Collateral Posting and Choice of Collateral Currency

Transcription

1 Collaeral Posing and Choice of Collaeral Currency -Implicaions for derivaive pricing and risk managemen- Masaaki Fujii, Yasufumi Shimada, Akihiko Takahashi KIER-TMU Inernaional Workshop on Financial Engineering 2010 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 friends 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 General Manager, Capial Markes Division, Shinsei Bank, Limied Graduae School of Economics, The Universiy of Tokyo 1 / 37

2 Oulines 1 Inroducion Texbook-syle IR Model has been obsolee OTC Marke and Collaeralizaion Imporan Insrumens and Marke Realiies 2 Term Srucure Model under Collaeralizaion and Basis spread Pricing under he Collaeralizaion Consrucion of Term Srucure 3 Choice of Collaeral Currency Single Eligible Collaeral Currency Muliple Eligible Collaeral Currencies 4 Conclusions 5 Appendix : HJM-framework under he collaeralizaion 2 / 37

3 Texbook-syle IR Model has been obsolee Texbook-syle IR Model has been obsolee Texbook-syle Ineres Rae Model (such as LMM) has been ou of use for many years, a leas, in major invesmen banks Unable o explain non-negligible basis spread in cross currency swap (CCS) marke Unable o consruc consisen muli-currency framework FX forward (CCS + IRS) Exended IR models (wih muli curves) have been used for long, especially by US banks, o reflec (and o exploi) funding cos asymmery (such as Japan premium) in derivaive prices 3 / 37

4 Texbook-syle IR Model has been obsolee Texbook-syle IR Model has been obsolee New marke realies afer he Financial Crisis Much more volaile CCS basis spread Non-negligible basis spreads even in he single currency marke Tenor swap spread, Libor-OIS spread, ec Widespread use of Collaeralizaion 4 / 37

5 OTC Marke and Collaeralizaion OTC Marke and Collaeralizaion Collaeralizaion The mos imporan credi risk miigaion ool margin call, selemen and associaed procedures legal specificaions are provided by CSA (Credi Suppor Annex ) Dramaic increase in recen years (ISDA [4]) 30%(2003) 70%(2009) in erms of rade volume for all OTC Coverage is up o 78% (for all OTC) and 84% (for fixed income) among major financial insiuions More han 80% of collaeral is Cash Abou half of he cash collaeral is USD Almos all he credi derivaives are collaeralized 5 / 37

6 OTC Marke and Collaeralizaion Impac of Collaeralizaion Impac of collaeralizaion : Reducion of Couner-pary Exposure Associaed change in CVA has been acively sudied Change of Funding Cos (opic of his alk) Require new mehodology for erm srucure consrucion Add cheapes-o-deliver opionaliy when here are muliple eligible collaeral Significan impac on derivaive pricing and risk managemen 6 / 37

7 OTC Marke and Collaeralizaion Source of Funding Cos Difference Unsecured Funding and Conrac (old picure) Cash Libor Cash=PV Loan A B opion paymen Libor is unsecured offer rae in he inerbank marke Libor discouning makes he presen value of Loan zero Libor discouning is appropriae for unsecured rades beween financial firms wih Libor credi qualiy Libor discouning (+ CVA) has been he sandard marke pracice 7 / 37

8 OTC Marke and Collaeralizaion Source of Funding Cos Difference Collaeralized (Secured) Conrac (curren picure) opion paymen A cash=pv collaeral B col rae loan No ourigh cash flow (collaeral=pv) No exernal funding is needed Funding is deermined by over-nigh (ON) rae Libor discouning is inappropriae 8 / 37

9 Imporan Insrumens and Marke Realiies Imporan Insrumens and Marke Realiies Overnigh Index Swap (OIS) OIS rae Compounded ON Floaing side: Daily compounded ON rae Usually, here is only one paymen for < 1yr Marke Quoe : fixed rae, called OIS rae 9 / 37

10 Imporan Insrumens and Marke Realiies Imporan Insrumens and Marke Realiies Hisorical behavior of IRS (1Y)-OIS (1Y) spreads (bps) Figure: Source:Bloomberg 10 / 37

11 Imporan Insrumens and Marke Realiies Imporan Insrumens and Marke Realiies Tenor Swap (TS) 1 Libor (shor enor) +spread Libor (long enor) Texbook-syle Implemenaion Zero spread Marke: Spread is quie significan and volaile since lae I is also common ha paymen of shor-enor Leg is compounded and paid a he same ime wih he oher Leg 11 / 37

12 Imporan Insrumens and Marke Realiies Imporan Insrumens and Marke Realiies Hisorical behavior of JPY TS spreads (bps) Figure: Source:Bloomberg 12 / 37

13 Imporan Insrumens and Marke Realiies Imporan Insrumens and Marke Realiies Hisorical behavior of USD TS spreads (bps) Figure: Source:Bloomberg 13 / 37

14 Imporan Insrumens and Marke Realiies Imporan Insrumens and Marke Realiies Hisorical behavior of EUR TS spreads (bps) Figure: Source:Bloomberg 14 / 37

15 Imporan Insrumens and Marke Realiies Imporan Insrumens and Marke Realiies Mark-o-Marke Cross Currency Swap N i = f x (i,j) () f (i,j) x N j 1 N i δl i N i δl i (N i = N i + f(i,j) x ) USD Libor is exchanged by Libor +spread of he oher currency USD leg noional is rese every sar of accrual period Texbook-syle Implemenaion Zero spread Marke: Spread is quie significan and volaile for long ime Drasic/Rapid change in recen years 15 / 37

16 Imporan Insrumens and Marke Realiies Imporan Insrumens and Marke Realiies Hisorical behavior of USDJPY CCS spreads (bps) Figure: Source:Bloomberg 16 / 37

17 Imporan Insrumens and Marke Realiies Imporan Insrumens and Marke Realiies Hisorical behavior of EURUSD CCS spreads (bps) Figure: Source:Bloomberg 17 / 37

18 Imporan Insrumens and Marke Realiies Imporan Insrumens and Marke Realiies I is very dangerous o use exbook-syle implemenaion of IR models, because mispricing of various fundamenal insrumens: Tenor Swap (TS) Cross Currency Swap (CCS) FX Overnigh Index Swap (OIS) Hence, all he producs are mispriced Poenial loss can be a few % of ousanding noional Unable o recognize he imporan dela exposure, such as o Libor-OIS spread Proper conrol of risk exposure is impossible 18 / 37

19 Crieria for Models Workable in Real Business Crieria Consisen discouning/forward curve consrucion Price all ypes of IR swaps correcly: OIS, IRS and TS Take collaeralizaion ino accoun Mainain consisency in muli-currency environmen CCS basis spreads need o be recovered Sochasic Modeling of Basis spreads Allow sysemaic calibraion procedures Flexible enough o allow non-rivial erm srucure of spreads 19 / 37

20 Pricing under he Collaeralizaion Pricing under he Collaeralizaion Assumpion Coninuous adjusmen of collaeral amoun Perfec collaeralizaion by Cash Zero minimum ransfer amoun Commens Daily margin call/selemen is becoming popular By making use of Repo / Reverse-Repo, oher collaeral asses can be convered ino he equivalen amoun of cash collaeral General Collaeral (GC) repo rae closely racks overnigh rae 20 / 37

21 Pricing under he Collaeralizaion Pricing under he Collaeralizaion Proposiion T -mauring European opion under he collaeralizaion is given by [ e ( T r (i) (s)ds T y e ) ] (j) (s)ds h (i) (T ) h (i) () = E Q i where, = D (i) (, T )E T c (i) [( e ) ] T y (i,j) (s)ds h (i) (T ) 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 c (i) [ ]: expecaion under he fwd measure associaed wih D (i) (, T ) 21 / 37

22 Pricing under he Collaeralizaion Pricing under he Collaeralizaion Collaeral amoun in currency j a ime s is given by h(i) (s) f (i,j) x (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 Noe ha X() = e 0 r (i) (s)ds h (i) ()+ e s 0 r (i) (u)du y (j) (s)h (i) (s)ds 0 is a Q i -maringale Then, he process of he opion value is wrien by ( ) dh (i) () = r (i) () y (j) () h (i) ()d + dm() wih some Q i -maringale M This esablishes he proposiion f x (i,j) (): Foreign exchange rae a ime represening he price of he uni amoun of currency j in erms of currency i 22 / 37

23 Pricing under he Collaeralizaion Pricing under he Collaeralizaion Corollary If paymen and collaeral currencies are he same, he opion value is given by h() = E Q [e ] T c(s)ds h(t ) = D(, T )E T c [h(t )] The discouning is deermined by collaeral rae, which is consisen wih he schemaic picure seen before 23 / 37

24 Consrucion of Term Srucure Building Blocks for IR Term Srucure Model Building Blocks c (i) (, T ) = T ln D(i) (, T ) B (i) (, T k ; τ ) = E T c k,(i) y (i,k) (, T ) = ( T ln E Q i ( ) 1 D (i) (, T k 1 ) δ (i) D (i) 1 (, T k ) k [e ]) T y (i,k) (s)ds [ ] L (i) (T k 1, T k ; τ ) These building blocks are enough o calibrae all he relevan OIS, IRS, TS and CCS 24 / 37

25 Consrucion of Term Srucure Consrucion of Term Srucure Term srucure consrucion procedures See, (Fujii, Shimada, Takahashi 2009) [1] for deails (1), OIS c (i) (, s) (2), IRS+TS+(1) B (i) (, s; τ ) (3), CCS+(1)+(2) y (i,j) (, s) Assume collaeralizaion in domesic currency for OIS, IRS and TS 2 Assume collaeralizaion in USD for CCS (USD crosses) No-arbirage dynamics of hese underlyings in HJM framework is given in (Fujii, Shimada, Takahashi 2009)[2] 2 Assumpion on collaeral currency has only minor impac on he marke par quoes 25 / 37

26 Consrucion of Term Srucure Consrucion of Term Srucure R OIS (T ) = ln(d(0, T ))/T E T m c 1 e R τ (T m 1 )T m 1 [L(T m 1, Tm; τ )] = 1 δm e Rτ (Tm)Tm 26 / 37

27 Single Eligible Collaeral Currency Choice of Collaeral Currency Role of y (i,j) Paymen currency i wih Collaeral currency j D (i) (, T ) E Qi [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) 27 / 37

28 Single Eligible Collaeral Currency Choice of Collaeral Currency R y(i,j) (T ) = 1 ( [ ln E Q i e T T y (i,j) ]) (s)ds = 1 T T 0 y(i,j) (0, s)ds 28 / 37

29 Single Eligible Collaeral Currency Choice of Collaeral Currency 29 / 37

30 Muliple Eligible Collaeral Currencies Choice of Collaeral Currency Role of y (i,j) Opimal behavior of collaeral payer can significanly change he derivaive value Paymen currency i wih muliple currencies as eligible collaeral choice C D (i) (, T ) E Qi [e ] T max j C{y (i,j) (s)}ds D (i) (, T ) Paymen currency and USD as eligible collaeral is relaively common D (i) (, T ) E Qi [e ] T max{y(i,usd) (s),0}ds D (i) (, T ) Volailiy of y (i,j) is an imporan deerminan 30 / 37

31 Muliple Eligible Collaeral Currencies Choice of Collaeral Currency Close relaionship o CCS basis spread 31 / 37

32 Muliple Eligible Collaeral Currencies Choice of Collaeral Currency Figure: 3M-Roll hisorical volailiy of y (EUR,USD) insananeous forward Annualized in absolue erms 32 / 37

33 Muliple Eligible Collaeral Currencies Choice of Collaeral Currency 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 / 37

34 Muliple Eligible Collaeral Currencies Choice of Collaeral Currency Figure: Modificaion of JPY discouning facors based on HW model for y (JP Y,USD) as of 2010/3/16 The mean-reversion parameer is 15%, and he volailiy is given a each label 34 / 37

35 Conclusions We proposed erm srucure model under he collaeralizaion consisenly wih cross currency marke Choice of collaeral currency is quie imporan Embedded cheapes-o-deliver opion can significanly change he effecive discouning facor Use of exbook-syle IR model leads o significan mispricing, and unable o provide imporan risk exposure 35 / 37

36 Appendix: HJM-framework under he collaeralizaion dc (i) (, s) = σ (i) c (, s) ( s dy (i,k) (, s) = σ y (i,k) (, s) db (i) (, T ; τ ) B (i) (, T ; τ ) df x (i,j) () f x (i,j) () ( s ) σ(i) c (, u)du ) σ(i,k) y (, u)du d + σ c (i) (, s) dw Q i d + σ y (i,k) (, s) dw Q i = σ (i) ( T B (, T ; τ ) σ (i) ) c (, s)ds d + σ (i) B (, T ; τ ) dw Q i = dc (j) (, s) = σ (j) c ( c (i) () c (j) () + y (i,j) ) () d + σ (i,j) X () dw Q i (, s) [( s dy (j,k) (, s) = σ (j,k) y (, s) db (j) (, T ; τ ) B (j) (, T ; τ ) ) σ(j) c (, u)du σ (i,j) [( s σ(j,k) y (, u)du ] X () ) = σ (j) [( T B (, T ; τ ) σ (j) ) c (, s)ds σ (i,j) ] X () d +σ (j) B (, T ; τ ) dw Q i d + σ c (j) (, s) dw Q i σ (i,j) ] X () d + σ y (j,k) (, s) dw Q i 36 / 37

37 Main References [1] Fujii, M, Shimada, Y, Takahashi, A, 2009, A noe on consrucion of muliple swap curves wih and wihou collaeral, CARF Working Paper Series F-154, available a hp://ssrncom/absrac= [2] Fujii, M, Shimada, Y, Takahashi, A, 2009, A Marke Model of Ineres Raes wih Dynamic Basis Spreads in he presence of Collaeral and Muliple Currencies, CARF Working Paper Series F-196, available a hp://ssrncom/absrac= [3] Fujii, M, Shimada, Y, Takahashi, A, 2010, Collaeral Posing and Choice of Collaeral Currency -Implicaions for Derivaive Pricing and Risk Managemen-, CARF Working Paper Series F-216, available a hp://ssrncom/absrac= [4] ISDA Margin Survey 2010, Preliminary Resuls Marke Review of OTC Derivaive Bilaeral Collaeralizaion Pracices 37 / 37