Modeling of Interest Rate Term Structures under Collateralization and its Implications

Transcription

1 Modeling of Ineres Rae Term Srucures under Collaeralizaion and is Implicaions Masaaki Fujii, Yasufumi Shimada, Akihiko Takahashi Firs version: 22 Sepember 2010 Curren version: 24 Sepember 2010 Absrac In recen years, we have observed dramaic increase of collaeralizaion as an imporan credi risk miigaion ool in over he couner (OTC marke [6]. Combined wih he significan and persisen widening of various basis spreads, such as Libor-OIS and cross currency basis, he praciioners have sared o noice he imporance of difference beween he funding cos of conracs and Libors of he relevan currencies. In his aricle, we inegrae he series of our recen works [1, 2, 4] and explain he consisen consrucion of erm srucures of ineres raes in he presence of collaeralizaion and all he relevan basis spreads, heir no-arbirage dynamics as well as heir implicaions for derivaive pricing and risk managemen. Paricularly, we have shown he imporance of he choice of collaeral currency and embedded cheapeso-deliver (CTD opion in a collaeral agreemen. Keywords : swap, collaeral, Libor, OIS, EONIA, Fed-Fund, cross currency, basis, HJM, CSA, CVA 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 view of Shinsei Bank, Limied or any oher insiuions. The auhors are no responsible or liable in any manner for any losses and/or damages caused by he use of any conens in his research. M.Fujii is graeful for friends and former colleagues of Morgan Sanley, especially in IDEAS, IR opion, and FX Hybrid desks in Tokyo for fruiful and simulaing discussions. The conens of he paper do no represen any views or opinions of Morgan Sanley. Graduae School of Economics, The Universiy of Tokyo. Capial Markes Division, Shinsei Bank, Limied Graduae School of Economics, The Universiy of Tokyo 1

2 1 Inroducion The recen financial crisis and he following liquidiy and credi squeeze have caused significan and persisen widening of various basis spreads 1. In paricular, we have winessed drasic movemen of cross currency swap (CCS, Libor-OIS, and enor swap 2 (TS basis spreads. In some occasions, he size of spreads has exceeded several ens of basis poins, which is far wider han he general size of bid/offer spreads. Furhermore, here has been a dramaic increase of collaeralizaion in financial conracs recen years, and i has become almos a marke sandard among he major financial insiuions, a leas [6]. The imporance of he collaeralizaion is mainly wofold: 1 reducion of he counerpary credi risk, and 2 change of he funding cos of he rade. The firs one is well recognized, and here exis a large number of sudies in he conex of credi value adjusmen (CVA. Alhough i is no as obvious as he firs one, he second effec is also imporan and i has sared o arac srong aenions among praciioners. As we will see laer, he deails of collaeralizaion specified in CSA (credi suppor annex, he exisence of various basis spreads change he effecive discouning rae appropriae for he specific conrac. Because of he characerisic of he producs, he effecs are mos relevan in ineres rae and long-daed foreign exchange (FX markes. These findings cas a srong warning o he financial insiuions using he sandard Libor Marke Model (LMM, which reas he Libor as a risk-free ineres rae and hence canno reflec he exisence of various basis spreads and collaeralizaion. These drawbacks make he LMM incapable of calibraing o he relevan swap markes nor heir dynamics, which is likely o cause he financial firms overlooking criical risk exposures. In his aricle, we provide a sysemaic soluion for he above marke developmens by inegraing our series works, Fujii, Shimada & Takahashi (2009, 2009, 2010 [1, 2, 4]. We firs presen he pricing formula of derivaives under he collaeralizaion, including he case where he paymen and collaeral currencies are differen. Based on his resul, we propose he procedures for he consisen erm srucure consrucion of collaeralized swaps in muli-currency environmen. Secondly, we provide he no-arbirage dynamics under Heah-Jarrow-Moron (HJM framework which is capable of calibraing o all he relevan swaps and allow full sochasic modeling of basis spreads. Thirdly, we explain he imporance of he choice of collaeral currency and he effecs of embedded cheapeso-deliver opion in collaeral agreemen when i allows he replacemen of collaeral currency. And hen, we conclude. As relaed works, we refer o Johannes & Sundaresan (2007 [7] (and is working paper as he firs work focusing on he cos of collaeral posing and sudying is effec on he swap par raes based on empirical analysis of he US reasury and swap markes. More recenly, Pierbarg (2010 [8] has used he similar formula given in [7] o ake he collaeral cos ino accoun for he opion pricing in general, bu he has no discussed he case where he paymen and collaeral currencies are differen, nor he erm srucure modeling of ineres rae and FX under he collaeralizaion. 1 A basis spread generally means he ineres rae differenials beween wo differen floaing raes. 2 I is a floaing-vs-floaing swap ha exchanges Libors wih wo differen enors wih a fixed spread in one side. 2

3 2 Pricing under he collaeralizaion This secion reviews [1], our resuls on pricing derivaives under he collaeralizaion. Le us make he following simplifying assumpions abou he collaeral conrac. 1. Full collaeralizaion (zero hreshold by cash. 2. The collaeral is adjused coninuously wih zero minimum ransfer amoun. In fac, he daily margin call is now quie popular in he marke and here is also an increasing endency o include a provision for add-hoc call, which allows he paricipans o call margin arbirary ime in addiion o he periodical regular calls. Furhermore, here is a wide movemen o reduce he hreshold and minimum ransfer amoun since he various sudies indicae ha he exisence of meaningful size of hese variables significanly reduces he effeciveness of credi risk miigaion of collaeralizaion, which resuls in higher CVA charge in urn. These developmens make our assumpions a good proxy for he marke realiy a leas for he sandard fixed income producs. Since he assumpions allow us o neglec he loss given defaul of he counerpary, we can rea each rade/paymen separaely wihou worrying abou he non-lineariy arising from he neing effecs and he asymmeric handling of exposure. We consider a derivaive whose payoff a ime T is given by h (i (T in erms of currency i. We suppose ha he currency j is used as he collaeral for he conrac. Noe ha he insananeous reurn (or cos when i is negaive by holding he cash collaeral a ime is given by y (j ( = r (j ( c (j (, (2.1 where r (j and c (j denoe he risk-free ineres rae and collaeral rae of he currency j, respecively. In he case of cash collaeral, he collaeral rae c (i is usually given by he overnigh rae of he corresponding currency, such as Fed-Fund rae for USD. If we denoe he presen value of he derivaive a ime by h (i ( (in ( erms of currency i, he collaeral amoun posed from he counerpary is given by h (i (/f x (i,j (, where f x (i,j ( is he foreign exchange rae a ime represening he price of he uni amoun of currency j in erms of currency i. These consideraions lead o he following calculaion for he collaeralized derivaive price, h (i ( = E Q i + f x (i,j (E Q j [e ] T r (i(sds h (i (T [ T e s r(j (udu y (j (s ( ] h (i (s f x (i,j ds (s, (2.2 where E Q i [ ] is he ime condiional expecaion under he risk-neural measure of currency i, where he money-marke accoun of currency i is used as he numeraire. Here, he second erm akes ino accoun he reurn/cos from holding he collaeral. By aligning he measure o Q i, we obain h (i ( = E Q i [ e T T r (i(sds h (i (T + e ] s r(i (udu y (j (sh (i (sds. (2.3 3

4 Now, i is easy o see ha X( := e 0 r(i (sds h (i ( + 0 e s 0 r(i (udu y (j (sh (i (sds (2.4 is a Q i -maringale under appropriae inegrabiliy condiions. This ells us ha he process of he opion value can be wrien as ( dh (i ( = r (i ( y (j ( h (i (d + dm( (2.5 wih some Q i -maringale M. As a resul, we have he following heorem: Theorem 1 3 Suppose ha h (i (T is a derivaive s payoff a ime T in erms of currency i and ha he currency j is used as he collaeral for he conrac. Then, he value of he derivaive a ime, h (i ( is given by where h (i ( = E Q i [e T = D (i (, T E T c (i ( r (i (sds T e y (j (sds h (i (T [e ] T y (i,j(sds h (i (T ] (2.6, (2.7 y (i,j (s = y (i (s y (j (s (2.8 wih y (i (s = r (i (s c (i (s and y (j (s = r (j (s c (j (s. Here, we have also defined he collaeralized zero-coupon bond of currency i as D (i (, T = E Q i [e ] T c (i (sds (2.9 and he collaeralized forward measure T(i c, where he collaeralized zero-coupon bond is used as he numeraire; hus, E T (i c [ ] denoes he ime condiional expecaion under he measure T(i c. As a corollary of he heorem, we have h( = E Q [e ] T c(sds h(t = D(, T E T c [h(t ] (2.10 when he paymen and collaeral currencies are he same. Finally, le us also define he forward collaeral rae of currency i, c (i (, T, for he convenience of laer discussion: or equivalenly c (i (, T = T ln D(i (, T, (2.11 ( T D (i (, T = exp c (i (, sds. ( Alhough we are dealing wih coninuous processes here, we obain he same resul as long as here is no simulaneous jump of underlying asses when he counerpary defauls even in more general seing. 4

5 3 Calibraion o Single Currency Swaps In his secion, we consruc he relevan yield curves in a single currency marke, where here exis hree differen ypes of ineres rae swap, which are overnigh index swap (OIS, sandard ineres rae swap (IRS, and enor swap (TS. In he following, we explain how we can exrac relevan forwards based on he resul of previous secion following he works [1, 3, 4]. Throughou his secion, we assume ha he relevan swap is collaeralized by he domesic (or paymen currency. 3.1 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. Here, le us assume ha he OIS iself is coninuously collaeralized by he domesic currency. In his case, using he Eq.(2.10, we ge he consisency condiion of T 0 -sar T N -mauring OIS rae as OIS N N n E Q [ e Tn 0 c(sds ] = E Q [ e ( T ] Tn n 0 c(sds T c(sds e n 1 1. (3.1 Here, OIS N is he marke quoe of par rae for he lengh-n OIS, and c( is he overnigh (and hence collaeral rae a ime of he domesic currency. and δ denoe day-coun fracions for fixed and floaing legs, respecively. We can simplify he above equaion ino he form OIS N N n D(0, T n = D(0, T 0 D(0, T N (3.2 by using he collaeralized zero-coupon bond. Now, from Eq.(3.2, we can easily obain he erm srucure of D(0, by he sandard boosrap and spline echniques [5]. 3.2 Ineres rae swap In he sandard ineres rae swap (IRS, wo paries exchange a fixed coupon and Libor for a cerain period wih a given frequency. The enor of Libor τ is deermined by he frequency of floaing paymens, i.e., 6m-enor for semi-annual paymens, for example. For a T 0 -sar T M -mauring IRS wih he Libor of enor τ, we have IRS M M m=1 m D(0, T m = M δ m D(0, T m E T m[l(t c m 1, T m ; τ] (3.3 m=1 as a consisency condiion. Here, IRS M is he marke IRS quoe, L(T m 1, T m ; τ is he Libor rae wih enor τ for a period of (T m 1, T m. Since we already have he erm srucure of D(0,, i is sraighforward o exrac he se of E T m c [L(Tm 1, T m ; τ] for each T m from Eq. (3.3. 5

6 3.3 Tenor swap A enor swap is a floaing-vs-floaing swap where he paries exchange Libors wih differen enors wih a fixed spread on one side, which we call TS basis spread in his aricle. Usually, he spread is added on op of he Libor wih shorer enor. For example, in a 3m/6m enor swap, quarerly paymens wih 3m Libor plus spread are exchanged by semi-annual paymens of 6m Libor fla. The condiion ha he enor spread should saisfy is δ n D(0, T n ( E T M n c [L(T n 1, T n ; τ S ] + TS N = δ m D(0, T m E T m c [L(T m 1, T m ; τ L ], m=1 (3.4 where T N = T M, m and n disinguish he difference of paymen frequency. TS N denoes he marke quoe of of he basis spread for he T 0 -sar T N -mauring enor swap. The spread is added on he Libor wih he shorer enor τ S in exchange for he Libor wih longer enor τ L. From he above relaion, one can exrac he forward Libor wih differen enors. Here, we have explained using slighly simplified erms of conrac. In he acual marke, i is more common ha he coupons of he Leg wih he shorer enor are compounded by Libor fla and being paid wih he same frequency of he oher Leg. However, he size of correcion from he above simplificaion can be shown negligibly small. 3.4 Calibraion Example Figure 1: USD zero rae curves of Fed-Fund rae, 3m and 6m Libors. In Fig. 1, we have given examples of calibraed yield curves for USD marke on 2009/3/3 and 2010/3/16, where R OIS, R 3m and R 6m denoe he zero raes for OIS (Fed- Fund rae, 3m and 6m forward Libor, respecively. R OIS ( is defined as R OIS (T = 6

7 ln(d(0, T /T. For he forward Libor, he zero-rae curve R τ ( is deermined recursively hrough he relaion ( E T m[l(t c m 1, T m ; τ] = 1 e Rτ (T m 1T m 1 δ m e R 1. (3.5 τ (T m T m In he acual calculaion of D(0,, we have used he Fed-Fund vs 3m-Libor basis swap, where he wo paries exchange 3m Libor and he compounded Fed-Fund rae wih spread, which seems more liquid and a larger number of quoes available han he usual OIS. In Fig. 2, one can see he hisorical behavior of he spread beween 1yr IRS and OIS for USD, JPY and EUR, where he underlying floaing raes of IRS are 3m-Libor for USD and EUR and 6m-Libor for JPY. Figure 2: Difference beween 1yr IRS and OIS. Underlying floaing raes are 3m-Libor for USD and EUR, and 6m-Libor for JPY. Remarks: In he above calculaions, we have assumed ha he condiions given in he previous secion are saisfied, and also ha all he insrumens are collaeralized by he cash of domesic currency which is he same as he paymen currency. Cauious readers may worry abou he possibiliy ha he marke quoes conain significan conribuions from marke paricipans who use a foreign currency as collaeral. However, he induced changes in IRS/TS quoes are very small and impossible o disinguish from he bid/offer spreads in normal circumsances, because he correcion appears boh in he fixed and floaing legs which keeps he marke quoes almos unchanged 4. However, as we will see in he laer secions, he presen values of off-he-marke swaps will be significanly affeced when he collaeral currency is differen. 4 As for cross currency swaps, he change can be a few basis poin, and hence comparable o he marke bid/offer spreads. 7

8 4 Calibraion o Cross Currency Swaps Afer compleing he calibraion o he single currency swaps, we should have obained he erms srucures of D (i (0, T and E T (i c [ L (i (T m 1, T m ; τ ] for each currency and enor. The remaining freedom of he model is he erm srucure of y (i,j ( for each relevan currency pair. As we will see, his is he mos imporan ingredien o deermine he value of CCS. 4.1 Mark-o-Marke Cross Currency Swap In his secion, we will discuss how o make he erm srucure consisen wih CCS (cross currency swap markes [1, 3]. The curren marke is dominaed by USD crosses where 3m USD Libor fla is exchanged wih 3m Libor of a differen currency wih addiional basis spread. There are wo ypes of CCS, one is CNCCS (Consan Noional CCS, and he oher is MMCCS (Mark-o-Marke CCS. In a CNCCS, he noionals of boh legs are fixed a he incepion of he rade and kep consan unil is mauriy. On he oher hand, in a MMCCS, he noional of USD leg is rese a he sar of every calculaion period of he Libor while he noional of he oher leg is kep consan hroughou he conrac period. Alhough he required calculaion becomes a bi more complicaed, we will use MMCCS for calibraion due o is beer liquidiy 5. Firs, le us define y (i,j (, s, he forward rae of y (i,j (s a ime as e T y (j,i (,sds = E Q j [e T y (j,i (sds ]. (4.1 We consider a MMCCS of (i, j currency pair, where he leg of currency i (inended o be USD needs noional refreshmens. We assume ha he collaeral is posed in currency i, which seems common in he marke. The value of j-leg of a T 0 -sar T N -mauring MMCCS is calculaed as P V j = D (j (0, T 0 E T c 0,(j + [ e ] T 0 [ 0 y (j,i (sds + D (j (0, T N E T n,(j c e ] T N 0 y (j,i (sds [ δ n (j D (j (0, T n E T n,(j c e ( ] Tn 0 y (j,i (sds L (j (T n 1, T n ; τ + B N, (4.2 where he basis spread B N is available as a marke quoe. In [2], we have assumed ha all of he {y (k ( } and hence {y (i,j ( } are deerminisic funcions of ime o make he curve consrucion more racable. Here, we slighly relax he assumpion allowing randomness of {y (i,j ( }. As long as we assume ha {y (i,j ( } is independen from he dynamics of Libors and collaeral raes, he procedures of boosrapping given in [2] can be applied in he same way 6. Under he assumpion of independence, we obain P V j = D (j (0, T 0 e T 0 0 y (j,i(0,sds + D (j (0, T N e T N 0 y (j,i (0,sds + δ n (j D (j (0, T n e ( Tn 0 y (j,i (0,sds E T n,(j[l c (j (T n 1, T n ; τ] + B N. (4.3 5 As for he deails of MMCCS and CNCCS, see [1, 3]. 6 In pracice, i would no be a problem even if here is a non-zero correlaion as long as i does no meaningfully change he model implied quoes compared o he marke bid/offer spreads. 8

9 where On he oher hand, he presen value of i-leg in erms of currency j is given by P V i = = + E Q i E Q i [ e ] T n 1 0 c (i(sds f x (i,j (T n 1 /f x (i,j (0 [ e ( ] Tn 0 c (i(sds f x (i,j (T n δ n (i L (i (T n 1, T n ; τ /f x (i,j (0 δ (i n D (i (0, T n E T c n,(i [ f (i,j x (T n 1 f (i,j x (0 B (i (, T k ; τ = E T k,(i c [ ] L (i (T k 1, T k ; τ B (i (T n 1, T n ; τ ( 1 D (i (, T k 1 δ (i D (i 1 (, T k k ], (4.4, (4.5 which represens a Libor-OIS spread. Since we found no persisen correlaion beween FX and Libor-OIS spread in hisorical daa, we have reaed hem as independen variables. Even if a non-zero correlaion exiss in a cerain period, he expeced correcion seems no numerically imporan due o he ypical size of bid/offer spreads for MMCCS (abou a few bps a he ime of wriing. Since 3-monh iming adjusmen of FX is safely negligible, an approximae value of i-leg is obained as P V i δ n (i D (i (0, T n D(j (0, T n 1 Tn 1 D (i (0, T n 1 e 0 y (j,i(0,sds B (i (0, T n ; τ. (4.6 Here, we have used he following expression of he forward FX collaeralized wih currency i: f x (i,j (, T = f x (i,j ( D(j (, T T D (i (, T e y (j,i(,sds. (4.7 Noice ha, afer calibraing o he single currency swaps for each currency, he only unknown in Eqs. (4.3 and (4.6 is y (j,i (0,. Therefore, one can easily see ha he consisency condiion P V i = P V j wih given marke spread B N for each mauriy allows us o boosrap he erm srucure of {y (i,j (0, }. Finally, le us menion he fac ha he (i, j-mmccs par spread is expressed as B N = δ n (i D (i D(j T n 1 T n e T n 1 0 y (j,i(0,sds B (i T n δ n (j D (j T n e Tn 0 y (j,i(0,sds B (j T n D (i T n 1 D (j T n 1 e T n 1 0 y (j,i (0,sds ( e Tn T n 1 y (j,i (0,sds 1 ] / δ n (j D (j T n e Tn 0 y (j,i(0,sds, in he above menioned approximaion, where we have shorened he noaions as D (k (0, T = D (k T and B (k (0, T ; τ = B (k T. (4.8 9

10 4.2 Calibraion Example and Hisorical Behavior In Fig. 3, we have given examples of calibraion for EUR/USD and USD/JPY MMCCS as of 2010/3/16. We have ploed he zero raes of y (j,i defined as ( ln E Q j [e ] T 0 y(j,i (sds R y(j,i (T = = 1 T y (j,i (0, sds (4.9 T T ogeher wih he erm srucure of MMCCS basis spreads. I is easy o expec ha here are significan conribuions from he second line of Eq. (4.8 o he CCS basis spreads from he similariies beween R y(x,usd and CCS quoes. The implied forward FXs derived from Eq. (4.7 were well wihin he bid/offer spreads 7. 0 Figure 3: MMCCS par spreads, R y(jp Y,USD and R y(eur,usd as of 2010/3/16. Le us also check he hisorical behavior of R y(eur,usd and R y(jp Y,USD given in Fig. 4 o 8 8. For boh cases, he erm srucures of R y have quie similar shapes and levels o hose of he corresponding CCS basis spreads. In Fig. 4, hisorical behavior of R y(x,usd (T = 5y (X = EUR, JPY and corresponding 5y-MMCCS spreads are given. One can see ha a significan porion of CCS spreads movemen sems from he change of y (i,j, raher han he difference of Libor-OIS spread beween wo currencies. The level (difference-correlaion beween R y and CCS spread is quie high, which is abou 93% (75% for EUR or abou 70% (92% for JPY for he hisorical series used in he figure, for example. The 3m-roll hisorical volailiies of y (EUR,USD insananeous forwards, which are annualized in absolue erms, are given in Fig. 9. In a calm marke, hey end o be 50 bps or so, bu hey were more han a percenage poin jus afer he marke crisis, 7 In any case, i is quie wide for long mauriies. 8 Due o he lack of OIS daa for JPY marke, we have only a limied daa for (JPY,USD pair. We have used Cubic Monoone Spline for calibraion alhough he figures are given in linear plos for ease. For spline echnique, see [5], for example. 10

11 Figure 4: R y(eur,usd (5y, R y(jp Y,SD (5y and corresponding quoes of 5y-MMCCS. which is reflecing a significan widening of he CCS basis spread o seek USD cash in he low liquidiy marke. Excep he CCS basis spread, y does no seem o have persisen correlaions wih oher variables such as OIS, IRS and FX forwards. A leas, wihin our limied daa, he 3m-roll hisorical correlaions wih hese variables flucuae mainly around ±20% or so. 5 No-arbirage dynamics in Heah-Jarrow-Moron Framework In his secion, we give he se of SDEs for he whole sysem 9. From he previous discussion, we have seen ha he relevan building blocks of erm srucures are given by or equivalenly {D (i (, T }, {E T c n,(i [ ] L (i (T n 1, T n ; τ }, and {y (i,j (, T }, (5.1 {c (i (, T }, {B (i (, T ; τ}, and {y (i,j (, T }, (5.2 for each mauriy T, enor τ, currency i, and currency pair (i, j. As seen in Sec.2, he collaeral rae plays a criical role as he effecive discouning rae. Thus, le us fix he base currency i, and consider he dynamics of c (i (, s. Suppose ha he dynamics of he forward collaeral rae under he Q i is given by dc (i (, s = α (i (, sd + σ (i c (, s dw Q i (, (5.3 where α (i (, s is a scalar funcion for is drif, and W Q i ( is a d-dimensional Brownian moion under he Q i -measure. σ c (i (, s is a d-dimensional vecor and he following 9 See [2, 3, 4] for deails. 11

12 abbreviaion have been used: σ (i c (, s dw Q i ( = d j=1 [ ] σ c (i (, s dw Q i j (. (5.4 j Here, we will no specify he deails of he volailiy process : I can depend on he collaeral rae iself, or any oher sae variables. Applying Iô s formula o Eq.(2.12, we have dd (i (, T D (i (, T = { c (i ( T α (i (, sds T σ (i c 2} ( T (, sds d σ c (i (, sds dw Q i. (5.5 On he oher hand, by definiion, he drif rae of D (i (, T should be c (i ( = c (i (,. Therefore, i is necessary ha α (i (, s = d j=1 [σ (i c = σ (i c (, s (, s] j ( s ( s σ c (i (, udu σ c (i (, udu j (5.6, (5.7 and as a resul, he process of c (i (, s under he Q i -measure is given by ( s dc (i (, s = σ c (i (, s σ c (i (, udu d + σ c (i (, s dw Q i (. (5.8 In exacly he same way, we obain ( s dy (i,j (, s = σ y (i,j (, s σ y (i,j (, udu d + σ y (i,k (, s dw Q i (. (5.9 Nex, le us consider he dynamics of Libor-OIS spread, B (i (, T ; τ. From he definiion in Eq. (4.5, i is clear ha B (i (, T ; τ is a maringale under he collaeralized forward measure T(i c, where he numeraire is given by D(i (, T. Using he Maruyama- Girsanov heorem, one can see ha Brownian moion under he forward measure T(i c, or W T (i, c is relaed o he W Q i as and hence, one easily obains ( T dw T (i( c = db (i (, T ; τ B (i (, T ; τ = σ(i B (, T ; τ ( T Since we have he relaion σ c (i (, sds d + dw Q i (, (5.10 σ c (i (, sds d + σ (i B (, T ; τ dw Q i (. (5.11 r (i ( r (j ( = c (i ( c (j ( + y (i,j ( (

13 he SDE for he spo FX process is given by df x (i,j (/f x (i,j ( = ( c (i ( c (j ( + y (i,j ( d + σ (i,j X ( dw Q i (. (5.13 Maruyama-Girsanov heorem ells us ha Brownian moions in wo differen currencies are relaed by he formula dw Q j ( = σ (i,j X (d + dw Q i (, (5.14 which allows us o derive he SDEs of hese building blocks under a differen base currency, oo. For example, he SDE of collaeral rae of he foreign currency j is given by [( s ] dc (j (, s = σ c (j (, s σ c (j (, udu σ (i,j X ( d + σ c (j (, s dw Q i (. (5.15 Pricing formulas for some of he vanilla opions are available in [2]. 6 Implicaions for Derivaive Pricing Alhough i is worh exploring various implicaions of collaeralizaion by using he dynamics given in he previous secion, he leading order effecs are expeced o arise from he change of he effecive discouning rae. In his secion, we discuss some of hese imporan implicaions by using he calibraed yield curves. 6.1 Choice of Collaeral Currency When he paymen and collaeral currencies are he same, he discouning facor is given by he collaeral rae which is under conrol of he relevan cenral bank as indicaed in Eq. (2.10. Tradiionally, among financial firms, he Libor curve has been widely used o discoun he fuure cash flows. However, his mehod would easily underesimae heir values by several percenage poins for long mauriies, even wih he curren level of Libor- OIS spread, or bps. Considering he mechanism of collaeralizaion, financial firms need o hedge he change of OIS in addiion o he sandard hedge agains he movemen of Libors. Especially, he risk of floaing-rae paymens needs o be checked carefully, since he overnigh rae can move in he opposie direcion o he Libor as was observed in his financial crisis. In Fig. 10, he presen values of Libor floaing legs wih final principal (= 1 paymen P V = δ n D(0, T n E T n c [L(T n 1, T n ; τ] + D(0, T N (6.1 are given for various mauriies. If radiional Libor discouning is being used, he sream of Libor paymens has he consan presen value 1, which is obviously wrong from our resuls. This poin is very imporan in risk-managemen, since financial firms may overlook he quie significan ineres-rae risk exposure when hey adop he radiional ineres rae model in heir sysem. 13

14 If a rade wih paymen currency j is collaeralized by foreign currency i, an addiional modificaion o he discouning facor appears ( See, Eq. ( : e T y (j,i(,sds = E Q j [e ] T y (j,i (sds. (6.2 From Figs. 6 and 8, one can see ha posing USD as collaeral ends o be expensive from he view poin of collaeral payers, which is paricularly he case when he marke is illiquid. For example, from Fig. 8, one can see ha he value of JPY paymen in 10 years ime is more expensive by around 3% when i is collaeralized by USD insead of JPY. The effecs should be more profound for emerging currencies where he implied CCS basis spread can easily be 100 bps or more. 6.2 Embedded Cheapes-o-Deliver Opion We now discuss he embedded CTD opion in a collaeral agreemen. In some cases, financial firms make conracs wih CSA allowing several currencies as eligible collaeral. Suppose ha he payer of collaeral has a righ o replace a collaeral currency whenever he wans. If his is he case, he collaeral payer should choose he cheapes collaeral currency o pos, which leads o he modificaion of he discouning facor as [e ] T max i C{y (j,i (s}ds, (6.3 E Q j where C is he se of eligible currencies. Noe ha, by he definiion of collaeral payers, hey wan o make ( P V (> 0 as small as possible. Alhough here is a endency oward a CSA allowing only one collaeral currency o reduce he operaional burden, i does no seem uncommon o accep he domesic currency and USD as eligible collaeral, for example. In his case, he above formula urns ou o be [e ] T max{y (j,usd (s,0}ds. (6.4 E Q j In Figs. 11 and 12, we have ploed he modificaions of discouning facors given in Eq. (6.4, for j = EUR and JPY as of 2010/3/16. We have used he Hull-Whie model for he dynamics of y (EUR,USD ( and y (JP Y,USD (, wih a mean reversion parameer 1.5% per annum and he se of volailiies, σ = 0, 25, 50 and 75 bps 11, respecively. As can be seen from he hisorical volailiies given in Fig. 9, σ can be much higher under volaile environmen. The curve labeled by USD (EUR, JPY denoes he modificaion of he discoun facor when only USD (EUR, JPY is eligible collaeral for he ease of comparison. One can easily see ha here is significan impac when he collaeral currency chosen opimally. For example, from Fig. 12, one can see if he paries choose he collaeral currency from JPY and USD opimally, i roughly increases he effecive discouning rae by around 50 bps annually even when he annualized volailiy of spread y (JP Y,USD is 50 bps. In he calculaion, we have used daily-sep Mone Carlo simulaion. Alhough we can expec ha here are various obsacles o implemen he opimal sraegy in pracice, he developmen of common infrasrucure for collaeral managemen, such as he elecronic auomaion of he margin call and collaeral delivery, will make he opimal choice of collaeral currency be an imporan issue in coming years. 10 Here, we are assuming independence of y from reference asses. 11 These are annualized volailiies in absolue erms. 14

15 7 Conclusions In his aricle, which inegraes he series of our recen works [1, 2, 4], we explain he consisen consrucion of muliple swap curves in he presence of collaeralizaion and cross currency basis spreads, heir no-arbirage dynamics, and implicaions for derivaive pricing. Especially, we have shown he imporance of he choice of collaeral currency and embedded cheapes-o-deliver (CTD opion in collaeral agreemens. We have also emphasized dangers o use he sandard LMM in acual financial business since i allows he financial firms o overlook poenially criical risk exposures. References [1] Fujii, M., Shimada, Y., Takahashi, A., 2009, A noe on consrucion of muliple swap curves wih and wihou collaeral, CARF Working Paper Series F-154, available a hp://ssrn.com/absrac= [2] Fujii, M., Shimada, Y., Takahashi, A., 2009, A Marke Model of Ineres Raes wih Dynamic Basis Spreads in he presence of Collaeral and Muliple Currencies, CARF Working Paper Series F-196, available a hp://ssrn.com/absrac= [3] Fujii, M., Shimada, Y., Takahashi, A., 2010, On he Term Srucure of Ineres Raes wih Basis Spreads, Collaeral and Muliple Currencies, based on he presenaions given a Inernaional Workshop on Mahemaical Finance a Tokyo, Japanese Financial Service Agency, and Bank of Japan. [4] Fujii, M., Shimada, Y., Takahashi, A., 2010, Collaeral Posing and Choice of Collaeral Currency-Implicaions for derivaive pricing and risk managemen, CARF Working Paper Series F-216, available a hp://ssrn.com/absrac= [5] Hagan, P.S. and Wes, G., 2006, Inerpolaion Mehods for Curve Consrucion, Applied Mahemaical Finance, Vol. 13, No. 2, [6] ISDA Margin Survey 2010, Preliminary Resuls, Marke Review of OTC Derivaive Bilaeral Collaeralizaion Pracices, ISDA Margin Survey [7] Johannes, M. and Sundaresan, S., 2007, The Impac of Collaeralizaion on Swap Raes, Journal of Finance 62, [8] Pierbarg, V., 2010, Funding beyond discouning : collaeral agreemens and derivaives pricing Risk Magazine. 15

16 Figure 5: Hisorical movemen of calibraed R y(eur,usd. Figure 6: Examples of R y(eur,usd erm srucure. 16

17 Figure 7: Hisorical movemen of calibraed R y(jp Y,USD. Figure 8: Examples of R y(jp Y,USD erm srucure. 17

18 Figure 9: 3M-Roll hisorical volailiy of y (EUR,USD insananeous forward. Annualized in absolue erms. Figure 10: Presen value of USD Libor sream wih final principal (= 1 paymen. 18

19 Figure 11: Modificaion of EUR discouning facors based on HW model for y (EUR,USD as of 2010/3/16. The mean-reversion parameer is 1.5%, and he volailiy is given a each label. Figure 12: Modificaion of JPY discouning facors based on HW model for y (JP Y,USD as of 2010/3/16. The mean-reversion parameer is 1.5%, and he volailiy is given a each label. 19