Choice of Collateral Currency

Transcription

1 Masaaki Fujii join work wih Akihiko Takahashi Counerpary Risk Froniers: May 4h, This research is suppored by CARF (Cener for Advanced Research in Finance) and he global COE program The research and raining cener for new developmen in mahemaics All he conens expressed in his research are solely hose of he auhors and do no represen he views of any insiuions The auhors are no responsible or liable in any manner for any losses and/or damages caused by he use of any conens in his research MFujii is graeful for YShimada, and former colleagues of Morgan Sanley, especially in IDEAS, IR opion, and FX Hybrid desks in Tokyo for fruiful and simulaing discussions The conens of he research do no represen any views or opinions of Morgan Sanley Graduae School of Economics, The Universiy of Tokyo Graduae School of Economics, The Universiy of Tokyo 1 / 46

2 Oulines 1 Inroducion Some facs on Collaeralizaion Disincion of IR raes Funding Spread of Currency 2 Pricing Framework Crieria Pricing under he full Collaeralizaion 3 in Single Currency in Muliple Currencies Cross Currency Swap 4 5 Impac of Collaeral Managemen 6 2 / 46

3 Some facs on Collaeralizaion Some facs on Collaeralizaion Collaeralizaion The mos imporan credi risk miigaion ool CSA gives he deails of collaeral agreemens Dramaic increase in recen years (ISDA Margin Survey) 30%(2003) 70%(2011) in erms of rade volume for all OTC Coverage goes up o 79% (for all OTC) and 88% (for fixed income) among major financial insiuions More han 80% of collaeral is Cash Abou half of he cash collaeral is USD Daily porfolio reconciliaion is he marke sandard for large dealers More sringen collaeral managemen for CCPs 3 / 46

4 Some facs on Collaeralizaion Impac of Collaeralizaion Impac of collaeralizaion : Reducion of Couner-pary Exposure CVA/DVA Change Clean Valuaion Framework (main opic of my alk) Assuming no meaningful counerpary risk and Funding Cos of Currency: Cross Currency Swap cheapes-o-deliver opion SCSA and USD Silo 4 / 46

5 Some facs on Collaeralizaion A Simple Schemaic Picure Collaeralized (Secured) Conrac (curren picure) opion paymen A cash=pv collaeral B collaeral rae loan No ourigh cash flow (collaeral=pv) No exernal funding is needed Funding is deermined by he collaeral rae Reference Rae : LIBOR Discouning Rae : OIS 5 / 46

6 Disincion of IR raes Disincion among LIBORs and OIS Hisorical behavior of IRS (1Y)-OIS (1Y) spreads (bps) Figure: Source:Bloomberg 6 / 46

7 Disincion of IR raes Disincion among LIBORs and OIS Hisorical behavior of JPY TS spreads (bps) Figure: Source:Bloomberg 7 / 46

8 Disincion of IR raes Disincion among LIBORs and OIS Hisorical behavior of USD TS spreads (bps) Figure: Source:Bloomberg 8 / 46

9 Disincion of IR raes Disincion among LIBORs and OIS Hisorical behavior of EUR TS spreads (bps) Figure: Source:Bloomberg 9 / 46

10 Funding Spread of Currency Funding Spread of Currency The origin of muli-curve Seup (?) Japan premium in lae 1990s Japanese financial firms had o pay exra premium o fund USD hrough Cross Currency Swap A leas, some of he firms sared o calculae JPY relaed conracs wih wo curves, one for discouning and he oher for reference raes around 1998 or so Currency Funding Spread Funding Cos in he Domesic Marke Funding Cos hrough Cross Currency Swap 10 / 46

11 Funding Spread of Currency Cross Currency Swap USD USD $Libor JP US $Libor Libor Libor+X ccs JPY JPY X ccs 0 X ccs < 0 for JPY, for example USD funding cos for Japanese firm is higher han USD Libor JPY funding cos for US firm is lower han JPY Libor 11 / 46

12 Funding Spread of Currency Cross Currency Swap Hisorical behavior of USDJPY CCS spreads (bps) Figure: Source:Bloomberg 12 / 46

13 Funding Spread of Currency Cross Currency Swap Hisorical behavior of EURUSD CCS spreads (bps) Figure: Source:Bloomberg 13 / 46

14 Funding Spread of Currency Recen hisory of 1Y CCS basis Cross Currency Swap Figure: Blue=JPY, Red=EUR, Green=GBP (Source:Bloomberg) 14 / 46

15 Funding Spread of Currency Cross Currency Swap Term Srucure of CCS basis (4/19/2012) Figure: Blue=JPY, Red=EUR, Green=GBP (Source:Bloomberg) 15 / 46

16 Crieria Crieria for Valuaion model for Clean Price Crieria Consisen discouning/forward curve consrucion Dynamic Price all ypes of IR swaps correcly under Collaeralizaion: OIS, IRS and TS (enor swap) Mainain consisency in muli-currency environmen CCS basis spreads need o be recovered FX Forward Cos of cash collaeral and is difference among major currencies should be aken ino accoun Inconsisency in pricing across differen curves and currencies may provide a heavenly environmen for rogue raders 16 / 46

17 Pricing under he full Collaeralizaion Assumpions for Collaeralizaion Assumpions Coninuous adjusmen of collaeral amoun Perfec collaeralizaion by Cash Zero minimum ransfer amoun Focus on Clean Price Commens By making use of Repo marke informaion, he same mehod can be applied o oher collaeral asses Longer erm quoes are no ypically available Liquidiy swap may provide informaion in coming years (?) 17 / 46

18 Pricing under he full Collaeralizaion Pricing under he full Collaeralizaion Pricing Formula PV of T -mauring payoff h (i) (T ) in currency (i) collaeralized in currency (j) h (i) () = E Q i where, [ e ( T r (i) (s)ds T y e ) ] (j) (s)ds h (i) (T ) [( e ) ] T y (i,j) (s)ds h (i) (T ) = D (i) (, T )E T (i) y (j) (s) = r (j) (s) c (j) (s), y (i,j) (s) = y (i) (s) y (j) (s) D (i) (, T ) = E Q i [e ] T c (i) (s)ds h (i) (T ): opion payoff a ime T in currency i collaeral is posed in currency j c (j) (s): insananeous collaeral rae of currency j a ime s r (j) (s): insananeous risk-free rae of currency j a ime s E T (i) [ ]: expecaion under he fwd measure associaed wih D (i) (, T ) 18 / 46

19 Pricing under he full Collaeralizaion Pricing under he full Collaeralizaion Corollary: Single Currency Case If paymen and collaeral currencies are he same (i), he opion value is given by h (i) () = E Q i [e ] T c(i) (s)ds h (i) (T ) [ ] h (i) (T ) = D (i) (, T )E T (i) The discouning is deermined by collaeral rae, which is consisen wih he schemaic picure seen before 19 / 46

20 Pricing under he full Collaeralizaion Pricing under he full Collaeralizaion f x (i,j) (): Foreign exchange rae a ime represening he price of he uni amoun of currency j in erms of currency i Collaeral amoun in currency j a ime s is given by h(i) (s) f x (i,j) (s), which is invesed a he rae of y (j) (s): [ h (i) () = E Q i e ] T r (i) (s)ds h (i) (T ) [ ( ) ] T +f x (i,j) ()E Q j e s r (j) (u)du y (j) h (i) (s) (s) f x (i,j) ds (s) [ = E Q i e T T r (i) (s)ds h (i) (T ) + e ] s r (i) (u)du y (j) (s)h (i) (s)ds This is a FBSDE, bu wih simple linear form 20 / 46

21 Pricing under he full Collaeralizaion Pricing under he full Collaeralizaion Economic Meanings of Spread y y (i) = r (i) c (i) effecive dividend yield from collaeral of ccy (i) cos of collaeral from he view poin of collaeral payer y (i,j) = y (i) y (j) Funding spread beween currency (i) and (j) Full Collaeralizaion Linear and Addiive Imperfec Collaeralizaion Credi Risk, Funding Asymmery Non-linear FBSDE 1 Zero-h order : Clean Price (full collaeralizaion) Firs order : Gaeaux Derivaive CCA, CVA, DVA ec Higher orders: Non-linear FBSDE Series of Linear FBSDEs 1 depends on assumpions abou collaeral value, recovery scheme, ec 21 / 46

22 in Single Currency in Single Currency Collaeralized Overnigh Index Swap Assumpions: paymen and collaeral currencies are he same collaeral rae is given by he overnigh rae Condiion for he lengh-n OIS rae: OIS (i) N N n E Q i n=1 = [ e Tn 0 c (i) (s)ds ] N n=1 E Q i [ e Tn 0 c (i) (s)ds ( )] T n T c (i) (s)ds e n 1 1 or, equivalenly, N nd (i) (0, T n) = D (i) (0, T 0 ) D (i) (0, T N ) OIS (i) N n=1 Then, he collaeralized ZCB price can be boosrapped as D (i) (0, T N ) = D(i) (0, T 0 ) OIS (i) N N 1 n=1 nd (i) (0, T n ) 1 + OIS (i) N N 22 / 46

23 in Single Currency IRS (i) M in Single Currency Collaeralized IRS M md (i) (0, T m) = m=1 M m=1 δ md (i) (0, T m)e T (i) m [L (i) (T m 1, T m; τ )] Collaeralized Tenor (marke basis) Swap 2 N ( [ ] ) δ n D (i) (0, T n ) E T n (i) L (i) (T n 1, T n ; τ S ) + T S (i) N n=1 = M m=1 δ md (i) (0, T m)e T m (i) [ ] L (i) (T m 1, T m; τ L ) Marke quoes of collaeralized OIS, IRS, TS, and proper spline mehod allow us o deermine [ ] {D (i) (0, T )}, {E T m (i) L (i) (T m 1, T m, τ ) } for all he relevan T, T m and enor τ of Libor of currency (i) 2 The shor-enor Leg may be compounded, and hen exis addiional small correcions 23 / 46

24 in Muliple Currencies : Muliple Currencies Collaeralized FX Forward: USD/JPY Suppose USD= (i), JPY= (j) and collaeral currency is USD Curren ime: Mauriy: T A T, one uni of (i) is exchanged for K (fixed a ) uni of (j) FX forward is he break-even value of K [e T (c (j) s +y s (j,i) KE Q j ] )ds = f x (j,i) ()E Q i ( T f x (j,i) (, T ; (i)) = f x (j,i) () D(i) (, T ) D (j) (, T ) exp where y (j,i) (, T ) = T ln ( E T (j) [e T y (j,i) s ds ]) FX Forward Forward curve of funding spread CCS for longer mauriies [e ] T c (i) s ds 1 ) y (j,i) (, u)du 24 / 46

25 in Muliple Currencies Term Srucure of Funding Spread (EUR USD) R y(i,j) (T ) = 1 ( ln E T [ (i) e T T y (i,j) ]) (s)ds = 1 T T 0 y(i,j) (0, s)ds 25 / 46

26 in Muliple Currencies Term Srucure of Funding Spread (JPY USD) 26 / 46

27 in Muliple Currencies CCS Basis and Funding Spread CCS Basis Spread Funding Spread y (i,j) LIBOR-OIS (and hence credi risk) seems o have only minor effecs 27 / 46

28 Cross Currency Swap Agains USD Consan Noional CCS and MM-CCS USD-LIBOR X-LIBOR + basis spread Consan Noional CCS (CNCCS) Noional of boh legs are kep consan Mark-o-Marke CCS (MMCCS) Noional of currency X is kep consan Noional of USD is readjused o f x (USD,X) N X a every sar of LIBOR accrual period Prices of wo conracs were quie close and he difference was no paid enough aenion Their quoes were no clearly disinguished on broker screens a a I am no sure he very recen siuaion 28 / 46

29 Cross Currency Swap Consan Noional CCS and MM-CCS USD-JPY CCS (Spo-sar, T N -mauring) USD: currency-(i), JPY: currency-(j) Xccs MM Xccs CN N n=1 δnd(i) (0, T n)e T n(i) = [( ) f x (j,i) (0) 1 f x (j,i) (T n 1 ) N n=1 δnd(j) (0, T n; i) ] B (i) (T n 1, T n) where D (j) (, T n; i) = E Q j [ e Tn 0 (c (j) s +y s (j,i) )ds ( ) B (i) (T n 1, T n ) = L (i) (T n 1, T n ) 1 1 δ n D (i) (T n 1, T 1 n) ] 29 / 46

30 HJM-framework under he collaeralizaion SDEs for HJM-framework dc (i) (, s) = σ (i) c (, s) df x (i,j) () f (i,j) = x () where ( s dy (i,k) (, s) = σ (i,k) y (, s) db (i) (, T ; τ ) B (i) (, T ; τ ) ) σ c (i) (, u)du ( s = σ(i) B (, T ; τ ) ( T ) σ y (i,k) (, u)du ( c (i) () c (j) () + y (i,j) () B (i) (, T k ; τ ) = E T k(i) is forward LIBOR-OIS spread d + σ c (i) (, s) dw Q i d + σ y (i,k) (, s) dw Q i ) σ c (i) (, s)ds d + σ (i) B (, T ; τ ) dw Q i ) d + σ (i,j) X () dw Q i ( ) [ ] L (i) (T k 1, T k ; τ ) 1 D (i) (, T k 1 ) δ (i) D (i) 1 (, T k ) k 30 / 46

31 Role of y (i,j) Paymen currency i wih Collaeral currency j D (i) (, T ) E T (i) [e T y(i,j) (s)ds ] D (i) (, T ) afer neglecing small correcions from possible non-zero correlaions To choose srong currency, such as USD, is expensive (for he collaeral payer) 31 / 46

32 Role of y (i,j) Opimal behavior of collaeral payer can significanly change he derivaive value Paymen currency: (i), Eligible Collaeral Se: C D (i) (, T ) E T (i) [e T max j C{y (i,j) (s)}ds ] D (i) (, T ) Paymen currency: (i), Eligible Collaeral Se: (i, USD) D (i) (, T ) E T (i) [e ] T max{y(i,usd) (s),0}ds D (i) (, T ) Volailiy of y (i,j) is an imporan deerminan 32 / 46

33 Figure: Modificaion of EUR discouning facors based on HW model for y (EUR,USD) as of 2010/3/16 The mean-reversion parameer is 15%, and he volailiy is given a each label 33 / 46

34 Impac of Collaeral Managemen Inefficien Collaeral Managemen Wha happens if an invesor can selec he cheapes collaeral bu he counerpary canno? Asymmeric CSA Symmeric CSA bu he counerpary has only limied access o he cheapes collaeral The counerpary is much less sophisicaed in collaeral managemen Maybe disasrous for he counerpary 34 / 46

35 Impac of Collaeral Managemen Collaeral Managemen Suppose he siuaion The invesor 1 can selec collaeral currency from he se C The counerpary 2 can only use he currency (i) Pricing Formula V = E Q [ exp ],T ] ( s ( ru µ(u, V u ) ) ) ] du dd s µ(u, V u ) = y 1 u1 {Vu <0} + y 2 u1 {Vu 0} y 1 u = min j C (r(j) c (j) ) yu 2 = r (i) c (i) V is he conrac value from he view poin of he invesor D denoes a cumulaive cash-flow of he conrac 35 / 46

36 Impac of Collaeral Managemen Collaeral Managemen One sees r u µ(u, V u ) = c (i) u Firs order approximaion: V V + E [ T + max j C e s c (i) u [ ] V = E e s c (i) u du dd s ],T ] y(i,j) u 1 {Vu<0} du [ V s] + max j C ] y(i,j) s ds Similar o CVA formula for an uncollaeralized conrac (from he counerpary poin of view) If he counerpary does no recognize he opionaliy, i may lose significan value 36 / 46

37 Impac of Collaeral Managemen Collaeral Managemen: Numerical Example For demonsraion, consider a simplisic sysem: USD: currency (i) JPY: currency (j) Home currency: (j) ( ) dc (j) = θ (j) () κ (j) c (j) d + σ c (j) dw ( ) dc (i) = θ (i) () σ c (i) σ x (j,i) κ (i) c (i) d + σ c (i) dw ( ) dy (j,i) = θ (j,i) () κ (j,i) y (j,i) d + σ y (j,i) dw ( d ln f x (j,i) () = c (j) c (i) + y (j,i) 1 2 σ(j,i) x 2) d + σ x (j,i) dw 37 / 46

38 Impac of Collaeral Managemen Collaeral Managemen: Numerical Example JPY OIS (10y): invesor: pary-1 can selec USD, JPY as collaeral counerpary: pary-2 can only use JPY as collaeral 38 / 46

39 Impac of Collaeral Managemen Collaeral Managemen: Numerical Example JPY OIS (20y): invesor: pary-1 can selec USD, JPY as collaeral counerpary: pary-2 can only use JPY as collaeral 39 / 46

40 Impac of Collaeral Managemen Collaeral Managemen: Numerical Example USD/JPY Cross Currency OIS (10y): invesor: pary-1 can selec USD, JPY as collaeral counerpary: pary-2 can only use USD as collaeral 40 / 46

41 Impac of Collaeral Managemen Collaeral Managemen: Numerical Example USD/JPY Cross Currency OIS (20y): invesor: pary-1 can selec USD, JPY as collaeral counerpary: pary-2 can only use USD as collaeral 41 / 46

42 Impac of Collaeral Managemen Collaeral Managemen Numerical example suggess I is bes o avoid o make a flexible agreemen on eligible collaerals if here is no capabiliy of choosing he cheapes collaeral Winning posiions wih flexible CSA may be suffering from significan negaive gammas 42 / 46

43 Impac of Collaeral Managemen Collaeral Managemen Some implicaions for neing Assume some regulariy condiions and perfec collaeralizaion Suppose ha, for each pary i, is collaeral funding cos y i does no explicily depend on he value process of conrac Le V a, V b, and V ab be, respecively, he value processes (from view poin of he pary 1) of conracs wih cumulaive dividend processes D a, D b, and D a + D b (ie, need porfolio) Then, we have, V ab V a + V b if y 1 y 2 V ab V a + V b if y 1 y 2 43 / 46

44 Sandard CSA Embedded opionaliy in CSA no price ransparency difficul o do unwinding and novaion of rades difficul o hedge Sandard Credi Suppor Annex (SCSA) 3 17 Currency Silos emerging currencies, (mos of?) muli-currency rades USD silo 3 No sure for he conens of final decisions 44 / 46

45 Remarks on USD Silo under he USD Silo No domesic OIS marke Separaion of c (i) and y (i,usd) is impossible However, wha we need are only he discouning and reference raes under USD collaeralizaion Simulaneous calibraion of USD-collaeralized domesic IRS and USD-collaeralized CCS provides relevan curves 45 / 46

46 Thank You! 46 / 46