A Note on Construction of Multiple Swap Curves with and without Collateral


 Tyler Briggs
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1 <FSA Research Review Vol.6(March 2010) > A Noe on Consrucion of Muliple Swap Curves wih and wihou Collaeral Masaaki Fujii, Yasufumi Shimada, Akihiko Takahashi Absrac There are now available wide variey of swap producs which exchange Libors wih differen currencies and enors. Furhermore, he collaeralizaion is becoming more popular due o he increased aenion o he couner pary credi risk. These developmens require clear disincion among differen ype of Libors and he discouning raes. This noe explains he mehod o consruc he muliple swap curves consisenly wih all he relevan swaps wih and wihou a collaeral agreemen. Keywords : Libor, swap curve, collaeral, overnigh index swap, basis spread 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 Shinsei Bank, Limied, Universiy of Tokyo, he Financial Services Agency or he Financial Research and Training Cener. 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. Graduae School of Economics, The Universiy of Tokyo General Manager, Capial Markes Division, Shinsei Bank, Limied Professor, Graduae School of Economics, The Universiy of Tokyo Special Research Fellow, Financial Research and Training Cener, Financial Services Agency
2 1 Inroducion Among he marke paricipans, Libor (London Iner Bank Offer Rae) has been widely used as a discouning rae of fuure cashflows. However, he basis spread observed in Cross Currency Swap (CCS) marke has been far from negligible in recen years. Even in he single currency marke, he enor swap (TS), which exchanges he wo Libors wih differen enors, requires nonzero basis spread o be added in eiher side. From hese facs, i is clear ha we canno rea all Libors equally as discouning raes in order o price he financial producs consisenly wih exising swap markes. Furhermore, we are winessing an increasing number of financial conracs are being made wih collaeral agreemens. Due o he recen financial crisis and increasing aenion o he counerpary credi risk, we can expec his endency will accelerae in coming years. As we will see, he exisence of he collaeral agreemen ineviably changes he funding cos of financial insiuions, which makes he use of Libor discouning inappropriae for he proper pricing and hedging of collaeralized conracs. In his brief noe, we explain he mehod o consruc he erm srucures of yield curves consisenly wih all he exising swap markes wih and wihou he collaeralizaion 1). 2 Swap curve consrucion wihou collaeral In his secion, we develop he mehod o consruc he erm srucures of yield curves consisenly wih he ineres rae swaps (IRS), cross currency swaps (CCS) and enor swaps (TS) wihou a collaeral agreemen. Here, we will concenrae on he radiional CCS, which keeps noional consan hroughou he conrac. The implicaion of he new ype of CCS (markomarke CCS), which reses noional periodically using he spo exchange rae, will be discussed in Sec.3.5 under he conex of collaeralized swaps. We choose a single Libor as a discouning rae, and derive muliple index 2) curves in addiion o he discouning curve o make he whole sysem consisen wih he observable swap markes. As we will see, choosing a proper Libor as a discouning rae is imporan in order o reflec he difference of funding cos of each financial insiuion o he markomarke of is porfolio. We will also discuss he implicaions of he exisence of muliple curves for he required hedges of ineres rae producs. 2.1 Case of Single IRS As a preparaion for laer discussions, we firs consider he siuaion where we have a single IRS marke of a single currency only. For simpliciy, le us assume ha he paymen daes of he fixed and floaing raes of he IRS are he same. Then, he condiion ha he presen value of he wo legs are equal when we use he marke swap rae as he coupon of he fixed leg is given by N C N n P,Tn = δ n E L(T n 1, T n )]P,Tn. (1) Here, C N is he swap rae of he lenghn IRS a ime, n and δ n are he daycoun fracions of he fixed and floaing legs, respecively. L(T n 1, T n ) is he Libor which is going o be rese a ime T n 1 1) Afer he compleion of he firs version of his noe, we were able o develop a fully dynamic erm srucure model of ineres raes where all he basis spreads are sochasic. Please consul A marke model of ineres raes wih dynamic basis spreads in he presence of collaeral and muliple currencies of Ref. 1]. 2) We call he marke raes (such as Libors) ha are no equivalen o discouning raes as index raes
3 and mauring T n. P,Tn denoes he ime value of he riskfree zero coupon bond mauring a T n. In he remainder of he paper, he expecaion E ] is assumed o be aken under he appropriae forward measure unless i is specially menioned. In order o deermine he se of {E L(T n 1, T n )]} and {P,Tn } uniquely, we need o impose some relaionship beween hese wo ype of variables since here is only one consrain of Eq.(1). Therefore, as we have menioned, le us assume ha he Libor is in fac he discouning rae. Then, he noarbirage condiion beween he zero coupon bond and he Libor floaing paymen gives E L(T n 1, T n )] = 1 ( ) P,Tn 1 1. (2) δ n P,Tn Using his relaion, we can wrie Eq.(1) as N C N n P,Tn = P,T0 P,TN. (3) Now we can deermine he se of discouning facor (and hence he forward Libors) sequenially, by ransforming he above equaion in he following form: P,TN = P,T0 C N ( N 1 np,tn ) 1 + C N N. (4) In he above formula, P,T0 is he discouning facor o he effecive dae, and can be deermined by he overnigh rae. Alhough, we need o carry ou delicae splining o ge a coninuous se of discouning facor and forward Libor, which is imporan for pracical applicaion o he generic pricing, we will no sep ino he echnical deails, and concenrae on he concepual undersanding of he curve consrucion. 2.2 Case of IRS and CCS (USD Libor base) In his secion, we discuss he simple siuaion where here exis IRS and CCS markes and ake he exising cross currency basis spread ino accoun. To make he sory concree, we adop USD and JPY as he relevan currencies and assume ha he USD 3mLibor as he discouning rae. This assumpion is useful for he high raed firms whose funding currency is USD. For furher simplificaion, we also assume ha he paymen frequency of he floaing leg is quarerly boh in he JPY IRS and USDJPY CCS 3). In his seup, he curve consrucion for USD can be done in exacly he same way as explained in he previous secion, since he Eq.(2) holds for he USD discouning facor and Libor. For JPY, his is no he case. The consisency condiions required from he JPY IRS and USDJPY CCS are 3) In realiy, JPY IRS has semiannual paymens and 6m enor of Libor. On he oher hand, he sandard USDJPY CCS exchanges USD 3mLibor fla agains JPY 3mLibor plus spread. The implicaions from he difference of enor will be discussed in he nex secion
4 given in he following forms, respecively. N C N n P,Tn = N JP Y { P,T0 + = f x () δ n E L(T n 1, T n )]P,Tn, (5) } δ n (E L(T n 1, T n )] + b N ) P,Tn + P,TN { P $,T 0 + } δ ne $ $ L $ (T n 1, T n )]P,T $ n + P,T $ N Here, he $index denoes ha he variable is relevan for USD, b N is he basis spread for lenghn CCS, N JP Y is he JPY noional per USD and f x () is he USDJPY exchange rae a ime 4). Since we are assuming ha he USD Libor as he discouning rae, he righhand side of Eq.(6) is acually zero. Therefore, eliminaing he floaing pars in Eqs.(5) and (6) gives us he simple formula: ( n C N + δ n b N ) P,Tn = P,T0 P,TN. (7) From he above equaion, jus as we did in he previous secion, we can deermine he se of {P,Tn } sequenially and make i coninuous wih he help of appropriae spline mehod. Once his is done, we ge he se of forward Libors by subsiuing he derived P,T ino he Eq.(5). As a resul, we are forced o have wo differen curves, one for discouning and he oher for he forward Libor index of JPY raes. One can see ha he effecive rae deermining he JPY discouning facor is approximaely given by C eff N C N + δ b N, (8) and i is clear ha he popular relaion given in Eq.(2) does no hold for JPY Libor as long as here exiss nonzero basis spread. Eq.(8) ells us ha he JPY discouning curve lies below he index curve by he size of basis spread, which is usually negaive b N < 0 in he curren marke. (6) 2.3 Case of IRS and CCS wih TS basis ino accoun (USD Libor base) In he previous secion, we have assumed he common paymen frequency and enor of JPY floaing raes boh in he IRS and CCS. However, in realiy, he JPY Libor used in CCS has 3m enor and quarerly paymens, bu i has 6m enor and semiannual paymens in JPY IRS. In addiion, here exiss 3m/6m enor swap, in which one pary pays 3mLibor plus spread quarerly in exchange for receiving 6mLibor semiannually, where he observed spread is ofen nonnegligible, say more han 10bps. In his secion, we coninue o rea USD 3mLibor as he discouning rae, bu exend he previous mehod o ake he observed TS basis spread ino accoun consisenly wih IRS and CCS. In he remainder of he paper, we disinguish semiannual and quarerly paymens and corresponding Libors by he ind of m and n, respecively. 4) A he incepion of he CCS, N JP Y is deermined by he spo exchange rae, which is he forward exchange rae mauring a he T + 2 effecive dae. Due o his fac, he curren f x() and N JP Y are slighly differen in realiy. However, we will neglec his small difference hroughou his paper since i does no affec he main discussion
5 The required condiions for he JPY raes are given as follows: M C M m P,Tm = M δ m E L(T m 1, T m )]P,Tm, (9) δ n (E L(T n 1, T n )] + τ N ) P,Tn = N JP Y { P,T0 + M δ m E L(T m 1, T m )]P,Tm, (10) } δ n (E L(T n 1, T n )] + b N ) P,Tn + P,TN { } = f x () P,T $ 0 + δ ne $ $ L $ (T n 1, T n )]P,T $ n + P,T $ N, (11) where, we have assumed N = 2M, and τ N denoes he ime marke spread of he lenghn 3m/6m enor swap. Since we are reaing USD 3mLibor as he discouning rae, he righ hand side of Eq.(11) is zero as before. Eliminaing he floaing pars from hese relaions, one can easily show he equaion M C M m P,Tm + δ n (b N τ N )P,Tn = P,T0 P,TN (12) holds among he JPY discouning facors. From his formula, i is sraighforward o derive he (coninuous) se of discouning facor by appropriae spline mehod as before. Then, using he deermined discouning facors, we can derive {E L(T, T + 3m)]} and {E L(T, T + 6m)]}, he se of 3m and 6m forward Libors, from Eqs.(11) and (9), respecively 5). Now ha, under he assumpion of USD 3mLibor being he discouning rae, we have derived he se of JPY discouning and wo index curves, which make i possible o carry ou JPY markomarke consisenly wih IRS, CCS and TS a he same ime. If here exiss a differen ype of TS marke, one can easily exend he mehod o derive forward Libors wih differen enors, such as 1m and 12m. We can also use he same mehod o derive JPY Tibor since here exiss a swap exchanging Libor wih Tibor plus spread. As for USD raes, we can use he mehod in sec.2.1 o derive he discouning facors and forward 3mLibors, and hen use USD TS informaion o derive he Libors wih differen enors. In order o undersand he relaion among he JPY discouning and index curves, i is convenien o use he following approximaion: m P,Tm By puing n = m /2, we can simplify he Eq.(12) as m 2 (P,T m 3m + P,Tm ). (13) { n C M + δ n (b N τ N )} P,Tn P,T0 P,TN (14) and hen we see he effecive swap rae implying he discouning facor is given by C eff M C M + δ (b N τ N ). (15) 5) We have no included he informaion available from USDJPY foreign exchange (FX) marke. Since he FX forward conracs can be replicaed by CCSs, he implied forward FX from he resulan discouning facors is mosly consisen wih he marke. Due o he liquidiy issues, i is also common o use forward FX conracs insead of CCSs in he shor end of he curve
6 I is clear from he above relaion ha JPY discouning facor depends no only on swap raes {C M } bu also on CCS and TS spreads, {b N, τ N }. Therefore, even if we have a posiion only in he sandard JPY IRS, we need o hedge he exposures o he sensiiviies of hese spreads. I is also insrucive o undersand he relaion among JPY discouning and wo index curves. If he marke quoes of IRS, TS and CCS are all fla, one can easily undersand he relaion L 3m = R discoun b, (16) L 6m = L 3m + τ = R discoun (b τ) (17) holds among he corresponding forward raes. Here, b and τ denoe he fla CCS and TS basis spreads, respecively. We have also negleced he difference in he daycoun fracions. 2.4 Case of IRS and CCS wih TS basis ino accoun (JPY Libor base) In he previous secions, we have assumed ha he USD 3mLibor is he discouning rae. However, for he financial insiuions which funding bases are locaed in Japan, i would be more appropriae o consider JPY Libor as he discouning rae. In his secions, we carry ou he same exercise under he assumpion ha JPY 3mLibor is he discouning rae 6). In his seup, Eq.(2) holds beween he JPY 3mLibor and he discouning facor, which allows us o rewrie Eq.(10) as M P,T0 P,TN + δ n τ N P,Tn = δ m E L(T m 1, T m )]P,Tm. (18) And hen, eliminaing he floaing pars from he above equaion using Eq.(9) yields he following formula: M C M m P,Tm δ n τ N P,Tn = P,T0 P,TN. (19) Once we calculae he se of {P,T } wih proper splining from he above equaion, we can easily recover he se of forward 3mLibors from he relaion given in Eq.(2), and ha of forward 6mLibors from Eq.(18). As was explained in he previous secion, i is easy o obain he forward Libors wih differen enors if here exis addiional TS markes. Now, le us consruc he USD curves consisenly wih he assumpion of JPY 3mLibor discouning. Noe ha Eq.(2) now holds for JPY 3mLibor, Eq.(11), which is he condiion from he CCS, is rewrien as ( N ) N $ δ n P,Tn = P,T $ 0 + δ ne $ $ L $ (T n 1, T n )]P,T $ n + P,T $ N, (20) b N where N $ = N JP Y f x (), (21) 6) I is sraighforward o apply he same mehodology for JPY Libor wih differen enors, or even Tibor as he discouning rae
7 and i is almos 1 and we rea i as a consan 7). We also have C K $ k=1 K $ k P,T $ k = δ ne $ $ L $ (T n 1, T n )]P,T $ n (22) as he consrain from USD IRS. Here, N = 4K and we have disinguished he annual paymen of fixed coupon by he index of k from he quarerly paymen in he floaing side in he sandard USD IRS. As before, by eliminaing he floaing pars from Eqs.(20) and (22), we ge he following equaion among he USD discouning facors: P,T $ 0 + P,T $ N + C K $ ( K $ k P,T $ k = N $ k=1 N b N δ n P,Tn ). (23) Since he righ hand side is already known, we can repea he same spline mehod o ge he se of he discouning facors, {P,T $ }. Then, forward 3mLibors can be obained from Eq.(22) by subsiuing he derived discoun facors, and forward Libors wih differen enors if here exis corresponding USD TS markes. Under he assumpion of JPY 3mLibor discouning, he inerdependence among discouning and index curves are quie differen from ha of he las secion. I is clear from Eq.(19) ha he basis spread in CCS does no affec he JPY discouing facors bu ha he USD discouning facors depend no only on he USD IRS quoes, bu also on he basis spreads in CCS and JPY TS. I is imporan o noice ha we need an aggregae risk managemen sysem o deal wih he inerdependence among USD and JPY ineres raes boh in he las and curren cases. 2.5 Implicaions from differen choice of discouning curve Le us consider he implicaion from he differen choice of Libor as a discouning rae. As is clear from he previous wo secions, differen choice leads o he differen discouning curves, which ineviably leads o differen presen values even for he same cashflow. Alhough i does allow he arbirage if he wo mehods coexis, we will now see ha i is precisely reflecing he asymmery of funding cos of financial firms. For concreeness, le us firs ake a look a a highraed financial firm locaed in he Unied Saes, which can borrow USD loan wih 3mLibor fla. In his case, he presen value of he iniial receip of USD noional followed by 3mLibor and he final noional repaymens should be zero in oal, which will make i convenien for his firm o use USD 3mLibor as he discouning rae. Now, we wan o know how much i coss o borrow JPY loan for he same firm. The firm can firs borrow USD loan in US marke, and hen swap i ino JPY loan by enering USDJPY CCS. The implied JPY funding cos is hen given by JPY 3mLibor + basis spread. Since in he USDJPY basis spread is usually negaive, i can borrow JPY cash a cheaper cos han he Japanese domesic marke. Therefore, he firm can make profi when i accesses he domesic marke o provide JPY loan wih JPY 3mLibor fla. One can see ha our curve consrucion based on USD Libor can explain his fac by making he JPY discouning rae displaced from he Libor by he CCS basis spread. On he oher hand, we have a quie differen sory for a highraed Japanese financial firm. Since is funding cos of JPY loan is JPY Libor, i canno raise any profi by lending a loan wih JPY Libor 7) Precisely speaking, i depends on he overnigh raes of USD and JPY, since N JP Y is usually deermined by he spo FX rae, which is acually T+2 forward rae as we have menioned before. We will neglec is rae dependency for simpliciy hroughou he paper
8 fla in he domesic marke. Now, le us consider he case where he firm wans o provide USD loan wih USD Libor fla o is clien. Since i does no have ample pool of USD cash, i needs o swap he JPY cash o USD by enering CCS marke. The firm pays he USD Libor o he CCS couner pary by passing he repaymens from he clien in reurn for receiving JPY Libor + basis spread. This essenially means ha he firm provided a loan a lower yield han is funding cos because of he negaive basis spread. Thus he firm has o recognize he loss from his conrac. If we use he JPY Libor as he discouning rae and follows he consrucion explained in he las secion, we can ake his fac ino accoun for he pricing of financial producs. As is now clear from he above examples, each financial firm needs o choose he appropriae reference as is discouning rae when consrucing he se of curves 8). I should be emphasized ha he coexisence of differen assumpions wihin he single firm needs o be avoided. I would allow he arbirage wihin he sysem, and make i impossible o carry ou consisen hedges agains he exposures o he various spreads in he marke 9). 3 Swap curve consrucion wih collaeral Up o now, we have assumed ha he swap conrac is made wihou a collaeral agreemen and explained he curve consrucion based on he specific Libor reaed as a discouning (or funding) rae. However, in recen years, more and more financial producs have been made wih collaeral agreemens due o he increased aenion o he couner pary credi risk. I is especially he case for major fixed income producs such as swaps 2]. I seems ha he endency will accelerae furher and will be applied o wider variey of producs as a fallou of he curren financial urmoil. As we will see laer, he exisence of collaeral no only reduces he credi risk bu also changes he funding cos significanly and hence affecs he valuaion of financial producs in an imporan fashion. In he remainder of he paper, we will discuss he implicaion of he exisence of collaeral for he swap curve consrucion. 3.1 Pricing of collaeralized producs In his secion, before going o he deails of curve consrucion, we will discuss he generic pricing of collaeralized rades. Under he collaeral agreemen, he firm receives he collaeral from he couner pary when he presen value of he conrac is posiive, and needs o pay he margin called collaeral rae on he ousanding collaeral o he payer. Alhough he deails can differ rade by rade, he 8) The differen Libor choice among marke paricipans is difficul o be recognized a he incepion of swaps, since i is common o ener he swap wih he ousanding par rae which resuls in zero presen value. However, we in fac experience some difficuly in he price agreemen when we close he posiion. 9) I does allow he arbirage among marke paricipans if hey have differen funding currencies. The siuaion is even more sriking in some emerging markes where he implied basis spreads are asonishingly large (and negaive). Alhough some of he foreign financial firms are acually aking advanage of he asymmery of funding cos among differen currencies o make profi, i seems ha he aciviies are no enough o make spreads disappear. Some possible reasons are various regulaions on foreign firms, heir limied peneraion in domesic markes, accouning rules making he recogniion of profi from hese aciviies difficul, and large USD demand o fulfill he hedge needs from domesic exporing companies and financial insiuions wih big foreign asse exposures. I would be imporan o sudy he economic reasons ha lead o he exisence of significan size of he currency basis spread
9 mos commonly used collaeral is a currency of developed counries, such as USD, EUR and JPY, and he markomarke of he conracs is o be made quie frequenly. In he case of cash collaeral, he overnigh rae for he collaeral currency, such as Fedfund rae for USD, is usually used as he collaeral rae. In general seup, carrying ou he pricing of collaeralized producs is quie hard due o he nonlineariy arising from he credi risk. In he remainder of he paper, in order o make he problem racable, we will assume he perfec and coninuous collaeralizaion wih zero hreshold by cash, which means ha markomarke and collaeral posing is o be made coninuously, and he posed amoun of cash is 100% of he conrac s presen value. Acually, he daily adjusmen of he collaeral should be he bes pracice in he marke and seems becoming popular, and hence he approximaion should no be oo far from he realiy. Under he above simplificaion, we can neglec he couner pary defaul risk and recover he lineariy among differen paymens. Therefore, we can decompose he cashflow of a collaeralized swap and rea hem as a porfolio of he independenly collaeralized srips of paymens. Le us consider he sochasic process of he collaeral accoun V () wih an appropriae selffinancing rading sraegy under he riskneural measure, following he mehod someime used in he pricing of fuures. Since one can inves he posed collaeral wih he riskfree ineres rae bu need o pay he collaeral rae, he process of he collaeral accoun is given by dv (s) = y(s)v (s)ds + a(s)dh(s), (24) where, y(s) = r(s) c(s) is he difference of he riskfree rae r(s) and he collaeral rae c(s) a ime s, h(s) denoes he imes value of he derivaive which maures a T wih he cashflow h(t ), and a(s) is he number of posiions of he derivaive. We ge V (T ) = e T T y(u)du V () + e T s y(u)du a(s)dh(s) (25) by inegraing Eq.(24). Adoping he rading sraegy specified by V () = h() ( s a(s) = exp ) y(u)du (26) allows us o rewrie Eq.(25) as V (T ) = e T y(s)ds h(t ). (27) Then, we see he presen value of he underlying derivaive is given by h() = E Q e ] T (r(s) y(s))ds h(t ) = E Q e ] T c(s)ds h(t ). (28) Here, E Q ] denoes he expecaion where he moneymarke accoun is being used as he numeraire 10). 10) Considering he coninuous and perfec collaeralizaion and is invesmen wih rae y(), we see h() = E Q should hold. From his equaion, one can show ha e T T ] r(s)ds h(t ) + e s r(u)du y(s)h(s)ds (29) X() = e 0 r(s)ds h() + 0 e s 0 r(u)du y(s)h(s)ds (30)
10 Nex, le us consider he case where he collaeral is posed by a foreign currency. In his case, he process of he collaeral accoun V f is dv f (s) = y f (s)v f (s)ds + a(s)dh(s)/f x (s)], (32) where f x (s) is he foreign exchange rae a ime s, and y f (s) = r f (s) c f (s) denoes he difference of he riskfree and collaeral rae of he foreign currency. Inegraing i, we obain V f (T ) = e T T yf (s)ds V f () + e T s yf (u)du a(s)dh(s)/f x (s)]. (33) This ime, we adop he rading sraegy which yields V f () = h()/f x () ( s ) a(s) = exp y f (u)du, (34) V f (T ) = e T yf (s)ds h(t )/f x (T ). (35) Then, we see he price of he derivaive in erms of he domesic currency is given by h() = V f ()f x () = E Q e ] T r(s)ds V f (T )f x (T ) = E Q e ( T r(s)ds e ) ] T (rf (s) c f (s))ds h(t ). (36) From he above discussion, i is now clear ha Libor discouning is no appropriae for he pricing of collaeralized rades. As we can see from Eq.(28), we have o discoun he fuure cashflow by he collaeral rae, which can be significanly lower han he Libor for he corresponding currency, especially under he disressed marke condiions. I is also useful o inerpre he resuls in erms of he funding cos for he possessed posiions. Firs, le us consider he case where here is a receip of cash a a fuure ime (hence, posiive presen value) from he underlying conrac. In his case, we are immediaely posed an equivalen amoun of cash as is collaeral, on which we need o pay he collaeral rae and reurn is whole amoun in he end. We consider i as a loan where we fund he posiion a he expense of he collaeral rae. On he oher hand, if here is a paymen of cash a fuure ime (negaive presen value), he required collaeral posing can be inerpreed as a loan provided o he couner pary wih he same rae. Therefore, compared o he noncollaeralized rade (and hence, Libor funding), we ge more in he case of posiive presen value since we can fund he loan cheaply, bu lose more in he case of negaive value due o he lower reurn from he loan len o he clien. 3.2 Overnigh Index Swap As we have seen in he previous secion, i is criical o deermine he forward curve of overnigh rae for he pricing of collaeralized swaps. Forunaely, here is a produc called overnigh index swap (OIS), which exchanges he fixed coupon and he dailycompounded overnigh rae. is a maringale process, which hen implies ha he price process of h() is expressed wih a cerain maringale process M() as dh() = c()h()d + dm(). (31) This would also leads o he formula given in Eq.(28)
11 Here, le us assume ha he OIS iself is coninuously and perfecly collaeralized wih zero hreshold, and approximae he daily compounding wih coninuous compounding 11). In his case, using he Eq.(28), we ge he condiion from he OIS as N S N n E Q e Tn c(s)ds ] = E Q e Tn ( T )] n c(s)ds c(s)ds T e n 1 1. (37) Here, S N is he ime par rae for he lenghn OIS, and c() is he overnigh ( and hence collaeral) rae a ime. We can simplify he above equaion ino he form N S N by defining he discouning facor of he collaeral rae: n D,Tn = D,T0 D,TN (38) D,T = E Q e T c(s)ds]. (39) Now, from Eq.(38), we can obain he coninuous se of {D,T } by appropriae splining as before. 3.3 Case of collaeralized swaps in single currency In he case of single currency, calculaion of he forward Libors is quie sraighforward. The consisency condiions from he collaeralized IRS and TS corresponding o Eqs.(9) and (10) are M C M m D,Tm = M δ m D,Tm E c L(T m 1, T m )], (40) δ n (E c L(T n 1, T n )] + τ N ) D,Tn = M δ m D,Tm E c L(T m 1, T m )], (41) where E c ] denoes he expecaion aken under he measure where D,T is used as he numeraire. Since all he relevan {D,T } are already known from he OIS marke, we can easily calculae he se of forward Libors from hese condiions. Here, we have assumed ha OIS swap marke is available up o necessary range o deermine he enire forward curve. 3.4 Case of collaeralized swaps in muliple currencies (wih Consan Noional CCS) In his secion, we consider he mehod o consruc he erm srucures of collaeralized swaps in he mulicurrency seup, where we coninue o use he consan noional CCS as a calibraion insrumen. We will discuss he implicaions of new ype of CCS, MarkoMarke CCS, in he nex secion. In he single currency case, i is common o use he same currency as he collaeral, and we can easily derive he relevan curves as we have seen in Secs.3.2 and 3.3. However, here ineviably appear he paymens wih differen currency from ha of he collaeral in CCS, which makes he deerminaion of he forward Libors complicaed due o he involvemen of he riskfree and collaeral rae a he same ime as indicaed by Eq.(36). In he acual marke, USD is being widely used as he collaeral for he rades including muliple currencies. 11) Typically, here is only one paymen a he very end for he swap wih shor mauriy (< 1yr) case, and oherwise periodical paymens, quarerly for example
12 As in he previous secions, le us use USD and JPY swaps o demonsrae he mehod. To make he problem simpler, we rea he FedFund rae, which is he collaeral rae for USD, o be he riskfree ineres rae. Then, we have he relaion D $,T = EQ$ e T c$ (s)ds ] = E Q$ The required condiions from JPYcollaeralized JPY swaps are given by N S N M C M e T r$ (s)ds ] = P $,T. (42) n D,Tn = D,T0 D,TN, (43) m D,Tm = M δ m D,Tm E c L(T m 1, T m )], (44) δ n (E c L(T n 1, T n )] + τ N ) D,Tn = and, hose of USDcollaeralized USD swaps are S N $ C K $ k=1 M δ m D,Tm E c L(T m 1, T m )], (45) $ np,t $ n = P,T $ 0 P,T $ N, (46) K $ k P,T $ k = δ $ n δ np $,T $ n E $ L $ (T n 1, T n )], (47) ( ) E $ L $ (T n 1, T n )] + τ N $ P,T $ n = M δ mp $,T $ m E $ L $ (T m 1, T m )], (48) where, he condiions are from OIS, IRS and TS, respecively. As before, we can add addiional TS condiion if exiss. One can now derive he discouning facors {D,T } and {P,T $ } from he OIS condiions, and hen he remaining forward Libors in urn. Now, le us consider he deerminaion of USDcollaeralized JPY ineres raes. If he USDJPY CCS is collaeralized by USD cash, which is he common pracice in he marke, we ge he following condiion by applying he resul in Eq.(36): Here, δ n (E L(T n 1, T n )] + b N ) P,Tn P,T0 + P,TN = V N. (49) { N } V N = δ ne $ $ L $ (T n 1, T n )]P,T $ n P,T $ 0 + P,T $ n /N $ (50) and i is given by he resul of previous calculaions for USD swaps. As you can see, i is impossible o deermine he JPY riskfree zero coupon bond price {P,Tn } and he forward Libors {E L(T n 1, T n )]} uniquely, from hese sandard se of swaps only. However, if here exis USDcollaeralized JPY IRS and TS markes 12), we ge he addiional informaion as C M M m P,Tm = M δ m P,Tm E L(T m 1, T m )], (51) δ n (E L(T n 1, T n )] + τ N ) P,Tn = M δ m P,Tm E L(T m 1, T m )]. (52) 12) In fac, i seems ha he US banks end o ask heir couner paries o pos USD collaeral even for he JPY IRS and TS
13 Here, CM and τ N denoe he par raes of he USDcollaeralized JPY swaps, which differ from C M and τ N, he par raes of JPY collaeralized swaps in general. We can now eliminae he floaing pars from Eqs.(49), (51) and (52), and obain δ n (b N τ N )P,Tn + C M M m P,Tm V N = P,T0 P,TN. (53) Then, as we did in Sec.2.3, we can deermine he se of {P,T } and he forward Libors wih he boh enors by applying an appropriae spline mehod. If i is difficul o obain he separae quoes for USDcollaeralized JPY swaps, we may no be able o use Eqs.(51) and (52) for he curve consrucion. If his is he case, one possible approach is o se E L(T n 1, T n )] = E c L(T n 1, T n )] (54) by neglecing he correcion arising from he change of numeraire. This approximaion would be reasonable if he dynamic properies of he JPY riskfree and he overnigh ineres raes are similar wih each oher. Once admiing he assumpion, one can deermine he se of discoun facors from Eq.(49). If here exiss enough liquidiy in he FX forward marke, hen using he FX forward quoes and he USD discouning facor o derive {P,T } is anoher possible way. Finally, le us menion he case where we have JPYcollaeralized USD swap markes. Since we have no assumed ha he JPY overnigh rae is riskfree, he difference beween he riskfree and collaeral raes appears in he expression of presen value as given in Eq.(36). The condiions from he JPYcollaeralized USD IRS is given by C K $ k=1 K $ k P,T $ k E $ e T k y(s)ds ] = δ np $,T $ n E $ e ] Tn y(s)ds L $ (T n 1, T n ), (55) where y(s) = r(s) c(s) is he difference beween he JPY riskfree and collaeral raes, and C $ K is he par rae of he lenghk IRS. In he same way, if here exiss JPYcollaeralized USDJPY CCS, we also have he following condiion: where, b N δ np $,T $ n E $ e ] Tn y(s)ds L $ (T n 1, T n ) ( N ) = N $ δ n (E c L(T n 1, T n )] + b N )D,Tn D,T0 + D,TN, (56) is he par spread of he lenghn CCS. Since he righ hand side of Eq.(56) and USD discoun facors are already known, we can deermine he se of E $ e ] Tn y(s)ds, E $ e ] Tn y(s)ds L $ (T n 1, T n ). (57) This complees he calculaion of whole se of curves, which are USDcollaeralized USD raes, JPYcollaeralized JPY raes, USDcollaeralized JPY raes, and JPYcollaeralized USD raes. 3.5 Case of collaeralized swaps in muliple currencies (wih MarkoMarke Cross Currency Swap) In his secion, we discuss a differen ype of swap called markomarke cross currency swap (MM CCS) and is implicaion o he curve consrucion. Similarly o he radiional CCS, he paricipans
14 exchange he Libor in one currency and he Libor plus spread in anoher currency wih noional exchanges. The differen feaure of he MMCCS is ha he noional on he currency paying Libor fla is adjused a he every sar of he Libor calculaion period based on he spo FX, and he difference beween he noional used in he previous period and he nex one is also paid or received a he rese ime. Here, he noional for he oher currency is kep consan hroughou he conrac. For pricing, we can consider i as a porfolio of he srips of he oneperiod radiional CCS wih he common noional and he spread for he side paying Libor plus spread. Here, he ne effec from he final noional exchange of he (i)h CCS and he iniial exchange of he (i+1)h CCS is equivalen o he noional adjusmen a he sar of (i+1)h period of MMCCS. Usually, we need o adjus he noional of he USD side, since i is he marke sandard o exchange USD Libor fla agains Libor plus spread in anoher currency. For concreeness, le us consider he case of USDJPY MMCCS wih USD collaeral, and coninue o idenify collaeral rae (FedFund rae) as he USD riskfree rae. I is simple o calculae he presen value in JPY side, since he noional is kep consan. Using he same noaion, he presen value from he view poin of JPY Libor receiver is given by P V JP Y = P,Tn 1 + P,Tn (1 + δ n (b N + E L(T n 1, T n )])) = P,T0 + P,TN + P,Tn δ n (b N + E L(T n 1, T n )]), (58) which is equivalen o he lef hand side of Eq.(49). On he oher hand, he presen value of USD side is expressed as e ] T n 1 P V USD = E Q$ r $ (s)ds e ] Tn r $ + E Q$ (s)ds (1 + δ nl $ $ (T n 1, T n )) f x (T n 1 ) f x (T n 1 ) P,T $ N = n 1 F X(, T n 1 ) + e ] Tn r $ E Q$ (s)ds (1 + δ nl $ $ (T n 1, T n )). (59) f x (T n 1 ) Here, F X(, T ) denoes he ime forward exchange rae mauring a T. If we assume ha he USD Libor is he riskfree rae, hen he second erm cancels he firs one and urns ou o be zero in oal, P V USD = 0 13). However, we are now making a disincion beween USD Libor and he riskfree FedFund rae, here ineviably appears a model dependen erm. To undersand i more clearly, le us decompose he marke Libor ino he riskfree par and he residual par: ( ) L $ (T n 1, T n ) = S(T δ n P T $ n 1, T n ), (60) n 1,T n where he second erm S(T n 1, T n ) denoes he residual par in he Libor L $ (T n 1, T n ) a ime T n 1. Then, we ge he USD side value as P V USD = E Q$ e Tn r $ (s)ds δ n S(T n 1, T n ) f x (T n 1 ) ], (61) which depends on he covariance of he riskfree zero coupon bonds and he FX rae even if he spread is deerminisic. The correcion from he forward value arises when we change he numeraire ino he 13) Therefore, if we consider he noncollaeralized swaps wih USD Libor as he discouning rae, we can repea exacly he same argumens in Secs.2.2 and
15 riskfree zero coupon bond wih mauriy T n. If we can evaluae his model dependen erm, i is possible o repea he same discussions following Eq.(49) afer replacing V N by f x () P V USD : δ n (E L(T n 1, T n )] + b N ) P,Tn P,T0 + P,TN = f x ()P V USD. (62) For simpliciy, le us assume he deerminisic spread 14) and he geomeric Brownian moion for boh of he forward FX and he USD forward riskfree Bond: F X(, T n 1 ) = f x () P $,T n 1 P,Tn 1, F B(, T n 1, T n ) = P $,T n 1 P $,T n. (63) We denoe heir deerminisic lognormal volailiies and he correlaion beween F X(, T n 1 ) and F B(, T n 1, T n ) as (σ F Xn 1 (), σ F Bn 1,n (), ρ n 1 ()), respecively. In his simples case, he USD side presen value can be evaluaed as ( P $ ),T P V USD = n δ n S(T n 1, T n ) Tn 1 exp ρ n 1 (s)σ F Xn 1 (s)σ F Bn 1,n (s)ds. (64) F X(, T n 1 ) Therefore, in his simple seup, he curve calibraion can be done in he following way. Firsly, consruc he USD FedFund rae curve and he collaeralized USD Libor curve as discussed in he las secion, and hen exrac he spread beween hem assuming ha i is deerminisic. Secondly, alhough he available mauriy is limied, we can exrac he FedFund rae volailiy from he OIS opion marke. The forward FX volailiy can be direcly read from he vanilla FX opion marke. As for he correlaion beween he USD riskfree bond and he forward FX, we need o use eiher he hisorical daa, or possibly make use of he informaion in quano producs. Now he las remaining ingredien is he FX forward rae. Of course, we can direcly read he quoes from he marke if here is enough liquidiy in he FX forward conracs. Even if his is no he case, here is a way around requiring only swap informaion. Since he mauriy of FX forward is shorer han ha of he MMCCS by one period, if we have he JPY discouning facor up o P,Tn 1, hen we can sequenially derive P,Tn by using Eq.(62) and he discussion following Eq.(49) in he las secion. Therefore, alhough he procedure is more complicaed, we can sill consruc he curves under he simplifying assumpions. Finally, le us check he case where he MMCCS is collaeralized by JPY cash. The presen value of he JPY side is P V JP Y = D,T0 + D,TN + D,Tn δ n ( b N + E c L(T n 1, T n )]), (65) where he b N denoes he JPYcollaeralized MMCCS spread. The USD side is now given by e T n 1 P V USD = E Q$ r $ (s)ds e ] T n 1 y(s)ds f x (T n 1 ) e Tn r $ + E Q$ (s)ds e ] Tn y(s)ds (1 + δ nl $ $ (T n 1, T n )), (66) f x (T n 1 ) where y(s) = r(s) c(s) denoes he difference of JPY riskfree rae and he collaeral rae. If we assume ha y(s) is deerminisic, or independen from he oher variables in addiion o he assumpion on 14) Precisely speaking, he independence of he spread moion from he riskfree USD rae and FX is enough o apply he following discussion
16 he residual spread of Libor, we can repea he same calculaion o derive he convexiy correcion. Following he similar discussion afer Eq.(56) in he las secion, we can obain he correcion o he forward USD Libor in he case of JPY collaeralizaion, which is he facor of exp( T y(s)ds) 15). 4 Imporance of appropriae curve consrucion Up o his poin, we have explained how o consruc muliple swap curves which can mark various swaps o he marke consisenly wih and wihou collaeral agreemens. Some of he readers may wonder if his is oally unnecessary complicaion o explain anyway very small basis spreads by inferring ha he spreads affec he profi/loss of he financial firms only hrough he proporion : spread size level of ineres rae. However, i is no a all he case since heir profi and loss are made only hrough he change of ineres rae insead of is level. Therefore, he poenial impac would be disasrous if he sysem canno recognize he exisence of basis spreads and if i is unable o risk manage he exposure o heir movemens. Basically, he exisence of basis spreads affecs he markomarke of he rades hrough he following wo roues: (1) Change of he forward expecaion of Libors; (2) Change of he discouning rae. In he following, le us explain each effec using simple examples so ha he readers can easily recognize he imporance of consisen curve consrucion. Le us sar from he firs case. Suppose here is one firm which does no recognize he enor swap spreads and working in srucured produc business; The firm pays he srucured payoff o is cliens and receives Libor (plus spread o cover he opionaliy premia) in reurn as is funding. Le us suppose he funding legs of he firm s porfolio conain he wo frequencies wih equal fracions, 3m and 6m JPYLibor, reflecing he differen demands among he cliens. If he firm consrucs he swap curve based on JPY IRS wih semiannual frequency, and if i is no able o handle he 3m/6m enor spread, boh of he 3m and 6m forward Libors are derived from he common discouning curve based on he IRS. In his case, he model implied 3m/6m enor spread is zero. As one can easily imagine, he firm is significanly overesimaing he value of 3mLibor funding legs. The easies way o esimae is impac is o conver he sream of 3mLibor paymens ino ha of 6mLibor by enering he 3m/6m JPYLibor enor swap as he payer side of 3mLibor. Since he firm needs o pay he 3m/6m enor spread on op of he 3mLibor, he loss of he firm from he mispricing of he funding legs can be esimaed as Loss Ousanding Noional PVO1(Average Duraion) ( 3m/6m enor spread ), where he PV01 denoes he annuiy of he corresponding swap, which is he sum of he discoun facors imes daycoun fracions. If he average duraion and he enor spread is around 10yr and 10bp respecively, he loss would be abou one percenage poin of he oal noional ousanding, which would be far from negligible for he firm. Of course, if he srucured payoffs are dependen on he 3mLibor, here will be addiional conribuions. Furhermore, when he firm is an acive 15) Under he assumpion ha y is a deerminisic funcion of ime, we can make he curve consrucion more sraighforward. Please see he relaed discussion in Ref. 1]
17 paricipan of IRS marke a he same ime, he poenial impac would be much worse. Since he sysem unable o recognize he spread gives he raders an incenive o ener he posiions as 3mLibor receivers, since hey can offer very compeiive prices relaive o heir compeiors while making heir profi posiive wihin he firm s fauly sysem. Now, le us discuss he impac from he second effec, or he change in he discouning facors. This is he dominan change when we properly ake he collaeralizaion ino accoun. Alhough he impac will be smaller han he direc change of he forward Libors, here would be quie significan impac especially from he cross currency rades, where we usually have final noional exchanges. In he presence of 10bp LiborOIS spread, he presen value of he noional paymen in 10yrs ime would be differen by around one percenage poin of is noional. For he whole porfolio, he impac from he difference beween he Libor and he collaeral rae of each currency can be remendous. In addiion, here is anoher roue hrough which he change of discouning facors affecs he firm s profi in an imporan fashion. If, as a more preliminary level, he firm is no capable of reaing he CCS basis spread correcly, he resulan discouning curves never reproduce he marke level of FX forwards 16). If hey are paricipaing in FX derivaives business wihou having developed he proper sysem, he effec hrough FX forward will be quie criical, if i is no faal. On he oher hand, even if he discouning curve of foreign currency is properly consruced o reproduce he FX forwards, if he sysem neglecs he difference beween he resulan discouning curve and he forward Libor of he corresponding currency, he value of fuure cash flow dependen on he foreign Libor will be oally wrong. This effec would be paricularly imporan for he FXIR hybrid producs, such as PRDCs. 5 Use of muliple curves in a rading sysem I is now clear ha we need a large number of Libor index and discouning curves o price he financial producs consisenly wih he observable swap markes. In he remaining par of he paper, we will discuss some imporan poins relaed o he use of he muliple curves in an acual rading sysem. 5.1 Use of curves for noncollaeralized producs Here, we will discuss he case of noncollaeralized producs. In his case, wha we need o do firs is o choose a single appropriae reference rae, which should reflec he funding cos of he relevan firm reasonably well, and also have good liquidiy in he marke, such as he Libor of he funding currency. I will be used as he base discouning rae when we consruc he muliple curves. Alhough he complexiy of hedge does depend on he choice, i should be unique hroughou he firm o avoid he arbirage wihin he sysem and o reain he consisency of hedges. Afer he choice of a single funding rae, we can uniquely deermine he discouning and forward Libor curves for each currency excep he freedom associaed wih he deails of spline mehod. For he pracical use, i would be convenien o creae following quaniies: {P 0,T }, {P 1m 0,T }, {P 3m 0,T }, {P 6m 0,T }, (67) 16) Noe ha he combinaion of IRS and CCS effecively replicae FX forward conracs
18 where he firs one is he discouning facor, and he ohers are recursively defined by he relaion ( ) P0,0 τ 1 P τ 0,T τ = 1, 1 = EL(T τ, T )]. (68) τ P τ 0,T The quaniy, P0,T τ, can be considered as he risky discouning facor reflecing he relaive risk among he Libors wih differen enors. Since i is naural o assume ha he relaive risk of he Libor changes smoohly in erms of is enor, we can approximae P0,T τ wih an arbirary τ by inerpolaing he se of (67). This would be quie useful for he pricing of overhecouner producs, which someimes require he Libor wih a enor which is no available in he liquid TS marke. The pricing of producs wihou opionaliy is hen carried ou sraighforwardly, by calculaing he appropriae forward rae using he inerpolaion of P τ if necessary, and hen muliplying he discouning facor of he paymen dae. Dela (and hence gamma) sensiiviies are calculable by using differen se of curves afer blipping he marke quoes of he relevan swaps, {C M, b N, τ N }. As we have seen, i is imporan o noice ha he movemen of quoes even in differen currencies can affec he hedges hrough he effec of CCS. 5.2 Use of curves for collaeralized producs Now le us discuss he case of collaeralized producs. Firsly, we need o choose a riskfree ineres rae o consruc he curves. Considering he available lengh of he OIS, he FedFund rae would be useful. Alhough he basic idea is he same, he operaion under he collaeralizaion is more complicaed han he noncollaeralized case. As we have seen, under he collaeralizaion, he effecive discouning facors and associaed expecaion of forward Libors depend on he collaeral currencies, and hence, i would be convenien o seup separae books for each of hem in he rading sysem. Ideally, we would like o have all he ypes of swaps for each collaeral currency, which hen allows o deermine he curves uniquely, and makes i possible o close he hedges wihin he swaps wih he same collaeral. However, i is no he case in general, and we are required o use he approximae relaion, such as Eq.(54), o relae he exposure o he available swaps. Excep hese complicaions, dealing wih he Libors wih differen enors and he hedge operaions are he same as hose in he noncollaeralized case. 5.3 Commens on Simulaion Scheme Finally, le us commen on he issue relaed o he simulaion scheme in he mulicurve seup. Generally speaking, we need o make all he curves dynamic if we wan o fully capure he opionaliy relaed o he spreads among differen Libors. However, as one can easily imagine, i would be a quie demanding ask o develop he sysem due o he complicaed calibraion mechanism even for he vanilla opions, and he need of delicae noise reducion o recover he observed swap prices wihin a reasonable calculaion ime. On he oher hand, despie he difficulies, we also know he imporance o incorporae he mulicurve seup ino he simulaion sysem so ha we can properly reflec he observed marke swap prices in he srucured derivaives, and appropriaely manage he exposures o he various spreads in he marke. The simples approach is o assume consan and imehomogeneous spreads among he discouning curve and he Libor index curves wihin each currency. Under he assumpion, we can simply adop he usual ineresrae erm srucure model o drive he discouning curve. For pricing, we
19 check he relevan enor of he reference rae, adding up he relevan spread o he simulaed discouning rae o ge he pahwise realizaion of he Libor index. We can look a he model wih he above simplificaion as he minimum requiremen for mos of he financial firms so ha hey can properly manage he exposure o he exising spreads in various swaps. Of course, however, here are a lo of poenially imporan problems arising from his simplificaion. Especially, he dynamics of he overnigh rae se by he cenral bank and he Libor index in he marke can be significanly differen especially when he credi condiion is igh, which suggess he need of independen modeling of hese wo underlyings. I is an imporan remaining research opic o develop he model which can handle muliple dynamic curves and is pracical calibraion scheme 17). References 1] Fujii, Masaaki, Shimada, Yasufumi and Takahashi, Akihiko, A Marke Model of Ineres Raes wih Dynamic Basis Spreads in he Presence of Collaeral and Muliple Currencies December Available a SSRN: hp://ssrn.com/absrac= CARF Working Paper Series CARFF ] ISDA Margin Survey and a/pdf/isdamarginsurvey2009.pdf 17) Recenly, afer he compleion of firs version of his noe, we have wrien he paper proposing a new framework of ineres rae model which allows fully sochasic basis spreads 1], where he resulan curves consruced in his noe are direcly used as iniial condiions of he simulaion. We also presened he more sraighforward curve consrucion in mulicurrency environmen under he collaeralizaion
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