A multiple-curve HJM model of interbank risk

Transcription

1 A muliple-curve HJM model of inerbank risk Séphane Crépey, Zorana Grbac and Hai-Nam Nguyen Laboraoire Analyse e probabiliés Universié d'évry Val d'essonne 9137 Évry Cedex, France January 9, 213 Absrac In he afermah of he nancial crisis, a variey of spreads have developed beween quaniies ha had been essenially he same unil hen, noably LIBOR-OIS spreads, LIBOR-OIS swap spreads, and basis swap spreads. By he end of 211, wih he sovereign credi crisis, hese spreads were again signican. In his paper we sudy he valuaion of LIBOR ineres rae derivaives in a muliple-curve seup, which accouns for he spreads beween a risk-free discoun curve and LIBOR curves. Towards his end we resor o a defaulable HJM mehodology, in which hese spreads are explained by an implied defaul inensiy of he LIBOR conribuing banks, possibly in conjuncion wih an addiional liquidiy facor. Markovian shor rae specicaions are given in he form of an exended CIR and a Lévy HullWhie model for a risk-free shor rae and a LIBOR shor spread. The use of Lévy drivers leads o he more parsimonious specicaion. Numerical values of he FRA spreads and he basis swap spreads compued wih he laer largely cover he ranges of values observed even a he peak of he crisis. Keywords: Ineres Rae Derivaives, LIBOR, HJM, Muliple Curve, Inerbank Risk, Lévy Processes. MSC: 91G3, 91G2, 6G51 JEL classicaion: G12, E43 1 Inroducion The recen nancial crisis caused a number of anomalies ha were no previously observed in he xed income markes. The ineres raes whose dynamics were very closely following each oher have sared o diverge subsanially. In paricular, he spreads beween he LIBOR raes and he OIS raes he swap raes of ineres rae swaps whose oaing-rae paymens are indexed o a compounded overnigh rae of he same mauriy have been far from negligible, as well as he spreads beween he swap raes of he LIBOR-indexed ineres The research of he auhors beneed from he suppor of he `Chaire Risque de crédi', Fédéraion Bancaire Française, and of he DGE. The auhors hank Jeroen Kerkhof, from Jeeries, London, for he graphs of Figure 1, and Alexander Herbersson, from Universiy of Gohenburg, Sweden, for his help in deailing he compuaions abou swapions.

2 2 rae swaps and he OIS raes. The former ype of spreads is known as he LIBOR-OIS spread and he laer as he LIBOR-OIS swap spread. We refer o Secion 2.1 for precise deniions of various ineres raes and o Secions 4.2 and 4.3 for deniions of ineres rae swaps and spreads. In Figure 1 lef he hisorical EURIBOR-EONIAswap spreads in he period are ploed for mauriies ranging from 1 monh o 12 monhs. Before he crisis hese spreads were pracically negligible, whereas a he peak of he crisis hey were greaer han 2 basis poins for some mauriies. The EURIBOR rae and he EONIAswap rae are analogs of he LIBOR rae and he OIS rae in he EUR xed income marke cf. Secion 2.1 for deails. Furhermore, since he nancial crisis he LIBOR raes of dieren mauriies have exhibied noably diverse behavior, which is reeced in he so-called basis swap spreads appearing when basis swaps are priced. In a basis swap wo sreams of oaing-rae paymens linked o LIBOR raes on dieren enors are exchanged cf. Secion 4.4 for deails. This is why praciioners nowadays end o produce dieren discoun curves for dieren enors; see Figure 1 righ, which displays discoun funcions relaed o he EONIAswap raes, he 3-monh and he 6-monh EURIBOR raes. All hese phenomena are described in Filipovi and Trolle 211 as he adven of a so-called inerbank risk Euribor - Eoniaswap spreads spread 1m spread 3m spread 6m spread 12m 15 bp Figure 1: Lef: Hisorical EURIBOR-EONIAswap spreads Righ: Discoun curves boosrapped on Sepember 2, 21. In addiion, when valuing and hedging ineres rae derivaives, he inerbank risk issue comes in combinaion wih he counerpary risk issue, which is he risk of a pary defauling in an OTC derivaive conrac. In his conex, he quesions such as which curve should be used as discouning curve, o which exen he choice of a given curve should be pu in relaion wih counerpary risk, or possibly hidden relaions beween bilaeral counerpary risk accouning for he defaul risk of boh paries and funding coss of funding a posiion in a conrac in a muliple-curve environmen, have become he subjec of endless debaes beween marke praciioners. In his paper we propose a model of inerbank risk for he pricing of LIBOR ineres rae derivaives in a muliple-curve seup. Noe ha his is a model of clean valuaion in he sense of Crépey 212, meaning clean of counerpary risk and excess funding coss above he risk-free rae in pracice: he OIS rae. However, a counerpary risk and excess funding coss correcion CVA for Credi Valuaion Adjusmen in he counerpary risk erminology can hen be obained as he value of an opion on his clean price process; see

3 3 for insance Crépey 212. Acually, he main moivaion for he presen work is o devise a model of clean valuaion of ineres rae derivaives wih inerbank risk, racable in iself, bu also from he perspecive of serving as an underlying model for CVA compuaions. This inegraion of he presen clean model ino a counerpary risky environmen will be considered in a follow-up paper. Resoring o he usual disincion beween shor rae, HJM and BGM or LIBOR marke models, one can classify he inerbank risk muliple-curve in his regard, ye clean in he above sense valuaion lieraure as follows. Kijima, Tanaka, and Wong 29 or Kenyon 21 propose shor rae approaches. Henrard 27, 21 derives correced Gaussian HJM formulas under he assumpion of deerminisic spreads beween he curves. Bianchei 21 resolves a wo-curve issue in a cross-currency mahemaical framework, deriving quano convexiy correcions o he usual BGM marke model valuaion formulas. Here he main ool is ha of a change of measure/numéraire. The LIBOR marke model approach is also exended in Mercurio 21b, 21a and Fujii, Shimada and Takahashi 211, 21 in such a way ha basis spreads for dieren enors are modeled as dieren processes. A hybrid HJM-LIBOR marke model is proposed in Moreni and Pallavicini 21, where he HJM framework is employed o obain a parsimonious model for muliple curves, using a single family of Markov driving processes. Finally, a credi risk approach is enaive in Morini 29. However, Morini concludes on page 43 ha in his model he credi risk alone does no explain he marke paerns. We recall ha LIBOR sands for London Inerbank Oered Rae and is produced for 1 currencies wih 15 mauriies, ranging from overnigh o 12 monhs, hus producing 15 raes each business day. The conribuing banks are seleced for each currency panel wih he aim of reecing he balance of he marke for a given currency based upon hree guiding principles: scale of marke aciviy, credi raing and perceived experise in he currency concerned. Each panel, ranging from 7 o 18 conribuors, is chosen by he independen Foreign Exchange and Money Markes Commiee FX&MM Commiee o give he bes represenaion of aciviy wihin he London money marke for a paricular currency. Twice a year he FX&MM Commiee underakes an assessmen of each LIBOR panel, evaluaing he conribuing banks and updaing he selecion if necessary. This rolling consrucion of he LIBOR conribuing group is inended o ensure ha, in principle, acual defauls canno occur wihin he group. However, he deerioraion of he credi qualiy of he LIBOR conribuors during he lengh of a LIBOR loan is greaer wih longer enors, resuling in a defaul spread beween he LIBOR markes of dieren enors OIS marke in he limiing case of an overnigh enor. Moreover, he economic fundamenals of inerbank risk are no only credi risk, bu also liquidiy risk, among oher facors such as sraegic game consideraions see Michaud and Upper 28, page 48, which migh from ime o ime incie a bank o declare as LIBOR conribuion a number slighly dieren from is inernal convicion regarding The rae a which an individual Conribuor Panel bank could borrow funds, were i o do so by asking for and hen acceping inerbank oers in reasonable marke size, jus prior o 11. London ime he heoreical deniion of he LIBOR rae. For hese inerpreaions and he relaed economeric aspecs we refer he reader o he quaniaive analysis of he erm srucure of inerbank risk which was recenly conduced by Filipovi and Trolle 211. Based on a daa se covering he period from Augus 27 unil January 211, heir resuls show ha he defaul componen is overall he main dominan driver of inerbank risk, excep for shor-erm conracs in he rs half of he sample see Figures 3 and 4 in heir paper. The second main driver is inerpreed as liquidiy risk, which is consisen wih he claims in Morini 29.

4 4 Here we make boh he credi and he liquidiy inerbank risk componens explici, in he mahemaical framework of a defaulable HeahJarrowMoron mehodology; see he seminal paper by Heah, Jarrow, and Moron 1992 and he defaulable exensions by Bielecki and Rukowski 2 and Eberlein and Özkan 23. Our moivaion for modeling he coninuously compounded forward raes using an HJM approach, insead of dealing direcly wih discreely compounded LIBOR raes in a BGM framework, is wofold. On he one side, i allows one o consider simulaneously he LIBOR raes for all possible enors recall ha one of he pos-crisis spreads sudied in his work is relaed o he LIBOR raes of various enors. The HJM framework is capable of producing a muli-curve model wih as many sochasic facors as LIBOR raes of dieren enors by increasing he dimension of he driving process, while sill reaining he racabiliy of he pricing formulas for any arbirary correlaion of sochasic facors. On he oher side, his is a unied approach for a very general class of ime-inhomogeneous Lévy driving processes. I is also imporan o menion ha various shor rae models can be accommodaed in his seup as special cases see Secion 3 for he exended CIR and he exended Hull-Whie model. As will be illusraed in a follow-up work, his direc link o he shor rae process r is useful in he conex of counerpary risk applicaions, where he model of his paper can be used as an underlying model for CVA compuaions. Numerical issues relaed o our model will be mainly considered in a follow-up paper. However, he las secion of his paper already makes clear ha, in conras o he conclusion of Morini 29 in his rs enaive credi risk approach, even a pure appropriae credi risk model wih liquidiy componen se o zero is in fac able o explain spreads very much in line wih he orders of magniude ha were observed in he marke even a he peak of he crisis. The res of he paper is organized as follows. In Secion 2, we apply an adapaion of he defaulable HJM approach o model he erm srucure of muliple ineres rae curves. Secion 3 presens a racable pricing model wihin his framework which we obain by choosing he class of nonnegaive mulidimensional Lévy processes as driving processes combined wih deerminisic volailiy srucures. In Secion 4 he basic ineres rae derivaives ied o he LIBOR rae are described and explici valuaion formulas are derived. Secion 5 presens numerical resuls illusraing he exibiliy of he model in producing a wide range of FRA spreads and basis swap spreads. In our view he main conribuions of his work are: a consisen and racable muliplecurve HJM erm srucure model of inerbank risk; low-dimensional exended CIR or Lévy HullWhie shor rae specicaions of he muliple-curve HJM seup, opening he door o he use of his model as an underlying model for ineres rae derivaive CVA compuaions; numerical evidence ha an appropriaely chosen credi risk seup is enough o accoun for even he mos exreme inerbank spreads observed in he marke. 2 Muliple-curve HJM seup 2.1 Noaion In his subsecion we inroduce he main noions and noaion we are going o work wih. The main reference rae for a variey of ineres rae derivaives is he LIBOR in he USD xed income marke and he EURIBOR in he EUR xed income marke. LIBOR resp. EURIBOR is compued daily as an average of he raes a which designaed banks belonging o he LIBOR resp. EURIBOR panel believe unsecured funding for periods of lengh up o one year can be obained by hem resp. by a prime bank. From now on we shall use

5 5 he erm LIBOR meaning any of hese wo raes. Anoher imporan reference rae in xed income markes is a so-called OIS Overnigh Indexed Swap rae, which is he swap rae of a swap whose oaing rae is obained by compounding an overnigh rae, i.e. a rae a which overnigh unsecured loans can be obained in he inerbank marke. In he USD xed income marke his rae is he FF Federal Funds rae and in he EUR marke i is he EONIA rae where EONIA sands for Euro Overnigh Index Average. From now on we shall use he generic erm OIS rae for boh xed income markes. The OIS rae is considered by praciioners o be he bes available marke proxy for he risk-free rae since he risk in an overnigh loan can be deemed almos negligible. On he oher hand, he LIBOR rae depends on he erm srucure of inerbank risk, which is reeced in he observed LIBOR-OIS and LIBOR-OIS swap spreads see he lef panel in Figure 1. In his paper we inroduce a defaul ime τ associaed wih he LIBOR reference curve via a given defaul inensiy γ. We emphasize ha τ is no mean o represen an acual defaul ime of any specic eniy recall ha he LIBOR panel is consanly being updaed. I is merely used as an implied model of defaul risk for he reference curve, o quanify he credi spread componen of inerbank risk on a mahemaically racable defaul inensiy scale. We shall work wih insananeous coninuously compounded forward raes, specifying he dynamics of he erm srucure of he risk-free OIS forward ineres raes f T and of he forward spreads g T corresponding o he risky LIBOR raes of he reference curve. We denoe by f T he insananeous coninuously compounded risky forward raes, so for every T, g T = f T f T, 1 where T [, T ] and T is a nie ime horizon. The corresponding shor raes r and r are given, for every [, T ], by We also dene he shor erm spread λ by r = f and r = f. 2 λ = g = r r. The discoun facors associaed wih our wo yield curves are denoed by B T and B T, respecively. These are ime- cumulaive prices and pre-defaul prices of risk-free and risky zero coupon bonds wih mauriy T, wih B T T = 1 and B T T = 1. The bond prices are relaed o he forward raes via he following formulas, for T, T B T = exp f udu and T B T = exp f udu. 3 The T -spo LIBOR rae L T T, T + δ is a simply compounded ineres rae xed a ime T for he ime inerval [T, T + δ], which will be dened in our seup as L T T, T + δ = 1 δ 1 B T T + δ 1. 4 We hus use in his deniion he risky bond prices B, where he reference eniy of he risky bond is o be inerpreed as consising of a sylized represenaive of he LIBOR conribuing banks.

6 6 2.2 Driving process We consider a complee probabiliy space Ω, F T, IP, where T is he nie ime horizon. Le E = E [, T ] denoe a lraion on his space saisfying he usual condiions. The driving process Y = Y T is assumed o be a process wih independen incremens and absoluely coninuous characerisics PIIAC in he sense of Eberlein, Jacod, and Raible 25, also called a ime-inhomogeneous Lévy process in Eberlein and Kluge 26a, or an addiive process in he sense of Deniion 1.6 in Sao The process Y is aken as an E-adaped, càdlàg, R n -valued process, saring from zero. The law of Y, [, T ], is described by he characerisic funcion, in which u denoes a row-vecor in R n : IE[e iuy ] = exp iub s 1 2 uc su 5 + e iux 1 iuhx F s dx ds, R n where b s R n, c s is a symmeric, nonnegaive denie real-valued n-dimensional marix and F s is a Lévy measure on R n, i.e. F s {} = and R n x 2 1F s dx <, for all s [, T ]. The funcion h : R n R n is a runcaion funcion for example hx = x1 { x 1}. Le denoe he norm on he space of real n-dimensional marices, induced by he Euclidean norm on R n. The following sanding assumpion is saised: Assumpion 2.1 i The riple b, c, F T saises T ii There exis consans K, ε > such ha b + c + 1 x 2 F dx d < ; R n T x >1 for every u [ 1 + εk, 1 + εk] n. expuxf dxd <, 6 Condiion 6 ensures he exisence of exponenial momens of he process Y. More precisely, 6 holds if and only if IE[expuY ] <, for all T and u [ 1 + εk, 1 + εk] n cf. Lemma 2.6 and Corollary 2.7 in Papapanoleon 27. Moreover, Y is hen a special semimaringale, wih he following canonical decomposiion cf. Jacod and Shiryaev 23, II.2.38, and Eberlein, Jacod, and Raible 25 Y = b s ds + cs dw s + R n xµ νds, dx, [, T ], 7 where µ is he random measure of jumps of Y, ν is he IP-compensaor of µ, c s is a measurable version of a square-roo of he symmeric, nonnegaive denie marix c s, and W is a IP-sandard Brownian moion. The riple of predicable semimaringale characerisics of Y wih respec o he measure IP, denoed by B, C, ν T, is B = b s ds, C = c s ds, ν[, ] A = [,] A F s dxds, 8

7 7 for every Borel se A BR n \ {}. The riple b, c, F T represens he local characerisics of Y. Any of hese riples deermines he disribuion of Y, as he LévyKhinchine formula 5 obviously dicaes wih hx = x, which is a valid choice for he runcaion funcion due o 6. We denoe by κ s he cumulan generaing funcion associaed wih he inniely divisible disribuion characerized by he Lévy riple b s, c s, F s. One can exend κ s o row-vecors of complex numbers z C n such ha Rz [ 1 + εk, 1 + εk] n. We have, for s [, T ], κ s z = zb s zc sz + e zx 1 zx F s dx. 9 R n Noe ha 5 can be wrien in erms of κ: IE[e iuy ] = exp κ s iuds. 1 If Y is a Lévy process, in oher words if Y is ime-homogeneous, hen b s, c s, F s, and hus also κ s, do no depend on s. In ha case, κ boils down o he log-momen generaing funcion of Y 1. For deails we refer o Papapanoleon 27, Lemma 2.8, Remark 2.9 and Remark Remark 2.2 The moivaion for he choice of ime-inhomogeneous Lévy processes as driving processes in our model is wofold. On he one side, hese processes are analyically racable, and on he oher side, hey posses a high degree of exibiliy, which allows for an adequae of he erm srucure of volailiy smiles, i.e. of he change in he smile across mauriies; see Eberlein and Kluge 26a, 26b and Eberlein and Koval 26 for applicaions of ime-inhomogeneous Lévy processes in ineres rae modeling. 2.3 Term srucure of ineres raes In his subsecion, we model he erm srucures of he risk-free and he risky ineres raes. We shall be concerned wih wo lraions on he sanding risk-neural probabiliy space Ω, F T, IP of his paper: he background lraion E = E T, and he full lraion F = F T conaining E and he informaion abou he defaul ime τ. The bond price processes B T T and B T T, and also he corresponding forward rae processes f T T and f T T, for any T [, T ], are all E-adaped. I is assumed ha τ is no an E-sopping ime, bu i is an F-sopping ime. Moreover, we assume ha immersion holds beween E and F, i.e. every E-local maringale is an F-local maringale. We assume ha τ possesses an E-hazard inensiy γ. Thus, is Azéma supermaringale is given by IPτ > E = e γ s ds, 11 where γ is an E-adaped, nonnegaive and inegrable process. Le us now specify he insananeous coninuously compounded forward raes f T and he insananeous forward spreads g T, which in urn provide he bond prices B T and B T via 3. We are going o make use of he resuls from Eberlein and Raible 1999 and Eberlein and Kluge 26b, where HJM models driven by ime-inhomogeneous Lévy processes were developed, and he resuls from Bielecki and Rukowski 2 and Eberlein and Özkan 23, where defaulable exensions of he HJM framework were inroduced.

8 8 Conrary o he laer wo papers, we choose here o model direcly he forward spreads g T insead of he risky forward raes f T, which is clearly equivalen due o 1. However, one should have B T B T, i.e. he risky bonds are cheaper han he risk-free bonds wih he same mauriy. This implies by 3 ha f T f T, or equivalenly, g T. Hence, we decide o model he forward spreads direcly and sudy heir nonnegaiviy in some special cases. In he nex subsecion wo racable nonnegaive examples are provided. Le us also menion here a paper by Chiarella, Maina, and Nikiopoulos 21, where a class of sochasic volailiy HJM models admiing nie dimensional Markovian srucures is proposed. They model he defaul-free forward raes and he forward spreads, whose dynamics are driven by correlaed Brownian moions. One of he examples in he sequel, he sochasic volailiy CIR model of Secion 3.1, can be ino his modeling framework Risk-free raes The dynamics of he risk-free forward raes f T, for T [, T ], is given by f T = f T + a s T ds + σ s T dy s, 12 where he iniial values f T are deerminisic, bounded and Borel measurable in T. Moreover, σ and a are sochasic processes dened on Ω [, T ] [, T ] aking values in R n and R, respecively. Le P and O respecively denoe he predicable and he opional σ-eld on Ω [, T ]. We recall ha he predicable σ-eld is he σ-eld on Ω [, T ] generaed by all càg adaped processes and he opional σ-eld is generaed by all càdlàg adaped processes considered as mappings on Ω [, T ]. The mappings ω; s, T a s ω; T and ω; s, T σ s ω; T are measurable wih respec o P B[, T ]. For s > T we have a s ω; T = and σ s ω; T =, as well as sup,t T a ω; T + σ ω; T <. These condiions ensure ha we can nd a join-version of all f T such ha ω;, T f ω; T 1 { T } is O B[, T ]-measurable see Eberlein, Jacod, and Raible 25. Then i follows cf. equaion 2.4 in Eberlein and Kluge 26b, for [, T ], ha where we se B T = B T exp r s A s T ds A s T := T s a s udu, Σ s T := T s Σ s T dy s, 13 σ s udu. 14 Insering T = ino 13, he risk-free discoun facor process β = β T, dened by β = exp r sds, can be wrien as β = B exp A s ds Σ s dy s. 15 Combining his wih 13 we obain he following useful represenaion for he bond price process B T = B T B exp A s A s T ds + Σ s Σ s T dy s. 16

9 9 We make a sanding assumpion ha he volailiy srucure is bounded in he sense ha one has Σ i st K 2 for every s T T and i {1, 2,..., n}, where K is he consan from Assumpion 2.1ii. Noe ha if Y is a Brownian moion, his assumpion holds wih K =. In oher words, he volailiy srucure in he Brownian case does no have o be bounded. As is well-known, he model is free of arbirage if he bond prices discouned a he risk-free rae, β B T T, are F-maringales wih respec o a risk-neural measure IP. Due o he immersion propery i suces ha hey are E-maringales. This is guaraneed by he following drif condiion, which is assumed henceforh: A s T = κ s Σ s T, s [, T ], 17 where κ s is he cumulan of Y dened in 9. This condiion can be found in Eberlein and Kluge 26b, see equaion 2.3 herein and commens hereafer. For more deailed compuaions, see Proposiion 2.2 of Kluge 25 in he case of deerminisic volailiy, and Theorem 7.9 and Corollary 7.1 of Raible 2 for a sochasic volailiy combined wih a ime-homogeneous Lévy driving process. If Y is a sandard Brownian moion, condiion 17 simplies o A s T = 1 2 Σ st 2, which is he classical HJM no-arbirage condiion Risky raes The dynamics of he forward spreads g T, [, T ], is given by g T = g T + a st ds + σ st dy s, 18 where he iniial values g T are deerminisic, bounded and Borel measurable in T. Moreover, a T and σ T saisfy he same measurabiliy and boundedness condiions as at and σt. The risky forward raes are hen given by where we se f T = f T + ā st ds + σ st dy s, 19 f T = f T + g T, ā st = a s T + a st, σ st = σ s T + σ st. The dynamics of he bond prices B T T can be obained exacly in he same way as he dynamics of B T T in equaion 13. Therefore, for [, T ], B T = B T exp rs Ā st ds Σ st dy s, 2 where Seing we have Ā st := A st := T s T s ā sudu and Σ s T := a sudu and Σ st := T s T s σ sudu. 21 σ sudu, Ā st = A s T + A st and Σ s T = Σ s T + Σ st. 22

10 1 Recall from 2 ha he shor rae r s is given by r s + λ s. Similarly o 16, we can rewrie he bond price B T as B T = B T B exp Ā s Ā st ds + Σ s Σ st dy s. 23 In defaulable HJM models no-arbirage requiremens yield a drif condiion relaing he drif erm Ā T and he volailiy erm Σ st. To see i le us emporarily assume ha our risky bond prices B T could be inerpreed as he pre-defaul prices of defaulable LIBOR zero-coupon bonds; le us hen sudy he consrains ha would correspond o precluding arbirage opporuniies relaed o dealing wih hese bonds, were such bonds raded in he marke. Noe ha such LIBOR bonds are acually no raded; no even synheically as averages of he defaulable bonds of LIBOR conribuors, since he LIBOR raes reeced in B T are only reference numbers and no ransacion quoes; see he deniion of he LIBOR rae in he inroducion. The defaulable bonds are assumed o pay a cerain recovery upon defaul. We adop he fracional recovery of a marke value scheme, which species ha in case of defaul of he bond issuer, he fracion of he pre-defaul value of he bond is paid a defaul ime. The value a mauriy of such a bond is given by BT T = 1 {τ >T } + 1 {τ T }R B τ T Bτ 1 T, where R [, 1] is he recovery and B T is he pre-defaul bond price dened in 3, for every [, T ]. Noe ha receiving he amoun 1 {τ T }R B τ T a τ is equivalen o receiving 1 {τ T }R B τ T a T. The ime- bond price can be wrien as T B 1 τ B T = 1 {τ >} B T + 1 {τ }R B τ T B 1 τ T B T. 24 The immersion propery implies ha B τ T = B τ T. Moreover, noe ha 1 {τ >} B T = 1 {τ >}B T, for every [, T ]. Le us now sudy he condiions which ensure he absence of arbirage, i.e. le us nd he condiions such ha B T discouned a he risk-free rae, β B T T, are F, IP-maringales, for all T [, T ]. For each T he maringale condiion is saised if B T R B T γ = B T ξ T, [, T ], 25 where ξ T := λ Ā T + κ Σ T and R is he erminal recovery process in he sense of Condiion HJM.8 in Secion of Bielecki and Rukowski 22. The proof of he above saemen is similar o he derivaion of Condiion in Bielecki and Rukowski 22, Secion in he Gaussian case. For similar condiions in ime-inhomogeneous Lévy driven models, we refer o Eberlein and Özkan 23 or Grbac 21, Secion 3.7. Under he recovery scheme assumed above i.e. he fracional recovery of a marke value, one ges a paricularly convenien form of he maringale condiion 25. The recovery process R akes he following form cf. 24 R := R B T B 1 T,

11 11 which insered ino 25 yields 1 R γ = ξ T, [, T ]. 26 Since condiion 26 mus be rue for all T [, T ], i is acually equivalen o he following wo condiions: 1 R γ = λ 27 and Ā T = κ Σ T. 28 Indeed, condiions 27 and 28 obviously imply 26. To see he converse, one has o inser T = ino 26 and noe ha Ā = and Σ = by 14. Moreover, κ = by 9, which yields 27. Condiion 28 now follows from 26 by insering T. Now, in our model, he risky bonds B T are mahemaical conceps which represen he inerbank risk of he LIBOR group and hey are neiher defaulable in he classical sense, nor hey are raded asses. Moreover, inerbank risk does no need o consis only of credi risk, i can also have a liquidiy componen. We hus relax he defaulable HJM drif condiion 28 ino a less sringen condiion Ā T = κ Σ T + α T, 29 where αt saises he same measurabiliy and boundedness condiions as AT and A T. Recall ha he LIBOR raes are dened by 4, where αt hen appears via 29 hrough 23. Since κ Σ T is relaed o he pure credi risk inerpreaion of he risky bonds, as shown above, we shall refer o i as he credi risk componen of inerbank risk. The remaining conribuion αt will be referred o as he liquidiy componen of inerbank risk, in reference o he economerically demonsraed explanaion of inerbank risk as a mixure of credi and liquidiy risk of he LIBOR conribuing banks cf. Filipovi and Trolle 211. However, in he remainder of his secion, as well as in Secions 3 and 5, for simpliciy we shall work wihou he liquidiy componen α T being explicily presen. In oher words, we shall work under he more specic credi assumpion 28. We emphasize ha Secion 4 does no rely on his assumpion and all he resuls herein are sill valid under condiion 29, provided ha he liquidiy componen α T is deerminisic in which case we wrie α T = α, T. Proposiion 2.3 i The forward rae f T is given by and he shor rae r by f T = f T + r = f + T κ s Σ s T ds + κ s Σ s ds + σ s T dy s, 3 σ s dy s. 31 ii The forward spread g T is given by g T = g T + T κ s Σ st Σ s T T κ s Σ s T ds + σ st dy s, 32

12 12 and he shor erm spread λ by λ = g + + κ s Σ s Σ s κ s Σ s ds σ sdy s. 33 iii The E-inensiy γ of he defaul ime τ is given by γ = 1 1 R g + κ s Σ s Σ s κ s Σ s ds + Proof. To prove i, noe ha from condiion 17 i follows ha a s T = T κ s Σ s T. σ sdy s. This immediaely yields 3 and 31. Similarly, o prove ii, we make use of 28 and obain a st = ā st a s T = T κ s Σ st T κ s Σ s T = T κ s Σ st Σ s T T κ s Σ s T. Hence, 32 and 33 follow. Finally, o prove iii we combine 27 and The model In his secion we focus our aenion on ime-homogeneous Lévy processes Y. The cumulan generaing funcion associaed wih Y is hen given by κz := zb zcz + e zx 1 zx F dx, R n where b, c, F is he Lévy riple of Y 1 compare 9. We sudy condiions ha ensure he nonnegaiviy of he risk-free ineres raes and he spreads, considering in paricular wo cases: a pure-jump Lévy process wih nonnegaive componens subordinaors combined wih deerminisic bond price volailiy srucures, and a wo-dimensional Brownian moion in combinaion wih sochasic volailiy srucures. We shall focus in paricular on he rs case, which urns ou o be very racable for valuaion purposes. Noe ha he general HJM model, as well as many shor rae models, does no necessarily produce nonnegaive ineres raes. The sandard argumen is ha he probabiliy of negaive ineres raes is sucienly small, and herefore his undesirable feaure is sill olerable. However, when ineres raes are small as in he recen years, he nonnegaiviy of ineres raes produced by a model becomes a pracically relevan issue.

13 Sochasic volailiy CIR Assume ha he driving process Y = Y 1, Y 2 is a wo-dimensional Brownian moion wih correlaion ϱ. The canonical decomposiion 7 of Y is given by Y = cw 1, W 2, where W 1, W 2 is a wo-dimensional sandard Brownian moion and he covariance marix c = [c i,j ] i,j=1,2 is such ha c 1,1 = c 2,2 = 1 and c 1,2 = c 2,1 = ϱ. The cumulan generaing funcion of Y is given by κz = 1 2 zcz, z R 2. In order o produce nonnegaive shor raes and shor erm spreads wih his driving process, he volailiies in he HJM model canno be deerminisic. We make use of he volailiy specicaions ha produce he CIR shor rae and he CIR shor erm spread wihin he HJM framework, as shown in Chiarella and Kwon 21. Thus, we impose he following assumpions on he volailiies σ s and σ s: σ s = ζs r s e s kudu,, σs =, ζ s λ s e s k udu, where ζ, ζ, k and k are deerminisic funcions cf. equaion 6.2 in Chiarella and Kwon 21. Noe ha he wo-dimensional volailiy srucure above is chosen in such a way ha he risk-free raes are driven only by he rs Brownian moion Y 1 =: W r and he forward spreads are driven solely by Y 2 =: W λ. Hence, we can apply direcly he resuls from Chiarella and Kwon 21, equaion 6.3 and obain he following SDE for he shor rae r dr = ρ kr d + ζ r dw r, 34 where ρ = f + kf + σ 2 sds. This is a one-dimensional exended CIR shor rae model. We emphasize, however, ha ρ is non-deerminisic since i depends on he non-deerminisic σ s. An addiional, auxiliary facor ı = σ 2 sds, dı = ζ 2 r 2kı d is needed o make he model Markovian in r, ı. The forward rae volailiy specicaion ha yields he exended CIR shor rae model in which k and ζ do no depend on ime, was sudied in Heah, Jarrow, and Moron 1992, Secion 8, bu in his case ρ in 34 is no available in explici form. Reasoning along he same lines as above yields he following SDE for he shor erm spread λ dλ = ρ κ λ d + ζ λ dw λ, where ρ is dened accordingly. Similarly, we also dene j = σ s 2 ds, dj = ζ 2 λ 2k j d. In Theorem 2.1 of Chiarella and Kwon 21 i was shown ha he risk-free exended CIR model possesses an ane erm srucure wih wo sochasic facors. More precisely,

14 14 he bond prices can be wrien as exponenial-ane funcions of he curren level of he shor rae r and he process ı: B T = B T γ, B exp T f γ, T r 12 γ2, T ı, 35 where γ, T = T e u kvdv du is a deerminisic funcion combine Theorem 2.1 wih 2.4 and 1.2 in Chiarella and Kwon 21. For risky bonds B T a similar expression involving in addiion λ and j can be obained by exacly he same reasoning and making use of he represenaion T B T = B T exp g udu, 36 which follows from 1 and Jumps and deerminisic volailiy In CVA applicaions see Crépey 212, Markovian specicaions are required. The previous Brownian specicaion of he general muliple-curve HJM seup, yields a four-dimensional Markov facor model in erms of he process X = r, λ, ı, j T. In he ques of a more parsimonious Markovian specicaion, we now assume ha he driving process Y is an n-dimensional Lévy process, whose componens are subordinaors, and ha he volailiies are deerminisic. We derive condiions ha ensure he nonnegaiviy of he ineres raes and he spreads in his seing. I is worhwhile menioning ha when Y is wo-dimensional as in he previous example, his yields a wo-dimensional Markov facor model in erms of X = r, λ, which makes his specicaion preferable for applicaions. Le Y be an n-dimensional nonnegaive Lévy process, such ha is Lévy measure saises Assumpion 2.1. Is cumulan generaing funcion is given by κz = zb + e zx 1 F dx 37 R n + for z R n such ha z [ 1 + εk, 1 + εk] n, where b denoes he drif erm and he Lévy measure F has is suppor in R n +. We refer o Theorem 21.5 and Remark 21.6 in Sao 1999 for one-dimensional subordinaors; for muli-dimensional nonnegaive Lévy processes see 3.15 in Barndor-Nielsen and Shephard 21. Noe ha subordinaors do no have a diusion componen and heir jumps can be only posiive. Examples of hese processes include a compound Poisson process wih posiive jumps, Gamma process, inverse Gaussian IG process, and generalized inverse Gaussian GIG processes. In he remainder of he paper we impose he following sanding assumpions on he volailiies Σ and Σ : Assumpion 3.1 The volailiies Σ and Σ are nonnegaive, deerminisic and saionary funcions. More precisely, hey are given as Σ s = S i s and Σ s = S,i s, 1 i n 1 i n for every s, such ha s T, where S i : [, T ] R + and S,i : [, T ] R +, i = 1,..., n, are deerminisic funcions bounded by K 2, where K is he consan from 6.

15 15 Proposiion 3.2 i The dynamics of he forward rae f T and he shor rae r are given by and f T = f T κ Σ T + κ Σ T + r = f + κ Σ + σ s T dy s 38 σ s dy s. 39 ii The dynamics of he forward spread g T and he shor spread λ are given by and g T = g T κ Σ T Σ T + κ Σ T Σ T +κ Σ T κ Σ T + λ = g + κ Σ Σ κ Σ + σ st dy s 4 σ sdy s. 41 Proof. We begin by noing ha T Si T s = s Si T s and for i = 1,..., n. Hence, Assumpion 3.1 implies and T κ Σ st = s κ Σ st T S,i T s = s S,i T s, 42 T κ Σ st Σ s T = s κ Σ st Σ s T, which follows from 42 by diereniaion. Therefore, we obain and similarly, T κ Σ st ds = s κ Σ st ds = κ Σ T κ Σ T, T κ Σ st Σ s T ds = κ Σ T Σ T κ Σ T Σ T. Insering hese expressions ino 3 and 32 yields 38 and 4, respecively. To show 39 and 41 we noe ha and κ Σ sds = κ Σ κ Σ s Σ s ds = κ Σ Σ,

16 16 due o κ Σ = κ = and κ Σ Σ =, which follows by 14 and 21 combined wih 37. In he nex wo proposiions we give necessary and sucien deerminisic condiions for he nonnegaiviy of he ineres raes and he spreads. Le us denoe µ, T := f T µ := r σ s T dy s = f T κ Σ T + κ Σ T σ s dy s = f + κ Σ, where he second equaliy in each line follows by 38 and 39, respecively. Noe ha µ, T and µ are hus deerminisic. Proposiion 3.3 i The shor rae r is nonnegaive if µ, for [, T ]. ii Assume ha he disribuion of he random vecor Y 1 has [, n as is suppor. Then he converse of i is also rue, i.e. if r, hen µ, for every [, T ]. Moreover, if r, for every [, T ], hen f T, for every T [, T ]. In words, he nonnegaiviy of he shor rae implies he nonnegaiviy of he forward rae. Proof. Since Y has nonnegaive componens and he volailiy σ is nonnegaive by assumpion, i is obvious ha µ implies r, for every. This proves i. In case when he suppor of Y 1 is [, n, we show he converse saemen by noing ha σ s dy s ω K 2 n i=1 dy i s ω = K 2 n i=1 Y i ω, 43 for every ω Ω. Noe ha since Y i, i = 1,..., n, are increasing process, here he sochasic inegrals coincide wih he Sieljes inegrals, and hence we are able o do he inegraion pahwise. Moreover, since Y 1 has he suppor [, n, so does Y. This implies ha IP ω Ω : n i=1 Y i ω < ε >, for every ε >. This combined wih 43 yields ha IP ω Ω : σ s dy s ω < ε >, for every ε >. Since µ is deerminisic, i follows ha r = µ + σ s dy s only if µ. Thus, we have proved he rs claim in ii. To show he second one, namely ha he nonnegaiviy of he shor rae r for all [, T ], implies he nonnegaiviy of he forward rae f T, noe ha µ, T = µt κ Σ T. Since we have jus proved ha r T implies µt, i suces o show ha κ Σ T o deduce ha µ, T. Bu his follows easily from Σ T combined wih 37. Thus, we have µ, T, which implies f T by deniion of µ, T.

17 17 Compleely analogously, we can derive condiions for he nonnegaiviy of he forward spread g T and he shor erm spread λ. Le us denoe µ, T := g T σ st dy s = g T κ Σ T Σ T + κ Σ T Σ T µ := λ + κ Σ T κ Σ T which follows by σ sdy s = g + κ Σ Σ κ Σ, Proposiion 3.4 i The shor erm spread λ is nonnegaive if µ, for every [, T ]. ii Assume ha he disribuion of Y 1 has [, n as is suppor. Then he converse of i is also rue, i.e. if λ, hen µ, for every. Moreover, if λ, for every [, T ], hen g T, for every T [, T ], i.e. he nonnegaiviy of he shor erm spread implies he nonnegaiviy of he forward spread. Le us now assume ha Y = Y 1, Y 2 is a wo-dimensional nonnegaive Lévy process. We shall sudy in more deail he dependence beween is componens. Bu before doing so, le us give an example of he volailiy srucures ha saisfy he condiions of his secion and produce nonnegaive raes and spreads. Example 3.5 Vasicek volailiy srucure Assume ha he volailiy of he forward raes f T and he volailiies of he forward spreads g T, for T [, T ], are of Vasicek ype, so for every s T T, σ s T = σe at s,, σ st =, σ e a T s, 44 where σ, σ > and a, a are real consans such ha µ and µ from Proposiions 3.3i and 3.4i are nonnegaive. Then T σ Σ T = σ udu = 1 e at,, Σ T =, σ a a 1 e a T and by 22 Σ T = Σ T + Σ T = σ 1 e at, σ a a 1 e a T. The volailiies Σ and Σ saisfy he sanding Assumpion 3.1. Moreover, insering hem ino Proposiion 3.2, we noe ha he forward raes f T and he shor rae r are driven solely by he rs subordinaor Y 1, whereas he forward spreads g T and he shor spread λ are driven by he second subordinaor Y 2. Wih his volailiy specicaion, one obains he Lévy HullWhie exended Vasicek model for he shor rae r cf. Corollary 4.5 and equaion 4.11 in he risk-free seup of Eberlein and Raible 1999 dr = aρ r d + σdy 1.

18 18 By similar reasoning, one can obain he Lévy HullWhie exended Vasicek model for he shor erm spread λ dλ = a ρ λ d + σ dy 2. The funcions ρ and ρ are deerminisic funcions of ime which are chosen is such a way ha he models he iniial erm srucures f T and g T observed in he marke. Insering he Vasicek volailiies ino equaion 39 for r and equaion 41 for λ, and diereniaing wih respec o ime, one obains ρ and ρ. We have and ρ = f + 1 a f + κ 1 σ a e a 1 κ 1 σ e a 1 σ a a e a, ρ = g + 1 a g κ 1 σ e a 1 + κ 1 σ a a σ +κ e a 1, σ a a e a σ a κ e a 1, σ a a e a 1, e a 1 σ a e a where κ 1 is he cumulan funcion of Y 1. Moreover, his model possesses an ane erm srucure. I means ha he risk-free bond prices can be wrien as exponenial-ane funcions of he curren level of he shor rae r, and he risky bond prices as exponenial-ane funcions of he shor rae r and he shor erm spread λ. We have where and B T m, T = log B [ κ 1 σ a B T = expm, T + n, T r, 45 [ n, T f + κ 1 σ a e at s 1 κ 1 σ a T n, T = e a e au du = 1 a e a 1 ] e at 1. ] e a s 1 ds This resul is proved in Raible 2, Theorem 4.8. For B T i follows, by exacly he same reasoning and using represenaion 36, ha m, T = log σ +κ B T = expm, T + n, T r + m, T + n, T λ, 46 T [ g udu n, T g κ 1 σ a ] a e a 1 e a 1, σ a σ [κ 2 a e a T s 1 e a 1 σ κ 2 a e a s 1 ] ds

19 19 and T n, T = e a where κ 2 is he cumulan funcion of Y 2. e a u du = 1 e a T a 1, Example 3.6 Dependen drivers In order o specify he dependence beween componens Y 1 and Y 2 of he driving process Y, we presen here a common facor model. Possible racable alernaives o creae dependence would be o subordinae wo independen Lévy processes wih an independen common subordinaor, or o use a Lévy copula, see Con and Tankov 23. Le us assume ha Y 1 and Y 2 are given as Y 1 = Z 1 + Z 3 and Y 2 = Z 2 + Z 3, where Z i, i = 1, 2, 3, are muually independen subordinaors wih drifs b Zi and Lévy measures F Zi. Then Y 1 and Y 2 are again subordinaors his follows by Proposiion 11.1 and Theorem 21.5 in Sao 1999 and hey are obviously dependen. The Lévy measures and he cumulan funcions for subordinaors Y 1 and Y 2, as well as for he wo-dimensional process Y = Y 1, Y 2, can be calculaed explicily, as shown below. Consider a hree-dimensional Lévy process Z = Z 1, Z 2, Z 3, consising of muually independen subordinaors Z i, as above. Applying Sao 1999, Exercise 12.1, page 67, he independence of Z 1, Z 2 and Z 3 implies ha he Lévy measure F Z of Z is given by F Z A = 3 F Zi A i, A BR 3 \ {}, 47 i=1 where for every i, A i = {x R : xe i A} wih e i a uni vecor in R 3 wih 1 in he i-h posiion and oher enries zero. Now we simply have o wrie Y, Y 1 and Y 2 as linear ransformaions of Z and apply Proposiion 11.1 in Sao For example, we have Y = UZ, where [ ] 1 1 U =. 1 1 Hence, b Y = Ub Z and he Lévy measure F Y is given, for B BR 2 \ {}, by F Y B = F Z x R 3 : Ux B = F Z x R 3 : x 1 + x 3, x 2 + x 3 B, which combined wih 47 yields F Y B = F Z1 x R : x, B + F Z2 x R :, x B + F Z3 x R : x, x B. The cumulan funcion κ Y of Y is given, for z R 2 such ha κ Zi, i = 1, 2, 3, below are well-dened, by κ Y z = κ Z1 z 1 + κ Z2 z 2 + κ Z3 z 1 + z 2. This can be derived direcly recalling ha κ Y z = log IE[e zy ] and using independence beween Z 1, Z 2 and Z 3. Similarly, wriing each Y i, i = 1, 2, as a linear ransformaion of Z, we obain is Lévy measure F Y i, for C BR \ {}, F Y i C = F Z x R 3 : x i + x 3 C = F Zi x R : x C + F Z3 x R : x C

20 2 and he drif b Y i = b Zi + b Z3, which shows ha Y i is indeed a subordinaor recall Theorem 21.5 in Sao The cumulan funcion κ Y i of Y i is given, for z R such ha κ Zi and κ Z3 below are well-dened, by κ Y i z = κ Zi z + κ Z3 z. To conclude his secion, we describe wo well-known subordinaors: an inverse Gaussian IG process and a Gamma process. In addiion, we recall an example of a subordinaor belonging o he CGM Y Lévy family, which was inroduced by Carr, Geman, Madan, and Yor 22. Noe ha hese processes have innie aciviy, which makes hem suiable drivers for he erm srucure of ineres raes in our model. Example 3.7 IG process According o Kyprianou 26, Secion 1.2.5, a process Z = Z obained from a sandard Brownian moion W by seing Z = inf{s > : W s + bs > }, where b >, is an inverse Gaussian IG process and has he Lévy measure given by F dx = 1 2πx 3 e b2 x 2 1{x>} dx. The disribuion of Z is IG b, 2. The Lévy measure F saises condiion 6 for any wo consans K, ε > such ha 1 + εk < b2 2. Hence, he cumulan funcion κ exiss for all z b2 2, b2 2 acually for all z, b2 2 since F is concenraed on, and is given by κz = b zb Example 3.8 Gamma process The Gamma process Z = Z α, β > is a subordinaor wih Lévy measure given by wih parameers F dx = βx 1 e αx 1 {x>} dx, see Kyprianou 26, Secion The disribuion of Z is Γβ, α. The Lévy measure F saises condiion 6 for any wo consans K, ε > such ha 1 + εk < α. Hence, he cumulan funcion κ is well-dened for all z, α and is given by κz = β log 1 z α Example 3.9 CGMY subordinaor The CGMY Lévy process Z = Z wih parameers G = and Y < 1 is a subordinaor by Theorem 21.5 in Sao Is Lévy measure is given by F dx = C exp M x x 1+Y 1 {x>} dx, where C, M > and Y < 1; see Raible 2, A.3.2. For an overview of he main properies of he class of CGMY Lévy processes we refer o Carr, Geman, Madan, and Yor 22 or Raible 2, A.3.2. Noe ha he cumulan funcion κ is known in closed form for Y < and given by κz = CΓ Y M z Y M Y, for all z, M; see Carr, Geman, Madan, and Yor 22, Theorem 1 and Raible 2, A.3.2..

21 21 4 Valuaion of ineres rae derivaives Here we give an overview of he basic ineres rae derivaives where he underlying rae is he LIBOR and calculae heir value in our seup. We work wih general ime-inhomogeneous Lévy processes and under he assumpions of deerminisic volailiies and drif erms in equaions 16 and 23, as well as under he assumpion 17. Before proceeding wih he valuaion of ineres rae derivaives, le us recall ha he forward maringale measure IP T associaed wih he dae < T T is a probabiliy measure dened on Ω, F T and equivalen o IP. I is characerized by he following densiy process dip T dip F = β B T B T, where T. In our seup his densiy process is given by cf. 13 dip T dip = exp F A s T ds Σ s T dy s. 49 Noe ha he densiy process is E-adaped. The payos of he derivaives ha we are going o sudy in he sequel are ypically some combinaions of deerminisic funcions of he LIBOR raes L T T, T + δ, which are E T -measurable random variables, for any T [, T δ]. Then we have IE[fL T T, T + δ F ] = IE[fL T T, T + δ E ], for any deerminisic, Borel measurable funcion f : R R. This propery is due o he immersion propery beween E and F see Bielecki and Rukowski 22, Secion 6.1.1, which by assumpion holds in our model. Moreover, he propery holds rue under any forward measure IP T as well, since he densiy process in 49 is E-adaped. Henceforh in all compuaions we shall auomaically replace F by E. B Finally, noe ha in a muliple-curve seup he process T B T T +δ is no a maringale under he forward measure IP T +δ. Consequenly, he forward LIBOR rae, if dened as L T, T + δ = 1 B T δ, B T +δ 1 would be dieren from a forward rae implied by a forward rae agreemen for he fuure ime inerval [T, T + δ], as we shall see below. In he one-curve seup, he forward LIBOR rae dened as L T, T + δ = 1 BT δ B T +δ 1 is precisely he FRA rae for [T, T + δ]. 4.1 Forward rae agreemens The simples ineres rae derivaive is a forward rae agreemen FRA wih incepion dae T and mauriy T + δ, where T T δ. Le us denoe he xed rae by K and he noional amoun by N. The payo of such an agreemen a mauriy T + δ is equal o P F RA T + δ; T, T + δ, K, N = NδL T T, T + δ K, where L T T, T + δ is he T -spo LIBOR rae. Thus, he value of he FRA a ime T is calculaed as he condiional expecaion wih respec o he forward measure IP T +δ associaed wih he dae T + δ and is given by P F RA ; T, T + δ, K, N = NδB T + δie IPT +δ [L T T, T + δ K E ].

22 22 We emphasize again ha he forward rae implied by his FRA, ha is he rae K such ha P F RA ; T, T + δ, K, N =, diers in he muliple-curve seup from he classical forward LIBOR rae. Le us derive he value of he FRA and calculae he forward rae K in he model. Using deniion 4 of he LIBOR rae L T T, T + δ we have P F RA ; T, T + δ, K, N = NB T + δie IPT +δ [ ] 1 B T T + δ K E, 5 where K = 1 + δk. The key issue is hus o compue condiional expecaions of he form [ v T,S 1 ] := IE IPS B T S E, T S. 51 Insering S = T + δ and 23 ino 51 we obain T,T +δ v = B T T B T + δ exp Ā st + δ Ā st ds T ] IE [exp IPT +δ Σ st + δ Σ st dy s E = c T,T +δ exp Σ st + δ Σ st dy s T ] IE [exp IPT +δ Σ st + δ Σ st dy s, 52 where c T,T +δ = B T T B T + δ exp Ā st + δ Ā st ds. For he second equaliy in 52 we use he fac ha Σ st + δ Σ st dy s is E - measurable. Moreover, since Y is a ime-inhomogeneous Lévy process under he measure IP T +δ, is incremens are independen cf. Proposiion 2.3 and Lemma 2.5 in Kluge 25. This combined wih he deerminisic volailiy srucure which is inegraed wih respec o Y yields he equaliy. The remaining expecaion can be calculaed making use of Proposiion 3.1 in Eberlein and Kluge 26b, which yields T ] IE [exp IPT +δ Σ st + δ Σ st dy s T = exp κ IPT +δ s Σ st + δ Σ st ds, 53 IPT +δ where κs denoes he cumulan funcion of Y under he measure IP T +δ. However, o obain he expression for his expecaion using he cumulan funcion κ s of Y under he

23 23 measure IP, we have he following sequence of equaliies T ] IE [exp IPT +δ Σ st + δ Σ st dy s = exp IE T [ exp T = exp IE [ exp A s T + δds T T Σ st + δ Σ st dy s κ s Σ s T + δds T Σ s T + δdy s ] [ ] IE exp Σ s T + δdy s ] Σ st + δ Σ st Σ s T + δdy s T = exp κs Σ st + δ Σ st Σ s T + δ κ s Σ s T + δ ds, where we have used equaion 49 for he rs equaliy, and he drif condiion 17 plus he independence of incremens of Y for he second one. The hird equaliy follows by Eberlein and Kluge 26b, Proposiion 3.1. Finally, we obain T,T +δ v = c T,T +δ exp Σ st + δ Σ st dy s T exp κs Σ st + δ Σ st Σ s T + δ κ s Σ s T + δ ds. IPT +δ Noe ha along he same lines one obains a formula for he cumulan funcion κs IPT +δ κs z = κ s z Σ s T + δ κ s Σ s T + δ, 56 for z R n such ha κ s z Σ s T + δ is well-dened. This follows by combining 53 and 54, for every [, T ] and for Σ st + δ Σ st replaced wih z. In paricular we proved he following Proposiion 4.1 The value of he FRA a ime = is given by [ P F RA T,T +δ ; T, T + δ, K, N = NB T + δ v K ], where T,T +δ v = B T B T + δ exp T Ā s T + δ Ā st κ s Σ s T + δ ds T exp κ s Σ st + δ Σ st Σ s T + δds. The forward rae K implied by his FRA is given by K = 1 [ ] T,T +δ v δ The spread above he one-curve forward rae given by 1 B T δ B T +δ 1, is equal o Spread F RA = 1 [ T,T +δ v B ] T. 58 δ B T + δ As soon as he driving process Y and he parameers of he model are specied, all hese values can be easily compued. We provide an example in Secion 5.

24 Ineres rae swaps An ineres rae swap is a nancial conrac beween wo paries o exchange one sream of fuure ineres paymens for anoher, based on a specied noional amoun N. Here we consider a xed-for-oaing swap, where a xed paymen is exchanged for a oaing paymen linked o he LIBOR rae. We assume, as is ypical, ha he LIBOR rae is se in advance and he paymens are made in arrears. The swap is iniiaed a ime T. Denoe by T 1 < < T n, where T 1 > T, a collecion of he paymen daes and by S he xed rae. Then he ime- value of he swap for he receiver of he oaing rae is given by, for T, P Sw ; T 1, T n = N = N = N n δ k 1 B T k IE IPTk [L Tk 1 T k 1, T k S E ] k=1 n P F RA ; T k 1, T k, S, 1 k=1 n k=1 B T k v T k 1,T k S k 1, 59 where δ k 1 = T k T k 1, Sk 1 = 1 + δ k 1 S, and v T k 1,T k is given by 55, for every k = 1,..., n. This formula follows direcly from 5 and 51. The swap rae S; T 1, T n is he rae ha makes he ime- value P Sw ; T 1, T n of he swap equal o zero. Therefore, Proposiion 4.2 The swap rae S; T 1, T n, for T, is given by S; T 1, T n = n k=1 B T k v T k 1,T k 1 n k=1 δ. 6 k 1B T k 4.3 Overnigh indexed swaps OIS In an overnigh indexed swap OIS he counerparies exchange a sream of xed-rae paymens for a sream of oaing-rae paymens linked o a compounded overnigh rae. Le us assume he same enor srucure as in he previous subsecion is given and denoe again he xed rae by S. Similarly o Filipovi and Trolle 211, Secion 2.5, equaion 11, he ime- value of he swap for he receiver of he oaing rae is given by, for T, n P OIS ; T 1, T n = N B T B T n S δ k 1 B T k. The OIS rae OIS; T 1, T n, for T, is given by k=1 OIS; T 1, T n = B T B T n n k=1 δ k 1B T k. 61 The LIBOR-OIS spread a ime T, for he inerval [T, T + δ], where T T δ, is hus obained as a dierence of 4 and 61 for a single paymen dae as L T T, T + δ OIST ; T + δ, T + δ = 1 1 δ B T T + δ B T T + δ