A defaultable HJM multiple-curve term structure model

Transcription

1 A defaulable HJM muliple-curve erm srucure model Séphane Crépey, Zorana Grbac and Hai-Nam Nguyen Laboraoire Analyse e probabiliés Universié d Évry Val d Essonne 9125 Évry Cedex, France Ocober 9, 211 Absrac In he afermah of he 27 9 financial 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. In his paper we sudy he valuaion of LIBOR ineres rae derivaives in a muliple-curve seup, which accouns for a discrepancy beween a risk-free discoun curve and a LIBOR fixing curve. Toward his end we resor o a defaulable HJM mehodology, in which his discrepancy is modeled by an implied defaul inensiy of he LIBOR conribuing banks. Keywords: Ineres Rae Derivaives, LIBOR, HJM, Muliple Curve, Inerbank Risk, Lévy Processes. MSC: 91G3, 91G2, 6G51 JEL classificaion: G12, E43 1 Inroducion In he afermah of he 27 9 financial 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 see Figure 1. This is reckoned in Filipović and Trolle 211 as he adven of a so-called inerbank risk. The research of he auhors benefied from he suppor of he Chaire Risque de crédi, Fédéraion Bancaire Française, and of he DGE. The auhors hank Jeroen Kerkhof, from he bank Jefferies, 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 Euribor - Eoniaswap spreads spread 1m spread 3m spread 6m spread 12m 15 bp Figure 1: Lef: Hisorical Euribor-Eonia swap spreads boosrapped on Sepember Righ: Discoun curves 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, 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 ino a conrac in a muliple-curve environmen, have become 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 211, meaning clean of counerpary risk and excess funding coss over he risk-free 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 for insance Crépey 211. Acually, he iniial moivaion for he presen work was o devise a model of clean valuaion of ineres rae derivaives wih inerbank risk, racable in iself and also in he perspecive of serving as 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, 29 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 29, 21 and Fujii, Shimada and Takahashi 29, 21 in such a way ha each basis spread is modeled as a differen process. 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

3 3 Markov driving processes. Finally, a firs 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. In his paper we also resor o a credi risk mahemaical seup. Le us add ha by credi risk here we do no mean counerpary risk of he paries of a conrac we acually deliberaely disregard counerpary risk in a clean valuaion perspecive. Wha we mean here is simply an inerpreaion or measuremen of LIBOR quoes which are he main inpu o mos ineres rae derivaive cash flows in excess over he risk-free rae, in he mahemaically racable scale of an implied defaul inensiy of he LIBOR conribuing banks. Noe ha a rolling consrucion of he LIBOR group is precisely inended o he effec ha, in principle, acual defauls canno occur wihin he LIBOR group. We are also fully aware ha 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 differen from is inimae 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 offers in reasonable marke size, jus prior o 11. London ime he heoreical definiion of he LIBOR. More precisely, we shall follow a defaulable Heah Jarrow Moron mehodology for modeling he erm srucure of muliple ineres raes; see he seminal paper by Heah, Jarrow, and Moron 1992 and he defaulable exensions by Bielecki and Rukowski 2 and Eberlein and Özkan 23. 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 counerpoin o Morini 29 conclusions in his firs enaive credi risk approach, an appropriae credi risk model 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. These findings are also in line wih a quaniaive analysis of he erm srucure of inerbank risk which was recenly conduced by Filipović 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 firs half of he sample see Figures 3 and 4 in heir paper. The second main driver is inerpreed as liquidiy risk, consisenly wih he claims in Morini 29. We poin ou in his regard ha, even hough we did no see he necessiy of i ye and herefore did no do i for he sake of parsimony of he model, a simple amendmen o our model allows o make explici also a non-defaul componen of inerbank risk. For his i is enough o add one more componen o he driver of our risky ineres rae and he HJM-ype valuaion formulas can be derived in exacly he same manner as below. Besides, our moivaion for modeling he coninuously compounded forward raes in a HJM fashion, insead of dealing direcly wih discreely compounded LIBORs in a BGM perspecive, is wofold. On he one side, i allows one o consider simulaneously he LI- BORs for all possible enors recall ha one of he pos-crisis spread sudied in his work is beween LIBORs of various enors. The HJM framework is capable of producing a mulicurve model wih as many sochasic facors as LIBORs of differen 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 unified 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

4 4 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. In our view he main conribuions of his work are: a consisen and racable defaulable HJM erm srucure model of inerbank risk; low-dimensional exended CIR or Lévy Hull Whie shor rae specificaions of he defaulable HJM seup, opening he door o he use of his model as underlying model o ineres rae derivaives CVA compuaions; empirical evidence ha an appropriaely chosen credi risk seup is enough o accoun for even he mos exreme inerbank spreads ever observed in he marke. The res of he paper is organized as follows. In Secion 2, we apply a 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 LIBOR are described and explici valuaion formulas are derived. Secion 5 presens numerical resuls illusraing he flexibiliy of he model in producing a wide range of FRA and basis swap spreads. 2 Defaulable HJM seup 2.1 Noaion In his subsecion we inroduce he main noions and noaion we are going o work wih. The basic reference rae for a variey of ineres rae derivaives is he LIBOR in he USD fixed income marke and he EURIBOR in he EUR fixed 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 he erm LIBOR meaning any of hese wo raes. Anoher imporan reference rae in fixed income markes is a so-called OIS Overnigh Indexed Swap rae, which is he rae a which overnigh unsecured loans can be obained in he inerbank marke. In he USD fixed income marke i is he FF Federal Funds rae and in he EUR marke i is he EONIA Euro Overnigh Index Average rae. From now on we shall use he generic erm OIS for any of hese raes. 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 depends on he erm srucure of inerbank risk, which is refleced 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 o he LIBOR reference curve via a given defaul inensiy γ. Again, τ is no mean o represen an acual defaul ime of some specific 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 inerbank risk in a mahemaically racable defaul inensiy scale. This being said, our credi risk formalism is consisen however wih he empirical evidence in Filipović and Trolle 211 ha defaul risk is a major componen of he inerbank risk. We shall work wih insananeous coninuously compounded forward raes, specifying he dynamics of he erm srucure of he risk-free forward ineres raes f T and of he

5 5 forward credi spreads g T corresponding o he risky 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 The corresponding shor raes r and r are given by r = f and r = f. 2 We also define he shor erm credi spread process λ by, for [, T ], 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, B T = exp T f udu and B T = exp T f udu. 3 The T -spo LIBOR L T T, T + δ is a simply compounded ineres rae fixed a ime T for he ime inerval [T, T + δ], which will be defined in our seup as L T T, T + δ = 1 1 δ B T T + δ 1. 4 We hus use in his definiion he pre-defaul 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. 2.2 Driving process We consider a filered probabiliy space Ω, F T, IP and a finie ime horizon T. Le E = E [, T ] denoe a filraion 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 Definiion 1.6 in Sao 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, non-negaive definie 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 e.g. 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 saisfied:

6 6 Assumpion 2.1 i The riple b, c, F saisfy 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[exp uy ] <, 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, 7 where μ is he random measure of he jumps of Y, ν is he IP-compensaor of μ, c s is a measurable version of a square-roo of he symmeric, non-negaive definie 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, is B = b s ds, C = c s ds, ν[, ] A = [, ] A F s dxds, 8 for every Borel se A BR n {}. The riple b, c, F represens he local characerisics of Y. Any of hese riples deermines he disribuion of Y, as he Lévy Khinchine 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 infiniely divisible disribuion characerized by he Lévy riple b s, c s, F s. For a row-vecor 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 2.16.

7 7 2.3 Term srucure of ineres raes In his subsecion, we model he risk-free and he risky erm srucure of ineres raes. We shall be concerned wih wo filraions on he sanding risk-neural probabiliy space Ω, F T, IP of his paper: he defaul-free filraion E = E T, and he full filraion F = F T conaining E and he informaion abou he defaul ime τ. The defaul-free bond price process B T, he pre-defaul bond price process B T, and he corresponding forward rae processes f T and f 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. We assume ha τ possesses an E-hazard inensiy γ. Thus, is Azéma supermaringale is given by Qτ > E = e γ s ds, 11 where γ is an E-adaped, non-negaive and inegrable process. The risky bonds are assumed o pay a cerain recovery upon defaul. We adop he fracional recovery of marke value scheme, which specifies ha in case of defaul of he bond issuer, he fracion of he pre-defaul value of he bond is paid a he 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 was defined in 3. 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 B 1 τ T a T. The ime- price of such a bond can be wrien as B T = 1 {τ >} B T + 1 {τ }R B τ T B 1 τ T B T. 12 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 specify he insananeous coninuously compounded forward raes f T and he insananeous forward credi spreads g T, which in urn provide he defaul-free bond prices B T and he pre-defaul bond prices 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. Conrary o he laer wo papers, we choose here o model direcly he forward credi spreads insead of he risky forward raes. Clearly, in order o model he pre-defaul erm srucure, i is equivalen o specify eiher he forward raes f T, or he forward credi spreads g T. However, by no-arbirage one has B T B T, i.e. he risky bonds are cheaper han he defaul-free bonds wih he same mauriy. This implies by 3 ha one should have f T f T, or equivalenly, g T. Hence, we decide o model he forward credi spreads direcly and sudy heir non-negaiviy in some special cases. In he nex subsecion we provide wo racable non-negaive examples. Le us also menion here a paper by Chiarella, Maina, and Nikiopoulos 21, where a class of sochasic volailiy HJM models admiing finie dimensional Markovian srucures is proposed. They model he defaul-free forward raes and he forward credi spreads, whose dynamics are driven by correlaed Brownian moions. One of our examples in he sequel, he sochasic volailiy CIR model of Secion 3.1, could be fi ino his modeling framework.

8 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, 13 where he iniial values f T are deerminisic, bounded and Borel measurable in T. Moreover, σ and a are sochasic processes defined on Ω [, T ] [, T ] aking values in R n and R, respecively. Le P and O respecively denoe he predicable and he opional σ-field 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 find a joinversion 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 ], B T = B T exp r s A s T ds Σ s T dy s, 14 where we se A s T := T s a s udu, Σ s T := T s σ s udu. 15 Insering T = ino 14, he risk-free discoun facor process β = β T, defined by β = exp r sds, can be wrien as β = B exp A s ds Σ s dy s. 16 Combining his wih 14 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. 17 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, βb T, are F-maringales wih respec o a risk-neural measure IP. Due o he immersion propery i suffices ha hey are E-maringales. This is guaraneed by he following drif condiion: A s T = κ s Σ s T, s [, T ], 18 where κ s is he cumulan of Y defined 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 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 18 simplifies o A s T = 1 2 Σ st 2, which is he classical HJM no-arbirage condiion.

9 Risky raes The dynamics of he forward credi spreads g T, [, T ], is given by g T = g T + a st ds + σ st dy s, 19 where he iniial values g T are deerminisic, bounded and Borel measurable in T. Moreover, a and σ saisfy he same measurabiliy and boundedness condiions as a and σ. The risky forward raes are hen given by where we se f T = f T + a st ds + σ st dy s, 2 f T = f T + g T, a st = a s T + a st, σ st = σ s T + σ st. The dynamics of he bond prices B T can be obained exacly in he same way as he dynamics of B T in equaion 14. Therefore, for [, T ], B T = B T exp rs A st ds Σ st dy s, 21 where A st := T s a sudu and Σ s T := T s σ sudu. 22 Recall from 2 ha he shor rae r s is given by r s + λ s. Similarly o 17, we can rewrie he bond prices process B T as follows B T = B T B exp A s A st ds + Σ s Σ st dy s. 23 For he sake of model parsimony we require in addiion ha he defaulable bond prices discouned a he risk-free rae, βb T, are F, IP-maringales, for all T [, T ]. Noe ha his consrain would correspond o precluding arbirage opporuniies relaed o dealing wih he risky bonds B T, were such risky bonds raded in he marke which hey are acually no, even no synheically as averages of risky bonds of LIBOR conribuors, since he LIBORs refleced in B T are only reference numbers and no ransacion quoes; see he definiion of he LIBOR in he inroducion. For each T his addiional maringale condiion is saisfied if where B T R B T γ = B T α T, [, T ], 24 α T := λ A 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.

10 1 Under he fracional recovery of marke value scheme which is assumed in his paper, one ges a paricularly convenien form of he drif condiion 24. The recovery process R akes he following form cf. 12 which insered ino 24 yields R := R B T B 1 T, 1 R γ = α T, [, T ]. 25 Since condiion 25 has o hold for all T [, T ], i is acually equivalen o he following wo condiions: 1 R γ = λ 26 and A T = κ Σ T. 27 Indeed, condiions 26 and 27 obviously imply 25. To see he converse, one has o inser = T ino 25 and noe ha A = and Σ = by 15. Moreover, κ = by 9. This yields 26. Condiion 27 now follows from 25 by insering = T. We work henceforh under he following Assumpion 2.2 The no-arbirage condiions 18, 26 and 27 are saisfied. Proposiion 2.3 i The forward rae f T is given by f T = f T + T κ s Σ s T ds + and he shor rae r by r = f + κ s Σ s ds + σ s T dy s, 28 σ s dy s. 29 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 + and he shor erm spread λ by λ = g + + σ st dy s, 3 κ s Σ s Σ s κ s Σ s ds σ sdy s. 31 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 + σ sdy s.

11 11 Proof. To prove i, noe ha from condiion 18 i follows ha a s T = T κ s Σ s T. This immediaely yields 28 and 29. Similarly, o prove ii, we make use of 27 and obain a st = a 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, 3 and 31 follow. Finally, o prove iii we combine 26 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 non-negaiviy of he risk-free ineres raes and he credi spreads, considering in paricular wo cases: a pure-jump Lévy process wih non-negaive componens subordinaors combined wih deerminisic bond price volailiy srucures, and a wo-dimensional Brownian moion combined wih sochasic volailiy srucures. We shall focus in paricular on he firs 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 non-negaive ineres raes. The sandard argumen is ha he probabiliy of negaive ineres raes is sufficienly small, and herefore his undesirable feaure is sill olerable. However, when ineres raes are small as in he recen years, he non-negaiviy of ineres raes produced by a model becomes a pracically relevan issue. 3.1 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 non-negaive shor raes and shor erm spreads wih his driving process, he volailiies in he HJM model canno be chosen deerminisic. We make use of he volailiy specificaions ha produce he CIR shor rae and he CIR shor erm spread wihin he HJM framework, as shown in

12 12 Chiarella and Kwon 21. Thus, we impose he following assumpions on he volailiies σ s and σs: 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 firs Brownian moion Y 1 =: W r and he credi 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, 32 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 r, ı Markov. The forward rae volailiy specificaion 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 32 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 defined accordingly. Similarly, we also define ȷ = σ s 2 ds, dȷ = ζ 2 λ 2k ȷ d. In Theorem 2.1 of Chiarella and Kwon 21 i was shown ha he risk-free exended CIR model possesses an affine erm srucure wih wo sochasic facors. More precisely, he bond prices can be wrien as exponenial-affine funcions of he curren level of he shor rae r and he process ı: where B T = B T B exp γ, T f γ, T r 12 γ2, T ı, 33 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 defaulable bonds B T a similar expression involving in addiion λ and ȷ can be obained by exacly he same reasoning and making use of he represenaion T B T = B T exp g udu, 34 which follows from 1 and 3.

13 Jumps and deerminisic volailiy In CVA applicaions see Crépey 211, Markovian specificaions are used. The previous Brownian specificaion of he general HJM defaulable seup, yields a four-dimensional Markov facor process X = r, λ, ı, ȷ. In he ques of a more parsimonious Markovian specificaion, 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 non-negaiviy of he ineres raes and he credi 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 process X = r, λ, which makes his specificaion preferable for applicaions. Le Y be an n-dimensional non-negaive Lévy process, such ha is Lévy measure saisfies Assumpion 2.1. Is cumulan generaing funcion is given by κz = zb + e zx 1 F dx 35 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 non-negaive Lévy processes see 3.19 in Barndorff-Nielsen and Shephard 2. Noe ha subordinaors do no have a diffusion 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 bond price volailiies Σ and Σ : Assumpion 3.1 Volailiies Σ and Σ are non-negaive, deerminisic and saionary funcions. More precisely, hey are given as follows 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. Proposiion 3.2 i The dynamics of he forward raes f T and r are given by and f T = f T κ Σ T + κ Σ T + r = f + κ Σ + ii The dynamics of he credi spreads g T and λ are given by and σ s T dy s 36 σ s dy s. 37 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 38 σ sdy s. 39

14 14 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, 4 T κ Σ st Σ s T = s κ Σ st Σ s T, which follows from 4 by differeniaion. Therefore, we obain T κ Σ st ds = s κ Σ st ds = κ Σ T κ Σ T, and similarly, T κ Σ st Σ s T ds = κ Σ T Σ T κ Σ T Σ T. Insering hese expressions ino 28 and 3 yields 36 and 38, respecively. To show 37 and 39 we noe ha and κ Σ sds = κ Σ κ Σ s Σ s ds = κ Σ Σ, due o κ Σ = κ = and κ Σ Σ =, which follows by 15 and 22 combined wih 35. In he nex wo proposiions we give necessary and sufficien deerminisic condiions for he non-negaiviy of he ineres raes and credi spreads. Noe ha by 36-37, one has ha f T r σ s T dy s = f T κ Σ T + κ Σ T =: μ, T σ s dy s = f + κ Σ =: μ, where μ, T and μ are hus deerminisic. Proposiion 3.3 i The shor rae r is non-negaive 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 non-negaiviy of he shor rae implies he non-negaiviy of he forward rae.

15 15 Proof. Since Y has non-negaive componens and he volailiy σ is non-negaive 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 n n σ s dy s ω K dys i ω = K Y i ω, 41 i=1 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 41 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 proved he firs claim in ii. To show he second one, namely ha he non-negaiviy of he shor rae r for all [, T ], implies he non-negaiviy 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 suffices o show ha κ Σ T o deduce ha μ, T. Bu his follows easily from Σ T combined wih 35. Thus, we have μ, T, which implies f T by definiion of μ, T. Compleely analogously, we can derive condiions for he non-negaiviy 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 + κ Σ T κ Σ T μ := μ, = λ which follows by i=1 σ sdy s = g + κ Σ Σ κ Σ, Proposiion 3.4 i The shor erm spread λ is non-negaive 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 non-negaiviy of he shor erm spread implies he non-negaiviy of he forward spread. Le us now assume ha Y is a wo-dimensional non-negaive 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 non-negaive raes and spreads.

16 16 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 are of he Vasicek ype, so for every s T T, σ s T = σe at s,, σst =, σ e a T s, 42 where σ, σ > and a, a = are real consans such ha μ and μ from Proposiions 3.3i and 3.4i are non-negaive. Then T σ Σ T = σ udu = 1 e at,, Σ T =, σ a a 1 e a T. These 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 firs 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 specificaion, one obains he Lévy Hull Whie exended Vasicek model for he shor rae r cf. Corollary 4.5 and equaion4.11 in he defaul-free seup of Eberlein and Raible 1999 dr = aρ r d + σdy 1. By similar reasoning, one can obain he Lévy Hull Whie 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 fi he iniial erm srucures f T and g T observed in he marke. Insering he Vasicek volailiies ino equaion 37 for r and equaion 39 for λ, and differeniaing wih respec o ime, one obains ρ and ρ. For ρ we have ρ = f + 1 a f + κ 1 σ a e a 1 κ 1 σ e a 1 σ a a e a, where κ 1 is he cumulan funcion of Y 1 compare equaions 4.1 and4.11 in Eberlein and Raible 1999, and ρ is derived in a similar fashion. Moreover, his model possesses an affine erm srucure. I means ha he defaul-free bond prices can be wrien as exponenial-affine funcions of he curren level of he shor rae r, and he pre-defaul defaulable bond prices as exponenial-affine funcions of he shor rae r and he shor erm spread λ: where B T m, T = log B [ κ 1 σ a and B T = expm, T + n, T r, 43 n, T f + e at s 1 κ 1 σ a T n, T = e a e au du = 1 a σ κ1 a e a s 1 ds e a s 1 e at 1. ] ds

17 17 This resul for defaul-free zero coupon bonds B T is proved in Raible 2, Theorem 4.8. For defaulable bonds B T i follows, by exacly he same reasoning and using represenaion 34, ha B T = expm, T + n, T r + m, T + n, T λ, 44 where he deerminisic funcions m, T and n, T can be defined similarly as m, T and n, T above Dependen drivers In order o specify he dependence beween componens Y 1 and Y 2 of he driving process Y, he simples way is a common facor model, ha we presen here an alernaive would be o use a Lévy copula, see Con and Tankov 23. Le us assume ha Y 1 and Y 2 are given as follows 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, 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 {}, 45 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 45 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-defined, 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.

18 18 Similarly, wriing each Y i 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 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-defined, 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. Noe ha hese processes have infinie aciviy, which makes hem suiable drivers for he erm srucure of ineres raes in our model. Example 3.6 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 saisfies 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 2. Example 3.7 Gamma process The Gamma process Z wih parameers α, β > is a subordinaor wih Lévy measure given by F dx = βx 1 e αx 1 {x>} dx, see Kyprianou 26, Secion The Lévy measure F saisfies condiion 6 for any wo consans K, ε > such ha 1 + εk < α. Hence, he cumulan funcion κ is well-defined for all z, α and is given by κz = β log 1 z. α Example 3.8 CGMY subordinaor The CGMY Lévy process Z wih parameers G = and Y 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 ; 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.

19 19 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 under he assumpions of Secion 2 and Secion 3.2 jumps and deerminisic volailiies. We emphasize ha our seup provides a versaile muli-curve model of LIBORs of differen enors, which is relevan for pricing of muli-enor derivaives such as basis swaps. For insance, if one wishes o have one sochasic driving facor for each enor and for he risk-free rae, hen i suffices o consider a hree-dimensional process Y, where he firs componen drives he risk-free raes and he remaining wo componens are reserved for he credi spread. Before proceeding wih he ineres rae derivaive valuaion, le us recall ha a forward maringale measure IP T associaed wih he dae < T T is a probabiliy measure defined 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. In our seup his densiy process is given by cf. 14 dip T = exp A s T ds dip F Σ s T dy s. 46 Noe ha he densiy process is E-adaped. The payoffs of he derivaives ha we are going o sudy in he sequel are ypically some combinaions of deerminisic funcions of he LIBORs L T T, T + δ, which is an E T -measurable random variable, 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 equivalen 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 46 is E-adaped. Henceforh in all compuaions we shall replace auomaically F by E. B Finally, noe ha in a muliple-curve seup he forward price process T B T +δ T is NOT a maringale under he forward measure IP T +δ. Consequenly, he forward LIBOR, which would be defined as L T, T + δ = 1 B T δ, B T +δ 1 is differen 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 defined 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. Le us denoe he fixed rae by K and he noional amoun by N. The payoff 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. Thus, he value of he FRA a ime is calculaed as he condiional expecaion wih respec o he forward measure IP T +δ associaed o he

20 2 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 ]. 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 =, is differen in he muliple-curve seup from he forward LIBOR. Le us derive he value of he FRA and calculae he forward rae K in our seup. Using definiion 4 of he LIBOR 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 ], 47 where K = 1 + δk. The key issue is hus o compue he condiional expecaion [ T,T +δ v := IE IPT +δ 1 ] B T T + δ E. 48 Insering 23 ino 48 we obain T,T +δ v = B T T B T + δ exp A st + δ A 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, 49 wih c T,T +δ = = B T T B T + δ exp A st + δ A st ds B T B T + δ exp T κ Σ s T + δ κ Σ st ds, where we used he drif condiion 27. For he second equaliy in 49 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, 5

21 21 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 direcly he cumulan funcion κ of Y under he measure IP, we have he following sequence of equaliies T ] IE [exp IPT +δ Σ st + δ Σ st dy s 51 = exp IE T [ exp T = exp IE T = exp [ exp A s T + δds T T Σ st + δ Σ st dy s κ 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 κ Σ s T + δ Σ st Σ s T + δ κ Σ s T + δ ds, where we have used equaion 46 for he firs equaliy, and he drif condiion 18 plus he independence of he 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 52 T exp κ Σ s T + δ Σ st Σ s T + δ κ Σ s T + δ ds. IPT +δ Noe ha as a by-produc we obain a formula for he cumulan funcion κs IPT +δ κs z = κz Σ s T + δ κ Σ s T + δ, 53 for z R n such ha κz Σ s T + δ is well-defined. This follows by combining 5 and 51, for every [, T ] and for Σ st + δ Σ st replaced wih z. Le us sum-up our findings in he form of 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 ], wih T,T +δ v = B T B T + δ exp T κ Σ s T + δ κ Σ st κ Σ s T + δ ds T exp κ 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

22 22 The spread wih respec o he one-curve forward rae given by 1 δ B T B T +δ 1, is equal o F RA Spread = 1 [ T,T +δ v B ] T. 55 δ B T + δ As soon as he driving process Y and he parameers of he model are specified, all hese values can be easily compued. We provide an example in Secion Ineres rae swaps An ineres rae swap is a financial conrac beween wo paries o exchange one sream of fuure ineres paymens for anoher, based on a specified noional amoun N. Here we consider a fixed-for-floaing swap, where a fixed paymen is exchanged for a floaing paymen linked o he LIBOR. We assume ha, as ypical, he LIBOR is se in advance and he paymens are done 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 fixed rae. Then he ime- value of he swap for he receiver of he floaing rae is given by 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, 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 52, for every k = 1,..., n. This formula follows direcly from 47 and 48. 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 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. 56 k 1B T k 4.3 Basis swaps A basis swap is an ineres rae swap, where wo floaing paymens linked o he LIBORs of differen enors are exchanged. For example, a buyer of such a swap receives semiannually a 6m-LIBOR and pays quarerly a 3m-LIBOR, boh se in advance and paid in arrears. Noe ha here exis also oher convenions regarding he paymens on he wo legs of a basis swap. A more deailed accoun on basis swaps can be found in Mercurio 21, Secion 5.2 and Filipović and Trolle 211, Secion 2.4 and Appendix F. Le us consider a basis swap wih he wo enor srucures denoed by T 1 = {T 1 <... < T 1 n 1 } and T 2 = {T 2 <... < T 2 n 2 }, where T 1 = T 2, T 1 n 1 = T 2 n 2 = ˆT, and T 1 T 2. The noional amoun is denoed by N and he swap is iniiaed a ime T 1, where he firs paymens are due a T 1 1 and T 2 1. The

23 23 ime- value of such an agreemen is given by P BSw ; ˆT, N = N n1 δi 1B 1 T 1 i=1 1 i IE IPT i [L T 1 i 1 T 1 i 1, T 1 i E ] n 2 δj 1B 2 Tj 2 2 IE IPT j [L T 2 T 2 j 1 j 1, Tj 2 E ]. j=1 Making use of 47 and 48 we obain Proposiion 4.3 The value of he basis swap a ime is given by P BSw ; ˆT n 1, N = N B Ti 1 v T i 1 1,T i 1 n B Tj 2 v T j 1 2,T j 2, 57 i=1 where v T i k 1,T i k is given by 52, for each enor srucure T i, i = 1, 2. Noe ha before he 27-9 credi crisis he value of such a swap was zero a any ime. Since he crisis, markes quoe posiive basis swap spreads ha have o be added o he smaller enor leg, which is consisenly accouned for in our seup; see Secion 5 for a numerical example. Le us check ha he value of he basis swap in he one-curve seup is indeed zero. We recall ha in his seup he forward LIBORs, which were defined using he defaul-free zero coupon bonds as L T, T + δ = 1 δ corresponding forward measures. We hus have P BSw ; ˆT, N = N n1 δi 1B 1 T 1 i=1 j=1 BT B T +δ T 1, are maringales under he 1 i IE IPT i [L T 1 i 1 T 1 i 1, T 1 i E ] n 2 δj 1B 2 Tj 2 2 IE IPT j [L T 2 T 2 j 1 j 1, Tj 2 E ] j=1 n 1 = N δi 1B 1 Ti 1 L Ti 1, 1 Ti 1 i=1 n 2 j=1 = N B T 1 B Tn 1 1 B T 2 B Tn 1 2 δj 1B 2 Tj 2 L Tj 1, 2 Tj 2 by iniial assumpions on he enor srucures. In he muliple-curve seup we canno use he same calculaion, since now he LIBORs are no maringales under he classical forward measures. Hence, one ends up wih formula 57, which in general yields a non-zero value of he basis swap and his value is exacly he basis swap spread cf. Tables 4 6 in Secion Caps and floors Recall ha an ineres rae cap respecively floor is a financial conrac in which he buyer receives paymens a he end of each period in which he ineres rae exceeds respecively falls below a muually agreed srike level. The paymen ha he seller has o make covers =,

24 24 exacly he difference beween he srike K and he ineres rae a he end of each period. Every cap respecively floor is a series of caples respecively floorles. The ime- price of a caple wih srike K and mauriy T, which is seled in arrears, is given by P Cpl ; T, K = δ B T + δie IPT +δ [ L T T, T + δ K + E ] [ ] = B T + δie IPT +δ 1 + B T K E T + δ where K = 1 + δk. I is worhwhile menioning ha he classical ransformaion of a caple ino a pu opion on a bond does no work in he muliple-curve seup. More precisely, he sill valid fac ha he payoff 1 + δl T T, T + δ K + seled a ime T + δ is equivalen o he payoff B T T + δ 1 + δl T T, T + δ K + seled a ime T will no yield he desired cancelaion of discoun facors. Since he LIBOR depends on he B T bonds and he defaul-free B T bonds are used for discouning, we have B T T + δ 1 + δl T T, T + δ K = BT T + δ K, T + δ which canno be simplified furher as in he one-curve case. Le us now calculae he value of he caple a ime = using he Fourier ransform mehod. We have [ P Cpl ; T, K = B T + δie IPT +δ 1 B T K T + δ B T = B T + δie IPT +δ [ e X K + ], where X is a random variable given by see 23 + ] X := log B T T B T + δ + A st + δ A st ds + T Σ st + δ Σ st dy s. T +δ Le us denoe by MX he momen generaing funcion of X under he measure IP T +δ., i.e. T +δ MX z = [ +δ IEIPT e zx ], for z R such ha he above expecaion is finie. We have T T +δ MX z = exp κ Σ s T + δds B exp z log T T B T + δ + 58 κ Σ s T + δ κ Σ st ds T exp κ z Σ s T + δ Σ st Σ s T + δ ds where κ is he cumulan funcion of Y under he measure IP. The derivaion of his formula follows along similar lines as he compuaions in Secion 4.1. In paricular, we have used equaions 46, 18, 27, and Proposiion 3.1 in Eberlein and Kluge 26b.,

25 25 Le us impose some condiions on he boundedness of he volailiy srucures Σ and Σ for he sake of he nex resul. We assume ha here exiss a posiive consan K < K 3 such ha Σ s T K and Σ st K componenwise and for all s, T [, T ] noe ha his is a slighly sronger boundedness condiion han he one in Assumpion 3.1. Now, applying Theorem 2.2 and Example 5.1 in Eberlein, Glau, and Papapanoleon 21 we obain Proposiion 4.4 The ime- price of a caple wih srike K and mauriy T is given by for any R 1, P Cpl ; T, K = B T + δ 2π K K. 2 K R K 1+iv R T +δ MX R iv dv, 59 iv R1 + iv R Proof. One has o apply Theorem 2.2 in Eberlein, Glau, and Papapanoleon 21 wih he Fourier ransform of he caple payoff funcion derived in Example 5.1 of he same paper, where oher prerequisies for Theorem 2.2 relaed o he payoff funcion are also checked. Noe ha he Fourier ransform of he caple payoff funcion is well-defined for T +δ K K any R 1, +. To ensure ha MX R iv is finie, i suffices o ake any R 1,. 2 K More precisely, for every i = 1,..., n, R Σi, i, s T + δ Σ s T Σ i st + δ R Σ i, s T + δ T +δ R2 K + K K K 2 K 2 K + K < K, i, Σ s T + Σ i st + δ and hus MX R < compare 58 and recall ha κ is well-defined for all z [ 1 + εk, 1 + εk] n. 4.5 Swapions A swapion is an opion o ener an ineres rae swap wih swap rae S and mauriy T n a a pre-specified dae T. Le us consider he swap from Secion 4.2. Recall ha a swapion can be seen as a sequence of fixed paymens δ j 1 ST ; T 1, T n S +, j = 1,..., n, ha are received a paymen daes T 1,..., T n, where ST ; T 1, T n is he swap rae of he underlying swap a ime T T 1. Hence, he value a ime of he swapion is given by P Swn ; T, T n, S = B T n [ δ j 1 IE IPT BT T j ST ; T 1, T n S + ] E ; j=1 see Musiela and Rukowski 25, Secion , p.482. A ime = we have n P Swn ; T, T n, S = B T IE IPT δ j 1 B T T j ST ; T 1, T n S + j=1 n = B T IE IPT B T T j v T j 1,T j T j=1 + n B T T j S j 1, j=1