CARF Working Paper CARF-F-196. A Market Model of Interest Rates with Dynamic Basis Spreads in the presence of Collateral and Multiple Currencies

Transcription

1 CARF Working Paper CARF-F-196 A Marke Model of Ineres Raes wih Dynamic Basis Spreads in he presence of Collaeral and Muliple Currencies Masaaki Fujii The Universiy of Tokyo Yasufumi Shimada Shinsei Bank, Limied Akihiko Takahashi The Universiy of Tokyo Firs Version: December 2009 Curren Version: April 2011 CARF is presenly suppored by Bank of Tokyo-Misubishi UFJ, Ld., Ciigroup, Dai-ichi Muual Life Insurance Company, Meiji Yasuda Life Insurance Company, Nippon Life Insurance Company, Nomura Holdings, Inc. and Sumiomo Misui Banking Corporaion in alphabeical order. This financial suppor enables us o issue CARF Working Papers. CARF Working Papers can be downloaded wihou charge from: hp:// Working Papers are a series of manuscrips in heir draf form. They are no inended for circulaion or disribuion excep as indicaed by he auhor. For ha reason Working Papers may no be reproduced or disribued wihou he wrien consen of he auhor.

2 A Marke Model of Ineres Raes wih Dynamic Basis Spreads in he presence of Collaeral and Muliple Currencies Masaaki Fujii, Yasufumi Shimada, Akihiko Takahashi Firs version: 12 November 2009 Curren version: 3 April 2011 Absrac The recen financial crisis caused dramaic widening and elevaed volailiies among basis spreads in cross currency as well as domesic ineres rae markes. Furhermore, he widespread use of cash collaeral, especially in fixed income conracs, has made he effecive funding cos of financial insiuions for he rades significanly differen from he Libor of he corresponding paymen currency. Because of hese marke developmens, he ex-book syle applicaion of a marke model of ineres raes has now become inappropriae for financial firms; I canno even reflec he exposures o hese basis spreads in pricing, o say nohing of proper dela and vega or kappa hedges agains heir movemens. This paper presens a new framework of he marke model o address all hese issues. Keywords : Marke Model, HJM model, Libor, enor, swap, curve, OIS, cross currency, basis spread, ineres rae model, derivaives, muli-currency 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

3 1 Inroducion The recen financial crisis and he following liquidiy and credi squeeze have caused significan widening and elevaed volailiies among various ypes of basis spreads 1. In paricular, we have winessed dramaic moves 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 a leas in he fixed income world 11. As seen laer, he exisence of collaeral agreemen reduces he discouning rae significanly relaive o he Libor of a given currency hrough frequen mark-o-marke and collaeral posings ha follow. Alhough he Libor Marke Model has been widely used among marke paricipans since is invenion, is ex-book syle applicaion does no provide an appropriae ool o handle hese new realiies; I can only rea one ype of Libor, and is unable o reflec he movemen of spreads among Libors wih differen enors. The discouning of a fuure cash flow is done by he same Libor, which does no reflec he exisence of collaerals and he funding cos differenials among muliple currencies in CCS markes 3. As a response o hese marke developmens, he invenion of a more sophisicaed financial model which is able o reflec all he relevan swap prices and heir behavior has risen as an urgen ask among academics and marke paricipans. Surprisingly, i is no a all a rivial ask even consrucing a se of yield curves explaining he various swap prices in he marke consisenly while keeping no-arbirage condiions inac. Amerano and Bianchei proposed a simple scheme ha is able o recover he level of each swap rae in he marke, bu gives rise o arbirage possibiliies due o he exisence of muliple discouning raes wihin a single currency. The model proposed by Bianchei using a muli-currency analogy does no seem o be a pracical soluion alhough i is a leas free from arbirage. The main problem of he model is ha he curve calibraion can no be separaed from he opion calibraion due o he enanglemen of volailiy specificaions, since i reas he usual Libor paymen as a quano of differen currencies wih a pegged FX rae. I also makes he daily hedge agains he move of basis spreads quie complicaed. In addiion, neiher of Bianchei 2008 and Amerano and Bianchei 2009 has discussed how o make he model consisen wih he collaeralizaion and cross currency swap markes. Our recen work, A Noe on Consrucion of Muliple Swap Curves wih and wihou Collaeral 6, have developed a mehod of swap-curve consrucion which allows us o rea overnigh index swap OIS, ineres rae swaps IRS, enor swaps TS, and cross currency swaps CCS consisenly wih explici consideraions of he effecs from collaeralizaion. The curren paper presens a framework of sochasic ineres rae models wih dynamic basis spreads addressing all he above menioned issues, where he oupu 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. 3 As for he cross currency basis spread, i has been an imporan issue for global financial insiuions for many years. However, here exiss no lieraure ha direcly akes is dynamics ino accoun consisenly in a muli-currency seup of an ineres rae model. 2

4 of curve calibraions in he work 6 can be direcly used as a saring poin of simulaion. In he mos generic seup in Ref.6, here remained a difficuly o calibrae all he parameers due o he lack of separae quoes of foreign-currency collaeralized swaps in he curren marke. This new work presens a simplified bu pracical way of implemenaion which allows exac fis o he domesic-currency collaeralized OIS, IRS and TS, ogeher wih FX forward and mark-o-marke CCS MMCCS wihou referring o he quoes of foreign collaeralized producs. Also, his paper adops an HJMHeah-Jarrow-Moron- ype framework jus for clariy of presenaion: Of course, i is quie sraighforward o wrie he model using a discreized ineres raes, which becomes an exension of he Libor and Swap marke models4,12. Since our moivaion is o explain he generic modeling framework, he deails of volailiy processes are no specified. Such as analyic expressions of vanilla opions and implicaions o he risk managemen for various ypes of exoics will be presened somewhere else in he fuure adoping a fully specified model. The organizaion of he paper is as follows: The nex secion firsly reminds readers of he pricing formula under he collaeral agreemen. Then, afer reviewing he fundamenal ineres rae producs, i presens he modeling framework wih sochasic basis spreads in a single currency environmen, which enables us o explain hese insrumens consisenly. Secion 3 exends he model ino he muli-currency environmen and explains how o make he model consisen wih he FX forward and MMCCS. Finally, afer Secion 4 briefly commens on inflaion modeling, Secion 5 concludes. 2 Single Currency Marke This secion develops a HJM-ype framework of an ineres rae model in a single currency marke. Our goal is o consruc a framework which is able o explain all he OIS, IRS and TS markes consisenly in an unified way. Here, i is assumed ha every rade has a collaeral agreemen using a domesic currency as collaeral Collaeralizaion Firsly, le us briefly explain he effecs of collaeralizaion. Under he collaeral agreemen, he firm receives he collaeral from he couner pary when he presen value of he ne posiion is posiive and needs o pay he margin called collaeral rae on he ousanding collaeral in exchange. On he oher hand, if he presen value of he ne posiion is negaive, he firm is asked o pos he collaeral o he couner pary and receives he collaeral rae in reurn. Alhough he deails can possibly differ rade by rade due o he OTC naure of he fixed income marke, he mos commonly used collaeral is a currency of developed counries, such as USD, EUR and JPY 11. In his case, he collaeral rae is usually fixed by he overnigh rae of he collaeral currency: for example, Fed-Fund rae, EONIA, and Muan for USD, EUR and JPY, respecively. In general seup, pricing of collaeralized producs is very hard due o he non-lineariy arising from he residual credi risk. Due o he neing procedures, he pricing of each produc becomes dependen on he whole conracs wih he couner pary, which makes he use of model unpracical for he daily pricing and hedging. In order o make he 4 I is easy o apply he similar mehodology o he unsecured or uncollaeralized rade by approximaely aking ino accoun he credi risk by using Libor as he effecive discouning rae. 3

5 problem racable, we will assume he perfec and coninuous collaeralizaion wih zero hreshold by cash, which means ha he mark-o-marke 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 mark-o-marke and adjusmen of collaeral amoun is he marke bes pracice, and he approximaion should no be oo far from he realiy. Under he above simplificaion, we can hink ha here remains no couner pary defaul risk and recover he lineariy among differen paymens. This means ha a generic derivaive is reaed as a porfolio of he independenly collaeralized srips of paymens. We would like o ask readers o consul Sec.3 of Ref. 6 for deails, bu he presen value of a collaeralized derivaive wih paymen ht a ime T is given by 5 h = E Q e T csds ht, 2.1 where E Q denoes he expecaion under he Money-Marke MM measure Q condiioned on he ime- filraion, and cs is he ime-s value of he collaeral rae. Noe ha cs is no necessarily equal o he risk-free ineres rae rs of a given currency. For he laer purpose, le us define he collaeralized zero-coupon bond D as D, T = E Q e T csds, 2.2 which is he presen value of he uni amoun of paymen under he conrac of coninuous collaeralizaion wih he same currency. In laer secions, we will frequenly use he expecaion E T c under he collaeralized-forward measure T c defined as e T csds ht = D, T E T c ht, 2.3 E Q where he collaeralized zero-coupon bond D, T is used as a numeraire. 2.2 Marke Insrumens Before going o discuss he modeling framework, his subsecion briefly summarizes he imporan swaps in a domesic marke as well as he condiions ha par swap raes have o saisfy. They are he mos imporan calibraion insrumens o fix he saring poins of simulaion Overnigh index swap An overnigh index swap OIS is a fixed-vs-floaing swap whose floaing rae is given by he daily compounded overnigh rae. Since he overnigh rae is same as he collaeral rae of he corresponding currency, he following relaion holds 6 : OIS N n E Q e Tn csds = E Q e Tn T n csds T csds e n 1 1, In his secion, he collaeral currency is he same as he paymen currency. 6 Typically, here is only one paymen a he very end for he swap wih shor mauriy < 1yr case, and oherwise here are periodical paymens, quarerly for example. 4

6 or equivalenly, OIS N n D, T n = D, T 0 D, T N, 2.5 where OIS N = OIS, T 0, T N is he marke quoe a ime of he T 0 -sar T N -mauring OIS rae, and T 0 is he effecive dae in he case of spo-sar OIS. Also n denoes he fixed leg day coun fracion for he period of T n 1, T n Ineres rae swap In an 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 D, T m = m=1 M m=1 δ m D, T m E T c m LT m 1, T m ; τ 2.6 as a consisency condiion. Here, IRS M = IRS, T 0, T M ; τ is he ime- value of he corresponding IRS quoe, LT m 1, T m ; τ is he Libor rae wih enor τ for a period of T m 1, T m, and δ m is is day coun fracion. In he remainder of he paper, we disinguish he difference of day coun convenions beween he fixed and floaing legs by and δ, respecively. Here, i is assumed ha he frequencies of boh legs are equal jus for simpliciy, and i does no affec our laer argumens even if his is no he case. Usually, IRS wih a specific choice of τ has dominan liquidiy in a given currency marke, such as 6m for JPY IRS and 3m for USD IRS. Informaion of forward Libors wih oher enors is provided by enor swaps, which will be explained nex 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 paper. 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 given by δ n D, T n E T n c LT n 1, T n ; τ S + T S = M m=1 δ m D, T m E T c m LT m 1, T m ; τ L, 2.7 where T N = T M, m and n disinguish he difference of paymen frequency. T S = T S, T 0, T N ; τ S, τ L denoes he ime- value of TS 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. Here, we have explained using slighly simplified erms of conrac. In he acual marke, he erms of conrac in which coupons of he Leg wih he shor enor are compounded by Libor fla and paid wih he same frequency of he oher Leg is more popular. However, 5

7 he size of correcion from he above simplified resul can be shown o be negligibly small. Please see Appendix for deails Underlying facors in he Model Using he above insrumens and he mehod explained in Ref. 6, we can exrac {D, T }, {E T c LT τ, T ; τ} 2.8 for coninuous ime T 0, T H where T H is he ime horizon of relevan pricing 7, and each relevan enor τ, such as 1m, 3m, 6m, 12m, for example 8. The nex secion will explain how o make hese underlying facors consisenly wih no-arbirage condiions in an HJM-ype framework. 2.3 Model wih Dynamic basis spreads in a Single Currency As seen in Sec.2.1, he collaeral rae plays a criical role as he effecive discouning rae, which leads us o consider is dynamics firs. Le us define he coninuous forward collaeral rae as c, T = ln D, T 2.9 T or, equivalenly D, T = e T c,sds, 2.10 where i is relaed o he spo rae as c, = c. Then, assume ha he dynamics of he forward collaeral rae under he MM measure Q is given by dc, s = α, sd + σ c, s dw Q, 2.11 where α, s is a scalar funcion for is drif, and W Q is a d-dimensional Brownian moion under he Q-measure. σ c, s is a d-dimensional vecor and he following abbreviaion have been used: d σ c, s dw Q = σ c, s j dw Q j j=1 As menioned in he inroducion, he deails of volailiy process will no be specified: I can depend on he collaeral rae iself, or any oher sae variables. Applying Iô s formula o Eq.2.10, we have dd, T D, T = { c T α, sds T 2} T σ c, sds d σ c, sds dw Q Basically, OIS quoes allow us o fix he collaeralized zero coupon bond values, and hen he combinaions of IRS and TS will give us he Libor forward expecaions. 8 We need o use proper spline echnique o ge smooh coninuous resul. See Hagan and Wes , for example. 6

8 On he oher hand, from he definiion of 2.2, he drif rae of D, T should be c. Therefore, i is necessary ha α, s = d s σ c, s j j=1 σ c, udu j 2.14 s = σ c, s σ c, udu, 2.15 and as a resul, he process of c, s under he Q-measure is obained by s dc, s = σ c, s σ c, udu d + σ c, s dw Q Now, le us consider he dynamics of Libors wih various enors. Mercurio has proposed an ineresing simulaion scheme 9. He follows he original idea of Libor Marke Model, and has modeled he marke observables or forward expecaions of Libors direcly, insead of considering he corresponding spo process as Ref.3. We will adop he Mercurio s scheme, bu separaing he spread processes explicily. Firsly, define he collaeralized forward Libor, and OIS forward as L c, T k 1, T k ; τ = E T k c LT k 1, T k ; τ, 2.17 L OIS, T k 1, T k = E T c k and also define he Libor-OIS spread process: 1 δ k = 1 δ k D, Tk 1 D, T k 1 DT k 1, T k , 2.19 B, T k ; τ = L c, T k 1, T k ; τ L OIS, T k 1, T k By consrucion, B, T ; τ is a maringale under he collaeralized forward measure T c, and is sochasic differenial equaion can be wrien as db, T ; τ = B, T ; τσ B, T ; τ dw T c, 2.21 where d-dimensional volailiy funcion σ B can depend on B or oher sae variables as before. Using Maruyama-Girsanov s heorem, one can see ha he Brownian moion under he T c -measure, W T c, is relaed o W Q by he following relaion: T dw T c = σ c, sds d + dw Q As a resul, he process of B, T ; τ under he Q-measure is obained by db, T ; τ T B, T ; τ = σ B, T ; τ σ c, sds d + σ B, T ; τ dw Q Exacly he same idea has been also adoped in inflaion modeling 2 as will be seen laer. 7

9 We need o specify B-processes for all he relevan enors in he marke, such as 1m, 3m, 6m, and 12m, for example. If one wans o guaranee he posiiviy for B, T ; τ L B, T ; τ S where τ L > τ S, i is possible o model his spread as Eq direcly. The lis of wha we need only consiss of hese wo ypes of underlyings. As one can see, here is no explici need o simulae he risk-free ineres rae in a single currency environmen if all he ineresed rades are collaeralized wih he same domesic currency. Le us summarize he relevan equaions: s dc, s = σ c, s σ c, udu d + σ c, s dw Q, 2.24 db, T ; τ B, T ; τ T = σ B, T ; τ σ c, sds d + σ B, T ; τ dw Q Since we already have {c, s} s, and {B, T ; τ} T each for he relevan enor, afer curve consrucion explained in Ref. 6, we can direcly use hem as saring poins of simulaion. If one needs an equiy process S wih an effecive dividend yield given by q wih he same collaeral agreemen, we can model i as ds/s = c q d + σ S dw Q, 2.26 and σ S and q can be sae dependen. Noe ha he effecive dividend yield q is no equal o he dividend yield in he non-collaeralized rade bu should be adjused by he difference beween he collaeral rae and he risk-free rae 10. In pracice, i is likely no a big problem o use he same value or process of he usual definiion of dividend yield. Here, we are no rying o reflec he deails of repo cos for an individual sock, bu raher ry o model a sock index, such as S&P500, for IR-Equiy hybrid rades. 2.4 Simple opions in a single currency This subsecion explains he procedures for simple opion pricing in a single currency environmen. In he following, suppose ha all he forward and opion conracs hemselves are collaeralized wih he same domesic currency Collaeralized overnigh index swapion As was seen from Sec.2.2.1, a T 0 -sar T N -mauring forward OIS rae a ime is given by OIS, T 0, T N = D, T 0 D, T N nd, T n When he lengh of OIS is very shor and here is only one final paymen, one can ge he correc expression by simply replacing he annuiy in he denominaor by N D, T N, a collaeralized zero coupon bond imes a day coun fracion for he fixed paymen. 10 The effecive dividend yield is given by q = q org r c wih he original dividend yield q org. In laer secions, we will use a simplified assumpion ha r c is a deerminisic funcion of ime. 8

10 Under he annuiy measure A, where he annuiy A, T 0, T N = nd, T n is being used as a numeraire, he above OIS rae becomes a maringale. Therefore, he presen value of a collaeralized payer opion on he OIS wih srike K is given by P V = A, T 0, T N E A OIST0, T 0, T N K +, 2.28 where one can show ha he sochasic differenial equaion for he forward OIS is given as follows under he A-measure: { D, T N TN dois, T 0, T N = OIS, T 0, T N σ c, sds D, T 0 D, T N T 0 1 Tn } + n D, T n σ c, sds dw A, 2.29 A, T 0, T N T 0 where W A is he Brownian moion under he A-measure, and is relaed o W Q as dw A = dw Q + 1 A, T 0, T N Tn n D, T n σ c, sds d We can derive an accurae approximaion of Eq.2.28 by applying asympoic expansion echnique 16, 17, 18, or ad hoc bu simpler mehods given, for example, in Brigo and Mercurio Collaeralized ineres rae swapion Nex, le us consider he usual swapion wih he collaeral agreemen. As we have seen in Sec.2.2.2, a T 0 -sar T N -mauring collaeralized forward swap rae is given by IRS, T 0, T N ; τ = δ nd, T n L c, T n 1, T n ; τ nd, T n 2.31 = D, T 0 D, T N nd, T n + δ nd, T n B, T n ; τ nd, T n 2.32 where we have defined IRS-OIS spread Sp OIS as = OIS, T 0, T N + Sp OIS, T 0, T N ; τ, 2.33 Sp OIS, T 0, T N ; τ = δ nd, T n B, T n ; τ nd, T n Noe ha we have slighly abused he noaion of OIS. In realiy, here is no guaranee ha he day coun convenions and frequencies are he same beween IRS and OIS, which may require appropriae adjusmens. 9

11 Sp OIS is a maringale under he A-measure, and one can show ha is sochasic differenial equaion is given by dsp OIS, T 0, T N ; τ = Sp OIS 1 Tj j D, T j σ c, sds A, T 0, T N T A sp, T 0, T N ; τ where we have defined j=1 δ n D, T n B, T n ; τ σ B, T n ; τ A sp, T 0, T N ; τ = Tn T 0 } σ c, sds dw A, 2.35 δ n D, T n B, T n ; τ Since IRS forward rae is a maringale under he annuiy measure A, he presen value of a T 0 ino T N collaeralized payer swapion is expressed as P V = A, T 0, T N E A OIST0, T 0, T N + Sp OIS T 0, T 0, T N ; τ K As in he previous OISwapion case, we can use asympoic expansion echnique or oher mehods o derive analyic approximaion for his opion Collaeralized enor swapion Finally, consider an opion on enor swap. From Sec.2.2.3, he forward TS spread for a collaeralized T 0 -sar T N = T M -mauring swap which exchanges Libors wih enor τ S and τ L is given by T S, T 0, T N ; τ S, τ L M m=1 = δ md, T m L c, T m 1, T m ; τ L δ nd, T n L c, T n 1, T n ; τ S δ, nd, T n = M m=1 δ md, T m B, T m ; τ L δ nd, T n δ nd, T n B, T n ; τ L δ, 2.38 nd, T n where we have disinguished he differen paymen frequencies by n and m. In he case of a 3m/6m enor swap, for example, N = 2M, τ S = 3m and τ L = 6m. Since he wo erms in Eq.2.38 are equal o Sp OIS excep he difference in day coun convenions, he enor swapion is basically equivalen o a spread opion beween wo differen Sp OIS s. The presen value of collaeralized payer enor swapion wih srike K can be expressed as N P V = δ n D, T n EÃ T ST0, T 0, T N ; τ S, τ L K Here, EÃ denoe he expecaion under he annuiy measure wih day coun fracion specified by ha of floaing leg, δ. 10

12 These opions explained in Secs , 2.4.2, and 2.4.3, can allow us o exrac volailiy informaion for our model. Considering he curren siuaion where here is no liquid marke of opions on he relevan basis spreads, we probably need o combine some hisorical esimaion for he volailiy calibraion. 3 Muliple Currency Marke This secion exends he framework developed in he previous secion ino muli-currency environmen. For laer purpose, le us define several variables firs. The T -mauring riskfree zero coupon bond of currency k is denoed by P k, T, and is calculaed from he equaion P k, T = E Q k e T r k sds, 3.1 where Q k and r k denoe he MM measure and risk-free ineres rae for he k-currency. Also define he insananeous risk-free forward rae by f k, T = T ln P k, T 3.2 as usual, and r k = f k,. As is well known, is sochasic differenial equaion under he domesic MM measure Q k is given by s df k, s = σ k, s σ k, udu d + σ k, s dw Q k, 3.3 where W Q k is he d-dimensional Brownian moion under he Q k -measure. The volailiy erm σ k is d-dimensional vecor and possibly depends on f k or any oher sae variables. Here, we have shown he risk-free ineres rae o make he srucure of he model easy o undersand hough our scheme does no direcly simulae i as will be seen laer. Le us also define he spo foreign exchange rae beween currency i and j : f i,j x. 3.4 I denoes he ime- value of uni amoun of currency j in erms of currency i. Then, define is dynamics under he Q i -measure as df x i,j /f x i,j = r i r j d + σ i,j X dw Q i. 3.5 The volailiy erm can depend on f i,j x or any oher sae variables. The Brownian moions of wo differen MM measures are conneced each oher by he relaion as indicaed by Maruyama-Girsanov s heorem. dw Q i = σ i,j X d + dw Q j,

13 3.1 Collaeralizaion wih foreign currencies Unil his poin, he collaeral currency have been assumed o be he same as he paymen currency of he conrac. However, his assumpion canno be mainained in mulicurrency environmen, since muli-currency rades conain differen currencies in heir paymens in general. In fac, his currency mismach is ineviable in a CCS rade whose paymens conain wo differen currencies, bu only one collaeral currency. Our previous work 6 have provided a pricing formula for a generic financial produc whose collaeral currency j is differen from is paymen currency k : h k = E Q k e T = P k, T E T k r k sds T e rj s c j sds h k T T e rj s c j sds h k T Here, h k is he presen value of a financial derivaive whose paymen h k T is o be made a ime T in k-currency. The collaeralizaion is assumed o be made coninuously by cash of j-currency wih zero hreshold, and c j is he corresponding collaeral rae. E T k denoes he expecaion under he risk-free forward measure of currency k, T k, where he risk-free zero coupon bond P k, T is used as a numeraire. As is clear from hese argumens, he price of a financial produc depends on he choice of collaeral currency. Le us check his impac for he mos fundamenal insrumens, i.e., FX forward conracs and Libor paymens in he nex secions FX forward and Currency riangle As is well known, he currency riangle relaion should be saisfied among arbirary combinaions of currencies j, k, l, f j,k x = f j,l x f l,k x 3.9 oherwise, he difference will soon be arbiraged away in he curren liquid foreign exchange marke. In he defaul-free marke wihou collaeral agreemen, his relaion should hold also in FX forward marke. However, i is no a rivial issue in he presence of collaeral as will be seen below 11. Le us consider a k-currency collaeralized FX forward conrac beween he currencies i, j. The FX forward rae f x i,j, T is given by he amoun of i-currency o be exchanged by he uni amoun of j-currency a ime T wih zero presen value: f x i,j, T P i, T E T T i e rk s c k sds and hence = f x i,j P j, T E T j f x i,j, T = f x i,j P j, T P i, T ET j E T i T e rk s c k sds, 3.10 e T rk s c k sds e T rk s c k sds FX forward conrac is usually included in he lis of rades for which neing and collaeral posings are o be made. 12

14 From he above equaion, i is clear ha he currency riangle relaion only holds among he rades wih he common collaeral currency, in general Libor paymen collaeralized wih a foreign currency Nex, le us consider he implicaions o a foreign-currency collaeralized Libor paymen. Using he resul of Sec.3.1, he presen value of a k-currency Libor paymen wih cash collaeral of j-currency is given by P V = δ n P k, T n E T n,k T n e r j s c jsds L k T n 1, T n ; τ Remind ha if he Libor is collaeralized by he same domesic currency k, he presen value of he same paymen is given by P V = δ n D k, T n E T n,k c L k T n 1, T n ; τ 3.13 = δ n P k, T n E T T n,k n e r k s c ksds L k T n 1, T n ; τ Here, he superscrip c in Tn,k c c of ET n,k denoes ha he expecaion is aken under he collaeralized forward measure insead of he risk-free forward measure. The above resuls sugges ha he price of an ineres rae produc, such as IRS, does depend on he choice of is collaeral currency Simplificaion for pracical implemenaion The findings of Secs and give rise o a big difficuly for pracical implemenaion. If all he relevan vanilla producs have separae quoes as well as sufficien liquidiy for each collaeral currency, i is possible o se up a separae muli-currency model for each choice of a collaeral currency. However, separae quoes for differen collaeral currencies are unobservable in he acual marke. Furhermore, closing he hedges wihin each collaeral currency is unrealisic. This is because one would like o use JPY domesic IR swaps o hedge he JPY Libor exposure in a complicaed muli-currency derivaives collaeralized by EUR, for example. The seup of a separae model for each collaeral currency will make hese hedges oo complicaed. In order o avoid hese difficulies, le us adop a very simple assumpion ha or σ k, s = σ k c, s y k, s = f k, s c k, s 3.16 is a deerminisic funcion of for each s and for every currency k. Here, σ c k is he volailiy erm defined for he forward collaeral rae of he k-currency as in Eq Under his assumpion, one can show ha for any s. Hence, i follows ha r k c k = f k s, c k s, 3.17 y k = r k c k

15 as a deerminisic funcion of ime. Under his assumpion, one can see ha he FX forward rae in Eq.3.11 becomes f x i,j, T = f x i,j P j, T P i, T 3.19 and i is independen from he choice of collaeral currency. Therefore, he relaion of cross currency riangle holds among FX forwards even when hey conain muliple collaeral currencies. In addiion, he collaeralized forward expecaion and he risk-free forward expecaion are equal for each currency k, E T k c = E T k 3.20 since he corresponding Radon-Nikodym derivaive becomes consan: e 0 rk s c k sds P k, T D k 0, T D k, T P k 0, T Now, Eq.3.12 urns ou o be P V = δ n P k, T n e = δ n D k, T n e T n y j sds E T n,k Tn y j s y ksds E T n,k L k T n 1, T n ; τ L k T n 1, T n ; τ Since i holds ha E T k c = E T k under he curren assumpion, even if he Libor paymen is collaeralized by a foreign j-currency, i is sraigh forward o calculae he exposure in erms of he sandard IRS collaeralized by he domesic currency. One can see ha all he correcions from our simplifying assumpion arise from eiher he convexiy correcion in E e T y k sds or from he covariance beween e T y k sds and oher sochasic variable such as Libor and FX raes. Considering he absolue size of he spread y and is volailiy, one can reasonably expec ha he correcions are quie small. Acually, he fac ha separae quoes of hese insrumens for each collaeral currency are unobservable indicaes ha he correcions induced from he assumpions are well wihin he curren marke bid/offer spreads. As will be seen in he following secions, he above assumpion will allow a flexible enough framework o address he issues described in he inroducion wihou causing unnecessary complicaions. 3.2 Model wih Dynamic basis spreads in Muliple Currencies Now, le us finally prese he modeling framework in he muli-currency environmen under he simplified assumpion given in Sec We have already se up he dynamics for he forward collaeral rae, Libor-OIS spread for each enor, and an equiy wih an effecive 14

16 dividend yield q for each currency as in Seq. 2.3: s dc i, s = σ c i, s σ c i, udu d + σ c i, s dw Q i, 3.24 db i, T ; τ T B i = σ i B, T ; τ, T ; τ σ c i, sds d + σ i B, T ; τ dw Q i, 3.25 ds i /S i = c i q i d + σ i S dw Q i We have he above se of sochasic differenial equaions for each currency i. The foreign exchange dynamics beween currency i and j 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, 3.27 where y i,j is defined as y i,j = y i y j 3.28 = r i r j c i c j, 3.29 which is a deerminisic funcion of ime. If a specific currency i is chosen o be a home currency for simulaion, he sochasic differenial equaions for oher currencies j i are given by dc j, s = σ j c, s db j, T ; τ B j, T ; τ ds j /S j = s = σ j B, T ; τ T c j q j σ c j, udu σ i,j X d + σ j c, s dw Q i, 3.30 σ c j, sds σ i,j X d + σ j B, T ; τ dw Q i, σ j S σi,j X d + σ j S dw Q i, where he relaion 3.6 has been used. muli-currency environmen. These are he relevan underlying facors for 3.3 Curve calibraion This secion explains how o se up he iniial condiions for he modeling framework explained in he previous secion. As will see, he spread curves {y i,j } for he relevan currency pairs can be boosrapped by fiing o he erm srucure of CCS basis spread, or equivalenly o he FX forwards. 15

17 3.3.1 Single currency insrumens Le us firs remind he seup of single currency secor of he model. As explained in Sec. 2.3, he collaeralized zero coupon bonds D, T and Libor expecaions E T k c LT k 1, T k ; τ can be exraced from he following se of equaions: OIS i N N IRS i M M m=1 i n D i, T n = D i, T 0 D i, T N 3.33 i m D i, T m = M m=1 δ i m D i, T m E T m,i c L i T m 1, T m ; τ δ n i D i, T n E T n,i c L i T n 1, T n ; τ S + T S i = M m=1 δ m i D i, T m E T m,i c L i T m 1, T m ; τ L from OIS, IRS, and TS conracs respecively. Using he relaions 3.34, 3.35 c i, s = s ln Di, s 3.36 and B i, T n ; τ = E T c n,i L i T n 1, T n ; τ 1 D i, T n 1 δ n i D i 1, T n, 3.37 one can ge he iniial condiions for he collaeral rae c, s, and he Libor-OIS spreads B, T ; τ for each currency FX forward Nex, le us consider FX forward conracs. In he curren seup, a FX forward conrac mauring a ime T beween currency i, j becomes f i,j x, T = f i,j x P j, T P i, T 3.38 = f x i,j Dj, T D i, T e T y i,jsds By he quoes of spo and forward FX raes, and he {D, T } derived in he previous secion, he value of T y i,j sds can be found. Based on he quoes for various mauriies T and proper spline echnique, y i,j s will be obained as a coninuous funcion of ime s. This can be done for all he relevan pairs of currencies. This will give anoher imporan inpu of he model required in Eq If one needs o assume ha he collaeral rae of a given currency i is acually he risk-free rae, he se of funcions {y j s} j i can be 16

18 obained by combinaion of he informaion of FX forwards wih y i s 0. Noe ha one canno assume he several collaeral raes are equal o he risk-free raes simulaneously since he model should be made consisen wih FX forwards and CCS. As menioned before, he curren seup does no recognize he differences among FX forwards from heir choice of collaeral currencies. I arises from our simplified assumpion ha he spread beween he risk-free and collaeral raes of a given currency is a deerminisic funcion of ime. This seems consisen wih he realiy, a leas in he curren marke Oher Vanilla Insrumens The insrumens explained in he previous secions and are sufficien o fix he iniial condiions of he curves used in he model. Nex, le us check oher fundamenal insrumens and he implicaions of he model European FX opion Calculaion of European FX opion is quie simple. Le us consider he T -mauring FX call opion for f x i,j collaeralized by k-currency. The presen value can be wrien as P V = E Q i e T r isds T + e y k sds f x i,j T K 3.40 = D i T, T e y k,isds E T i c + f x i,j T, T K The FX forward f x i,j in our assumpion, and is sochasic differenial equaion is given by T c i df i,j x, T f i,j x, T, T is a maringale under he forward measure T i or equivalenly = σ i,j T c F X, T dw i 3.42 = { T σ i,j X + T σ c i, sds } σ c j, sds dw T i c, 3.43 under he same forward measure. I is sraighforward o obain an analyical approximaion of Eq Consan noional cross currency swap A consan noional CCS CNCCS of a currency pair i, j is a floaing-vs-floaing swap where he wo paries exchange he i-libor fla vs j-libor plus fixed spread periodically for a cerain period. There are boh he iniial and final noional exchanges, and he noional for each leg is kep consan hroughou he conrac. The currency i, in which 12 Noe however ha he choice of collaeral currency does affec he presen value of a rade. As can be seen from Eq. 3.23, he presen value of a paymen a ime T in j-currency collaeralized wih i- currency is proporional o D j, T e T y i,j sds, and hence he payer of collaeral may wan o choose T he collaeral currency i for each period in such a way ha i minimizes y i,j sds. 17

19 Libor is paid in fla is dominaed by USD in he marke. CNCCS has been used o conver a loan denominaed in a given currency o ha of anoher currency o reduce is funding cos. Due o is significan FX exposure, mark-o-marke CCS MMCCS, which will be explained in he nex secion, has now become quie popular. The informaion in CNCCS is equivalen o he one exraced from FX forwards, since CNCCS combined wih IRS and TS wih he same collaeral currency can replicae a FX forward conrac. Here, we will provide he formula for he CNCCS of a currency pair i, j, jus for compleeness. Assume ha he collaeral is posed in i-currency. Then, he presen value of i-leg for uni noional is given by P V i = = δ n i D i, T n E T n,i c L i T n 1, T n ; τ D i, T 0 + D i, T N δ i n D i, T n B i, T n ; τ, 3.44 where T 0 is he effecive dae of he conrac. On he oher hand, he presen value of j-leg wih a spread BN CCS = BCCS N, T 0, T N ; τ for he uni noional is P V j = E Q j e T 0 r j s y i sds + E Q j e T N r j s y i sds + δ n j E Q j e Tn r j s y i sds L j T n 1, T n ; τ + BN CCS and using he assumpion of he deerminisic spread y leads o, 3.45 P V j = + δ j n D j, T n e D j, T n 1 e T n Tn 1 y i,j sds B j, T n ; τ + BN CCS T y i,j n sds T y e i,j sds n Le us denoe he noional of i-leg per uni amoun of j-noional as N i. Usually, i is fixed by he forward FX a he ime of incepion of he conrac as N i = f x i,j, T 0, and hen he oal presen value of i-leg in erms of currency j is given by N i f i,j x P V i = = δ i n N i δ i n N i f i,j x Di, T n B i, T n ; τ 3.47 f i,j x, T n Dj, T n e T n y i,j sds B i, T n ; τ Hence, he following expression of he T 0 -sar T N -mauring CNCCS basis spread is ob- 18

20 ained: B CCS N, T 0, T N ; τ = δ n j D j, T n e D j, T n 1 e T n y i,j sds Tn 1 { δ i n δ j n N i f i,j x, T n Bi, T n ; τ B j, T n ; τ T y i,j n sds T y e i,j sds n 1 1 / } δ n j D j Tn, T n e y i,jsds. One can also ge a formula for differen collaeral currency by repeaing similar calculaion. Noe ha he BN CCS, T 0, T N ; τ in Eq.3.49 is a maringale under he annuiy measure  where he i-collaeralized j-annuiy δj n D j, T n e Tn y i,jsds is used as he numeraire. Therefore, he presen value of a T 0 -sar T N -mauring consan-noional cross currency payer swapion wih srike spread K is given as 3.49 P V = δ j n D j, T n e T n y i,jsds E B CCS N T 0, T 0, T N ; τ K +, 3.50 where he noional of j-leg is assumed o be he uni amoun of a corresponding currency. Once every volailiy process is specified, i will be edious bu possible o derive an analyic approximaion by, for example, applying asympoic expansion echnique Mark-o-Marke cross currency swap Mark-o-Marke cross currency swap MMCCS is a similar conrac o he aforemenioned CNCCS excep ha he noional of he Leg which pays Libor fla is refreshed a he every sar of he Libor calculaion period based on he spo FX a ha ime. The noional for he oher leg is kep consan hroughou he conrac. More specifically, le us consider a MMCCS for i, j currency pair where j-libor plus spread is exchanged for i-libor fla. In his case, he noional of he i-leg is going o be se a f x i,j imes he noional of j-leg a beginning of every period and he amoun of noional change is exchanged a he same ime. Due o he noional refreshmen, a i, j-mmccs can be considered as a porfolio of one-period i, j-cnccs, where he noional of j-leg of every conrac is he same. Here, he ne effec from he final noional exchange of he n-h CNCCS and he iniial exchange of he n + 1-h CNCCS is equivalen o he noional adjusmen a he sar of he n + 1-h period of he MMCCS. Le us assume he collaeral currency is i as before. The presen value of j-leg can be calculaed exacly in he same way as CNCCS, and is given by P V j = + δ j n D j, T n e D j, T n 1 e T n Tn 1 y i,j sds B j, T n ; τ + BN MM Tn y i,j sds T y e i,j sds n 1 1,

21 where BN MM = BMM N, T 0, T N ; τ is he ime- value of he MMCCS basis spread for his conrac. On he oher hand, he presen value of i-leg can be calculaed as P V i = = + E Q i E Q i e T n 1 e Tn δ i n D i, T n E T c n,i c isds f x i,j T n 1 As a resul, he MMCCS basis spread is given by BN MM, T 0, T N ; τ = δ n j D j, T n e D j, T n 1 e T n y i,j sds c isds f x i,j T n δ n i L i T n 1, T n ; τ { Tn 1 y i,j sds and, afer some calculaion, we ge BN MM, T 0, T N ; τ = δ n j D j, T n e D j, T n 1 e Here, Y i,j n Y i,j n = E T c n,i where T n y i,j sds Tn 1 y i,j sds is defined by { Tn 1 exp + Tn 1 δ i n δ j n f x i,j T n 1 B i T n 1, T n ; τ E T c n,i } f x i,j T n 1 f x i,j, T n Bi T n 1, T n ; τ B j, T n ; τ Tn T y e i,j sds n 1 1 / { δ i n δ j n f i,j x, T n 1 f i,j x δ n j D j T n, T n e y i,jsds,, T n Bi, T n ; τy n i,j B j, T n ; τ Tn T y e i,j sds n 1 1 / σ i,j F X s, T n 1 σ i B s, T n; τ 3.53 } δ n j D j T n, T n e y i,jsds. Tn σ Xn s dw T c n,is 1 2 σ X n s 2 ds σ c i T n 1 } s, udu ds, 3.55 σ Xn = σ i,j F X, T n 1 + σ i B, T n; τ If we have liquid markes for FX forward and CNCCS, volailiy and correlaion parameers involved in he expression of Y n i,j needs o be adjused o make he model consisen wih he MMCCS. However, considering he populariy of MMCCS and limied liquidiy of FX forwards wih long mauriies, i may be more pracical o calibrae

22 {y i,j } using MMCCS direcly. One can see easily ha approximaing Y n i,j 1 allows us sraighforward boosrapping of {y i,j }. As is he case in CNCCS, he forward MMCCS basis spread given in Eq.3.53 is a maringale under he annuiy measure Â, where i-collaeralized j-annuiy, δj n is used as he numeraire. Therefore, a T 0 -sar T N -mauring mark-o-marke cross currency payer swapion wih srike spread K is calculaed as P V = δ j n D j, T n e T n y i,jsds EÂ B MM N T 0, T 0, T N ; τ K +, 3.57 where we have used he uni amoun of j-leg noional. A similar formula for a differen collaeral currency case can be also derived. One can see ha forward MMCCS basis spread has much smaller volailiy han ha of CNCCS due o he cancellaion of FX exposure hanks o is noional refreshmens. By comparing he expression in Eq. 3.49, we can also derive he difference of i- collaeralized CNCCS and MMCCS basis spread as follows: BN MM δi =, T 0, T n ; τ BN CCS, T 0, T n ; τ n D j, T n e Tn y i,j sds { f i,j x,t n 1 δj n f x i,j,t Bi, T n ; τy i,j n D j, T n e Tn y i,j sds n N i f i,j D j, T n e Tn y i,j sds } x,t Bi, T n ; τ n One can check ha he difference of FX exposure and he correcion erm Y n i,j o he gap beween he wo CCS s. give rise 4 Commens on Inflaion Modeling Before closing he paper, le us briefly commen on he inflaion modeling in he presence of collaeral. Alhough i is sraighforward o use he muli-currency framework as was proposed in he work of Jarrow and Yildirim 13, i requires he simulaion of unobservable real ineres raes. I is quie difficul o esimae he real rae volailiies and is correlaions o he oher underlying facors. Here, le us presen he mehod by which he collaeralized forward CPI is direcly simulaed in he same way as for he Libor-OIS spreads. This is a simple exension of he model proposed by Belgrade and Benhamou 2 for collaeralized conracs. Firs, define he forward CPI as he fixed amoun of paymen which is exchanged for IT unis of he corresponding currency a ime T. Here, IT is he ime-t CPI index. Le us consider CPI of i-currency coninuously collaeralized by j-currency. Then, he forward CPI I i, T should saisfy I i, T E Q i e T r isds T e y j sds = E Q i e T r isds T e y jsds IT. 4.1 Under he assumpion of deerminisic spread y j, i becomes I i, T = E T i IT = E T c i IT,

23 and is independen from he collaeralized currency as for he muli-currency example in he previous secion. The presen value of a fuure CPI paymen of he currency i collaeralized by he foreign currency j is expressed by using he forward CPI as P V i = D i T, T e y j,isds I i, T, 4.3 where y j,i s is available afer he muli-currency curve calibraion. The forward CPI can be easily exraced from a se of zero coupon inflaion swap ZCIS, which is he mos liquid inflaion produc in he curren marke. The break-even rae K N of he N-year zero coupon inflaion swap saisfies 1 + KN N 1 E T c IT N D, T N = 1 D, T N, 4.4 I and hence I, T N = I1 + K N N. 4.5 Here, he collaeral currency is assumed o be he same as he paymen currency. I is sraighforward o consruc a smooh forward CPI curve using appropriae spline echnique. Alhough we are no going ino deails, i is also quie imporan o esimae monh-on-monh MoM seasonaliy facors using hisorical daa. As is clear from is propery, i should no be reaed as a diffusion process, and hence i should be added on op of he simulaed forward CPI based on he smooh YoY rend process. Since I, T is a maringale under he T c measure, is sochasic differenial equaion under he MM measure Q can be specified as follows: T di, T = σ I, T σ c, sds d + σ I, T dw Q. 4.6 This should be undersood as he rend forward CPI process, and needs o be adjused properly by he use of seasonaliy facors o derive a forward CPI wih odd period. As a summary, necessary sochasic differenial equaions for IR-Inflaion Hybrids are given by s dc, s = σ c, s σ c, udu d + σ c, s dw Q, 4.7 db, T ; τ B, T ; τ 5 Conclusions T = σ B, T ; τ σ c, sds d + σ B, T ; τ dw Q, 4.8 T di, T = σ I, T σ c, sds d + σ I, T dw Q. 4.9 This paper has presened a new framework of ineres rae models which reflecs he exisence as well as dynamics of various basis spreads in he marke. I has also explicily aken he impacs from he collaeralizaion ino accoun, and provided is exension for muli-currency environmen consisenly wih FX forwards and MMCCS in he firs ime. I has also commened on he inflaion modeling in he presence of collaeral. 22

24 Finally, le us provide a possible order of calibraion in his framework. 1, Calibrae domesic swap curves and exrac {D, T } and {B, T ; τ} following he mehod in Ref. 6 for each currency. 2, Calibrae domesic ineres rae opions, such as swapions and caps/floors, and deermine he volailiy curves or surface of IR secor for each currency. For he seup of correlaion srucure, opion implied informaion or hisorical daa can be used. If one has a se of calibraed swap curves for a cerain period of hisory, i is sraigh forward o carry ou he principal componen analysis and exrac he several dominan facors. See he explanaion given, for example, in he work of Rebonao 15. 3, Calibrae FX forwards or CNCCS and exrac he se of {y i,j s} for all he relevan currency pairs. 4, Calibrae he vanilla FX opions and deermine he spo FX volailiy for all he relevan currency pairs. The resulan spo FX volailiy does depend on he correlaion srucure beween he spo FX and collaeral raes of he wo currencies. I should be esimaed using quano producs and/or hisorical daa. 5, Calibrae MMCCS and deermine he correlaion curve beween spo FX and Libor- OIS spread. Considering he size of correcion, one will have quie a good fi afer he calibraion of FX forwards, hough. There remain various ineresing opics for he pracical implemenaion of his new framework; Analyic approximaion for vanilla opions will be necessary for fas calibraion and for he use as regressors for Bermudan/American ype of exoics. Because of he separaion of discouning curve and Libor-OIS spread, here will be some imporan implicaions o he price of convexiy producs, such as consan-mauriy swap CMS. I is also an imporan problem o consider he mehod o obain sable aribuion of vega kappa exposure o each vanilla opions for generic exoics 13. A Compounding in Tenor Swap As we have menioned in Sec.2.2.3, here is a sligh complicaion in TS due o he compounding in he Leg wih he shor enor. For example, in a USD 3m/6m-enor swap, coupon paymens from he 3m-Leg occur semiannually where he previous coupon 3m- Libor plus enor spread is compounded by 3m-Libor fla. As a resul, he presen value 13 Afer compleion of he original version of his paper, we have published several new works for he relaed issues: Fujii and Takahashi 2010,2011 7, 8, 9, which include improvemens and furher exensions as well as some numerical examples. 23

25 of he 3m-Leg is calculaed as P V τs M = m=1 E Q e T 2m csds {δ 2m 1 LT 2m 2, T 2m 1 ; τ S + T S 1 + δ 2m LT 2m 1, T 2m ; τ S + δ 2m LT 2m 1, T 2m ; τ S + T S} 2M = D, T n δ n E T n c LT n 1, T n ; τ S + T S + + M δ 2m 1 δ 2m D, T 2m T SB, T 2m ; τ S m=1 M m=1 δ 2m 1 δ 2m D, T 2m E T c 2m LT 2m 2, T 2m 1 ; τ S BT 2m 1, T 2m ; τ S, A.1 where τ S = 3m. Noe ha he second and hird erms are correcion o he lef-hand side of Eq.2.7. Since he size of Libor-OIS and enor spreads have similar sizes, he correcion erm can no affec he calibraion meaningfully. Considering he bid/offer spread, one can safely neglec he compounding effecs in mos siuaions. References 1 Amerano, F. and Bianchei, M., 2009, Boosrapping he illiquidiy: Muliple yield curves consrucion for marke coheren forward raes esimaion, o be published in Modeling Ineres Raes: Laes advances for derivaives pricing, edied by F.Mercurio, Risk Books. 2 Belgrade, N. and Benhamou, E., 2004 Reconciling year on year and zero coupon inflaion swap: A marke model approach. 3 Bianchei, M., 2008, Two curves, one price: Pricing and hedging ineres rae derivaives using differen yield curves for discouning and forwarding, Working paper. 4 Brace, A., Gaaek, M. and Musiela, M., 1997, The Marke Model of Ineres Rae Dynamics, Mahemaical Finance, Vol. 7, No.2, Brigo, D. and Mercurio, F., 2006, Ineres Rae Models-Theory and Pracice, 2nd ediion, Springer. 6 Fujii, M., Shimada, Y. and 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= Fujii, M. and Takahashi, A., 2010, Modeling of Ineres Rae Term Srucures under Collaeralizaion and is Implicaions, Forhcoming in Proceedings of KIER-TMU Inernaional Workshop on Financial Engineering CARF Working Paper Series F-230, available a hp://ssrn.com/absrac=