Two Curves, One Price :Pricing & Hedging Interest Rate Derivatives Decoupling Forwarding and Discounting Yield Curves

Transcription

1 MPRA Munich Personal RePEc Archive Two Curves, One Price :Pricing & Heging Ineres Rae Derivaives Decoupling Forwaring an Discouning Yiel Curves Marco Bianchei Banca Inesasanpaolo 14. November 2008 Online a hp://mpra.ub.uni-muenchen.e/22022/ MPRA Paper No , pose 10. April :59 UTC

2 Two Curves, One Price: Pricing & Heging Ineres Rae Derivaives Decoupling Forwaring an Discouning Yiel Curves Marco Bianchei Firs version: 14 Nov. 2008, his version: 24 Jan Absrac We revisi he problem of pricing an heging plain vanilla single-currency ineres rae erivaives using muliple isinc yiel curves for marke coheren esimaion of iscoun facors an forwar raes wih ifferen unerlying rae enors. Wihin such ouble-curve-single-currency framework, aope by he marke afer he crei-crunch crisis sare in summer 2007, sanar single-curve noarbirage relaions are no longer vali, an can be recovere by aking properly ino accoun he forwar basis boosrappe from marke basis swaps. Numerical resuls show ha he resuling forwar basis curves may isplay a richer microerm srucure ha may inuce appreciable effecs on he price of ineres rae insrumens. By recurring o he foreign-currency analogy we also erive generalise noarbirage ouble-curve marke-like formulas for basic plain vanilla ineres rae erivaives, FRAs, swaps, caps/floors an swapions in paricular. These expressions inclue a quano ajusmen ypical of cross-currency erivaives, naurally originae by he change beween he numeraires associae o he wo yiel curves, ha carries on a volailiy an correlaion epenence. Numerical scenarios confirm ha such correcion can be non negligible, hus making unajuse ouble-curve prices, in principle, no arbirage free. Boh he forwar basis an he quano ajusmen fin a naural financial explanaion in erms of counerpary risk. JEL Classificaions: E43, G12, G13. Risk Managemen, Marke Risk, Pricing an Financial Moeling, Banca Inesa Sanpaolo, piazza P. Ferrari 10, Milan, Ialy, marco.bianchei(a)inesasanpaolo.com. The auhor acknowleges fruiful iscussions wih M. De Prao, M. Henrar, M. Joshi, C. Maffi, G. V. Mauri, F. Mercurio, N. Moreni, colleagues in he Risk Managemen an paricipans a Quan Congress Europe A paricular menion goes o M. Morini an M. Pucci for heir encouragemen, an o F. M. Amerano an he QuanLib communiy for he open-source evelopmens use here. The views expresse here are hose of he auhor an o no represen he opinions of his employer. They are no responsible for any use ha may be mae of hese conens. 1

3 Keywors: liquiiy, crisis, counerpary risk, yiel curve, forwar curve, iscoun curve, pricing, heging, ineres rae erivaives, FRAs, swaps, basis swaps, caps, floors, swapions, basis ajusmen, quano ajusmen, measure changes, no arbirage, QuanLib. Conens 1 Inroucion 2 2 Noaion an Basic Assumpions 4 3 Pre an Pos Crei Crunch Marke Pracices for Pricing an Heging Ineres Rae Derivaives Single-Curve Framework Muliple-Curve Framework No Arbirage an Forwar Basis 9 5 Foreign-Currency Analogy an Quano Ajusmen Forwar Raes Swap Raes Double-Curve Pricing & Heging Ineres Rae Derivaives Pricing Heging No Arbirage an Counerpary Risk 25 8 Conclusions 25 1 Inroucion The crei crunch crisis sare in he secon half of 2007 has riggere, among many consequences, he explosion of he basis spreas quoe on he marke beween single-currency ineres rae insrumens, swaps in paricular, characerise by ifferen unerlying rae enors (e.g. Xibor3M 1, Xibor6M, ec.). In fig. 1 we show a snapsho of he marke quoaions as of Feb. 16h, 2009 for he six basis swap erm srucures corresponing o he four Euribor enors 1M, 3M, 6M, 12M. As one can see, in he ime inerval 1Y 30Y he basis spreas are monoonically ecreasing from 80 o aroun 2 basis poins. Such very high basis reflec he higher liquiiy risk suffere by financial insiuions an he corresponing preference for receiving paymens wih higher frequency (quarerly insea of semi-annually, ec.). 1 We enoe wih Xibor a generic Inerbank Offere Rae. In he EUR case he Euribor is efine as he rae a which euro inerbank erm eposis wihin he euro zone are offere by one prime bank o anoher prime bank (see 2

4 basis sprea (bps) EUR Basis swaps 3M vs 6M 1M vs 3M 1M vs 6M 6M vs 12M 3M vs 12M 1M vs 12M Y 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y 10Y 11Y 12Y 15Y 20Y 25Y 30Y Figure 1: quoaions (basis poins) as of Feb. 16h, 2009 for he six EUR basis swap curves corresponing o he four Euribor swap curves 1M, 3M, 6M, 12M. Before he crei crunch of Aug he basis spreas were jus a few basis poins (source: Reuers ICAPEUROBASIS). There are also oher inicaors of regime changes in he ineres rae markes, such as he ivergence beween eposi (Xibor base) an OIS 2 (Eonia 3 base for EUR) raes, or beween FRA 4 conracs an he corresponing forwar raes implie by consecuive eposis (see e.g. refs. [AB09], [Mer09], [Mor08], [Mor09]). These fricions have hus inuce a sor of segmenaion of he ineres rae marke ino sub-areas, mainly corresponing o insrumens wih 1M, 3M, 6M, 12M unerlying rae enors, characerize, in principle, by ifferen inernal ynamics, liquiiy an crei risk premia, reflecing he ifferen views an ineress of he marke players. Noice ha marke segmenaion was alreay presen (an well unersoo) before he crei crunch (see e.g. ref. [TP03]), bu no effecive ue o negligible basis spreas. Such evoluion of he financial markes has riggere a general reflecion abou he mehoology use o price an hege ineres rae erivaives, namely hose financial insrumens whose price epens on he presen value of fuure ineres rae-linke cashflows. In his paper we acknowlege he curren marke pracice, assuming he exisence of a given mehoology (iscusse in eail in ref. [AB09]) for boosrapping muliple homogeneous forwaring an iscouning curves, characerize by ifferen unerlying rae enors, an we focus on he consequences for pricing an heging ineres rae eriva- 2 Overnigh Inexe Swaps. 3 Euro OverNigh Inex Average, he rae compue as a weighe average of all overnigh raes corresponing o unsecure lening ransacions in he euro-zone inerbank marke (see e.g. 4 Forwar Rae Agreemen. 3

5 ives. In paricular in sec. 3 we summarise he pre an pos crei crunch marke pracices for pricing an heging ineres rae erivaives. In sec. 2 we fix he noaion, we revisi some general concep of sanar, no arbirage single-curve pricing an we formalize he ouble-curve pricing framework, showing how no arbirage is broken an can be formally recovere wih he inroucion of a forwar basis. In sec. 5 we use he foreign-currency analogy o erive a single-currency version of he quano ajusmen, ypical of cross-currency erivaives, naurally appearing in he expecaion of forwar raes. In sec. 6 we erive he no arbirage ouble-curve marke-like pricing expressions for basic single-currency ineres rae erivaives, such as FRA, swaps, caps/floors an swapions. Conclusions are summarise in sec. 8. The opic iscusse here is a cenral problem in he ineres rae marke, wih many consequences in raing, financial conrol, risk managemen an IT, which sill lacks of aenion in he financial lieraure. To our knowlege, similar opics have been approache in refs. [FZW95], [BS05], [KTW08], [Mer09], [Hen09]an [Mor08], [Mor09]. In paricular W. Boenkos an W. Schmi [BS05] iscuss wo mehoologies for pricing cross-currency basis swaps, he firs of which (he acual pre-crisis common marke pracice), oes coincie, once reuce o he single-currency case, wih he ouble-curve pricing proceure escribe here 5. Recenly M. Kijima e al. [KTW08] have exene he approach of ref. [BS05] o he (cross currency) case of hree curves for iscoun raes, Libor raes an bon raes. Finally, simulaneously o he evelopmen of he presen paper, M. Morini is approaching he problem in erms of counerpary risk [Mor08], [Mor09], F. Mercurio in erms of an exene Libor Marke Moel [Mer09], an M. Henrar using an axiomaic moel [Hen09]. The presen work follows an alernaive roue wih respec o hose cie above, in he sense ha a) we aop a boom-up praciioner s perspecive, saring from he curren marke pracice of using muliple yiel curves an working ou is naural consequences, looking for a minimal an ligh generalisaion of well-known frameworks, keeping hings as simple as possible; b) we show how no-arbirage can be recovere in he ouble-curve approach by aking properly ino accoun he forwar basis, whose erm srucure can be exrace from available basis swap marke quoaions; c) we use a sraighforwar foreign-currency analogy o erive generalise ouble-curve marke-like pricing expressions for basic single-currency ineres rae erivaives, such as FRAs, swaps, caps/floors an swapions. 2 Noaion an Basic Assumpions Following he iscussion above, we enoe wih M x, x = {, f 1,..., f n } muliple isinc ineres rae sub-markes, characerize by he same currency an by isinc bank accouns B x an yiel curves x in he form of a coninuous erm srucure of iscoun facors, x = {T P x ( 0, T ), T 0 }, (1) 5 hese auhors were puzzle by he fac ha heir firs mehoology was neiher arbirage free nor consisen wih he pre-crisis single-curve marke pracice for pricing single-currency swaps. Such objecions have now been overcome by he marke evoluion owars a generalize ouble-curve pricing approach (see also [TP03]). 4

6 where 0 is he reference ae of he curves (e.g. selemen ae, or oay) an P x (, T ) enoes he price a ime 0 of he M x -zero coupon bon for mauriy T, such ha P x (T, T ) = 1. In each sub-marke M x we posulae he usual no arbirage relaion, P x (, T 2 ) = P x (, T 1 ) P x (, T 1, T 2 ), T 1 < T 2 (2) where P x (, T 1, T 2 ) enoes he M x forwar iscoun facor from ime T 2 o ime T 1, prevailing a ime. The financial meaning of expression 2 is ha, in each marke M x, given a cashflow of one uni of currency a ime T 2, is corresponing value a ime < T 2 mus be unique, boh if we iscoun in one single sep from T 2 o, using he iscoun facor P x (, T 2 ), an if we iscoun in wo seps, firs from T 2 o T 1, using he forwar iscoun P x (, T 1, T 2 ) an hen from T 1 o, using P x (, T 1 ). Denoing wih F x (; T 1, T 2 ) he simple compoune forwar rae associae o P x (, T 1, T 2 ), reseing a ime T 1 an covering he ime inerval [T 1 ; T 2 ], we have P x (, T 1, T 2 ) = P x (, T 2 ) P x (, T 1 ) = F x (; T 1, T 2 ) τ x (T 1, T 2 ), (3) where τ x (T 1, T 2 ) is he year fracion beween imes T 1 an T 2 wih aycoun c x, an from eq. 2 we obain he familiar no arbirage expression [ ] 1 1 F x (; T 1, T 2 ) = τ x (T 1, T 2 ) P x (, T 1, T 2 ) 1 = P x (, T 1 ) P x (, T 2 ) τ x (T 1, T 2 ) P x (, T 2 ). (4) Eq. 4 can be also erive (see e.g. ref. [BM06], sec. 1.4) as he fair value coniion a ime of he Forwar Rae Agreemen (FRA) conrac wih payoff a mauriy T 2 given by FRA x (T 2 ; T 1, T 2, K, N) = Nτ x (T 1, T 2 ) [L x (T 1, T 2 ) K], (5) L x (T 1, T 2 ) = 1 P x (T 1, T 2 ) τ x (T 1, T 2 ) P x (T 1, T 2 ) where N is he nominal amoun, L x (T 1, T 2, c x ) is he T 1 -spo Xibor rae for mauriy T 2 an K he (simply compoune) srike rae (sharing he same aycoun convenion for simpliciy). Inroucing expecaions we have, T 1 < T 2, FRA x (; T 1, T 2, K, N) = P x (, T 2 ) E QT 2 x [FRA (T 2 ; T 1, T 2, K, N)] { } = NP x (, T 2 ) τ x (T 1, T 2 ) E QT 2 x [L x (T 1, T 2 )] K (6) = NP x (, T 2 ) τ x (T 1, T 2 ) [F x (; T 1, T 2 ) K], (7) where Q T 2 x enoes he M x -T 2 -forwar measure corresponing o he numeraire P x (, T 2 ), E Q [.] enoes he expecaion a ime w.r.. measure Q an filraion F, encoing he marke informaion available up o ime, an we have assume he sanar maringale propery of forwar raes F x (; T 1, T 2 ) = E QT 2 x [F x (T 1 ; T 1, T 2 )] = E QT 2 x [L x (T 1, T 2 )] (8) 5

7 o hol in each ineres rae marke M x (see e.g. ref. [BM06]). We sress ha he assumpions above imply ha each sub-marke M x is inernally consisen as he whole ineres rae marke before he crisis. This is surely a srong hypohesis, ha coul be relaxe in more sophisicae frameworks. 3 Pre an Pos Crei Crunch Marke Pracices for Pricing an Heging Ineres Rae Derivaives We escribe here he evoluion of he marke pracice for pricing an heging ineres rae erivaives hrough he crei crunch crisis. We use consisenly he noaion escribe above, consiering a general single-currency ineres rae erivaive wih m fuure coupons wih payoffs π = {π 1,..., π m }, wih π i = π i (F x ), generaing m cashflows c = {c 1,..., c m } a fuure aes T = {T 1,..., T m }, wih < T 1 <... < T m. 3.1 Single-Curve Framework The pre-crisis sanar marke pracice can be summarise in he following working proceure (see e.g. refs. [Ron00], [HW06], [An07] an [HW08]): 1. selec one finie se of he mos convenien (i.e. liqui) ineres rae vanilla insrumens rae in real ime on he marke an buil a single yiel curve using he preferre boosrapping proceure; for insance, a common choice in he EUR marke is a combinaion of shor-erm EUR eposis, meium-erm Fuures/FRA on Euribor3M an meium-long-erm swaps on Euribor6M; 2. for each ineres rae coupon i {1,..., m} compue he relevan forwar raes using he given yiel curve as in eq. 4, F (; T i 1, T i ) = P (, T i 1 ) P (, T i ) τ (T i 1, T i ) P (, T i ) T i 1 < T i ; (9) 3. compue cashflows c i as expecaions a ime of he corresponing coupon payoffs π i (F ) wih respec o he T i -forwar measure Q T i, associae o he numeraire P (, T i ) from he same yiel curve, c i = c (, T i, π i ) = E QT i [π i (F )] ; (10) 4. compue he relevan iscoun facors P (, T i ) from he same yiel curve ; 5. compue he erivaive s price a ime as he sum of he iscoune cashflows, π (; T) = m P (, T i ) c (, T i, π i ) = i=1 m i=1 P (, T i ) E QT i [π i (F )] ; (11) 6. compue he ela sensiiviy by shocking one by one he marke pillars of yiel curve an hege he resuling ela risk using he suggese amouns (hege raios) of he same se of vanillas. 6

8 For insance, a 5.5Y mauriy EUR floaing swap leg on Euribor1M (no irecly quoe on he marke) is commonly price using iscoun facors an forwar raes calculae on he same epo-fuures-swap curve cie above. The corresponing ela risk is hege using he suggese amouns (hege raios) of 5Y an 6Y Euribor6M swaps 6. Noice ha sep 3 above has been formulae in erms of he pricing measure Q T i associae o he numeraire P (, T i ). This is convenien in our conex because i emphasizes ha he numeraire is associae o he iscouning curve. Obviously any oher equivalen measure associae o ifferen numeraires may be use as well. We sress ha his is a single-currency-single-curve approach, in ha a unique yiel curve is buil an use o price an hege any ineres rae erivaive on a given currency. Thinking in erms of more funamenal variables, e.g. he shor rae, his is equivalen o assume ha here exis a unique funamenal unerlying shor rae process able o moel an explain he whole erm srucure of ineres raes of all enors. I is also a relaive pricing approach, because boh he price an he hege of a erivaive are calculae relaively o a se of vanillas quoe on he marke. We noice also ha i is no sricly guaranee o be arbirage-free, because iscoun facors an forwar raes obaine from a given yiel curve hrough inerpolaion are, in general, no necessarily consisen wih hose obaine by a no arbirage moel; in pracice bi-ask spreas an ransacion coss hie any arbirage possibiliies. Finally, we sress ha he key firs poin in he proceure is much more a maer of ar han of science, because here is no an unique financially soun recipe for selecing he boosrapping insrumens an rules. 3.2 Muliple-Curve Framework Unforunaely, he pre-crisis approach ouline above is no longer consisen, a leas in is simple formulaion, wih he presen marke coniions. Firs, i oes no ake ino accoun he marke informaion carrie by basis swap spreas, now much larger han in he pas an no longer negligible. Secon, i oes no ake ino accoun ha he ineres rae marke is segmene ino sub-areas corresponing o insrumens wih isinc unerlying rae enors, characerize, in principle, by ifferen ynamics (e.g. shor rae processes). Thus, pricing an heging an ineres rae erivaive on a single yiel curve mixing ifferen unerlying rae enors can lea o iry resuls, incorporaing he ifferen ynamics, an evenually he inconsisencies, of isinc marke areas, making prices an hege raios less sable an more ifficul o inerpre. On he oher sie, he more he vanillas an he erivaive share he same homogeneous unerlying rae, he beer shoul be he relaive pricing an he heging. Thir, by no arbirage, iscouning mus be unique: wo ienical fuure cashflows of whaever origin mus isplay he same presen value; hence we nee a unique iscouning curve. In principle, a consisen crei an liquiiy heory woul be require o accoun for he ineres rae marke segmenaion. This woul also explain he reason why he asymmeries cie above o no necessarily lea o arbirage opporuniies, once counerpary an liquiiy risks are aken ino accoun. Unforunaely such a framework is no easy o consruc (see e.g. he iscussion in refs. [Mer09], [Mor09]). In pracice an empirical 6 we refer here o he case of local yiel curve boosrapping mehos, for which here is no sensiiviy elocalizaion effec (see refs. [HW06], [HW08]). 7

9 approach has prevaile on he marke, base on he consrucion of muliple forwaring yiel curves from plain vanilla marke insrumens homogeneous in he unerlying rae enor, use o calculae fuure cash flows base on forwar ineres raes wih he corresponing enor, an of a single iscouning yiel curve, use o calculae iscoun facors an cash flows presen values. Consequenly, ineres rae erivaives wih a given unerlying rae enor shoul be price an hege using vanilla ineres rae marke insrumens wih he same unerlying rae enor. The pos-crisis marke pracice may hus be summarise in he following working proceure: 1. buil one iscouning curve using he preferre selecion of vanilla ineres rae marke insrumens an boosrapping proceure; 2. buil muliple isinc forwaring curves f1,..., fn using he preferre selecions of isinc ses of vanilla ineres rae marke insrumens, each homogeneous in he unerlying Xibor rae enor (ypically wih 1M, 3M, 6M, 12M enors) an boosrapping proceures; 3. for each ineres rae coupon i {1,..., m} compue he relevan forwar raes wih enor f using he corresponing yiel curve f as in eq. 4, F f (; T i 1, T i ) = P f (, T i 1 ) P f (, T i ) τ f (T i 1, T i ) P f (, T i ), T i 1 < T i ; (12) 4. compue cashflows c i as expecaions a ime of he corresponing coupon payoffs π i (F f ) wih respec o he iscouning T i -forwar measure Q T i, associae o he numeraire P (, T i ), as c i = c (, T i, π i ) = E QT i [π i (F f )] ; (13) 5. compue he relevan iscoun facors P (, T i ) from he iscouning yiel curve ; 6. compue he erivaive s price a ime as he sum of he iscoune cashflows, π (; T) = m P (, T i ) c (, T i, π i ) = i=1 m i=1 P (, T i ) E QT i [π i (F f )] ; (14) 7. compue he ela sensiiviy by shocking one by one he marke pillars of each yiel curve, f1,..., fn an hege he resuling ela risk using he suggese amouns (hege raios) of he corresponing se of vanillas. For insance, he 5.5Y floaing swap leg cie in he previous secion 3.1 is currenly price using Euribor1M forwar raes calculae on he 1M forwaring curve, boosrappe using Euribor1M vanillas only, plus iscoun facors calculae on he iscouning curve. The ela sensiiviy is compue by shocking one by one he marke pillars of boh 1M an curves an he resuling ela risk is hege using he suggese amouns (hege raios) of 5Y an 6Y Euribor1M swaps plus he suggese amouns of 5Y an 6Y insrumens from he iscouning curve (see sec. 6.2 for more eails abou he heging proceure). 8

10 Such muliple-curve framework is consisen wih he presen marke siuaion, bu - here is no free lunch - i is also more emaning. Firs, he iscouning curve clearly plays a special an funamenal role, an mus be buil wih paricular care. This pre-crisis obvious sep has become, in he presen marke siuaion, a very suble an conroversial poin, ha woul require a whole paper in iself (see e.g. ref. [Hen07]). In fac, while he forwaring curves consrucion is riven by he unerlying rae homogeneiy principle, for which here is (now) a general marke consensus, here is no longer, a he momen, general consensus for he iscouning curve consrucion. A leas wo ifferen pracices can be encounere in he marke: a) he ol pre-crisis approach (e.g. he epo, Fuures/FRA an swap curve cie before), ha can be jusifie wih he principle of maximum liquiiy (plus a lile of ineria), an b) he OIS curve, base on he overnigh rae 7 (Eonia for EUR), jusifie wih collaeralize (riskless) counerparies 8 (see e.g. refs. [Ma08], [GS009]). Secon, builing muliple curves requires muliple quoaions: many more boosrapping insrumens mus be consiere (eposis, Fuures, swaps, basis swaps, FRAs, ec., on ifferen unerlying rae enors), which are available on he marke wih ifferen egrees of liquiiy an can isplay ransiory inconsisencies (see [AB09]). Thir, non rivial inerpolaion algorihms are crucial o prouce smooh forwar curves (see e.g. refs. [HW06]-[HW08], [AB09]). Fourh, muliple boosrapping insrumens implies muliple sensiiviies, so heging becomes more complicae. Las bu no leas, pricing libraries, plaforms, repors, ec. mus be exene, configure, ese an release o manage muliple an separae yiel curves for forwaring an iscouning, no a rivial ask for quans, risk managers, evelopers an IT people. The saic muliple-curve pricing & heging mehoology escribe above can be exene, in principle, by aoping muliple isinc moels for he evoluion of he unerlying ineres raes wih enors f 1,..., f n o calculae he fuure ynamics of he yiel curves f1,..., fn an he expece cashflows. The volailiy/correlaion epenencies carrie by such moels imply, in principle, boosrapping muliple isinc variance/covariance marices an heging he corresponing sensiiviies using volailiy- an correlaionepenen vanilla marke insrumens. Such more general approach has been carrie on in ref. [Mer09] in he conex of generalise marke moels. In his paper we will focus only on he basic maer of saic yiel curves an leave ou he ynamical volailiy/correlaion imensions. 4 No Arbirage an Forwar Basis Now, we wish o unersan he consequences of he assumpions above in erms of no arbirage. Firs, we noice ha in he muliple-curve framework classic single-curve no 7 he overnigh rae can be seen as he bes proxy o a risk free rae available on he marke because of is 1-ay enor. 8 collaeral agreemens are more an more use in OTC markes, where here are no clearing houses, o reuce he counerpary risk. The sanar ISDA conracs (ISDA Maser Agreemen an Crei Suppor Annex) inclue neing clauses imposing compensaion. The compensaion frequency is ofen on a aily basis an he (cash or asse) compensaion amoun is remunerae a overnigh rae. 9

11 arbirage relaions such as eq. 4 are broken up, being P f (, T 1, T 2 ) = P f (, T 2 ) P f (, T 1 ) = F f (; T 1, T 2 ) τ f (T 1, T 2 ) F (; T 1, T 2 ) τ (T 1, T 2 ) = P (, T 2 ) P (, T 1 ) = P (, T 1, T 2 ). (15) No arbirage beween isinc yiel curves an f can be immeiaely recovere by aking ino accoun he forwar basis, he forwar counerpary of he quoe marke basis of fig. 1, efine as P f (, T 1, T 2 ) := F (; T 1, T 2 ) BA f (, T 1, T 2 ) τ (T 1, T 2 ), (16) or hrough he equivalen simple ransformaion rule for forwar raes F f (; T 1, T 2 ) τ f (T 1, T 2 ) = F (; T 1, T 2 ) τ (T 1, T 2 ) BA f (, T 1, T 2 ). (17) From eq. 17 we can express he forwar basis as a raio beween forwar raes or, equivalenly, in erms of iscoun facors from an f curves as BA f (, T 1, T 2 ) = F f (; T 1, T 2 ) τ f (T 1, T 2 ) F (; T 1, T 2 ) τ (T 1, T 2 ) = P (, T 2 ) P f (, T 1 ) P f (, T 2 ) P f (, T 2 ) P (, T 1 ) P (, T 2 ). (18) Obviously he following alernaive aiive efiniion is compleely equivalen P f (, T 1, T 2 ) := [ F (; T 1, T 2 ) + BA f (, T 1, T 2 ) ] τ (T 1, T 2 ), (19) BA f (, T 1, T 2 ) = F f (; T 1, T 2 ) τ f (T 1, T 2 ) F (; T 1, T 2 ) τ (T 1, T 2 ) τ (T 1, T 2 ) [ 1 Pf (, T 1 ) = τ (T 1, T 2 ) P f (, T 2 ) P ] (, T 1 ) P (, T 2 ) = F (; T 1, T 2 ) [BA f (, T 1, T 2 ) 1], (20) which is more useful for comparisons wih he marke basis spreas of fig. 1. Noice ha if = f we recover he single-curve case BA f (, T 1, T 2 ) = 1, BA f (, T 1, T 2 ) = 0. We sress ha he forwar basis in eqs is a sraighforwar consequence of he assumpions above, essenially he exisence of wo yiel curves an no arbirage. Is avanage is ha i allows for a irec compuaion of he forwar basis beween forwar raes for any ime inerval [T 1, T 2 ], which is he relevan quaniy for pricing an heging ineres rae erivaives. In pracice is value epens on he marke basis sprea beween he quoaions of he wo ses of vanilla insrumens use in he boosrapping of he wo curves an f. On he oher sie, he limi of expressions is ha hey reflec 10

12 he saical 9 ifferences beween he wo ineres rae markes M, M f carrie by he wo curves, f, bu hey are compleely inepenen of he ineres rae ynamics in M an M f. Noice also ha he approach can be invere o boosrap a new yiel curve from a given yiel curve plus a given forwar basis, using he following recursive relaions P,i = = P f,i = = P f,i BA f,i P f,i 1 P f,i + P f,i BA f,i P,i 1 P f,i P f,i 1 P f,i BA f,i τ P,i 1, (21),i P,i P f,i 1 P,i + (P,i 1 P,i ) BA f,i P,i P,i + P,i 1 BA f,i τ P f,i 1, (22),i where we have invere eqs. 18, 20 an shorene he noaion by puing τ x (T i 1, T i ) := τ x,i, P x (, T i ) := P x,i, BA f (, T i 1, T i ) := BA f,i. Given he yiel curve x up o sep P x,i 1 plus he forwar basis for he sep i 1 i, he equaions above can be use o obain he nex sep P x,i. We now iscuss a numerical example of he forwar basis in a realisic marke siuaion. We consier he four ineres rae unerlyings I = {I 1M, I 3M, I 6M, I 12M }, where I = Euribor inex, an we boosrap from marke aa five isinc yiel curves = {, 1M, 3M, 6M, 12M }, using he firs one for iscouning an he ohers for forwaring. We follow he mehoology escribe in ref. [AB09] using he corresponing open-source evelopmen available in he QuanLib framework [Qua09]. The iscouning curve is buil following a pre-crisis raiional recipe from he mos liqui eposi, IMM Fuures/FRA on Euribor3M an swaps on Euribor6M. The oher four forwaring curves are buil from convenien selecions of epos, FRAs, Fuures, swaps an basis swaps wih homogeneous unerlying rae enors; a smooh an robus algorihm (monoonic cubic spline on log iscouns) is use for inerpolaions. Differen choices (e.g. an Eonia iscouning curve) as well as oher echnicaliies of he boosrapping escribe in ref. [AB09] obviously woul lea o slighly ifferen numerical resuls, bu o no aler he conclusions rawn here. In fig. 2 we plo boh he 3M-enor forwar raes an he zero raes calculae on an 3M as of 16h Feb cob 10. Similar paerns are observe also in he oher 1M, 6M, 12M curves (no shown here, see ref. [AB09]). In fig. 3 (upper panels) we plo he erm srucure of he four corresponing muliplicaive forwar basis curves f calculae hrough eq. 18. In he lower panels we also plo he aiive forwar basis given by eq. 20. We observe in paricular ha he higher shor-erm basis ajusmens (lef panels) are ue o he higher shor-erm marke basis spreas (see fig. 1). Furhermore, he meium-long-erm 6M basis (ash-oe green lines in he righ panels) are close o 1 an 0, respecively, as expece from he common use of 6M swaps in he wo curves. A similar, bu less evien, behavior is foun in he shor-erm 3M basis 9 we remin ha he iscoun facors in eqs are calculae on he curves, f following he recipe escribe in sec. 3.2, no using any ynamical moel for he evoluion of he raes. 10 close of business. 11

13 5.0% 4.5% 4.0% 3.5% 3.0% 2.5% 2.0% 1.5% 1.0% Zero raes Forwar raes EUR iscouning curve Feb 09 Feb 12 Feb 15 Feb 18 Feb 21 Feb 24 Feb 27 Feb 30 Feb 33 Feb 36 Feb % 4.5% 4.0% 3.5% 3.0% 2.5% 2.0% 1.5% 1.0% Zero raes Forwar raes EUR forwaring curve 3M Feb 09 Feb 12 Feb 15 Feb 18 Feb 21 Feb 24 Feb 27 Feb 30 Feb 33 Feb 36 Feb 39 Figure 2: EUR iscouning curve (upper panel) an 3M forwaring curve 3M (lower panel) a en of ay Feb. 16h Blue lines: 3M-enor forwar raes F ( 0 ;, + 3M, ac/360 ), aily sample an spo ae 0 = 18h Feb. 2009; re lines: zero raes F ( 0 ;, ac/365 ). Similar paerns are observe also in he 1M, 6M, 12M curves (no shown here, see ref. [AB09]). (coninuous blue line in he lef panels), as expece from he common 3M Fuures an he uncommon eposis. The wo remaining basis curves 1M an 12M are generally far from 1 or 0 because of ifferen boosrapping insrumens. Obviously such eails epen on our arbirary choice of he iscouning curve. Overall, we noice ha all he basis curves f reveal a complex micro-erm srucure, no presen eiher in he monoonic basis swaps marke quoes of fig. 1 or in he smooh yiel curves x. Such effec is essenially ue o an amplificaion mechanism of small local ifferences beween he an f forwar curves. In fig. 4 we also show ha smooh yiel curves are a crucial inpu for he forwar basis: using a non-smooh boosrapping (linear inerpolaion on zero raes, sill a iffuse marke pracice), he zero curve apparenly shows no paricular problems, while he forwar curve isplays a sagsaw shape inucing, in urn, srong an unnaural oscillaions in he forwar basis. We conclue ha, once a smooh an robus boosrapping echnique for yiel curve 12

14 Forwar Basis (muliplicaive) 1M vs Disc 3M vs Disc 6M vs Disc 12M vs Disc Forwar Basis (muliplicaive) 1M vs Disc 3M vs Disc 6M vs Disc 12M vs Disc 80 Forwar Basis (aiive) 7 Forwar Basis (aiive) basis poins M vs Disc 3M vs Disc 6M vs Disc 12M vs Disc basis poins Feb-09 May-09 Aug-09 Nov-09 Feb-10 May-10 Aug-10 Nov-10 Feb-11 May-11 Aug-11 Nov-11 Feb-12 Feb-12 Feb-15 Feb-18 Feb-21 Feb-24 Feb-27 Feb-30 Feb-33 Feb-36 Feb-39 Feb-09 May-09 Aug-09 Nov-09 Feb-10 May-10 Aug-10 Nov-10 Feb-11 May-11 Aug-11 Nov-11 Feb-12 Feb-12 Feb-15 Feb-18 Feb-21 Feb-24 Feb-27 Feb-30 Feb-33 Feb-36 Feb-39 1M vs Disc 3M vs Disc 6M vs Disc 12M vs Disc Figure 3: upper panels: muliplicaive basis ajusmens from eq. 18 as of en of ay Feb. 16h, 2009, for aily sample 3M-enor forwar raes as in fig. 2, calculae on 1M, 3M, 6M an 12M curves agains aken as reference curve. Lower panels: equivalen plos of he aiive basis ajusmen of eq. 20 beween he same forwar raes (basis poins). Lef panels: 0Y-3Y aa; Righ panels: 3Y-30Y aa on magnifie scales. The higher shor-erm ajusmens seen in he lef panels are ue o he higher shor-erm marke basis sprea (see Figs. 1). The oscillaing erm srucure observe is ue o he amplificaion of small ifferences in he erm srucures of he curves. consrucion is use, he richer erm srucure of he forwar basis curves provies a sensiive inicaor of he iny, bu observable, saical ifferences beween ifferen ineres rae marke sub-areas in he pos crei crunch ineres rae worl, an a ool o assess he egree of liquiiy an crei issues in ineres rae erivaives prices. I is also helpful for a beer explanaion of he profi&loss encounere when swiching beween he singlean he muliple-curve worls. 5 Foreign-Currency Analogy an Quano Ajusmen A secon imporan issue regaring no-arbirage arises in he muliple-curve framework. From eq. 13 we have ha, for insance, he single-curve FRA price in eq. 7 is generalise ino he following muliple-curve expression { FRA (; T 1, T 2, K, N) = NP (, T 2 ) τ f (T 1, T 2 ) E QT 2 } [L f (T 1, T 2 )] K. (23) 13

15 6% EUR forwaring curve 3M 0Y-30Y 5% 4% 3% 2% 1% zero raes forwar raes Feb-09 Feb-11 Feb-13 Feb-15 Feb-17 Feb-19 Feb-21 Feb-23 Feb-25 Feb-27 Feb-29 Feb-31 Feb-33 Feb-35 Feb-37 Feb Forwar Basis 3Y-30Y Feb-12 Feb-15 Feb-18 Feb-21 Feb-24 Feb-27 Feb-30 Feb-33 Feb-36 basis poins Feb-39 1M vs Disc 3M vs Disc 6M vs Disc 12M vs Disc Figure 4: he effec of poor inerpolaion schemes (linear on zero raes, a common choice, see ref. [AB09]) on zero raes (upper panel, re line) 3M forwar raes (upper panel, blue line) an basis ajusmens (lower panel). While he zero curve looks smooh, he sag-saw shape of he forwar curve clearly show he inaequacy of he boosrap, an he oscillaions in he basis ajusmen allow o furher appreciae he arificial ifferences inuce in similar insrumens price on he wo curves. Insea, he curren marke pracice is o price such FRA simply as FRA (; T 1, T 2, K, N) NP (, T 2 ) τ f (T 1, T 2 ) [F f (; T 1, T 2 ) K]. (24) Obviously he forwar rae F f (; T 1, T 2 ) is no, in general, a maringale uner he iscouning measure Q T 2, so eq. 24 iscars he ajusmen coming from his measure mismach. Hence, a heoreically correc pricing wihin he muliple-curve framework requires he compuaion of expecaions as in eq. 23 above. This will involve he ynamic properies of he wo ineres rae markes M an M f, or, in oher wors, i will require o moel he ynamics for he ineres raes in M an M f. This ask is easily accomplishe by resoring o he naural analogy wih cross-currency erivaives. Going back o he beginning of sec. 2, we can ienify M an M f wih he omesic an foreign markes, an f wih he corresponing curves, an he bank accouns B (), B f () wih he 14

16 corresponing currencies, respecively 11. Wihin his framework, we can recognize on he r.h.s of eq. 16 he forwar iscoun facor from ime T 2 o ime T 1 expresse in omesic currency, an on he r.h.s. of eq. 23 he expecaion of he foreign forwar rae w.r. he omesic forwar measure. Hence, he compuaion of such expecaion mus involve he quano ajusmen commonly encounere in he pricing of cross-currency erivaives. The erivaion of such ajusmen can be foun in sanar exbooks. Anyway, in orer o fully appreciae he parallel wih he presen ouble-curve-single-currency case, i is useful o run hrough i once again. In paricular, we will aap o he presen conex he iscussion foun in ref. [BM06], chs. 2.9 an Forwar Raes In he ouble curve-ouble-currency case, no arbirage requires he exisence a any ime 0 T of a spo an a forwar exchange rae beween equivalen amouns of money in he wo currencies such ha c () = x f () c f (), (25) X f (, T ) P (, T ) = x f () P f (, T ), (26) where he subscrips f an san for foreign an omesic, c () is any cashflow (amoun of money) a ime in unis of omesic-currency an c f () is he corresponing cashflow a ime (he corresponing amoun of money) in unis of foreign currency. Obviously X f (, T ) x f () for T. Expression 26 is sill a consequence of no arbirage. This can be unersoo wih he ai of fig. 5: saring from op righ corner in he ime vs currency/yiel curve plane wih an uniary cashflow a ime T > in foreign currency, we can eiher move along pah A by iscouning a ime on curve f using P f (, T ) an hen by changing ino omesic currency unis using he spo exchange rae x f (), ening up wih x f () P f (, T ) unis of omesic currency; or, alernaively, we can follow pah B by changing a ime T ino omesic currency unis using he forwar exchange rae X f (, T ) an hen by iscouning on using P (, T ), ening up wih X f (, T ) P (, T ) unis of omesic currency. Boh pahs sop a boom lef corner, hence eq. 26 mus hol by no arbirage. Now, our ouble-curve-single-currency case is immeiaely obaine from he iscussion above by hinking o he subscrips f an as shorhans for forwaring an iscouning an by recognizing ha, having a single currency, he spo exchange rae mus collapse o 1. We hus have x f () = 1, (27) X f (, T ) = P f (, T ) P (, T ). (28) Obviously for = f we recover he single-currency, single-curve case X f (, T ) = 1, T. The inerpreaion of he forwar exchange rae in eq. 28 wihin his framework is sraighforwar: i is nohing else ha he counerpary of he forwar basis in eq. 17 for 11 noice he lucky noaion use, where sans eiher for iscouning or omesic an f for forwaring or foreign, respecively. 15

17 Figure 5: Picure of no-arbirage inerpreaion for he forwar exchange rae in eq. 26. Moving, in he yiel curve vs ime plane, from op righ o boom lef corner hrough pah A or pah B mus be equivalen. Alernaively, we may hink o no-arbirage as a sor of zero circuiaion, sum of all raing evens following a close pah saring an sopping a he same poin in he plane. This escripion is equivalen o he raiional able of ransacion picure, as foun e.g. in fig. 1 of ref. [TP03]. iscoun facors on he wo yiel curves an f. Subsiuing eq. 28 ino eq. 17 we obain he following relaion BA f (, T 1, T 2 ) = X f (, T 2 ) τ f (T 1, T 2 ) τ (T 1, T 2 ) P (, T 1 ) P (, T 2 ) P (, T 1 ) X f (, T 1 ) P (, T 2 ) X f (, T 1 ). (29) Noice ha we coul forge he foreign currency analogy above an sar by posulaing X f (, T ) as in eq. 28, name i forwar basis an procee wih he nex sep. We procee by assuming, accoring o he sanar marke pracice, he following (rifless) lognormal maringale ynamic for f (foreign) forwar raes F f (; T 1, T 2 ) F f (; T 1, T 2 ) = σ f () W T 2 f (), T 1, (30) where σ f () is he volailiy (posiive eerminisic funcion of ime) of he process, uner he probabiliy space ( ) Ω, F f, Q T 2 f wih he filraion F f generae by he brownian 16

18 moion W T 2 f uner he forwaring (foreign) T 2 forwar measure Q T 2 f, associae o he f (foreign) numeraire P f (, T 2 ). Nex, since X f (, T 2 ) in eq. 28 is he raio beween he price a ime of a (omesic) raable asse (x f () P f (, T 2 ) in eq. 26, or P f (, T 2 ) in eq. 28 wih x f () = 1) an he numeraire P (, T 2 ), i mus evolve accoring o a (rifless) maringale process uner he associae iscouning (omesic) T 2 forwar measure Q T 2, X f (, T 2 ) X f (, T 2 ) = σ X () W T 2 X (), T 2, (31) where σ X () is he volailiy (posiive eerminisic funcion of ime) of he process an W T 2 X is a brownian moion uner QT 2 such ha W T 2 f () W T 2 X () = ρ fx (). (32) Now, in orer o calculae expecaions such as in he r.h.s. of eq. 23, we mus swich from he forwaring (foreign) measure Q T 2 f associae o he numeraire P f (, T 2 ) o he iscouning (omesic) measure Q T 2 associae o he numeraire P (, T 2 ). In our oublecurve-single-currency language his amouns o ransform a cashflow on curve f o he corresponing cashflow on curve. Recurring o he change-of-numeraire echnique (see refs. [BM06], [Jam89], [GKR95]) we obain ha he ynamic of F f (; T 1, T 2 ) uner Q T 2 acquires a non-zero rif F f (; T 1, T 2 ) F f (; T 1, T 2 ) = µ f () + σ f () W T 2 f (), T 1, (33) µ f () = σ f () σ X () ρ fx (), (34) an ha F f (T 1 ; T 1, T 2 ) is lognormally isribue uner Q T 2 by E QT 2 [ ln F f (T 1 ; T 1, T 2 ) F f (; T 1, T 2 ) Var QT 2 ] = T1 [ ln F f (T 1 ; T 1, T 2 ) F f (; T 1, T 2 ) wih mean an variance given [ µ f (u) 1 ] 2 σ2 f (u) u, (35) ] = T1 We hus obain he following expressions, for 0 < T 1, E QT 2 σ 2 f (u) u. (36) [F f (T 1 ; T 1, T 2 )] = F f (; T 1, T 2 ) QA f (, T 1, σ f, σ X, ρ fx ), (37) QA f (, T 1, σ f, σ X, ρ fx ) = exp = exp T1 [ µ f (u) u ] σ f (u) σ X (u) ρ fx (u) u, (38) where QA f (, T 1, σ f, σ X, ρ fx ) is he (muliplicaive) quano ajusmen. We may also efine an aiive quano ajusmen as T1 E QT 2 [F f (T 1 ; T 1, T 2 )] = F f (; T 1, T 2 ) + QA f (, T 1, σ f, σ X, ρ fx ), (39) QA f (, T 1, σ f, σ X, ρ fx ) = F f (; T 1, T 2 ) [QA f (, T 1, σ f, σ X, ρ fx ) 1], (40) 17

19 where he secon relaion comes from eq. 37. Finally, combining eqs. 37, 39 wih eqs. 18, 20 we may erive a relaion beween he quano an he basis ajusmens, T 2 BA f (, T 1, T 2 ) QA f (, T 1, σ f, σ X, ρ fx ) = EQ [L (T 1, T 2 )], (41) E QT 2 [L f (T 1, T 2 )] BA f (, T 1, T 2 ) QA f (, T 1, σ f, σ X, ρ fx ) = E QT 2 [L (T 1, T 2 )] E QT2 [L f (T 1, T 2 )] (42) for muliplicaive an aiive ajusmens, respecively. We conclue ha he foreign-currency analogy allows us o compue he expecaion in eq. 23 of a forwar rae on curve f w.r.. he iscouning measure Q T 2 in erms of a well-known quano ajusmen, ypical of cross-currency erivaives. Such ajusmen naurally follows from a change beween he T -forwar probabiliy measures Q T 2 f an Q T 2, or numeraires P f (, T 2 ) an P (, T 2 ), associae o he wo yiel curves, f an, respecively. Noice ha he expression 38 epens on he average over he ime inerval [, T 1 ] of he prouc of he volailiy σ f of he f (foreign) forwar raes F f, of he volailiy σ X of he forwar exchange rae X f beween curves f an, an of he correlaion ρ fx beween F f an X f. I oes no epen eiher on he volailiy σ of he (omesic) forwar raes F or on any sochasic quaniy afer ime T 1. The laer fac is acually quie naural, because he sochasiciy of he forwar raes involve ceases a heir fixing ime T 1. The epenence on he cashflow ime T 2 is acually implici in eq 38, because he volailiies an he correlaion involve are exacly hose of he forwar an exchange raes on he ime inerval [T 1, T 2 ]. Noice in paricular ha a non-rivial ajusmen is obaine if an only if he forwar exchange rae X f is sochasic (σ X 0) an correlae o he forwar rae F f (ρ fx 0); oherwise expression 38 collapses o he single curve case QA f = 1. The volailiies an he correlaion in eq. 38 can be exrace from marke aa. In he EUR marke he volailiy σ f can be exrace from quoe cap/floor opions on Euribor6M, while for oher rae enors an for σ X an ρ fx one mus resor o hisorical esimaes. Conversely, given a forwar basis erm srucure, such ha in fig. 3, one coul ake σ f from he marke, assume for simpliciy ρ fx 1 (or any oher funcional form), an boosrap ou a erm srucure for he forwar exchange rae volailiy σ X. Noice ha in his way we are also able o compare informaion abou he inernal ynamics of ifferen marke sub-areas. We will give some numerical esimae of he quano ajusmen in he nex secion Swap Raes The iscussion above can be remappe, wih some aenion, o swap raes. Given wo increasing aes ses T = {T 0,..., T n }, S = {S 0,..., S m }, T 0 = S 0 an an ineres rae swap wih a floaing leg paying a imes T i, i = 1,.., n, he Xibor rae wih enor [T i 1, T i ] fixe a ime T i 1, plus a fixe leg paying a imes S j, j = 1,.., m, a fixe rae, he corresponing fair swap rae S f (, T, S) on curve f is efine by he following equilibrium (no arbirage) relaion beween he presen values of he wo legs, n S f (, T, S) A f (, S) = P f (, T i ) τ f (T i 1, T i ) F f (; T i 1, T i ), T 0 = S 0, (43) i=1 18

20 where A f (, S) = m P f (, S j ) τ f (S j 1, S j ) (44) j=1 is he annuiy on curve f. Following he sanar marke pracice, we observe ha, assuming he annuiy as he numeraire on curve f, he swap rae in eq. 43 is he raio beween a raable asse (he value of he swap floaing leg on curve f ) an he numeraire A f (, S), an hus i is a maringale uner he associae forwaring (foreign) swap measure Q S f. Hence we can assume, as in eq. 30, a rifless geomeric brownian moion for he swap rae uner Q S f, S f (, T, S) S f (, T, S) = ν f (, T, S) W T,S f (), T 0, (45) where υ f (, T, S) is he volailiy (posiive eerminisic funcion of ime) of he process an W T,S f is a brownian moion uner Q S f. Then, mimicking he iscussion leaing o eqs , he following relaion m P (, S j ) τ (S j 1, S j ) X f (, S j ) = x f () j=1 m P f (, S j ) τ f (S j 1, S j ) j=1 = A f (, S) (46) mus hol by no arbirage beween he wo curves f an. Defining a swap forwar exchange rae Y f (, S) such ha A f (, S), = we obain he expression m P (, S j ) τ (S j 1, S j ) X f (, S j ) j=1 = Y f (, S) m P (, S j ) τ (S j 1, S j ) = Y f (, S) A (, S), (47) j=1 Y f (, S) = A f (, S) A (, S), (48) equivalen o eq. 28. Hence, since Y f (, S) is he raio beween he price a ime of he (omesic) raable asse x f () A f (, S) an he numeraire A (, S), i mus evolve accoring o a (rifless) maringale process uner he associae iscouning (omesic) swap measure Q S, Y f (, S) Y f (, S) = ν Y (, S) W S Y (), T 0, (49) where v Y (, S) is he volailiy (posiive eerminisic funcion of ime) of he process an WY S is a brownian moion uner QS such ha W T,S f () W S Y () = ρ fy (, T, S). (50) 19

21 Now, applying again he change-of-numeraire echnique of sec. 5.1, we obain ha he ynamic of he swap rae S f (, T, S) uner he iscouning (omesic) swap measure Q S acquires a non-zero rif S f (, T, S) S f (, T, S) = λ f (, T, S) + ν f (, T, S) W T,S f (), T 0, (51) λ f (, T, S) = ν f (, T, S) ν Y (, S) ρ fy (, T, S), (52) an ha S f (, T, S) is lognormally isribue uner Q S wih mean an variance given by [ E QS ln S ] f (T 0, T, S) T0 [ = λ f (u, T, S) 1 ] S f (, T, S) 2 ν2 f (u, T, S) u, (53) [ Var QS ln S ] f (T 0, T, S) T0 = νf 2 (u, T, S) u. (54) S f (, T, S) We hus obain he following expressions, for 0 < T 0, E QS f [S f (T 0, T, S)] = S f (, T, S) QA f (, T, S, ν f, ν Y, ρ fy ), (55) QA f (, T, S, ν f, ν Y, ρ fy ) = exp = exp T0 [ λ f (u, T, S) u ] ν f (u, T, S) ν Y (u, S) ρ fy (u, T, S) u T0 The same consieraions as in sec. 5.1 apply. In paricular, we observe ha he ajusmen in eqs. 55, 57 naurally follows from a change beween he probabiliy measures Q S f an QS, or numeraires A f (, S) an A (, S), associae o he wo yiel curves, f an, respecively, once swap raes are consiere. In he EUR marke, he volailiy ν f (u, T, S) in eq. 56 can be exrace from quoe swapions on Euribor6M, while for oher rae enors an for ν Y (u, S) an ρ fy (u, T, S) one mus resor o hisorical esimaes. An aiive quano ajusmen can also be efine as before (56) E QS [S f (T 0, T, S)] = S f (, T, S) + QA f (, T, S, ν f, ν Y, ρ fy ), (57) QA f (, T, S, ν f, ν Y, ρ fy ) = S f (, T, S) [QA f (, T, S, ν f, ν Y, ρ fy ) 1]. (58) 6 Double-Curve Pricing & Heging Ineres Rae Derivaives 6.1 Pricing The resuls of sec. 5 above allows us o erive no arbirage, ouble-curve-single-currency pricing formulas for ineres rae erivaives. The recipes are, basically, eqs or The simples ineres rae erivaive is a floaing zero coupon bon paying a ime T a single cashflow epening on a single spo rae (e.g. he Xibor) fixe a ime < T, ZCB (T ; T, N) = Nτ f (, T ) L f (, T ). (59) 20

22 Being he price a ime T is given by L f (, T ) = 1 P f (, T ) τ f (, T ) P f (, T ) = F f (;, T ), (60) ZCB (; T, N) = NP (, T ) τ f (, T ) E QT [F f (;, T )] = NP (, T ) τ f (, T ) L f (, T ). (61) Noice ha he forwar basis in eq. 61 isappears an we are lef wih he sanar pricing formula, moifie accoring o he ouble-curve framework. Nex we have he FRA, whose payoff is given in eq. 5 an whose price a ime T 1 is given by { FRA (; T 1, T 2, K, N) = NP (, T 2 ) τ f (T 1, T 2 ) E QT 2 } [F f (T 1 ; T 1, T 2 )] K = NP (, T 2 ) τ f (T 1, T 2 ) [F f (; T 1, T 2 ) QA f (, T 1, ρ fx, σ f, σ X ) K]. (62) Noice ha in eq. 62 for K = 0 an T 1 = we recover he zero coupon bon price in eq. 61. For a (payer) floaing vs fixe swap wih paymen aes vecors T, S as in sec. 5.2 we have he price a ime T 0 Swap (; T, S, K, N) n = N i P (, T i ) τ f (T i 1, T i ) F f (; T i 1, T i ) QA f (, T i 1, ρ fx,i, σ f,i, σ X,i ) i=1 m N j P (, S j ) τ (S j 1, S j ) K j. (63) j=1 For consan nominal N an fixe rae K he fair (equilibrium) swap rae is given by S f (, T, S) = where n P (, T i ) τ f (T i 1, T i ) F f (; T i 1, T i ) QA f (, T i 1, ρ fx,i, σ f,i, σ X,i ), (64) A (, S) i=1 A (, S) = m P (, S j ) τ (S j 1, S j ) (65) j=1 is he annuiy on curve. For caple/floorle opions on a T 1 -spo rae wih payoff a mauriy T 2 given by cf (T 2 ; T 1, T 2, K, ω,n) = NMax {ω [L f (T 1, T 2 ) K]} τ f (T 1, T 2 ), (66) he sanar marke-like pricing expression a ime T 1 T 2 is moifie as follows cf (; T 1, T 2, K, ω,n) = NE QT 2 [Max {ω [L f (T 1, T 2 ) K]} τ f (T 1, T 2 )] = NP (, T 2 ) τ f (T 1, T 2 ) Bl [F f (; T 1, T 2 ) QA f (, T 1, ρ fx, σ f, σ X ), K, µ f, σ f, ω], (67) 21

23 where ω = +/ 1 for caples/floorles, respecively, an Bl [F, K, µ, σ, ω] = ω [ F Φ ( ω +) KΦ ( ω )], (68) µ (, T ) = ± = ln F + µ (, T ) ± 1 K 2 σ2 (, T ), (69) σ (, T ) T µ (u) u, σ 2 (, T ) = T σ 2 (u) u, (70) is he sanar Black-Scholes formula. Hence cap/floor opions prices are given a T 0 by CF (; T, K, ω, N) = = n cf (T i ; T i 1, T i, K i, ω i,n i ) i=1 n N i P (, T i ) τ f (T i 1, T i ) i=1 Bl [F f (; T i 1, T i ) QA f (, T i 1, ρ fx,i, σ f,i, σ X,i ), K i, µ f,i, σ f,i, ω i ], (71) Finally, for swapions on a T 0 -spo swap rae wih payoff a mauriy T 0 given by Swapion (T 0 ; T, S, K, N) = NMax [ω (S f (T 0, T, S) K)] A (T 0, S), (72) he sanar marke-like pricing expression a ime T 0, using he iscouning swap measure Q S associae o he numeraire A (, S) on curve, is moifie as follows Swapion (; T, S, K, N) = NA (, S) E QS {Max [ω (S f (T 0, T, S) K)]} = NA (, S) Bl [S f (, T, S) QA f (, T, S, ν f, ν Y, ρ fy ), K, λ f, ν f, ω]. (73) where we have use eq. 55 an he quano ajusmen erm QA f (, T, S, ν f, ν Y, ρ fy ) is given by eq. 56. When wo or more ifferen unerlying ineres-raes are presen, pricing expressions may become more involve. An example is he sprea opion, for which he reaer can refer o, e.g., ch in ref. [BM06]. The calculaions above show ha also basic ineres rae erivaives prices inclue a quano ajusmen an are hus volailiy an correlaion epenen. In fig. 6 we show some numerical scenario for he quano ajusmen in eqs. 38, 40. We see ha, for realisic values of volailiies an correlaion, he magniue of he aiive ajusmen may be non negligible, ranging from a few basis poins up o over 10 basis poins. Time inervals longer han he 6M perio use in fig. 6 furher increase he effec. Noice ha posiive correlaion implies negaive ajusmen, hus lowering he forwar raes ha eners he pricing formulas above. Pricing ineres rae erivaives wihou he quano ajusmen hus leaves, in principle, he oor open o arbirage opporuniies. In pracice he correcion epens on financial variables presenly no quoe on he marke, making virually impossible o se up arbirage posiions an lock oay expece fuure posiive gains. Obviously one may be on his/her personal views of fuure realizaions of volailiies an correlaion. 22

24 Quano Ajusmen (muliplicaive) 1.02 Quano aj Sigma_f = 20%, Sigma_X = 5% Sigma_f = 30%, Sigma_X = 10% Sigma_f = 40%, Sigma_X = 20% Correlaion Quano Ajusmen (aiive) Quano aj. (bps) Sigma_f = 20%, Sigma_X = 5% Sigma_f = 30%, Sigma_X = 10% Sigma_f = 40%, Sigma_X = 20% Correlaion Figure 6: Numerical scenarios for he quano ajusmen. Upper panel: muliplicaive (from eq. 38); lower panel: aiive (from eq. 40). In each figure we show he quano ajusmen corresponing o hree ifferen combinaions of (fla) volailiy values as a funcion of he correlaion. The ime inerval is fixe o T 1 = 0.5 an he forwar rae enering eq. 40 o 4%, a ypical value in fig. 2. We see ha, for realisic values of volailiies an correlaion, he magniuo of he ajusmen may be imporan. 6.2 Heging Heging wihin he muli-curve framework implies aking accoun muliple boosrapping an heging insrumens. We assume o have a porfolio Π fille wih a variey of ineres rae erivaives wih ifferen unerlying rae enors. The firs issue is how o calculae he ela sensiiviy of Π. In principle, he answer is sraighforwar: having recognize ineres-raes wih ifferen enors as ifferen unerlyings, an having consruce muliple yiel curves = {, 1 f,..., } N f using homogeneous marke insrumens, we mus coherenly calculae he sensiiviy wih respec o he marke rae r B = { } r1 B,..., rn B B of 23