser-based exension of he generalized Poisson loss dynamics and consisency wih single names. IJTAF, 10, Also in Lipon and Rennie (Ediors), Credi Correlaion Life Afer copulas", World Scienific, Brigo, D., Pallavicini, A., Torresei, R. (2009) Credi Models and he Crisis, or: How I learned o sop worrying and love he CDOs. Available a ssrn.com. Vasly exended and updaed version in Credi Models and he Crisis: A journey ino CDOs, copulas, Correlaions and Dynamic Models". Wiley, Chicheser, Brunnermeier, M., Pedersen, L. (2009) Marke Liquidiy and Funding Liquidiy. Review of Financial Sudies, 22 (6). Burgard, C., Kjaer, M. (2010) PDE Represenaions of Opions wih Bilaeral Counerpary Risk and Funding Coss Available a ssrn.com. Burgard, C., Kjaer, M. (2011) In he Balance Available a ssrn.com. A. Pallavicini Funding Coss 29 May / 86

82 Seleced References Seleced references IV Casagna, A., Fede, F. (2013) Measuring and Managing Liquidiy Risk Wiley. Con, R. and Jessen, C. (2009) Consan Proporion Deb Obligaions (CPDOs): Modeling and Risk Analysis. Available a ssrn.com. Con, R. and Kokholm, T.(2013) Cenral Clearing of OTC Derivaives: bilaeral vs mulilaeral neing. Available a ssrn.com. Con, R., Mondescu, R.P. Yu, Y. (2011) Cenral clearing of ineres rae swaps: A comparison of offerings. Available a ssrn.com. Crépey, S. (2011) Bilaeral Counerpary Risk under Funding Consrains. Forhcoming in Mahemaical Finanace. Working paper available a auhor s home page. Crépey S., Grbac, Z., Ngor, N. (2012) A muliple-curve HJM model of inerbank risk. Mahemaics and Financial Economics 6(3) Crépey S., Gerboud, R., Grbac, Z., Ngor, N. (2012) Counerpary Risk and Funding: The Four Wings of he TVA. Available a arxiv.org. Duan, J.C. (2009) Clusered Defauls. Available a ssrn.com. Duffie, D., Singleon, K.J. (1997) An Economeric Model of he Term Srucure of Ineres-Rae Swap Yields. The Journal of Finance 52, Duffie, D. and Zhu, H. (2011) Does a Cenral Clearing Counerpary Reduce Counerpary Risk? Sanford Universiy Working Paper. Durand, C. and Rukowski, M. (2013) CVA for Bilaeral Counerpary Risk under Alernaive Selemen Convenions Available a arxiv.org. El Karoui, N. and Peng, S. and Quenez M. (1997) Backward sochasic differenial equaions in finance. Mahemaical Finance, 7, A. Pallavicini Funding Coss 29 May / 86