Curve Building and Swap Pricing in the Presence of Collateral and Basis Spreads SIMON GUNNARSSON

Transcription

1 Curve Building and Swap Pricing in he Presence of Collaeral and Basis Spreads SIMON GUNNARSSON Maser of Science Thesis Sockholm, Sweden 2013

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3 Curve Building and Swap Pricing in he Presence of Collaeral and Basis Spreads SIMON GUNNARSSON Maser s Thesis in Mahemaical Saisics (30 ECTS credis) Maser Programme in Mahemaics (120 credis) Supervisor a KTH was Boualem Djehiche Examiner was Boualem Djehiche TRITA-MAT-E 2013:19 ISRN-KTH/MAT/E--13/19--SE Royal Insiue of Technology School of Engineering Sciences KTH SCI SE Sockholm, Sweden URL: wwwkhse/sci

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5 Absrac The erupion of he financial crisis in 2008 caused immense widening of boh domesic and cross currency basis spreads Also, as a majoriy of all fixed income conracs are now collaeralized he funding cos of a financial insiuion may deviae subsanially from he domesic Libor In his hesis, a framework for pricing of collaeralized ineres rae derivaives ha accouns for he exisence of non-negligible basis spreads is implemened I is found ha losses corresponding o several percen of he ousanding noional may arise as a consequence of no adaping o he new marke condiions Keywords: Curve building, swap, basis spread, cross currency, collaeral

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7 Acknowledgemens I wish o hank my supervisor Boualem Djehiche as well as Per Hjorsberg and Jacob Niburg for inroducing me o he subjec and for providing helpful feedback along he way I also wish o express graiude owards my family who has suppored me hroughou my educaion Finally, I am graeful ha Marcus Josefsson managed o devoe a few hours o proofread his hesis

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9 Conens 1 Inroducion 1 11 The Libor-OIS and TED Spreads 2 12 Tenor Basis Spreads 4 13 Cross Currency Basis Spreads 5 14 Previous Research 6 15 FRA and Swap Pricing Before he Financial Crisis 7 2 Theoreical Background Curve Consrucion wihou Collaeral A Single IRS Marke An IRS and TS Marke Inroducing he Consan Noional CCS Curve Consrucion wih Collaeral Pricing of Collaeralized Derivaives Inroducing he OIS Curve Consrucion in a Single Currency Curve Consrucion in Muliple Currencies 15 3 Implemenaion Building he USD Curves The USD Discouning Curve The USD 3m Forward Curve The USD 1m Forward Curve The USD 6m Forward Curve Building he EUR Curves The EUR Discouning Curve The EUR 6m Forward Curve The EUR 1m Forward Curve The EUR 3m Forward Curve The EUR 1y Forward Curve The Case of USD Collaeral 26 4 Resuls The Case of USD The Case of EUR Comparing he Currencies 35

10 5 Conclusions Criique Suggesions for Furher Research 37 References 38 Appendices 41 A The Forward Measure 41 B Mark-o-Marke Cross Currency Swaps 42 C Day Coun Convenions 44 D Swap Convenions 45 E Cubic Spline Inerpolaion 46 F Proof of Proposiion G Tables 50 G1 Tables of USD Daa 50 G2 Tables of EUR Daa 52

11 1 Inroducion The global financial meldown during 2008 ineviably caused a lo of change on he financial markes Companies were faced wih increased credi and liquidiy problems and for banks his siuaion affeced heir rading abiliies Henceforh i became vial o accoun for credi and liquidiy premia when pricing financial producs The effecs were paricularly apparen in he marke for ineres rae producs, ie FRAs, swaps, swapions ec, and as a consequence professionals sared developing new pricing frameworks ha would correcly accoun for he increased credi and liquidiy premia More specifically, basis spreads beween differen enors and currencies ha were negligible (ypically smaller han he bid/ask spread) before he crisis were now much wider A new pricing framework would have o accoun for he magniudes of hese spreads and produce consisen prices ha are arbirage-free In pracice his enails ha one should esimae one forward curve for each enor, insead of using one universal forward curve for all enors Also, as mos over-he-couner ineres rae producs are nowadays collaeralized he quesion of how o correcly discoun fuure cash flows mus be raised In ligh of his, he purpose of his hesis is o implemen a pricing framework ha accouns for non-negligible basis spreads beween enors and currencies ha is also able o price collaeralized producs in a desirable manner The approach will be empirical, ie forward raes and discoun facors will be exraced from available marke quoes and we will no develop and implemen a framework ha models he erm srucures of basis spreads We will assume basic knowledge of sochasic calculus as covered in Øksendal (2003) [22] Maringale pricing of financial derivaives is also assumed a prerequisie and an inroducion is given in Björk (2009) [3] Geman e al (1995) [12] provides an exensive discussion on he imporan echnique of changing he numéraire Also, Friedman (1983) [8] rigorously presens various essenial conceps of analysis, such as fundamenal measure heory and Radon-Nikodym derivaives The res of his paper is srucured as follows The remainder of his chaper provides a summary of he various spreads ha widened during he financial crisis and concludes wih an inroducion o swap pricing in he absence of basis spreads Chaper 2 presens a framework ha accouns for he prevailing basis spreads beween enors and currencies, wih and wihou he presence of collaeral This framework is laer implemened in Chaper 3, where echnical deails are covered o a greaer exen Resuls are discussed in Chaper 4 and Chaper 5 concludes 1

12 11 The Libor-OIS and TED Spreads The USD London Inerbank Offered Rae (Libor from now on) is an average of he raes a which banks hink hey can obain unsecured funding I is managed by he Briish Bankers Associaion (BBA) o which he paricipaing banks 1 submi heir esimaed funding coss The European equivalen o he Libor is he European Inerbank Offered Rae (Euribor), which is managed by he European Banking Federaion While he Libor is an average of he perceived funding coss of he paricipaing banks, he Euribor is an average of he raes a which banks believe a prime bank can ge unsecured funding Boh raes are quoed for a range of enors, where he 3m and 6m are he mos widely moniored An overnigh indexed swap is a conrac beween wo paries in which one pary pays a fixed rae (he OIS rae) agains receiving he geomeric average of he (compound) overnigh rae over he erm of he conrac In he US, he overnigh rae is he effecive Federal Funds rae whereas in Europe i is he Euro Overnigh Index Average (Eonia) rae 2 The OIS rae can now be viewed upon as a measure of he marke s expecaion on he overnigh rae unil mauriy (Thornon, 2009) [29] Because no principal is exchanged and since funds ypically are exchanged only a mauriy here is very lile defaul risk inheren in he OIS marke Due o he low risk of defaul associaed wih he OIS rae he spread beween he Libor and he OIS rae should give an indicaion of he defaul risk in he inerbank marke In fac, he Libor-OIS spread is considered a much wider measure of he healh of he banking sysem, for example Morini (2009) [21] emphasizes ha liquidiy risk 3 also is explanaory for he Libor-OIS spread Cui e al (2012) [6] moreover suggess ha increased overall marke volailiy and indusry-specific problems may cause he Libor- OIS spread o widen A general fligh o safey whereby banks are relucan o ie-up liquidiy over longer periods of ime may also cause he spread o increase, as menioned in he Swedish Riksbank s survey of he Swedish financial markes (2012) [28] Recenly, Filipovic and Trolle (2012) [7] suggesed an approach of decomposing he Libor-OIS spread ino defaul and non-defaul componens Figure 11 depics he Libor-OIS and Euribor-OIS spreads for 3m raes As menioned in Sengupa (2008) [27], he Libor-OIS spread spiked a 365 basis poins on Ocober 10h 2008, presumably due o he broad illiquidiy wave ha followed he bankrupcy of 1 A lis of he banks conribuing o he Libor fixing is found a hp://wwwbbaliborcom/panels/usd 2 The effecive Federal Funds rae is compued as a ransacion-weighed average of he raes on overnigh unsecured loans ha banks make beween each oher The banks in quesion do no enirely coincide wih he Libor panel, which is he case for he Eonia rae 3 Liquidiy risk is defined as he risk ha banks canno conver heir asses ino cash 2

13 Lehman Brohers on Sepember 15h Boh he Libor-OIS and Euribor-OIS spreads hen narrowed o a relaively sable level below 40 basis poins Unil mid-2011 he spreads were highly correlaed and ended o follow each oher, however in he second half of ha year he siuaion in Europe deerioraed and he Euribor-OIS spread peaked a 100 basis poins As of oday, he spreads have ye again narrowed and lie beween basis poins Libor/Euribor-OIS Spread (bps) Libor-OIS Spread Euribor-OIS Spread Figure 11: The 3m Libor-OIS and 3m Euribor-OIS spreads over a 5 year period (Source: Daasream) Anoher, and complemenary o he Libor-OIS spread, measure of credi risk is he TED-spread In he US, i is defined as he difference of he 3m Libor and he 3m T-bill rae whereas in Europe i equals he difference of he 3m EUR Libor (no o be confused wih he Euribor) and he average 3m spo rae on AAA-raed European governmen bonds The TED spread is hus a measure of he risk premium required by banks for lending o oher banks insead of o he governmen Hence, when he TED spread widens i is a sign of higher perceived counerpary risk, causing Libor raes o increase and governmen yields o decrease (fligh o safey) Figure 12 picures he American and European TED spread over a 5 year inerval By comparing wih Figure 11, i is seen ha he TED spreads and Libor-OIS spreads are ighly correlaed 3

14 USD TED Spread EUR TED Spread TED Spread (bps) Figure 12: The American and European TED spreads over a 5 year period (Source: Daasream) 12 Tenor Basis Spreads A enor basis swap is a floaing for floaing swap where he paymens are linked o indices of differen enors The paymens may for example be 6m Libor semiannually on he firs leg and 3m Libor quarerly on he oher Tuckman and Porfirio (2003) [30] shows ha in a defaul-free environmen, a enor basis swap should rade fla This means ha lenders are indifferen beween receiving he 6m rae semiannually or he 3m rae rolled over every quarer, and he same goes for oher enors In realiy, he Libor raes have buil in credi premia and i is an acceped fac ha hese premia differ beween enors For example, lending a 6m Libor is associaed wih more counerpary risk han rolling lending a 3m Libor In order o clear markes, he 6m Libor mus hus be se higher han he rae implied by he 3m Libor in order o compensae for he higher counerpary risk However, in a enor basis swap counerpary risk can be eliminaed wih collaeralizaion and he advanage of receiving 6m Libor is miigaed by a spread added o he leg paying 3m Libor Hence, in he presence of credi risk enor basis swaps do no rade fla, bu wih a spread added o he leg wih he shorer enor Morini (2009) [21] suggess some explanaions as o why lending a a longer enor is associaed wih more counerpary risk as compared o rolling lending a a shorer enor Firsly, in case of defaul in he 3m-6m period, he 6m lender loses all his ineres whereas he 3m roller receives ineres for he firs 3 monhs Even hough boh lenders lose he 4

15 noional, he 3m roller is beer off Also, if he credi condiions of he counerpary worsen during he firs 3 monhs he 3m roller can exi a par and move on o anoher counerpary The 6m lender insead has o unwind he posiion a a cos ha incorporaes he increased risk of defaul Compared o he 3m roller ha exis a par, he 6m lender is worse off However, in he opposie siuaion, ie ha he credi condiions for he counerpary improve, he 6m lender may be beer off han he 3m roller This suggess ha here is no overall gain for he 3m roller, bu since here are commercial reasons for no unwinding a conrac when i is convenien for he lender, he 3m roller has an advanage Prior o Augus 2007 he spreads in he enor basis swap marke (enor basis spreads) were never higher han 10 basis poins The spreads sared widening during he fall of 2007 and spiked during he Lehman crash in Sepember 2008 As he enor basis spreads and Libor/Euribor-OIS spreads o some exen boh measure counerpary risk i is no surprising ha hey are posiively correlaed As of oday, he USD enor basis spread is a mos 15 basis poins (3m vs 6m Libor) a shor mauriies The enor basis spreads end o decrease as he mauriy increases and he difference in enor becomes less imporan, and for mauriies greaer han 10 years i rarely exceeds 10 basis poins 13 Cross Currency Basis Spreads A (consan noional) cross currency swap (CCS) exchanges he floaing rae in one currency for he floaing rae in anoher currency, plus he noionals a iniiaion and expiraion On November 14h 2012 one USD was worh 0787 EUR A ypical CCS could hus look as follows: Exchange 1 USD for 0787 EUR a iniiaion Exchange 3m Libor on 1 USD for 3m Euribor less 21 basis poins on 0787 EUR quarerly for 10 years Exchange 1 USD for 0787 EUR a expiraion In due course i will be eviden where he spread of 21 basis poins comes from Imagine a CCS ha exchanges he defaul-free Eonia rae for he defaul-free Federal Funds rae Tuckman and Porfirio (2003) [30] shows ha such swap should rade fla Indeed, paying 1 USD oday, receiving he defaul-free Federal Funds rae on 1 USD and finally receiving 1 USD a expiraion should be worh 1 USD oday Since a similar argumen can be made wih he EUR leg he swap should rade wihou a spread However, quoed cross currency swaps exchange Libor raes ha are no defaul-free One may hus decompose a CCS ino a porfolio of hree swaps; a cross currency swap 5

16 ha exchanges he defaul-free Eonia rae for he defaul-free Federal Funds rae, a USD enor basis swap ha exchanges he Federal Funds rae for he 3m Libor and a EUR enor basis swap ha exchanges he Eonia for he 3m Euribor I is now apparen ha he cross currency basis spread derives from he difference beween local enor basis spreads Now assume ha he 3m Euribor has more credi risk han he 3m Libor In a collaeralized swap wihou defaul risk a sream of 3m Euribor would hen be worh more han a sream of 3m Libor To compensae for his advanage a negaive spread is added o he leg paying EUR The siuaion above is exacly wha prevails on he markes Figure 13 shows how he 3m USDEUR cross currency basis spread has been negaive during he las hree years I is clearly seen ha he spread reached 150 basis poins in he laer half of 2011, presumably caused by he hen worsening siuaion in he Euro area As he markes calmed he spread narrowed and i is now less han 30 basis poins for all mauriies USDEUR Cross Currency Basis Spread (bps) USDEUR Spread Figure 13: The 3m USDEUR cross currency basis spread over a 3 year period (Source: Daasream) 14 Previous Research The works of Hull (2011) [17] and Ron (2000) [26] cover how o price ineres rae swaps in a marke absen of basis spreads The focus lies on boosrapping a single yield curve ha is used boh for discouning and exracing forward raes This approach is briefly 6

17 covered in Secion 15 Once equipped wih a discree se of yields, various inerpolaion echniques for obaining a coninuous yield curve are discussed in Hagan and Wes (2006) [13] and Hagan and Wes (2008) [14] To avoid arbirage, he inerpolaion scheme needs o produce posiive forward raes I is also desired ha he obained forward raes are sable and ha he inerpolaion funcion only changes nearby if an inpu is changed (ie i is local) Henrard (2007) [15] akes one sep owards refining he convenional pricing framework by addressing he effecs from changing he discouning curve In Henrard (2010) [16], he furher proposes a valuaion framework where one forward curve is buil for each Libor enor Similar work is done in Amerano and Bianchei (2009) [1], where a scheme ha is able o recover he marke swap raes is developed However, as muliple discoun raes exis wihin he same currency, heir model is subjec o arbirage The arbirage-free model proposed in Bianchei (2008) [2] is in one sense an improvemen, bu as noed in Fujii e al (2009a) [9] curve calibraion canno be separaed from opion calibraion, which makes he model somewha impracical Mercurio (2009) [20] inroduces a new Libor marke model ha is based on modeling he join evoluion of implied forward raes and FRA raes, where he log-normal case wih and wihou sochasic volailiy is analyzed Johannes and Sundaresan (2009) [19] and Whiall (2010a) [32] furher develop he muli-curve pricing framework by considering he impac of collaeralizaion on swap raes, whereas Whiall (2010b) [31] discusses which discoun rae o use in an uncollaeralized agreemen Oher works on he same opic include Morini (2009) [21] and Chibane e al (2009) [5] Fujii e al (2010) [11] presens a mehod ha consisenly reas ineres rae swaps, enor basis swaps, overnigh indexed swaps and cross currency basis swaps, where he effecs from collaeralizaion are explicily addressed This framework is refined in Fujii e al (2009a) [9], where a model of dynamic basis spreads is inroduced Finally, Filipovic and Trolle (2012) [7] proposes a erm srucure of inerbank risk ha is derived from observed basis spreads Moreover, he erm srucure is decomposed ino defaul (credi) and non-defaul componens I is shown ha defaul risk increases wih mauriy whereas he non-defaul componen is more dominan in he shor erm 15 FRA and Swap Pricing Before he Financial Crisis The forward Libor rae conraced a ime for [T n 1, T n ] is defined by L(, T n 1, T n ) = 1 ( ) Z(, Tn 1 ) 1, δ n 1,n Z(, T n ) 7

18 where δ n 1,n is he day coun facor and Z(, T i ) he ime- price of a defaul-free zero coupon bond mauring a T i Similarly, he spo Libor for [T n 1, T n ] is given by L(T n 1, T n ) = 1 ( ) 1 δ n 1,n Z(, T n ) 1 Le Q Tn be he forward measure wih Z(, T n ) as numéraire (an inroducion o he forward measure is given in Appendix A) and le E Tn [ ] = E QTn [ F ] I now holds ha E Tn [L(T n 1, T n )] = 1 [ ] Z(Tn 1, T E Tn n 1 ) 1 = 1 ( ) Z(, Tn 1 ) 1 δ n 1,n Z(T n 1, T n ) δ n 1,n Z(, T n ) = L(, T n 1, T n ) A forward rae agreemen (FRA) is a conrac in which one pary receives L(T n 1, T n ) a T n whereas he counerpary receives a fixed rae K simulaneously The ne payoff a T n is hus V Tn = δ n 1,n (L(T n 1, T n ) K) Hence, he value a some arbirary < T n equals V = E Q [δ n 1,n (L(T n 1, T n ) K)Z(, T n )] = δ n 1,n (E Tn [L(T n 1, T n )] K)Z(, T n ) = δ n 1,n (L(, T n 1, T n ) K)Z(, T n ) A iniiaion i mus hold ha V 0 = 0 and we can hen solve for he fixed rae K An ineres rae swap (IRS) is no more han a porfolio of forward rae agreemens and can herefore be priced similarly Since he presen value a iniiaion has o equal zero we ge C(, T N ) δm 1,mZ(, fi T m ) = m=1 δn 1,nE fl Q [L(T n 1, T n )Z(, T n )], where C(, T N ) is he ime- fixed rae for a swap mauring a T N numéraire o Z(, T i ) if n = i we ge C(, T N ) δm 1,mZ(, fi T m ) = m=1 δ fl n 1,nE Tn [L(T n 1, T n )]Z(, T n ) = By changing he δn 1,nL(, fl T n 1, T n )Z(, T n ) 8

19 By he definiion of he forward Libor rae we arrive a C(, T N ) δm 1,mZ(, fi T m ) = (Z(, T n 1 ) Z(, T n )) = Z(, T 0 ) Z(, T N ), m=1 and i is now a simple maer o deermine he swap rae C(, T N ) A more in-deph inroducion o he convenional way of pricing swaps using only one forward curve is given in Björk (2009) [3] 9

20 2 Theoreical Background In his secion i is described how o price ineres rae swaps (IRS), enor basis swaps (TS) and cross currency basis swaps (CCS) consisenly wih each oher in a muli currency seup, boh wih and wihou collaeralizaion The heory is primarily based on Fujii e al (2010) [11], Fujii e al (2009a) [9] and Fujii e al (2009b) [10] 21 Curve Consrucion wihou Collaeral Using observable quoes on he swap marke we derive a discouning curve as well as several (index-linked) forward curves under he assumpion ha no collaeral agreemen is in place Available insrumens include IRS, TS and he radiional CCS where he noional is consan unil mauriy (an inroducion o he newer kind of CCS, he marko-marke CCS, is found in Appendix B) We assume a Libor ha accuraely reflecs he funding cos of he insiuion a hand as discouning rae, for simpliciy he USD 3m Libor The resul will be a se of curves ha can price any uncollaeralized swap and ha is consisen wih observed marke quoes 211 A Single IRS Marke A firs we consider a single currency (USD) marke where only one kind of USD IRS is available A iniiaion i holds ha C(, T N ) δm 1,mZ(, fi T m ) = δn 1,nE fl [L(T n 1, T n )]Z(, T n ), m=1 where C(, T N ) is he ime- fair swap rae for an IRS of lengh T N, δ fi m 1,m and δ fl n 1,n are day coun facors of he fixed and floaing legs, respecively Z(, T n ) is he ime- price of a defaul free discoun bond mauring a T n and L(T n 1, T n ) is he USD 3m Libor from T n 1 o T n Surveys of day coun and swap convenions are found in Appendices C and D, respecively Unless menioned oherwise, E [] is assumed o be aken under he appropriae forward measure Since he available swaps have floaing legs linked o he USD 3m Libor and since he same rae is used for discouning, a simple no-arbirage argumen gives ha E [L(T n 1, T n )] = 1 ( ) Z(, Tn 1 ) 1 δn 1,n fl Z(, T n ) Using his relaion, he swap marke condiion becomes C(, T N ) δm 1,mZ(, fi T m ) = Z(, T 0 ) Z(, T N ), m=1 10

21 where Z(, T 0 ) is he discouning facor from ime- o he firs fixing dae (and can be deermined by he ON-rae) The discouning facors can now be uniquely deermined by sequenially solving Z(, T m ) = Z(, T 0) C(, T m ) m 1 i=1 δfi i 1,iZ(, T i ) 1 + C(, T m )δm 1,m fi This procedure requires ha all necessary mauriies are in fac raded and he difficuly ha arises when his is no he case is furher reaed in Secion 3 Also, inerpolaion has o be carried ou in order ge a coninuous curve of discouning facors and corresponding forward USD 3m-Libor raes This opic is furher covered in Secion 3 and more deeply in Appendix E 212 An IRS and TS Marke We now consider a (sill single currency) marke where TS as well as IRS wih floaing legs linked o USD Libor raes of varying enor are available To price an IRS wih a floaing leg linked o, for example, he USD 1m Libor, we canno due o he exisence of enor basis spreads use he USD 3m Libor forward curve I is hence necessary o deermine a se of USD 1m Libor forward raes This can be done by using he quoed USD 1m/3m TS, where one pary pays USD 1m Libor plus a spread monhly and receives USD 3m Libor quarerly The resuling condiions become C(, T N ) k=1 δm 1,mZ(, fi T m ) = m=1 δ 3m n 1,nE [L 3m (T n 1, T n )]Z(, T n ), δ 1m k 1,k(E [L 1m (T k 1, T k )] + T S(, T N ))Z(, T k ) = δ 3m n 1,nE [L 3m (T n 1, T n )]Z(, T n ), where T S(, T N ) is he ime- 1m/3m enor basis spread a mauriy T N The discoun facors and corresponding USD 3m Libor raes are compued as in Secion 211 Through he basis swaps and proper inerpolaion i is hen possible o compue a coninuous se of USD 1m Libor raes I is also sraighforward o derive forward curves of differen enors (6m, 1y for example) by adding more TS 213 Inroducing he Consan Noional CCS In his secion, we expand he model o allow for muliple currencies and for he exisence of a consan noional CCS More specifically, USD and EUR are he relevan currencies and he USD 3m Libor is sill he discouning rae Curve consrucion for US-based insiuions is done as in Secions 211 and 212, however for European insiuions one has o accoun for he cross currency basis spread inheren in he CCS Thus, he 11

22 condiions for he EUR raes (Euribor) become C(, T N ) l=1 δ fi,e l 1,l Ze (, T l ) = m=1 δ 6m,e m 1,mE e [L 6m,e (T m 1, T m )]Z e (, T m ), δ 3m,e n 1,n(E e [L 3m,e (T n 1, T n )] + T S(, T N ))Z e (, T n ) N e ( Z e (, T 0 ) + = m=1 δ 6m,e m 1,mE e [L 6m,e (T m 1, T m )]Z e (, T m ), δ 3m,e n 1,n(E e [L 3m,e (T n 1, T n )] + CCS(, T N ))Z e (, T n ) + Z e (, T N ) = f() ( Z $ (, T 0 ) + δ 3m,$ n 1,n(E $ [L 3m,$ (T n 1, T n )]Z $ (, T n ) + Z $ (, T N ) where CCS(, T N ) is he ime- USDEUR cross currency basis spread a mauriy T N, N e is he EUR noional per USD and f() is he ime- USDEUR exchange rae The e- and $-indices indicae ha he variable is relevan for EUR and USD, respecively Since we sill rea he USD 3m-Libor as he discouning rae, he USD floaing leg of he CCS equals zero and i holds ha δ 3m,e n 1,nE e [L 3m,e (T n 1, T n )]Z e (, T n ) = Z e (, T 0 ) Z e (, T N ) CCS(, T N ) Afer furher eliminaion of floaing pars we easily arrive a C(, T N ) l=1 δ fi,e l 1,l Ze (, T l ) + (CCS(, T N ) T S(, T N )) δ 3m,e n 1,nZ e (, T n ) δ 3m,e n 1,nZ e (, T n ) = Z e (, T 0 ) Z e (, T N ), ) ), and i is now possible o sequenially compue he EUR discouning facors Using he quoed IRS and TS one can hen derive he 3m- and 6m- forward Euribor curves By adding more TS, i is of course possible o derive forward Euribor curves wih oher enors Evidenly, he EUR discouning facors also depend on he enor basis spreads and cross currency basis spreads, and no only on he swap raes Therefore, if holding a simple EUR IRS, one also has o hedge for sensiiviies inheren in hese spreads Throughou his survey, he USD 3m Libor has been considered he discouning rae Using anoher discouning rae poses no problem, as he mehodology of deriving discoun facors and 12

23 forward raes will be analogous o wha has been covered herein 22 Curve Consrucion wih Collaeral According o he ISDA Margin Survey [18], close o 80% of all rades wih fixed income derivaives during 2012 were collaeralized For large dealers, his number approaches 90% As he exisence of a collaeral agreemen subsanially reduces he credi risk inheren in he rade i becomes quesionable o apply sandard Libor discouning when pricing a cerain produc In his secion, i is explained how o price a collaeralized produc and more specifically how collaeralizaion affecs curve consrucion for swap pricing 221 Pricing of Collaeralized Derivaives In a collaeralized rade, he pary whose conrac has a posiive presen value receives collaeral from he counerpary To compensae for his he pary has o pay a cerain margin called collaeral rae on he ousanding collaeral In case of cash collaeral, he collaeral rae is usually he overnigh rae for he collaeral currency, ie he Federal Funds rae for USD or he Eonia rae for EUR To avoid problems wih non-lineariy, we assume ha mark-o-marke and collaeral posing is made coninuously Also, he posed cash collaeral is assumed o cover 100% of he conrac s presen value As collaeral posing is commonly done on a daily basis, hese simplificaions are probably no oo far from realiy, a leas no for liquid currencies Since counerpary defaul risk can now be negleced, i is possible o recover a linear relaionship among paymens Wih collaeral posed in domesic currency and collaeral rae c(s) a ime s, he ime- value h() of a derivaive h mauring a T is given by he following proposiion Proposiion 21 h() = E Q [e ] T c(s)ds h(t ), where E Q [] is he expecaion wih he money-marke accoun as numéraire For a proof we refer o Appendix F If collaeral is posed in foreign currency, he value a ime of he derivaive is furhermore given by h() = E Q [e T r(s)ds (e ) ] T (rf (s) c f (s))(d)s h(t ), where r(s) and r f (s) are he domesic and foreign risk-free raes, respecively c f (s) is he collaeral rae on collaeral posed in foreign currency I can now be seen ha in a collaeralized rade fuure cash flows should be discouned by he collaeral rae As 13

24 he overnigh rae can differ significanly from he Libor, i becomes eviden ha Libor discouning is no longer appropriae 222 Inroducing he OIS Under he assumpion ha he collaeral rae on cash equals he overnigh rae one can deermine collaeralized discouned facors by using quoed overnigh indexed swaps (OIS) An OIS exchanges a fixed coupon for a daily compounded overnigh rae, where he daes of he wo paymens ypically coincide Hence, beween wo paymen daes T l 1 and T l he floaing leg pays T l s=t l 1 (1 + δ s c(s)) 1 muliplied by he noional Here, δ s is he daily accrual facor and c(s) is he collaeral rae a ime s By approximaing daily compounding wih coninuous compounding, we ge T l s=t l 1 (1 + δ s c(s)) 1 e Tl T l 1 c(s)ds 1 If we furher assume ha he OIS is perfecly collaeralized wih 100% cash i holds ha (as shown in Secion 221) S(, T N ) l=1 δ fi l 1,lE Q [ e T l ] c(s)ds = l=1 E Q [ e T l ( Tl )] c(s)ds T c(s)ds e l 1 1, where S(, T N ) is he ime- fair swap rae for an OIS of lengh T N By denoing he collaeralized discoun facors wih D(, T l ) = E Q [ e T l ] c(s)ds we arrive a S(, T N ) δl 1,lD(, fi T l ) = D(, T 0 ) D(, T N ) l=1 I is now a simple maer o sequenially derive he discoun facors by D(, T l ) = D(, T 0) S(, T l ) l 1 i=1 δfi i 1,iD(, T i ), 1 + S(, T l )δl 1,l fi and a coninuous discoun curve is obained by appropriae splining Informaion on common marke convenions for overnigh indexed swaps is found in Appendix D 14

25 223 Curve Consrucion in a Single Currency In a single currency, he consrucion of forward Libor curves of differen enors is very similar o ha of Secion 212 Afer deriving he collaeralized discoun curve as in Secion 222, one can compue, le s say, 1m and 3m Libor forward raes hrough he condiions C(, T N ) k=1 δm 1,mD(, fi T m ) = m=1 δ 3m n 1,nE c [L 3m (T n 1, T n )]D(, T n ), δ 1m k 1,k(E c [L 1m (T k 1, T k )] + T S(, T N ))D(, T k ) = δ 3m n 1,nE c [L 3m (T n 1, T n )]D(, T n ), where E c [] is he expecaion wih he appropriae D(, T n ) as numéraire I is of course possible o add more TS o derive forward curves wih oher enors 224 Curve Consrucion in Muliple Currencies Unlike he single-currency seup, where collaeral and swap paymens are in he same currency, we mus now allow for collaeral and swap paymens o be of differen currencies As in Secion 213 he consan noional CCS will be used as calibraion insrumen (how o use he mark-o-marke CCS for curve calibraion is covered in Appendix B) and he relevan currencies will be USD and EUR Also, he Federal Funds rae will be reaed as he risk-free rae Since i is also he collaeral rae for USD, i now holds ha D $ (, T ) = E Q$ [e T ] c $ (s)ds = E Q$ [e ] T r $ (s)ds = Z $ (, T ) Condiions for USD-collaeralized USD swaps (wih he USD 1m/3m TS) are hus S $ (, T N ) C $ (, T N ) k=1 l=1 m=1 δ fi,$ l 1,l Z$ (, T l ) = Z $ (, T 0 ) Z $ (, T N ), δ fi,$ m 1,mZ $ (, T m ) = δ 1m,$ k 1,k (E$ [L 1m,$ (T k 1, T k )] + T S $ (, T N ))Z $ (, T k ) = δ 3m,$ n 1,nE $ [L 3m,$ (T n 1, T n )]Z $ (, T n ), δ 3m,$ n 1,nE $ [L 3m,$ (T n 1, T n )]Z $ (, T n ) 15

26 Similarly, condiions for EUR-collaeralized EUR swaps (wih he EUR 3m/6m TS) are S e (, T N ) C e (, T N ) l=1 l=1 δ fi,e l 1,l De (, T l ) = D e (, T 0 ) D e (, T N ), δ fi,e l 1,l De (, T l ) = m=1 δ 6m,e m 1,mE c,e [L 6m,e (T m 1, T m )]D e (, T m ), δ 3m,e n 1,n(E c,e [L 3m,e (T n 1, T n )] + T S e (, T N ))D e (, T n ) = m=1 Of course, more TS condiions can be added if needed δ 6m,e m 1,mE c,e [L 6m,$ (T m 1, T m )]D e (, T m ) We now urn our aenion o USD-collaeralized EUR swaps Assume he exisence of a USD cash-collaeralized USDEUR consan noional CCS Wih he resuls of Secion 221 i holds ha 4 Z e (, T 0 ) + = N $ f() ( δ 3m,e n 1,n(E e [L 3m,e (T n 1, T n )] + CCS(, T N ))Z e (, T n ) + Z e (, T N ) Z $ (, T 0 ) + δ 3m,$ n 1,nE $ [L 3m,$ (T n 1, T n )]Z $ (, T n ) + Z $ (, T N ) where he righ-hand side is previously known I is however no possible o derive boh he EUR zero coupon bond prices and he EUR forward raes hrough his condiion only Ideally quoes for USD-collaeralized EUR IRS and TS are available, which would allow us o easily derive he ses of discoun facors and forward raes Alernaively, one could assume ha E e [L e (T n 1, T n )] = E c,e [L e (T n 1, T n )] In his approach we hus neglec he change of numéraire, and he approximaion is reasonable if he EUR risk-free and collaeral raes have similar dynamic properies This 4 The sum in he LHS is given by [ δ 3m,e n 1,n EQe e Tn ( r e (s)ds e ) ] Tn (r $ (s) c $ (s))ds (L(T n 1, T n ) + CCS(, T N )) = = [ δ 3m,e n 1,n EQe e ] Tn r e (s)ds (L(T n 1, T n ) + CCS(, T N )) δ 3m,e n 1,n (Ee [L 3m,e (T n 1, T n )] + CCS(, T N ))Z e (, T n ) ), 16

27 enables us o sequenially derive he EUR zero coupon bond prices We finally consider he case of EUR-collaeralized USD swaps The condiions for EUR-collaeralized USD IRS and consan noional CCS are C $ (, T N ) m=1 [ δm 1,mZ fi,$ $ (, T m )E $ = e Tm ] (r e (s) c e (s))ds [ δn 1,nZ 3m,$ $ (, T n )E $ e ] Tn (r e (s) c e (s))ds L 3m,$ (T n 1, T n ) = N $ ( D e (, T 0 ) + [ δn 1,nZ 3m,$ $ (, T n )E $ e ] Tn (r e (s) c e (s))ds L 3m,$ (T n 1, T n ), δ 3m,e n 1,n(E c,e [L 3m,e (T n 1, T n )] + CCS(, T N ))D e (, T n ) + D e (, T N ) where CCS(, T N ) and C $ (, T N ) are he fair raes for he EUR-collaeralized CCS and USD IRS, respecively Wih hese insrumens a hand, i is possible o deermine [ e Tm ] (r e (s) c e (s))ds and [ e ] Tn (r e (s) c e (s))ds L 3m,$ (T n 1, T n ) ), for each m and n By adding EUR-collaeralized USD enor basis swaps, i is also possible o derive Libor curves of oher enors 17

28 3 Implemenaion This secion describes how he various discouning and forward curves are derived We deal wih daa on collaeralized (in domesic currency) swaps as found in Appendix G and consider he USD and EUR markes separaely The implemenaion is done in Pyhon, where he SWIG-bindings for he C++ library QuanLib 5 are used for calendar and day couner classes 31 Building he USD Curves The discouning curve is firs buil hrough he USD OIS marke Being equipped wih he he relevan discouning facors we are allowed o exrac he USD 3m forward raes hrough IRS quoes Tenor basis spreads are hen used o derive he USD 1m and 6m forward curves 311 The USD Discouning Curve The relevan daa is found in Table G3 For swaps of lengh less han one year here is only one paymen a mauriy The discoun facors of shores mauriies are hence given by D $ (0, i) = S $ (0, i)δ fi,$ 0,i, i = 1d, 1w, 2w, 3w, 1m, 2m,, 11m For mauriies greaer han 1 year, here is one paymen a he end of each year As shown in Secion 222, he relevan condiion is S $ (, T N ) l=1 δ fi,$ l 1,l D$ (, T l ) + D $ (, T N ) = 1, where we have assumed ha D $ (, T 0 ) = 1 To be able o solve for each discoun facor requires ha here is a liquid marke in yearly mauriies for 1 year up o 50 years Since his is no he case we make an approximaion by using inerpolaion wih cubic splines o esimae he necessary quoes Anoher way of dealing wih his issue is covered in Hagan and Wes (2006) [13], where insead a mehod of ieraion is applied Having esimaed 5 For documenaion see hp://quanliborg/docsshml 18

29 all required OIS raes, we are able o solve he following sysem of equaions: S $ (0, 1y)δ fi,$ 0,1y S $ (0, 2y)δ fi,$ 0,1y S $ (0, 2y)δ fi,$ 1y,2y S $ (0, 50y)δ fi,$ 0,1y S $ (0, 50y)δ fi,$ 49y,50y + 1 D $ (0, 1y) D $ (0, 2y) D $ (0, 50y) 1 1 = We are now supplied wih esimaes of yearly discoun facors from 1 year up o 50 years However, we only use hose wih mauriies corresponding o quoed overnigh index swaps To obain a coninuous se of discoun facors, inerpolaion wih cubic splines is applied o his subse 312 The USD 3m Forward Curve To build he USD 3m forward curve we use he 3m spo Libor in Table G1 ogeher wih he IRS quoes in Table G3 The relevan condiion is now C $ (, T N ) m=1 δ fi,$ m 1,mD $ (, T m ) = δ 3m,$ n 1,nE c,$ [L 3m,$ (T n 1, T n )]D $ (, T n ), and is previously known from Secion 224 As we have already buil he discouning curve i is now possible o exrac he 3m forward raes However, jus as in Secion 311 we need o inerpolae he swap curve o obain esimaes of all necessary swap raes Also, since he fixed leg pays semiannually and he floaing leg quarerly he resuling sysem of equaions would become underdeermined To miigae his problem we assume ha he forward raes are piecewise fla, ie ha 1 E c,$ E c,$ [L 3m,$ (6m, 9m)] = E c,$ [L 3m,$ (9m, 12m)], [L 3m,$ (12m, 15m)] = E c,$ [L 3m,$ (15m, 18m)] 19

30 and so on Since we already know he 3m spo Libor, he resuling sysem of equaions is δ 3m,$ 3m,6m D$ (0, 6m) 0 0 E c,$ [L 3m,$ (3m, 6m)] δ 3m,$ 3m,6m D$ (0, 6m) 12m i=9m δ3m,$ i 3m,i D$ (0, i) 0 E c,$ [L 3m,$ (9m, 12m)] 0 δ 3m,$ 3m,6m D$ (0, 6m) 12m i=9m δ3m,$ i 3m,i D$ (0, i) 600m i=597m δ3m,$ i 3m,i D$ (0, i) E c,$ [L 3m,$ (597m, 600m)] C $ (0, 6m) 6m n=6m δfi,$ n 6m,n D$ (0, n) δ 3m,$ 0,3m Ec,$ [L 3m,$ (0, 3m)]D $ (0, 3m) C $ (0, 1y) 12m n=6m δfi,$ n 6m,n D$ (0, n) δ 3m,$ 0,3m Ec,$ [L 3m,$ (0, 3m)]D $ (0, 3m) = C $ (0, 50y) 600m n=6m δfi,$ n 6m,n D$ (0, n) δ 3m,$ 0,3m Ec,$ [L 3m,$ (0, 3m)]D $ (0, 3m) By solving his sysem we obain an array of USD 3m forward Libors, however we choose o discard hose wih mauriies ha do no coincide wih he mauriies of quoed ineres rae swaps Inerpolaion wih cubic splines on he remainder hen gives us he coninuous 3m forward curve 313 The USD 1m Forward Curve To consruc he USD 1m forward curve we use he 1m spo Libor in Table G1, he quoed enor basis spreads in Table G2 and he quoed 1m IRS in Table G3 The 1m implied swap rae is firs compued by C $ (0, 1m) = δ1m,$ 0,1m δ fi,$ 0,1m E c,$ [L 1m,$ (0, 1m)], and is added o he array of quoed IRS wih mauriies up o 12 monhs We exrac he forward raes wih mauriies 12m hrough he condiion C $ (, T N ) k=1 δ fi,$ k 1,k D$ (, T k ) = k=1 δ 1m,$ k 1,k Ec,$ [L 1m,$ (T k 1, T k )]D $ (, T k ), 20

31 and arrive a he following sysem of equaions: δ 1m,$ 0,1m D$ (0, 1m) 0 0 δ 1m,$ 0,1m D$ (0, 1m) δ 1m,$ 1m,2m D$ (0, 2m) 0 0 δ 1m,$ 0,1m D$ (0, 1m) δ 1m,$ 1m,2m D$ (0, 2m) δ 1m,$ 11m,12m D$ (0, 12m) = C $ (0, 1m) 1m m δfi,$ C $ (0, 2m) 2m m δfi,$ C $ (0, 12m) 12m m δfi,$ n 1m,n D$ (0, n) n 1m,n D$ (0, n) n 1m,n D$ (0, n) E c,$ [L 1m,$ (0, 1m)] E c,$ [L 1m,$ (1m, 2m)] [L 1m,$ (11m, 12m)] E c,$ Noe ha we do no have o inerpolae he swap curve in his case Nex, we compue he implied enor basis spreads for mauriies 3m, 6m, 9m and 12m by using he derived shor-end of he forward curve These spreads are hen added o he array of quoed spreads Since we have already obained he discouning curve and he 3m forward curve, we can now use he enor basis spreads and exrac he 1m forward curve hrough k=1 δ 1m,$ k 1,k (Ec,$ [L 1m,$ (T k 1, T k )] + T S $ (, T N ))D $ (, T k ) = δ 3m,$ n 1,nE c,$ [L 3m,$ (T n 1, T n )]D $ (, T n ), where inerpolaion of he basis spreads will be necessary Due o he same issue as in Secion 312 we assume piecewise fla forward raes, ie E c,$ E c,$ [L 1m,$ (12m, 13m)] = E c,$ [L 1m,$ (15m, 16m)] = E c,$ [L 1m,$ (13m, 14m)] = E c,$ [L 1m,$ (14m, 15m)], [L 1m,$ (16m, 17m)] = E c,$ [L 1m,$ (17m, 18m)], which allows us o build a solvable sysem of equaions 21

32 Wih we arrive a A = 12m i=1m δ 1m,$ i 1m,i Ec,$ [L 1m,$ (i 1m, i)]d $ (0, i) 15m i=13m δ1m,$ i 1m,i D$ (0, i) 0 0 E c,$ [L 1m,$ (14m, 15m)] 15m i=13m δ1m,$ i 1m,i D$ (0, i) 18m i=16m δ1m,$ i 1m,i D$ (0, i) 0 E c,$ [L 1m,$ (17m, 18m)] 0 15m i=13m δ1m,$ i 1m,i D$ (0, i) 18m i=16m δ1m,$ i 1m,i D$ (0, i) 600m i=598m δ1m,$ i 1m,i D$ (0, i) E c,$ [L 1m,$ (599m, 600m)] 15m n=3m δ3m,$ n 3m,n Ec,$ [L 3m,$ (n 3m, n)]d $ (0, n) T S $ (0, 15m) 15m i=1m δ1m,$ i 1m,i D$ (0, i) A 18m n=3m δ3m,$ n 3m,n Ec,$ [L 3m,$ (n 3m, n)]d $ (0, n) T S $ (0, 18m) 18m i=1m δ1m,$ i 1m,i D$ (0, i) A = 600m n=3m δ3m,$ n 3m,n Ec,$ [L 3m,$ (n 3m, n)]d $ (0, n) T S $ (0, 600m) 600m i=1m δ1m,$ i 1m,i D$ (0, i) A By solving his and applying inerpolaion wih cubic splines we have successfully derived he coninuous USD 1m forward curve 314 The USD 6m Forward Curve We firs compue he implied 6m 3m/6m enor basis spread by using he 6m spo Libor in Table G1 and he derived USD 3m forward curve, and hen add his spread o he array of quoed spreads in Table G2 Afer inerpolaing he basis spreads o esimae semiannual quoes we can successfully solve δ 6m,$ 0,6m D$ (0, 6m) 0 0 E c,$ [L 6m,$ (0, 6m)] δ 6m,$ 0,6m D$ (0, 6) δ 6m,$ 6m,12m D$ (0, 12m) 0 E c,$ [L 6m,$ (6m, 12m)] 0 δ 6m,$ 0,6m D$ (0, 6) δ 6m,$ 6m,12m D$ (0, 12m) δ 6m,$ 594m,600m D$ (0, 600m) E c,$ [L 6m,$ (594m, 600m)] 6m n=3m δ3m,$ n 3m,n (Ec,$ [L 3m,$ (n 3m, n)] + T S $ (0, 6m))D $ (0, n) 12m n=3m δ3m,$ n 3m,n (Ec,$ [L 3m,$ (n 3m, n)] + T S $ (0, 12m))D $ (0, n) = 600m n=3m δ3m,$ n 3m,n (Ec,$ [L 3m,$ (n 3m, n)] + T S $ (0, 600m))D $ (0, n) 22

33 Nex, we filer ou he forward raes wih mauriies ha correspond o mauriies of quoed insrumens Afer performing inerpolaion (again wih cubic splines) on his subse we have successfully compleed he consrucion of he USD 6m forward curve 32 Building he EUR Curves In his secion i is described how he EUR discouning curve is consruced hrough he OIS marke, how EUR 1m, 3m, 6m and 1y forward curves are buil using quoed ineres rae and enor basis swaps and, finally, how o derive he discouning curve for USD-collaeralized EUR swaps 321 The EUR Discouning Curve The consrucion of he EUR discouning curve is analogous o ha of he USD discouning curve For mauriies < 1y we ge he discoun facors by D e (0, i) = S e (0, i)δ fi,e 0,i, i = 1d, 1w, 2w, 3w, 1m, 2m,, 11m For mauriies 1y he condiion S e (, T N ) l=1 δ fi,e l 1,l De (, T l ) + D e (, T N ) = 1 holds, and we hus end up wih he following se of equaions: S e (0, 1y)δ fi,e 0,1y S e (0, 2y)δ fi,e 0,1y S e (0, 2y)δ fi,e 1y,2y S e (0, 50y)δ fi,e 0,1y S e (0, 50y)δ fi,e 49y,50y + 1 D e (0, 1y) D e (0, 2y) D e (0, 50y) 1 1 = To obain a coninuous se of discoun facors we apply he same procedure as in Secion 311 The inpu daa is found in Table G7 322 The EUR 6m Forward Curve The EUR 6m forward curve is buil in a similar manner as he USD 3m forward curve, only ha he ineres rae swaps now pay fixed annually and floaing semiannually The relevan condiion is hus C e (, T N ) l=1 δ fi,e l 1,l De (, T l ) = m=1 δ 6m,e m 1,mE c,e [L 6m,e (T m 1, T m )]D e (, T m ), 23 1

34 where he inpu swap raes and 6m Euribor are found in Tables G7 and G4, respecively Having deermined he appropriae discoun facors and assuming ha (compare wih Secion 312) E c,e E c,e he 6m forward raes are compued by solving [L 6m,e (12m, 18m)] = E c,e [L 3m,e (18m, 24m)], [L 6m,e (24m, 30m)] = E c,e [L 6m,e (30m, 36m)], δ 6m,e 6m,12m De (0, 12m) 0 0 E c,e [L 6m,e (6m, 12m)] δ 6m,e 6m,12m De (0, 12m) 24m i=18m δ6m,e i 6m,i De (0, i) 0 E c,e [L 6m,e (18m, 24m)] 0 δ 6m,e 6m,12m De (0, 12m) 24m i=18m δ6m,e i 6m,i De (0, i) 600m i=594m δ6m,e i 6m,i De (0, i) E c,e [L 6m,e (594m, 600m)] C e (0, 1y) 12m 2m δfi,e n 12m,n De (0, n) δ 6m,e 0,6m Ec,e [L 6m,e (0, 6m)]D e (0, 6m) C e (0, 2y) 24m 2m δfi,e n 12m,n De (0, n) δ 6m,e 0,6m Ec,e [L 6m,e (0, 6m)]D e (0, 6m) = C e (0, 50y) 600m 2m δfi,e n 12m,n De (0, n) δ 6m,e 0,6m Ec,e [L 6m,e (0, 6m)]D e (0, 6m) Appropriae splining (a procedure familiar by now) renders he coninuous EUR 6m forward curve 323 The EUR 1m Forward Curve To build he EUR 1m forward curve we use he 1m Euribor spo rae in Table G4, he quoed 1m/6m enor basis spreads in Table G6 and he quoed 1m ineres rae swaps in Table G7 The procedure is very similar o ha of he USD 1m forward curve in Secion 313, apar from a few differences This ime quoed 1m IRS of mauriies up o wo years are available, bu since hey are no so on a monhly basis inerpolaion of he swap curve becomes necessary Wih he inerpolaed 1m swap curve we can boosrap he shor-end of he forward curve (ie wih mauriies 2 years) Subsequenly, he implied 6m, 1y, 18m and 2y enor basis spreads are compued wih he aid of he boosrapped 1m forward curve By iniially assuming ha he long-end of he 1m forward curve is piecewise fla in inervals of 6 monhs (compare wih Secion 313) we can successfully compue forward raes for mauriies > 2 years As usual, forward raes ha correspond o mauriies 24

35 of quoed insrumens are filered ou before he coninuous EUR 1m forward curve is obained hrough inerpolaion wih cubic splines 324 The EUR 3m Forward Curve The shor-end of he EUR 3m forward curve is buil using he 3m Euribor in Table G4 and he Euribor fuures in Table G5 As Euribor conracs are available up o, and including, MAR we can successfully esimae forward raes wih mauriies up o 30 monhs using inerpolaion Afer compuing he implied 6m, 1y, 18m, 2y and 30m enor basis spreads and adding hese o he quoed 3m/6m spreads in Table G6 we end up wih (afer cusomary inerpolaion of he basis spreads and assuming piecewise fla forward raes in inervals of 6 monhs) 36m i=33m δ3m,e i 3m,i De (0, i) 0 0 E c,e [L 3m,e (33m, 36m)] 36m i=33m δ3m,e i 3m,i De (0, i) 42m i=39m δ3m,e i 3m,i De (0, i) 0 E c,e [L 3m,e (39m, 42m)] 0 36m i=33m δ3m,e i 3m,i De (0, i) 42m i=39m δ3m,e i 3m,i De (0, i) 600m i=597m δ3m,e i 3m,i De (0, i) E c,e [L 3m,e (597m, 600m)] 36m n=6m δ6m,e n 6m,n Ec,e [L 6m,e (n 6m, n)]d e (0, n) T S e (0, 36m) 36m i=3m δ3m,e i 3m,i De (0, i) A 42m n=6m δ6m,e n 6m,n Ec,e [L 6m,e (n 6m, n)]d e (0, n) T S e (0, 42m) 42m i=3m δ3m,e i 3m,i De (0, i) A =, 600m n=6m δ6m,e n 6m,n Ec,e [L 6m,e (n 6m, n)]d e (0, n) T S e (0, 600m) 600m i=3m δ3m,e i 3m,i De (0, i) A where A = 30m δ 3m,e i 3m,i Ec,e i=3m [L 3m,e (i 3m, i)]d e (0, i) Now ha boh he shor- and he long-end are esimaed we have finished he consrucion of he EUR 3m forward curve 325 The EUR 1y Forward Curve The EUR 1y forward curve is consruced using he 1y Euribor spo rae in Table G4 and he 6m/1y enor basis spreads in Table G6 Apar from all inervals in ime being wice as long, his curve is buil in exacly he same way as he USD 6m forward curve We herefore refer o Secion 314 for more deails 6 This refers o a conrac saring on he IMM dae in March 2015 and mauring on he IMM dae in June 2015 The IMM (Inernaional Moneary Marke) daes are he hird Wednesday of March, June, Sepember and December 25

36 326 The Case of USD Collaeral In his secion we derive he discouning curve for EUR derivaives ha are collaeralized in USD, where we use daa on USDEUR cross currency basis spreads as found in Table G8 As he cross currency swaps are only available wih mauriies 30 years we assume a consan basis spread for swaps of lengh 30 years Drawing from Secion 224 and assuming ha N $ = 1 and Z e (0, T 0 ) = 1 we arrive a δ 3m,e 0,3m (Ec,e [L 3m,e (0, 3m)] + b 0,3m ) δ 3m,e 0,3m (Ec,e [L 3m,e (0, 3m)] + b 0,6m ) δ 3m,e 3m,6m (Ec,e [L 3m,e (3m, 6m)] + b 0,6m ) + 1 δ 3m,e 0,3m (Ec,e [L 3m,e (0, 3m)] + b 0,600m ) V 3m + 1 V 6m + 1 =, V Z e (0, 3m) Z e (0, 6m) Z e (0, 600m) where b 0,i = CCS(0, i) and V i = f(0) ( 1 + i n=3m δ 3m,$ n 3m,nE c,$ [L 3m,$ (n 3m, n)]d $ (0, n) + D $ (0, i) where an exchange rae of f(0) = is used Having derived he discouning curve for EUR insrumens ha are collaeralized in USD we have successfully consruced a se of discouning and forward curves ha are able o price USD-collaeralized USD swaps, EUR-collaeralized EUR swaps and USD-collaeralized EUR swaps Secion 4 presens some of he resuls ha arise from incorporaing basis spreads in he pricing of ineres rae derivaives ), 26

37 4 Resuls In his secion we presen he discouning and forward curves as derived in Secion 3, where a discussion on how well hese curves are able o replicae he prices of quoed insrumens is included Moreover, we address he impac basis spreads have on swap pricing and he imporance of correcly adjusing for he exisence of such spreads The cases of USD and EUR are deal wih separaely, where a comparison beween he wo currencies concludes he chaper 41 The Case of USD The discouning and forward curves are shown in Figures 41 and 42 To be able o compare he differen forward curves we show he implied 6m forward raes for each enor For example, he 6m implied forward rae wih basis 1m is compued as E c [L 1m (i, i + 6m)] = ) 6 j=1 (1 + δi+(j 1)m,i+jm 1m Ec [L 1m (i + (j 1)m, i + jm)] δ 1m i,i+6m 1 I is seen ha he 6m forward curve lies above he 3m forward curve (and he 3m curve above he 1m curve) This is supposedly due o he higher liquidiy and credi risk associaed wih lending a 6m Libor as compared o rolling lending a 3m Libor, and is wha one should expec Wih hese curves a hand we can price a wide array of USD-collaeralized USD swaps, where he mos ineresing quesion is how imporan i is o adjus for basis spreads when pricing such swaps In Figure 43 he compued 1m, 3m and 6m swap curves are picured along wih he marke quoes for 3m swaps The 3m curve seems fairly good a replicaing he quoed swap raes, however as furher seen in Table 41 here are some discrepancies Mauriy (bps) 1y 048 2y 112 3y 177 5y y y 279 Table 41: The difference beween quoed and esimaed USD 3m swap raes (a selecion) Tha he 3m swap curve on average underesimaes he quoed 3m swap rae by 3 basis poins is explained by he assumpion we made when consrucing he forward curve 27

38 in Secion 312 As he USD IRS pays floaing quarerly bu fixed only semiannually we assumed ha he forward curve was piecewise fla in inervals of 6 monhs This assumpion was furher disregarded when inerpolaing he boosrapped forward raes (since only forward raes ha corresponded o mauriies of quoed insrumens were used for inerpolaion) and hus some of he esimaed forward raes are lower han implied by our assumpion and he quoed swap raes Consequenly, he value of he floaing leg, and in urn he swap rae, is slighly underesimaed Observe ha his would no be an issue for uncollaeralized derivaives, where we could use he relaionship beween he discouning facors and forward raes o (almos) perfecly replicae he swap curve Figure 43 displays how here is a posiive spread beween he 6m and 3m swap curves (and similarly for he 3m and 1m curves) This resul agrees wih heory, since by he definiion of he enor basis spread we ge ha C 6m (, T ) C 3m (, T ) T S 3m,6m (, T ) > 0, and so on The reason for his no being an equaliy is ha day coun convenions and/or paymen frequencies migh differ beween he fixed legs of he ineres rae swaps The fac ha he swap curves do no coincide raises he quesion of how imporan i is o accoun for basis spreads when pricing ineres rae swaps Assume for example ha a pary uses a single (3m forward) curve when pricing derivaives This pary would hus be willing o pay a higher fixed rae in an IRS wih a floaing leg linked o he 1m Libor han wha is jusified by he basis spreads This loss can be approximaed wih Loss = ( C 3m (, T ) C 1m (, T ) ) P V (Discoun Facors) N, where N is he noional and P V (Discoun Facors) is he presen value of all discoun facors imes year fracions Table 42 displays he percenage loss/gain of he noional ha would incur if basis spreads in he USD marke are no accouned for properly For mauriies > 10 years he loss is close o/above 1% for he 1m and 6m swaps, respecively This amoun is clearly significan and underlines he imporance of a pricing framework ha correcly and accuraely accouns for basis spreads Tha he loss is greaer for 6m swaps is in his sense naural, since he 3m/6m basis spreads are greaer han he corresponding 1m/3m spreads 28