Weierstraß-Institut. für Angewandte Analysis und Stochastik. Leibniz-Institut im Forschungsverbund Berlin e. V. Preprint ISSN

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1 Weerstraß-Insttut für Angewandte Analyss und Stochastk Lebnz-Insttut m Forschungsverbund Berln e V Preprnt ISSN Affne LIBOR models wth multple curves: Theory, eamples and calbraton Zorana Grbac, Antons Papapantoleon 2, John G M Schoenmakers 3, Davd Skovmand 4 submtted: May 5, 24 Unversté Pars Dderot Cede Pars France E-Mal: grbac@mathunv-pars-dderotfr 3 Weerstrass Insttute 2 Technsche Unversät Berln Straße des 7 Jun Berln Germany E-Mal: papapan@mathtu-berlnde 4 Copenhagen Busness School Mohrenstr 39 7 Berln Germany E-Mal: JohnSchoenmakers@was-berlnde Solbjerg Plads 3 2 Frederksberg Denmark E-Mal: dgsf@cbsdk No 95 Berln 24 2 Mathematcs Subject Classfcaton 9G3, 9G2, 6G44 Key words and phrases Multple curve models, LIBOR, OIS, bass spread, affne LIBOR models, caps, swaptons, bass swaptons, calbraton We are grateful to Fabo Mercuro and Steven Shreve for valuable dscussons and suggestons We also thank semnar partcpants at Cass Busness School, Imperal College London, the London Mathematcal Fnance Semnar and Carnege Mellon Unversty for ther comments All authors gratefully acknowledge the fnancal support from the DFG Research Center MATHEON, Projects E5 and E3

2 Edted by Weerstraß-Insttut für Angewandte Analyss und Stochastk WIAS Lebnz-Insttut m Forschungsverbund Berln e V Mohrenstraße 39 7 Berln Germany Fa: E-Mal: preprnt@was-berlnde World Wde Web:

3 INTRODUCTION The recent fnancal crss has led to paradgm shftng events n nterest rate markets because substantal spreads have appeared between rates that used to be closely matched; see Fgure for an llustraton We can observe, for eample, that before the credt crunch the spread between the three month LIBOR and the correspondng Overnght Indeed Swap OIS rate was non-zero, however t could be safely dsregarded as neglgble The same s true for the three month vs s month bass swap spread However, snce August 27 these spreads have been evolvng randomly over tme, are substantally too large to be neglected, and also depend on the tenor length Therefore, the assumpton of a sngle nterest rate curve that could be used both for dscountng and for generatng future cash flows was serously challenged, whch led to the ntroducton of the so-called multple curve nterest rate models In the multple curve framework, one curve s used for dscountng purposes, where the usual choce s the OIS curve, and then as many LIBOR curves as market tenors eg m, 3m, 6m and y are bult for generatng future cash flows The dfference between the OIS and each LIBOR rate s usually called bass spread or smply spread There are several ways of modelng the curves and dfferent defntons of the spread One approach s to model the OIS and LIBOR rates drectly whch leads to tractable prcng formulas, but the sgn of the spread s more dffcult to control and may become negatve Another approach s to model the OIS and the spread drectly and nfer the dynamcs of the LIBOR; ths grants the postvty of the spread, but prcng formulas are generally less tractable We refer to Mercuro 2b, pp -2 for a detaled dscusson of the advantages and dsadvantages of each approach Moreover, there est varous defntons of the spread: an addtve spread s used eg by Mercuro 2a, a multplcatve spread was proposed by Henrard 2, whle an nstantaneous spread was used by Andersen and Pterbarg 2; we refer to Mercuro and Xe 22 for a dscusson of the merts of each defnton The lterature on multple curve models s growng rapdly and the dfferent models proposed can be classfed n one of the categores descrbed above Moreover, dependng on the modelng approach, one can also dstngush between short rate models, Heath Jarrow Morton HJM models and LIBOR market models LMM wth multple curves The spreads appearng as modelng quanttes n the short rate and the HJM models are, by the very nature of these models, nstantaneous and gven n addtve form We refer to Banchett and Morn 23 for a detaled overvew of several multple curve models In the short rate framework, we menton Kenyon 2, Kjma, Tanaka, and Wong 29 and Morno and Runggalder 24, where the addtve short rate spread s modeled, whch leads to multplcatve adjustments for nterest rate dervatve prces HJM-type models have been proposed eg by Fuj, Shmada, and Takahash 2, Crépey, Grbac, and Nguyen 22, Moren and Pallavcn 24 and Cuchero, Fontana, and Gnoatto 24 The models by Mercuro 29, Banchett 2 where an analogy wth the cross-currency market has been eploted and Henrard 2 are developed n the LMM setup Typcally, multple curve models address the ssue of dfferent nterest rate curves under the same currency, however, the paper by Fuj et al 2 studes a multple curve model n a cross-currency setup Flpovć and Trolle 23 offer a thorough econometrc analyss of the multple curve phenomena and decompose the spread nto a credt rsk and a lqudty rsk component In

4 2 2 3m EUR LIBOR OIS EONIA spread Spot 3 year EUR 3m/6m Bass Swap Spread bass ponts FIGURE Spread Development from January 24 to Aprl 24 recent work, Galltschke, Müller, and Sefred 24 construct a structural model for nterbank rates, whch provdes an endogenous eplanaton for the emergence of bass spreads Let us also menton that there est varous other frameworks n the lterature where dfferent curves have been modeled smultaneously, for eample when dealng wth cross-currency markets cf eg Amn and Jarrow 99 or when consderng credt rsk cf eg the book by Beleck and Rutkowsk 22 The models n the multple curve world often draw nspraton from these frameworks The am of ths paper s to develop a multple curve LIBOR model that combnes tractable model dynamcs and sem-analytc prcng formulas wth postve nterest rates and bass spreads The framework of the affne LIBOR models proposed by Keller-Ressel, Papapantoleon, and Techmann 23 turns out to be talor-made for ths task, snce t allows us to model drectly LIBOR rates that are greater than ther OIS counterparts In other words, the non-negatvty of spreads s automatcally ensured Smultaneously, the dynamcs are drven by the wde and fleble class of affne processes Smlar to the sngle curve case, the affne property s preserved under all forward measures, whch leads to sem-analytcal prcng formulas for lqud nterest rate dervatves In partcular, the prcng of caplets s as easy as n the sngle curve setup, whle the model structure allows to derve effcent and accurate appromatons for the prcng of swaptons and bass swaptons usng a lnearzaton of the eercse boundary In addton, the model offers adequate calbraton results to a system of caplet prces for varous strkes and maturtes The paper s organzed as follows: n Secton 2 we revew the man propertes of affne processes and the constructon of ordered martngales greater than one Secton 3 ntroduces the multple curve nterest rate settng The multple curve affne LIBOR model s presented n Secton 4 and ts man propertes are dscussed, n partcular the ablty to produce postve rates and spreads and the analytcal tractablty e the preservaton of the affne property In Secton 5 we study the connecton

5 between the class of affne LIBOR models and the class of LIBOR market models drven by semmartngales Sectons 6 and 7 are devoted to the valuaton of the most lqud nterest rate dervatves such as swaps, caps, swaptons and bass swaptons In Secton 8 we construct a multple curve affne LIBOR model where rates are drven by common and dosyncratc factors and calbrate ths to market data Moreover, we test numercally the accuracy of the swapton and bass swapton appromaton formulas Fnally, Append A provdes an eplct formula for the termnal correlaton between LIBOR rates 3 2 AFFINE PROCESSES Ths secton provdes a bref revew of the man propertes of affne processes and the constructon of ordered martngales greater than one More detals and proofs can be found n Keller-Ressel et al 23 and the references theren Let Ω, F, F, IP denote a complete stochastc bass, where F = F t t,t ] and T denotes some fnte tme horzon Consder a stochastc process X satsfyng: Assumpton A Let X = X t t,t ] be a conservatve, tme-homogeneous, stochastcally contnuous Markov process wth values n D = R d, and IP D a famly of probablty measures on Ω, F, such that X =, IP -almost surely for every D Settng we assume that I T := { u R d : IE e u,x T ] <, for all D }, 2 I T, where I T denotes the nteror of I T ; the condtonal moment generatng functon of X t under IP has eponentally-affne dependence on ; that s, there est functons φ t u :, T ] I T R and ψ t u :, T ] I T R d such that for all t, u,, T ] I T D IE ep u, Xt ] = ep φ t u + ψ t u,, 22 Here, denotes the nner product on R d and IE the epectaton wth respect to IP The functons φ and ψ satsfy the followng system of ODEs, known as generalzed Rccat equatons t φ tu = F ψ t u, φ u =, 23a t ψ tu = Rψ t u, ψ u = u, 23b for t, u, T ] I T The functons F and R are of Lévy Khntchne form: F u = b, u + e ξ,u mdξ, 24a R u = β, u + D α 2 u, u + D e ξ,u u, h ξ µ dξ, 24b

6 4 where b, m, α, β, µ d are admssble parameters and h : R d R d are sutable truncaton functons The functons φ and ψ also satsfy the sem-flow equatons φ t+s u = φ t u + φ s ψ t u ψ t+s u = ψ s ψ t u for all t + s T and u I T, wth ntal condton 25a 25b φ u = and ψ u = u 26 We refer to Duffe, Flpovć, and Schachermayer 23 for all the detals The followng results and defntons wll be used n the sequel Inequaltes nvolvng vectors are nterpreted componentwse and :=,,, Lemma 2 The functons φ and ψ satsfy the followng: φ t = ψ t = for all t, T ] 2 I T s a conve set Moreover, for each t, T ], the functons I T u φ t u and I T u ψ t u are componentwse conve 3 φ t and ψ t are order-preservng: let t, u, t, v, T ] I T, wth u v Then φ t u φ t v and ψ t u ψ t v 27 4 ψ t s strctly order-preservng: let t, u, t, v, T ] I T, wth u < v Then ψ tu < ψ t v Defnton 22 Let X be a process satsfyng Assumpton A Defne γ X := sup IE ] e u,x T 28 u I T R d > An essental ngredent n affne LIBOR models s the constructon of parametrzed martngales whch are greater than or equal to one and ncreasng n ths parameter see also Papapantoleon 2 Lemma 23 Consder an affne process X satsfyng Assumpton A and let u I T R d Then the process M u = M u t t,t ] wth M u t = ep φ T t u + ψ T t u, X t, 29 s a martngale, greater than or equal to one, and the mappng u Mt u t, T ] s ncreasng, for every Proof Consder the random varable YT u := e u,x T Snce u I T R d u we have that YT s greater than one and ntegrable Then, from the Markov property of X, 22 and the tower property of condtonal epectatons we deduce that M u t = IE e u,x T F t ] = ep φt t u + ψ T t u, X t, 2 s a martngale Moreover, t s obvous that M u t follows from the orderng of Y u T and the representaton M u t for all t, T ], whle the orderng u v = M u t M v t t, T ], 2 = IEY u T F t]

7 5 T T T 2 T 3 T 4 T 5 T 6 T n T N T T T 2 T N T T 2 T 2 N 2 FIGURE 32 Illustraton of dfferent tenor structures 3 A MULTIPLE CURVE LIBOR SETTING We begn by ntroducng the notaton and the man concepts of multple curve LIBOR models We wll follow the approach ntroduced n Mercuro 2a, whch has become the ndustry standard n the meantme The fact that LIBOR-OIS spreads are now tenor-dependent means that we cannot work wth a sngle tenor structure any longer Hence, we start wth a dscrete, equdstant tme structure T = { = T < T < < T N }, where T k, k K := {,, N}, denote the maturtes of the assets traded n the market Net, we consder dfferent subsets of T wth equdstant tme ponts, e dfferent tenor structures T = { = T < T < < T N }, where X := {, 2,, n } s a label that ndcates the tenor structure Typcally, we have X = {, 3, 6, 2} months We denote the tenor length by δ = T k T k, for every X Let K := {, 2,, N } denote the collecton of all subscrpts related to the tenor structure T We assume that T T and T N = T N for all X A graphcal llustraton of a possble relaton between the dfferent tenor structures appears n Fgure 32 Eample 3 A natural constructon of tenor structures s the followng: Let T = { = T < T < < T N } denote a dscrete tme structure, where T k = kδ for k =,, N and δ > Let X = { =, 2,, n } N, where we assume that Net, set for every X k N, for all k =,, n T k = k δ =: kδ, for k =,, N := N/, where obvously T k = T k Then, we can consder dfferent subsets of T, e dfferent tenor structures T = { = T < T < < T N }, whch satsfy by constructon T T = T and also T N = N δ = T N, for all X We consder the OIS curve as dscount curve, followng the standard market practce of fully collateralzed contracts The market prces for caps and swaptons consdered n the sequel for model calbraton are ndeed quoted under the assumpton of full collateralzaton A detaled dscusson on the choce of the dscount curve n the multple curve settng can be found eg n Mercuro 2a and n Hull and Whte 23 The dscount factors B, T are strpped from market OIS rates and defned for

8 6 every possble maturty T T va T B, T = B OIS, T We denote by Bt, T the dscount factor, e the prce of a zero coupon bond, at tme t for maturty T, whch s assumed to concde wth the correspondng OIS-based zero coupon bond for the same maturty We also assume that all our modelng objects lve on a complete stochastc bass Ω, F, F, IP N, where IP N denotes the termnal forward measure, e the martngale measure assocated wth the numerare B, T N The correspondng epectaton s denoted by IE N Then, we ntroduce forward measures IP k assocated to the numerare B, T k for every par, k wth X and k K The correspondng epectaton s denoted by IE k The forward measures IP k are absolutely contnuous wth respect to IP N, and defned n the usual way, e va the Radon Nkodym densty dip k = B, T N dip N B, Tk BTk, T N 3 Remark 32 Snce T T there ests an l K and a k K such that T l = Tk Therefore, the correspondng numerares and forward measures concde, e B, T l = B, Tk and IP l = IP k See agan Fgure 32 Net, we defne the two rates that are the man modelng objects n the multple curve LIBOR settng: the OIS forward rate and the LIBOR rate We also defne the addtve and the multplcatve spread between these two rates Let us denote by LTk, T k the spot LIBOR rate at tme T k for the tme nterval Tk, T k ], whch s an F Tk -measurable random varable on the gven stochastc bass Defnton 33 The tme-t OIS forward rate for the tme nterval Tk, T k ] s defned by Fk t := Bt, T k δ Bt, Tk 32 Defnton 34 The tme-t LIBOR rate for the tme nterval Tk, T k ] s defned by L kt = IE k LT k, T k F t ] 33 The LIBOR rate s the fed rate that should be echanged for the future spot LIBOR rate so that the forward rate agreement has zero ntal value Hence, ths rate reflects the market epectatons about the value of the future spot LIBOR rate Notce that at tme t = Tk we have that L ktk = IE k LT k, Tk F T k ] = LT k, Tk, 34 e ths rate concdes wth the spot LIBOR rate at the correspondng tenor dates Remark 35 In the sngle curve setup, 32 s the defnton of the forward LIBOR rate However, n the multple curve setup we have that LTk, Tk δ BTk, T k, hence the OIS and the LIBOR rates are no longer equal

9 7 Defnton 36 The spread between the LIBOR rate and the OIS forward rate s defned by S k t := L kt F k t 35 Let us also provde an alternatve defnton of the spread based on a multplcatve, nstead of an addtve, decomposton Defnton 37 The multplcatve spread between the LIBOR rate and the OIS forward rate s defned by + δ Rkt := + δ L k t + δ Fk 36 t A model for the dynamc evoluton of the OIS and LIBOR rates, and thus also of ther spread, should satsfy certan condtons whch stem from economc reasonng, arbtrage requrements and ther respectve defntons These are, n general, consstent wth market observatons We formulate them below as model requrements: M F k t and F k MIP k, for all X, k K, t, T k ] M2 L k t and L k MIP k, for all X, k K, t, T k ] M3 S k t, for all X, k K, t, T k ] Here MIP k denotes the set of IP k -martngales Remark 38 If the addtve spread s postve the multplcatve spread s also postve and vce versa 4 THE MULTIPLE CURVE AFFINE LIBOR MODEL We descrbe net the affne LIBOR model for the multple curve nterest rate settng and analyze ts man propertes In partcular, we show that t satsfes the modelng requrements M M3 presented above and that t s analytcally tractable In ths framework, OIS forward rates and LIBOR rates are modeled n the sprt of the affne LIBOR model ntroduced by Keller-Ressel et al 23 Let X be an affne process defned on Ω, F, F, IP N, satsfyng Assumpton A and startng at the canoncal value Consder a fed X and the assocated tenor structure T We construct two famles of parametrzed martngales followng Lemma 23: take two sequences of vectors u k k K and v k k K, and defne the IP N -martngales M u k and M v k va and M u k t = ep φ TN tu k + ψ TN tu k, X t, 4 M v k t = ep φ TN tv k + ψ TN tv k, X t 42 The multple curve affne LIBOR model postulates that the OIS and the LIBOR rates assocated wth the -tenor evolve accordng to + δ Fk t = M u k t M u k t for every k = 2,, N and t, T k ] and + δ L kt = M v k t M u k t, 43

10 8 In the followng three propostons, we show how to construct a multple curve affne LIBOR model from any gven ntal term structure of OIS and LIBOR rates Proposton 4 Consder the tme structure T, let B, T l, l K, be the ntal term structure of non-negatve OIS dscount factors and assume that Then the followng statements hold: B, T B, T N If γ X > B, T /B, T N, then there ests a decreasng sequence u u 2 u N = n I T R d, such that M u l = B, T l B, T N for all l K 44 In partcular, f γ X =, the multple curve affne LIBOR model can ft any ntal term structure of OIS rates 2 If X s one-dmensonal, the sequence u l l K s unque 3 If all ntal OIS rates are postve, the sequence u l l K s strctly decreasng Proof See Proposton 6 n Keller-Ressel et al 23 After fttng the ntal term structure of OIS dscount factors, we want to ft the ntal term structure of LIBOR rates, whch s now tenor-dependent Thus, for each k K, we set u k := u l, 45 where l K s such that T l = T k ; see Remark 32 In general, we have that l = kt /T, whle n the settng of Eample 3 we smply have l = k, e u k = u k Proposton 42 Consder the settng of Proposton 4, the fed X and the correspondng tenor structure T Let L k, k K, be the ntal term structure of non-negatve LIBOR rates and assume that for every k K The followng statements hold: L k δ B, T k B, T k = Fk 46 If γ X > ma k K +δ L k B, T k /B, T N, then there ests a sequence v, v 2,, v N = n I T R d, such that v k u k and M v k = + δ L k+m u k+, for all k K \{N } 47 In partcular, f γ X =, then the multple curve affne LIBOR model can ft any ntal term structure of LIBOR rates 2 If X s one-dmensonal, the sequence vk k K s unque 3 If all ntal spreads are postve, then vk > u k, for all k K \{N } Proof Smlarly to the prevous proposton, by fttng the ntal LIBOR rates we obtan a sequence vk k K whch satsfes 3 The nequalty v k u k follows drectly from 46 Proposton 43 Consder the settng of the prevous propostons Then we have:

11 9 = u N u N v N u k v k u k u v = u N u N u k u v v N v k FIGURE 43 Two possble orderngs of u k and v k F k and L k are IP k -martngales, for every k K 2 L k t F k t, for every k K, t, T k ] Proof Snce M u k and M vk are IPN -martngales and the densty process relatng the measures IP N and IP k s provded by dip we get from 43 Smlarly, k Bt, T N = M u k t Ft = B, T N Bt, Tk dip N B, Tk M u k, 48 + δ F k MIP k because + δ F k M u k = M u k MIPN 49 + δ L k MIP k because + δ L km u k = M v k MIPN 4 The monotoncty of the sequence u k together wth 2 yelds that M u k M u k Moreover, from the nequalty vk u k together wth 2 agan, t follows that M v k M u k, for all k K Hence, + δ L k + δ Fk Therefore, the OIS forward rates, the LIBOR rates and the correspondng spreads are non-negatve IP k -martngales Remark 44 Let us now look more closely at the relatonshp between the sequences vk and u k Propostons 4 and 42 mply that u k u k and v k u k for all k K However, we do not know the orderng of vk and u k, or whether the sequence v k s monotone or not The market data for LIBOR spreads ndcate that n a normal market stuaton vk u k, u k ] More precsely, on the one sde, we have vk u k because the LIBOR spreads are nonnegatve On the other sde, f v k > u k, then the LIBOR rate would be more than two tmes hgher than the OIS rate spannng an nterval twce as long, startng at the same date Ths contradcts normal market behavor, hence vk u k, u k ] and consequently the sequence vk wll also be decreasng Ths orderng of the parameters v k and u k s llustrated n Fgure 43 top graph However, the normal market stuaton alternates wth an etreme stuaton, where the spread s hgher than the OIS rate In the bottom graph of Fgure 43 we plot another possble orderng of the parameters vk and u k correspondng to such a case of very hgh spreads Intutvely speakng, the value of the correspondng model spread depends on the dstance between the parameters vk and u k, although n a non-lnear fashon The net result clarfes an mportant property of the multple curve affne LIBOR model, namely ts analytcal tractablty n the sense that the model structure s preserved under dfferent forward measures More precsely, the process X remans affne under any forward measure, although ts characterstcs

12 are now tme-dependent We refer to Flpovć 25 for tme-nhomogeneous affne processes Ths property plays a crucal role n the dervaton of tractable prcng formulas for nterest rate dervatves n the forthcomng sectons Proposton 45 The process X s a tme-nhomogeneous affne process under the measure IP k, for every X and k K In partcular IE k ] e w,x t = ep φ k, t w + ψ k, t w, X, 4 where for every w I k, wth φ k, t w := φ t ψtn tu k + w φ t ψtn tu k, 42a ψ k, t w := ψ t ψtn tu k + w ψ t ψtn tu k, 42b I k, := { w R d : ψ TN tu k + w I T } 43 Proof Usng the densty process between the forward measures, see 48, we have that e w,x t ] ] Fs = IEN e w,xt M u k t /M u k s Fs IE k ep φ TN tu k + ψ TN tu k + w, X t Fs ]/M u k s = ep φ TN tu k φ TN su k + φ t s ψ TN tu k + w ep ψ t s ψ TN tu k + w ψ TN su k, X s, 44 = IE N where the above epectaton s fnte for every w I k, ; recall 2 Ths shows that X s a tmenhomogeneous affne process under IP k, whle 4 follows by substtutng s = n 44 and usng the flow equatons 25 Remark 46 Sngle curve and determnstc spread The multple curve affne LIBOR model easly reduces to ts sngle curve counterpart cf Keller-Ressel et al 23 by settng vk = u k for all X and k K Another nterestng queston s whether the spread can be determnstc or, smlarly, whether the LIBOR rate can be a determnstc transformaton of the OIS rate Consder, for eample, a 2-dmensonal drvng process X = X, X 2 where X s an arbtrary affne process and X 2 the constant process e X 2 t X 2 Then, by settng where u,k, v 2,k > we arrve at u k = u,k, and v k = u,k, v 2,k + δ L kt = + δ F k t e v 2,k X2 Therefore, the LIBOR rate s a determnstc transformaton of the OIS rate, although the spread Sk as defned n 35 s not determnstc In that case, the multplcatve spread Rk defned n 36 s obvously determnstc

13 Remark 47 The multple curve affne LIBOR model fulflls requrements M M3, whch are consstent wth the typcal market observatons of nonnegatve nterest rates and nonnegatve spreads However, negatve rates and negatve spreads have been observed for short tme ntervals, n partcular when a tenor of one month s consdered for the LIBOR Snce these occurrences are lmted to spot rates of the shortest avalable tenor and occur for perods of a sngle day usually, we consder them to be of perpheral nterest for our modelng framework Nevertheless, negatve nterest rates and spreads can be easly accommodated n ths setup by consderng, for eample, affne processes on R d nstead of R d or shfted postve affne processes where suppx a, d wth a < 5 CONNECTION TO LIBOR MARKET MODELS In ths secton, we wll clarfy the relatonshp between the affne LIBOR models and the classcal LIBOR market models, cf Sandmann, Sondermann, and Mltersen 995 and Brace, Ga tarek, and Musela 997 More precsely, we wll embed the multple curve affne LIBOR model n the framework of Jamshdan 997 and derve the dynamcs of OIS forward rates and LIBOR rates We wll concentrate on affne dffuson processes for the sake of smplcty, n order to epose the deas wthout too many techncal detals The generalzaton to affne processes wth jumps s straghtforward and left to the nterested reader An affne dffuson process on the state space D = R d s the soluton X = X of the SDE dx t = b + BX t dt + σx t dw t, X =, 5 where W = W N s a d-dmensonal IP N -Brownan moton The coeffcents b, B = β,, β d and σ have to satsfy the admssblty condtons for affne dffusons on R d, see Flpovć 29, Ch That s, the drft vectors satsfy b R d, β R and β j R for all, j d, 52 whle the dffuson matr σ satsfes σσ T = d α, 53 where α are symmetrc, postve semdefnte matrces for all d, such that = α R and α jk = for all, j, k d 54 Therefore, the affne dffuson process X s componentwse descrbed by dx t = b + BX t dt + X t σ T dw t, 55 for all =,, d, where σ = α e wth e the unt vector 5 OIS dynamcs We start by computng the dynamcs of OIS forward rates As n the prevous secton, we consder a fed X and the assocated tenor structure T Usng the structure of the IP N -martngale M u k n 4, we have that dm u k t = M u k t ψ TN tu k T dx t + dt 56

14 2 Hence, applyng Itô s product rule to 43 and usng 56 yelds that dfk t = d M u k t δ M u k t = M u k t ψtn tu δ k ψ M u TN tu k T dxt + dt k t = δ + δ F k t ψ TN tu k ψ TN tu k T dxt + dt Therefore, the OIS rates satsfy the followng SDE df k t F k t = + δ F k t δ F k t ψtn tu k ψ TN tu k T dxt + dt 57 for all k = 2,, N Now, usng the dynamcs of the affne process X from 55 we arrve at dfk t Fk t = + δ Fk t δ Fk t = + δ F k t δ F k t d = d ψ TN tu k ψt N tu k dxt + dt = ψ TN tu k ψ T N tu k X t σ T dw N t + dt =: Γ T,kt dw N t + dt, 58 where we defne the volatlty structure Γ,k t = + δ F k t δ F k t d ψ TN tu k ψt N tu k X t σ R d 59 = On the other hand, we know from the general theory of dscretely compounded forward rates cf Jamshdan 997 that the OIS forward rate should satsfy the followng SDE under the termnal measure IP N dfk t N Fk t = l=k+ δ Fl t + δ F,ltΓ l,k t dt + Γ T,kt dw N tγt t, 5 for the volatlty structure Γ,k gven n 59 Therefore, we get mmedately that the IP k -Brownan moton W,k s related to the termnal Brownan moton W N va the equalty W,k := W N = W N N l=k+ N d δ F l t + δ F l tγ,lt dt l=k+ = ψ TN tu l ψ T N tu l X t σ dt 5

15 Moreover, the dynamcs of X under IP k take the form dxt = b + BX t dt + Xt σ T dw,k t N d + σ T X t ψ j T N t u l ψ j T N t u l X j t σ j dt = l=k+ j= b + BX t + N l=k+ ψ TN tu l ψ T N tu l X t σ 2 + X t σ T dw,k t, 52 for all =,, d The last equaton provdes an alternatve proof to Proposton 45 n ths settng, snce t shows eplctly that X s a tme-nhomogeneous affne dffuson process under IP k One should also note from 5, that the dfference between the termnal and the forward Brownan moton does not depend on other forward rates as n classcal LIBOR market models The same property s shared by forward prce models, see eg Eberlen and Kluge 27 Thus, we arrve at the followng IP k-dynamcs for the OIS forward rates df k t F k t = ΓT,kt dw,k t 53 wth the volatlty structure Γ,k provded by 59 The structure of Γ,k shows that there s a bult-n shft n the model, whereas the volatlty structure s determned by ψ and σ dt 3 52 LIBOR dynamcs Net, we derve the dynamcs of the LIBOR rates assocated to the same tenor Usng 43, 42 and repeatng the same steps as above, we obtan the followng dl k t L k t = = δ L k tdm M δ L k t v k t M u k t vk t M u k t = + δ L k t δ L k t d = ψtn tv k ψ TN tu k T dxt + dt ψ TN tv k ψ T N tu k X t σ T dw N t + dt, for all k = 2,, N Smlarly to 59 we ntroduce the volatlty structure Λ,k t := + δ L k t d ψ δ L k t TN tvk ψ T N tu k X t σ R d, 54 = and then obtan for L k the followng IP k -dynamcs dl k t L k t = ΛT,kt dw,k t, 55 where W,k s the IP k -Brownan moton gven by 5, whle the dynamcs of X are provded by 52

16 4 53 Spread dynamcs Usng that Sk = L k F k, the dynamcs of LIBOR and OIS rates under the forward measure IP k n 53 and 55, as well as the structure of the volatltes n 59 and 54, after some straghtforward calculatons we arrve at { dsk t = Sk tυ T t vk, u k + + δ Fk t } Υ T t v δ k, u k dw,k t, where Υ t w, y := d ψ TN tw ψt N ty X t σ 56 = 54 Instantaneous correlatons The dervaton of the SDEs that OIS and LIBOR rates satsfy allows to provde quckly formulas for varous quanttes of nterest, such as the nstantaneous correlatons between OIS and LIBOR rates or LIBOR rates wth dfferent maturtes or tenors We have, for eample, that the nstantaneous correlaton between the LIBOR rates maturng at Tk and T l s heurstcally descrbed by therefore we get that Corr t L k, L l = Corr t L k, L l ] = dl k t dl k t L k t L k t dl l t L l t dl k t dl l t dl L k t L l t l t L l t ] 55 = ΛT,k Λ,l Λ,k Λ,l ψ TN t v k ψ TN t u k ψt N t v l ψ TN t u l X σ 2 d = ψ TN t v k ψ TN t u k 2 X σ 2 d v l ψ TN t u l 2 X σ 2 d = = ψ TN t Smlar epressons can be derved for other nstantaneous correlatons, eg ] Corr t F k, L k or Corr t L k, ] L 2 k Instantaneous correlatons are mportant for descrbng the nstantaneous nterdependences between dfferent LIBOR rates In the LIBOR market model for nstance, the rank of the nstantaneous correlaton matr determnes the number of factors eg Brownan motons that s needed to drve the model Eplct epressons for termnal correlatons between LIBOR rates are provded n Append A 6 VALUATION OF SWAPS AND CAPS 6 Interest rate and bass swaps We start by presentng a fed-for-floatng payer nterest rate swap on a notonal amount normalzed to, where fed payments are echanged for floatng payments lnked to the LIBOR rate The LIBOR rate s set n advance and the payments are made n arrears The swap s ntated at tme Tp, where X and p K The collecton of payment

17 dates s denoted by Tpq := {Tp+ < < Tq }, and the fed rate s denoted by K Then, the tme-t value of the swap, for t Tp, s gven by S t K, T pq = q k=p+ = δ q k=p+ Thus, the far swap rate K t T pq s gven by K t T pq = ] δ Bt, Tk IE k LT k, Tk K F t Bt, T k L kt K 6 q k=p+ Bt, T k L k t q k=p+ Bt, T k 62 5 Bass swaps are new products n nterest rate markets, whose value reflects the dscrepancy between the LIBOR rates of dfferent tenors A bass swap s a swap where two streams of floatng payments lnked to the LIBOR rates of dfferent tenors are echanged For eample, n a 3m 6m bass swap, a 3m-LIBOR s pad receved quarterly and a 6m-LIBOR s receved pad semannually We assume n the sequel that both rates are set n advance and pad n arrears; of course, other conventons regardng the payments on the two legs of a bass swap also est A more detaled account on bass swaps can be found n Mercuro 2b, Secton 52 or n Flpovć and Trolle 23, Secton 24 and Append F Note that n the pre-crss setup the value of such a product would have been zero at any tme pont, due to the no-arbtrage relaton between the LIBOR rates of dfferent tenors; see eg Crépey, Grbac, and Nguyen 22 Let us consder a bass swap assocated wth two tenor structures denoted by T pq := {T < < T q } and T 2 pq := {T 2 < < T 2 q 2 }, where T = T 2, T q = T 2 q 2 and T 2 pq T pq The notonal amount s agan assumed to be and the swap s ntated at tme T, whle the frst payments are due at tmes T + and T 2 + respectvely The bass swap spread s a fed rate S, whch s added to the payments on the shorter tenor length More precsely, for the -tenor, the floatng nterest rate LT, T at tenor date T s replaced by LT, T + S, for every { +,, q } The tme-t value of such an agreement s gven, for t T = T 2, by BS t S, T pq, T 2 pq = q 2 = + = δ 2 Bt, T 2 IE 2 LT 2, T ] 2 F t q = + q 2 = + δ Bt, T IE LT, T ] + S F t δ 2 Bt, T 2 L 2 t q = + δ Bt, T L t + S 63 We also want to compute the far bass swap spread S t T pq, T 2 pq Ths s the spread that makes the value of the bass swap equal zero at tme t, e t s obtaned by solvng BS t S, T pq, T 2 pq = We

18 6 get that S t T pq, T 2 pq = q2 = + δ 2 Bt, T 2 L 2 t q = + δ Bt, T L t q = + δ Bt, T 64 The formulas for the far swap rate and bass spread can be used to bootstrap the ntal values of LIBOR rates from market data, see Mercuro 2b, Caps The valuaton of caplets, and thus caps, n the multple curve affne LIBOR model s an easy task, whch has complety equal to the complety of the valuaton of caplets n the sngle-curve affne LIBOR model There are two reasons for ths: on the one hand, the LIBOR rate s modeled drectly compare eg wth Mercuro 2a where the LIBOR rate s modeled mplctly as the sum of the OIS rate and the spread On the other hand, the drvng process remans affne under any forward measure, cf Proposton 45, whch allows the applcaton of Fourer methods for opton prcng Proposton 6 Consder an -tenor caplet wth strke K that pays out δ LTk, T k K+ at Tk The tme- prce s provded by C K, T k = B, T k 2π R K R+w Θ W k R w dw, 65 R wr w for R, Ĩk,, where K = + δ K, Θ W k s gven by 67, whle the set Ĩk, s defned as } Ĩ k, = {z R : zψ TN T k uk + zψ TN T k vk I T Proof Usng 33 and 43 the tme- prce of the caplet equals C K, Tk = δ B, Tk IE k LT k, Tk K +] = δ B, Tk IE k L k Tk K +] + ] = B, Tk IE k M v k T /M u k k T K k + ] = B, Tk IE k e W k K, where Wk = log M v k T k /M u k Tk = φ TN T k v k φ TN T k u k + ψ TN T k v k ψ TN T k u k, X T k =: A + B, X T k 66 Now, usng Eberlen, Glau, and Papapantoleon 2, Thm 22, E 5, we arrve drectly at 65, where Θ W k denotes the IP k -moment generatng functon of the random varable W k, e for

19 7 z Ĩk,, ] Θ W k z = IE k e zwk = IE k ep za + B, XT ] k = ep za + φ k, T zb + ψ k, k T X k zb, 67 The last equalty follows from Proposton 45, notng that z Ĩk, mples zb I k, 7 VALUATION OF SWAPTIONS AND BASIS SWAPTIONS Ths secton s devoted to the prcng of optons on nterest rate and bass swaps, n other words, to the prcng of swaptons and bass swaptons In the frst part, we provde general epressons for the valuaton of swaptons and bass swaptons makng use of the structure of multple curve affne LIBOR models In the followng two parts, we derve effcent and accurate appromatons for the prcng of swaptons and bass swaptons by further utlzng the model propertes n partcular, the preservaton of the affne structure under forward measures and applyng the lnear boundary appromaton developed by Sngleton and Umantsev 22 Let us consder frst a payer swapton wth strke rate K and eercse date Tp on a fed-for-floatng nterest rate swap startng at Tp and maturng at T q ; ths was defned n Secton 6 A swapton can be regarded as a sequence of fed payments δ K T p Tpq K + that are receved at the payment dates Tp+,, Tq ; see Musela and Rutkowsk 25, Secton 32, p 524 Here K Tp T pq s the swap rate of the underlyng swap at tme Tp, cf 62 Note that the classcal transformaton of a payer resp recever swapton nto a put resp call opton on a coupon bond s not vald n the multple curve setup, snce LIBOR rates cannot be epressed n terms of zero coupon bonds; see Remark 35 The value of the swapton at tme t T p s gven by S + t K, T pq = Bt, T p Therefore, at tme t = we have S + K, T pq = B, T p IE p q =p+ = B, T p IE p q =p+ δ IE p BT p, T K T p T pq K + Ft ] δ BT p, T K T p T pq K + q =p+ δ L T p BT p, T ] q =p+ δ KBT p, T + ] snce the swap rate K T p Tpq s gven by 62 for t = Tp Usng 32, 43 and a telescopng product, we get that / BTp, T = M u u Tp M p T p

20 8 Moreover, usng 43 and 48, we obtan that where K := + δ K S + K, T pq = B, T p IE p q =p+ = B, T N IE N q =p+ M v T p M u p T p M v T p q =p+ q =p+ + M u Tp K M u p Tp + ] K M u T p, 7 Net, we move on to the prcng of bass swaptons A bass swapton s an opton to enter a bass swap wth spread S We consder a bass swap as defned n secton 6, whch starts at T = T 2 and ends at T q = T 2 q 2, whle we assume that the eercse date s T The payoff of a bass swap at tme T s gven by 63 for t = T Therefore, the prce of a bass swapton at tme t = s provded by BS + S, T pq, T 2 pq = B, T IE q = + q2 = + Along the lnes of the dervaton for swaptons and usng M u 2 T 2 BS + S, T pq, T 2 pq = where S := δ S = B, T IE q = + M v T = B, T N IE N q2 = + q2 = + /M u T M v 2 T 2 δ 2 L 2 T 2 BT 2, T 2 δ L T + S BT, T M v 2 T 2 /M u 2 T 2 S M u T M u 2 T 2 + ] = M u T cf 45, we arrve at /M u M u 2 T 2 q = + T /M u + ] M v T 2 T 2 S M u + ] T, 72 7 Appromaton formula for swaptons We wll now derve an effcent appromaton formula for the prcng of swaptons The man ngredents n ths formula are the affne property of the drvng process under forward measures and the lnearzaton of the eercse boundary Numercal results for ths appromaton wll be reported n Secton 83

21 We start by presentng some techncal tools and assumptons that wll be used n the sequel We defne the probablty measures IP k, for every k K, va the Radon Nkodym densty dip k Ft = M v k t 73 dip N The process X s obvously a tme-nhomogeneous affne process under every IP k More precsely, we have the followng result whch follows drectly from Proposton 45 Corollary 7 The process X s a tme-nhomogeneous affne process under the measure IP k, for every X, k K, wth IE k ] e w,x t = ep φ k, t w + ψ k, t w, X, 74 M v k 9 where φ k, t w := φ t ψtn tv k + w φ t ψtn tv k, 75a ψ k, t w := ψ t ψtn tv k + w ψ t ψtn tv k, 75b for every w I k, wth I k, := { w R d : ψ TN tv k + w I T } 76 Net, we defne the functon f : R d R by q fy = ep φ TN Tp v + ψ TN Tp v, y =p+ q =p+ K ep φ TN T p u + ψ TN T p u, y 77 Ths functon determnes the eercse boundary for the prce of the swapton, as wll become clear below Snce we cannot compute the characterstc functon of fx T p eplctly, we wll follow Sngleton and Umantsev 22 and appromate f by a lnear functon Appromaton S We appromate fx T p fx T p := A + B, X T p, 78 where the constants A, B are determned accordng to the lnear regresson procedure descrbed n Sngleton and Umantsev 22, pp The lne B, X T p = A appromates the eercse boundary, hence A, B are strke-dependent The followng assumpton wll be used for the prcng of swaptons and bass swaptons Assumpton CD The cumulatve dstrbuton functon of X t s contnuous for all t, T N ] Let Iz denote the magnary part of a comple number z C Now, we state the man result of ths subsecton

22 2 Proposton 72 Assume that A, B are determned by Appromaton S and that Assumpton CD s satsfed The prce of a payer swapton wth strke K, opton maturty Tp, and swap maturty T q, s appromated by S + K, T pq = B, T N K q =p+ q =p+ M v B, T 2 + π 2 + π I ξ z dz z I ζ z dz, 79 z where ζ and ξ are defned by 73 and 74 respectvely Proof Recall that the prce of a swapton s provded by 7 Usng 4 and 42 and the defnton of f n 77, we can epress the swapton prce as follows q ] q S + K, Tpq = B, T N IE N M v T K p M u T p {fxt p } =p+ = B, T N q =p+ IE N =p+ M v T p {fxt p } K q =p+ ] IE N M u T {fx T p }] Moreover, usng the relaton between the termnal measure IP N and the measures IP k and IP k n 48 and 73, we get that S + K, T pq = B, T N q =p+ ] M v IE {fxt p } K q =p+ ] B, T IE {fxt p } 7 In addton, from the nverson formula of Gl-Pelaez 95 and usng Assumpton CD, we get that for each K, where we defne IE {fxt p } ] = 2 + π IE {fxt p } ] = 2 + π Iζ z dz, 7 z Iξ z dz, 72 z ζ z := IE ep zfxt p ] and ξ z := IE ep zfxt p ]

23 However, the above characterstc functons cannot be computed eplctly, n general, thus we wll lnearze the eercse boundary as descrbed by Appromaton S Therefore, we appromate the unknown characterstc functons wth ones that admt an eplct epresson due to the affne property of X under the forward measures Indeed, from 78, Proposton 45 and Corollary 7 we get that ζk z ζ k z := IE k ep z fxt ] p = ep za + φ k, T zb + ψ k, p T zb, X, p 73 ξkz ξ kz := IE k ep z fxt ] p = ep za + φ k, T zb + ψ k, p T zb, X p 74 After nsertng 7 and 72 nto 7 and usng 73 and 74 we arrve at the appromaton formula for swaptons 79 Remark 73 The prcng of swaptons s nherently a hgh-dmensonal problem The epectaton n 7 corresponds to a d-dmensonal ntegral, where d s the dmenson of the drvng process However, the eercse boundary s non-lnear and hard to compute, n general See, eg Brace et al 997, Eberlen and Kluge 26 or Keller-Ressel et al 23, 72, 83 for some eceptonal cases that admt eplct solutons Alternatvely, one could epress a swapton as a zero strke basket opton wrtten on 2q p underlyng assets and use Fourer methods for prcng; see Hubalek and Kallsen 25 or Hurd and Zhou 2 Ths leads to a 2q p-dmensonal numercal ntegraton Instead, the appromaton derved n ths secton requres only the evaluaton of 2q p unvarate ntegrals together wth the computaton of the constants A, B Ths reduces the complety of the problem consderably 2 72 Appromaton formula for bass swaptons In ths subsecton, we derve an analogous appromate prcng formula for bass swaptons Numercal results for ths appromaton wll be reported n secton 84 Smlar to the case of swaptons, we defne the functon gy = q 2 = + q 2 + = + q = + q = + ep φ TN T 2 v 2 + ψ T N T 2 v 2, y ep φ TN T 2 u 2 + ψ TN T 2 u 2, y ep φ TN T v + ψ T N T v, y S ep φ TN T u + ψ TN T u, y, 75 whch determnes the eercse boundary for the prce of the bass swapton Ths wll be appromated by a lnear functon followng agan Sngleton and Umantsev 22

24 22 Appromaton BS We appromate where C and D are determned va a lnear regresson gx T gx T := C + D, X T, 76 Proposton 74 Assume that C, D are determned by Appromaton BS and that Assumpton CD s satsfed The prce of a bass swapton wth spread S, opton maturty T = T 2, and swap maturty T q = T 2 q 2, s appromated by BS + q 2 S, T pq, T 2 pq = B, T N M v I ξ 2 z dz π z = + q 2 B, T I ζ 2 z dz π z = + q B, T N M v 2 + I ξ z dz 77 π z = + q + S B, T 2 + I ζ z dz, π z where ζ l = + and ξ l are defned by 78 and 79 for l =, 2 Proof Usng the defnton of g n 75 we can rewrte the prce of a bass swapton 72 as follows: BS + S, T pq, T 2 pq = { q2 = B, T N = + q = + IE N M v 2 ] T 2 p {gx 2 } ] IE N M u 2 2 T T 2 p {gx 2 } 2 T IE N M v T {gxt } ] S IE N ] } M u T {gxt } Then we follow the same steps as n the prevous secton: Frst, we use the relaton between the termnal measure IP N and the measures IP k, IP k to arrve at an epresson smlar to 7 Second, we appromate g by g n 76 Thrd, we defne the appromate characterstc functons, whch can be computed eplctly: ζ l z := IE l ep z gxt l ] p l = ep zc + φ, l T pl zd + ψ, l T pl zd, X, 78 ξ l z := IE k ep z gxt l ] p l = ep zc + φ, l T pl zd + ψ, l T pl zd, X, 79

25 for l =, 2 Fnally, puttng all the peces together we arrve at the appromaton formula 77 for the prce of a bass swapton 23 8 NUMERICAL EXAMPLES AND CALIBRATION The am of ths secton s twofold: on the one hand, we demonstrate how the multple curve affne LIBOR model can be calbrated to market data and, on the other hand, we test the accuracy of the swapton and bass swapton appromaton formulas We start by dscussng how to buld a model whch can smultaneously ft caplet volatltes when the optons have dfferent underlyng tenors Net, we test numercally the swapton and bass swapton appromaton formulas 79 and 77 usng the calbrated models and parameters In the last subsecton, we buld a smple model and compute eact and appromate swapton and bass swapton prces n a setup whch can be easly reproduced by nterested readers 8 A specfcaton wth dependent rates There are numerous ways of constructng models and the trade-off s usually between parsmony and fttng ablty We opt here for a heavly parametrzed approach that focuses on the fttng ablty, as we beleve t best demonstrates the utlty of our model In addton, t s usually easer to move from a comple specfcaton towards a smpler one, than the converse More precsely, we provde below a model specfcaton where LIBOR rates are drven by common and dosyncratc factors, whch s sutable for sequental calbraton to market data The startng pont s to revst the epresson for the LIBOR rates n 43: + δ L kt = M v k t /M u k t 8 = ep φ TN tvk φ TN tu k + ψ TN tvk ψ TN tu k, X t Accordng to Proposton 42, when the dmenson of the drvng process s greater than one, then the vectors vk and u k are not fully determned by the ntal term structure Therefore, we can navgate through dfferent model specfcatons by alterng the structure of the sequences u k and v k Remark 8 The followng observaton allows to create an eponental lnear factor structure for the LIBOR rates wth as many common and dosyncratc factors as desred Consder an R d -valued affne process and denote the vectors v k, u k Rd by X = X,, X d, 82 v k = v,k,, v d,k and u k = u,k,, u d,k 83 Select a subset J k {,, d}, set v,k = u,k for all J k, and assume that {X } Jk are ndependent of {X j } j {,,d}\jk Then, t follows from 8 and Keller-Ressel 28, Prop 47 that L k wll also be ndependent of {X } Jk and wll depend only on {X j } j {,,d}\jk Let, 2 X and consder the tenor structures T, T 2 where T 2 T The dataset under consderaton contans caplets maturng on M dfferent dates for each tenor, where M s less than

26 24 the number of tenor ponts n T and T 2 In other words, only M maturtes are relevant for the calbraton The dynamcs of OIS and LIBOR rates are drven by tuples of affne processes dx t = λ X t θ dt + 2η X t dw t + dz t, 84 dx c t = λ c X c t θ c dt + 2η c X c t dw c t, 85 for =,, M, where X c denotes the common and X the dosyncratc factor for the -th maturty Here X R, λ, θ, η R for = c,,, M, and W c, W,, W M, are ndependent Brownan motons Moreover, Z are ndependent compound Posson processes wth constant ntensty ν and eponentally dstrbuted jumps wth mean values µ, for =,, M Therefore, the full process has dmenson M + : X = X c, X,, X M 86 The affne processes X c, X,, X M are mutually ndependent hence, usng Proposton 47 n Keller-Ressel 28, the functons φ X c,x, respectvely ψ X c,x, are known n terms of the functons φ X c and φ X, respectvely ψ X c and ψ X, for all {,, M} The latter are provded, for eample, by Grbac and Papapantoleon 23, E 23 In order to create a dagonal plus common factor structure, where each rate for each tenor s drven by the common factor X c and an dosyncratc factor X, we wll use Remark 8 The proposed structures for the u s and v s are descrbed n Fgures 84 and 85, where elements of u below a certan dagonal are coped nto v In partcular, we start from the longest maturty and add one ndependent factor at each caplet maturty date, e at each l k, k =,, M Here l k := k/δ for k =,, M, e ths functon maps caplet maturtes nto tenor ponts Smultaneously, the values of u for the subsequent factor are frozen to the latest-set value The constructon of v s analogous, where now the frozen values are coped from u ; see agan Remark 8 Moreover, all elements n these matrces are non-negatve and u N = v N = R M+ The boed elements are the only ones that matter n terms of prcng caplets, when these are not avalable at every tenor date of T The role of the common factor s determned by the dfference between ṽ k and ũ k If we set ṽ k = ũ k, t follows from 8 that L k wll be ndependent of the common factor X c and thus determned solely by the correspondng dosyncratc factor X, wth k = l If the values of ṽ k and ũ k are fed a pror, the remanng values ū k k=,,n and v k k=,,n are determned unquely by the ntal term structure of OIS and LIBOR rates; see agan Propostons 4 and 42 Ths model structure s consstent wth v k u k u k f and only f the sequences ũ and ū are decreasng, ṽ k ũ k and v k ū k for every k =,, N Moreover, ths structure wll be consstent wth the normal market stuaton descrbed n Remark 44 f, n addton, ṽ k ũ k, ũ k ] and v k ū k, ū k ] for every k =,, N The correspondng matrces for the 2 tenor are constructed n a smlar manner More precsely, u 2 s constructed by smply copyng the relevant rows from u Smultaneously, for v 2 the elements v 2 k k=,,n 2 are ntroduced n order to ft the 2 ntal LIBOR term structure, as well as the elements ṽ 2 k k=,,n 2 whch determne the role of the common factor We present only four rows from these matrces n Fgures 86 and 87, for the sake of brevty 82 Calbraton to caplet data The data we use for calbraton are from the EUR market on 27 May 23 collected from Bloomberg Bloomberg provdes synthetc zero coupon bond prces for EURIBOR

27 25 u l M = ũ l M ū l M u l M = ũ l M ū u l M 2 = ũ l M 2 ū u l M 3 = ũ l M 3 ū u l M = ũ l M ū l M ū u l M = ũ l M ū l M ū u l M 2 = ũ l M 2 ū l M 2 ū u l M 3 = ũ l M 3 ū l M 3 ū l M l M 2 l M 3 l M 3 l M 3 l M 3 l M 3 u l M 2 = ũ l M 2 ū l M 2 ū l M 3 ū l M 3 u l = u = ũ l ū l ū l 2 3 ū l M 2 3 ū l M 3 ū l M 3 ũ ū ū l 2 3 ū l M 2 3 ū l M 3 ū l M 3 FIGURE 84 The sequence u encompasses the proposed dagonal plus common factor structure In ths partcular eample, = 3 months and caplets mature on entre years v l M = ṽ l M v l M v l M = ṽ l M v v l M 2 = ṽ l M 2 v v l M 3 = ṽ l M 3 v v l M = ṽ l M v l M ū v l M = ṽ l M v l M ū v l M 2 = ṽ l M 2 v l M 2 ū v l M 3 = ṽ l M 3 v l M 3 ū v l M 2 = ṽ l M 2 v l M 2 ū l M 3 v l M l M 2 l M 3 l M 3 l M 3 l M 3 l M 3 l M 3 v l M 2 = ṽ l M 2 v l M 2 ū l M 3 v l M 3 v l = v l = v = ṽ l v l ū l 2 3 ū l M 2 3 ū l M 3 ū l M 3 ṽ l v l ū l 2 3 ū l M 2 3 ū l M 3 ū l M 3 ṽ v ū l 2 3 ū l M 2 3 ū l M 3 ū l M 3 FIGURE 85 The sequence v s constructed analogously to u In ths partcular eample, = 3 months and caplets mature on entre years

28 26 u 2 l 2 M = ũ l M ū l M u 2 l 2 M = ũ l M 2 ū u 2 l 2 M = ũ l M ū l M ū l M 2 l M 3 u 2 l 2 M = ũ l M 2 ū l M 2 ū l M 3 FIGURE 86 The frst four rows of u 2 In ths partcular eample, 2 = 6 months and caplets mature on entre years v 2 l 2 M = ṽ 2 l 2 M v 2 l 2 M v 2 l 2 M = ṽ 2 l 2 M v 2 v 2 l 2 M = ṽ 2 l 2 M v 2 l 2 M ū l 2 M l M 3 v 2 l 2 M = ṽ 2 l 2 M v 2 l 2 M ū l M 3 FIGURE 87 The frst four rows of v 2 In ths partcular eample, 2 = 6 months and caplets mature on entre years EONIA 3m EURIBOR 6m EURIBOR Intal Zero Coupon Rate FIGURE 88 Zero coupon rates, EUR market, 27 May 23 rates, as well as OIS rates constructed n a manner descrbed n Akkara 22 In our eample, we wll focus on the 3 and 6 month tenors only The zero coupon bond prces are converted nto zero coupon rates and plotted n Fgure 88 Cap prces are converted nto caplet mpled volatltes usng the algorthm descrbed n Levn 22 The mpled volatlty s calculated usng OIS dscountng when nvertng the Black 976 formula The caplet data we have at our dsposal correspond to 3- and 6- month tenor structures More precsely, n the EUR market caps wrtten on the 3-month tenor are quoted only up to a maturty of 2 years, whle 6-month tenor caps are quoted from maturty 3 years and onwards Moreover, we have opton prces only for the maturtes correspondng to entre years and not for every tenor pont We have a fed grd of 4 strkes rangng from % to % We calbrate to caplet data for maturtes, 2,, years and the OIS zero coupon bond B, 5 defnes the termnal measure We f n advance the values of the parameters ũ l and ṽ l, l =, 2, as well We found that the model performs slghtly better n calbraton usng ths numerare, than when choosng years as the termnal maturty