Parsimonious HJM Modelling for Multiple Yield-Curve Dynamics. Nicola Moreni Andrea Pallavicini

Transcription

1 Parsimonious HJM Modelling for Muliple Yield-Curve Dynamics Nicola Moreni Andrea Pallavicini Firs Version: July 16, This version: Ocober 28, 2010 Absrac For a long ime ineres-rae models were buil on a single yield curve used boh for discouning and forwarding. However, he crisis ha has affeced financial markes in he las years led marke players o revise his assumpion and accommodae basis-swap spreads, whose remarkable widening can no longer be negleced. In recen lieraure we find many proposals of muli-curve ineres-rae models, whose calibraion would ypically require marke quoes for all yield curves. A presen his is no possible since mos of he quoes are missing or exremely illiquid. Thanks o a suiable exension of he HJM framework, we propose a parsimonious model based on observed raes ha deduces yield-curve dynamics from a single family of Markov processes. Furhermore, we deail a specificaion of he model reporing numerical examples of calibraion o quoed marke daa. JEL classificaion code: G13. AMS classificaion codes: 60J75, 91B70 Keywords: Yield Curve Dynamics, Muli-Curve Framework, Gaussian Models, HJM Framework, Ineres Rae Derivaives, Basis Swaps, Counerpary Risk, Liquidiy Risk. Banca IMI, nicola.moreni@bancaimi.com Banca Leonardo, andrea.pallavicini@bancaleonardo.com 1

2 Conens 1 Inroducion 3 2 Muli-curve relevan feaures Risk-free raes Libor raes Credi risk premium and liquidiy issues Basis-swap spreads Exending he HJM framework Generalized dynamics Consrains on he volailiy process Dynamics of sae variables Exac calibraion and sensiiviies Eonia simple raes Swap raes Volailiy dynamics Model calibraion and numerical examples The Weighed Gaussian model Benchmark models Iniial forwarding and discouning curves Swapion pricing formula Calibraion examples Conclusions and furher developmens 20 A Appendix: vecor and marix noaion 22 The opinions here expressed are solely hose of he auhors and do no represen in any way hose of heir employers. 2

3 1 Inroducion Classical ineres-rae models were formulaed o saisfy by consrucion no-arbirage relaionships, which allow o hedge forward-rae agreemens in erms of zero-coupon bonds. As a direc consequence, hese models predic ha forward raes of differen enors are relaed o each oher by srong consrains. In pracice, hese no-arbirage relaionships migh no hold. An example is provided by basis-swap spread quoes, which are significanly non-zero, while hey should be equal o zero if such consrains held. This is wha happened saring from summer 2007, wih he raising of he credi crunch, where marke quoes of forward raes and zero-coupon bonds began o violae he usual noarbirage relaionships in a macroscopic way, under boh he pressure of a liquidiy crisis, which reduced he credi lines, and he possibiliy of a sysemic break-down suggesing ha counerpary risk could no be considered negligible any more. The resuling picure, as suggesed by Henrard (2007), describes a money marke where each forward rae seems o ac as a differen underlying asse. There are empirical sudies supporing he idea ha Libor rae levels canno be uerly jusified by counerpary credi risk argumens. In a European Cenral Bank working paper, Eisenschmid and Tapking (2009) compare he spread of he Euribor over he general collaeral repo rae o he spread of banking-secor credi defaul swaps of he same enor during he crisis period. Auhors found ha here is evidence of a large, persisen and ime varying componen of he Euribor-Eurepo spread ha canno no be explained by counerpary credi risk. In figure 1 we show he hisorical series of Euribor-Eurepo spread for a rae enor of one year and of a synheic index composed by senior one-year CDS spread of a baske of welve European banks represenaive of he Libor panel. Surely he wo series have some common qualiaive characerisics. Ye, we find ha he sharp rise in he Euribor-Eurepo spread of Sepember 2008 is only found hree-four monhs laer in he CDS spread series, confirming ha a liquidiy crisis needs ime o evolve as credi crisis. Hence, counerpary risk is only one of he Libor dynamics driving facors, as discussed in Heidere al. (2009). Recenly in he lieraure some auhors sared o deal wih hese issues, mainly concerning he valuaion of cross currency swaps as Boenkos and Schmid (2005), Kijima e al. (2009). In hese papers, as in Henrard (2007, 2009), he problem is faced in a pragmaic way by considering each forward rae as a single asse wihou invesigaing he microscopical dynamics implied by liquidiy and credi risks. Aemps in his differen direcion are made in Morini (2009), Morini and Prampolini (2010) and Fries (2010). In paricular, we refer o Moreni and Morini (2010) where Libor raes of differen enors are microscopically associaed o differen shor raes, which, in urn, are obained by adding an insananeous credi spread o he risk-free shor rae. Besides microscopic approaches, many auhors exended yield-curve boosrapping o a muli-curve seing, evenually resuling in new pricing models. These laers are ofen inspired by oher asse classes, as Bianchei (2009), Chibane and Sheldon (2009), Kijima e al. (2009), Mercurio (2009), Marìnez (2009), Kenyon (2010), or Pallavicini and Tarenghi (2010). We cie also a slighly differen approach by Fujii e al. (2010) and Mercurio (2010), 3

4 Figure 1: Hisorical series of Euribor-1y minus Eurepo-1y spread (black line) and a synheic index formed by senior one-year CDS of a baske of welve represenaive European banks (red line) ranging from May 2007 up o March Values are in basis poins. The figures are obained from Bloomberg R plaform. where each basis spread is modelled as a differen process. However, he hypohesis of inroducing differen underlying asses may lead o overparamerizaion issues ha affec he calibraion procedure. Indeed, he presence of swap and basis-swap quoes on many differen yield curves is no sufficien, as he marke quoes swapion premia only on few yield curves. For insance, even if he Euro marke quoes one-, hree-, six- and welve-monh swap conracs, liquidly raded swapions are only hose indexed o he hree-monh (mauriy one-year) and he six-monh (mauriies from wo o hiry years) Euribor raes. Swapions referring o oher Euribor enors or o Eonia are no acively quoed. A similar line of reasoning holds also for caps/floors and oher ineres-rae opions. In his paper we wish o inroduce a parsimonious model which is able o describe a muli-curve seing by saring from a limied number of (Markov) processes. Among he classical single yield-curve models, his goal is achieved by he HJM framework by Heah, Jarrow and Moron (1992), and by he funcional Markov models by Hun, Kennedy and Pelsser (2000), where a single family of Markov processes is used o drive all he ineresrae derived quaniies. Our proposal is o exend he logic of he former (HJM) o describe wih a family of Markov processes all he curves we are ineresed in. The srucure of he paper is he following: Secion 2 reviews he fundamenal moneymarke conceps ha underlie he consrucion of a muli-curve framework; in Secion 3 4

5 we describe an original exended HJM framework able o handle many yield curves; in Secion 4, we deail a simple ye relevan specificaion (dubbed he Weighed Gaussian Model) of he model ha allows for simple evaluaion of plain vanilla opions, ogeher wih he resuls of is calibraion o marke daa; finally, Secion 5 reviews our conribuions and hins for furher developmens. 2 Muli-curve relevan feaures In order o moivae our modelling choices, i is useful o summarize he changes ha occurred because of he credi crunch and he crucial issues a muli-curve framework should face. In his secion we sar idenifying he risk-neural measure, i.e. he risk-free discoun erm-srucure, wih he one coming from Overnigh Indexed Swaps (OIS), and hen we inroduce risky raes (Libor). We also discuss, supporing our argumens wih empirical analysis, he monooniciy properies of basis spreads and muli-enor Libors. 2.1 Risk-free raes Firs of all we assume ha he marke is arbirage free, hence posulaing he exisence of a risk-neural measure. Under his measure every (risk-free) radable asse insananeously increases is value a he risk-free rae r. Furhermore, we inroduce (risk-free) zero-coupon bond prices and insananeous forward raes as [ T ] P (T ) := E du r u (1) f (T ) := E T [ r T ] where he firs expecaion is aken under risk-neural measure, and he las expecaion is aken under a measure whose numeraire is P (T ) (hereafer simply T -forward measure). As usual, we wish o link our risk-free raes o marke quoes. In classical single-curve ineres-rae models, zero-coupon bond prices observed a ime = 0 form a erm srucure T P 0 (T ) which can be made consisen wih a selecion of quoes (deposis, fuures and ineres-rae swaps). However, since he beginning of he crisis, many of hem have been carrying a relevan amoun of credi and/or liquidiy risk and canno be considered as belonging o he risk-neural economy. Thus, he subse of he insrumens o boosrap he risk-free erm srucure from has o be carefully chosen. A closer look a he Euro money marke makes clear ha quoed insrumens are indexed on hree reference indices 1 : Eonia, Euribor and Eurepo. 1 See European Banking Federaion sie a hp:// 5

6 Eonia is an effecive rae calculaed from he weighed average of all overnigh unsecured lending ransacions underaken in he inerbank marke. Euribor(s) are offered raes a which Euro inerbank erm deposis of differen mauriies are raded by one prime bank o anoher one. Eurepo(s) are offered raes a which Euro inerbank secured money marke ransacions are raded. Eonia and Euribor raes are unsecured, so ha hey incorporae he defaul risk of he counerpary of he ransacion, while Eurepo raes are secured and free of credi risk. Thus, Eurepo raes could seem he naural proxy for risk-free raes 2. The main issue wih Eurepo is ha he longes quoed insrumen has a mauriy of one year. Longer mauriies Euro money marke deals are only indexed on Euribor and Eonia indices. In paricular, we found Eonia swap conracs (OIS) up o hiry years. Because of he pluraliy of available OIS insrumens and of he reduced credi/liquidiy exposure on overnigh deposis, o many exen Eonia raes are he bes available proxy for risk-free raes. This poin has been sressed by many auhors, and we refer o Fujii e al. (2010) for more deailed argumens. 2.2 Libor raes I is a common habi o refer o unsecured deposi raes over he period [, T ] as Libor raes (L (T )). In his paper we follow his nomenclaure and we reserve he erm Euribor for he index used as reference rae for deposis in he Euro area. As usual we can inroduce he forward raes F (T, x) defined as F (T, x) := E T [ L T x (T ) ]. (2) Forward raes F (T, x) are by consrucion maringales under he T -forward measure and each of hem represens he par rae seen a for a swaple accruing over [T x, T ] and paying a T a fixed rae in exchange for L T x (T ). Noice ha accordingly o wha said in he previous secion, we consider one-day deposis as being risk-free, while he longer he enor, he greaer will be he credi charge on unsecured deposi raes. In oher words we are hinking Eonia raes as (non-quoed) one-day-enor Libor raes reducing as much as possible he deposi risks. By pushing his analogy furher we inerpre Libor raes as microscopic raes a he same level of he shor-rae, and wrie r = lim L ( + x), (3) x 0 which, given Eqs. (1) and (2), also reads f (T ) = lim x 0 F (T, x). (4) 2 See for insance Eisenshmid and Tapking (2009) where he Euribor-Eurepo spread is used as an indicaor of credi risk. 6

7 Figure 2: Hisorical series of Euribor-6m minus Euribor-3m spread (black line) OIS-6m minus OIS-3m spread (red line) ranging from Augus 2007 up o February The figures are obained from Bloomberg R plaform Credi risk premium and liquidiy issues The usual no-arbirage relaionship beween (risk-free) zero-coupon bond prices and Libor raes holds only for non-defaulable counerparies and insrumens wihou liquidiy risk. Hence, if L (x) is a Libor rae relaed o he period [, + x], we ge in general L ( + x) 1 ( ) 1 x P ( + x) 1, x > 0. Hence, when he presence of credi and liquidiy risks invalidae he possibiliy of replicaing Libor indexed deposis wih non-risky bonds P (T ), hen ineres-rae modelling should consider Libor raes of differen enors as differen asses. Ye, hey should no move apar in a random way. A firs glance, credi risk argumens imply ha deposis wih longer enor mus be charged for a higher risk premium, so ha, if he risk-free yield curve is non decreasing, forward-raes should be a non-decreasing funcion of x. For insance, le us consider he EUR money marke and focus on he risk-free yield curve boosrapped from Eonia indexed producs, such as OIS up o one year of mauriy. We idenify risk-free linearly compounding raes wih single-period OIS raes defined as E (T, x) := 1 ( { T } ) exp du f (u) 1. x T x 7

8 Figure 3: Hisorical series of basis-swap spread beween Euribor-12m and Euribor-6m ranging from Augus 2009 up o Augus The figures are obained from Bloomberg R plaform. If he risk-free curve is no invered, credi risk premium argumens should lead o E 0 (x, x) > E 0 (x, x ) = L 0 (x) > L 0 (x ), x > x. However, liquidiy issues may invalidae such relaionships. Acually le us recall ha boh Eonia and Euribor raes refer o unsecured conracs, bu Euribor raes do no represen effecive ransacions, while Eonia rae does. As an example of violaions, we plo in figure 2 he daily hisorical values of spreads s E := E 0 (6m) E 0 (3m) and s L = L 0 (6m) L 0 (3m). We noice ha in periods of grea urmoil, as he las rimeser of 2007, even if he risk-free yield curve was ofen non-invered, sill he s L happened o be negaive. As a consequence, in he following we will no impose direc consrains on Libor or forward raes, focusing on relaionships o link forward-rae volailiies Basis-swap spreads The saring poin of our analysis was he raise of basis-swap spreads afer he credi crisis. Once again simple credi risk argumens would require basis-swap spreads o be posiive, bu liquidiy issues should also be considered. In he Euro area basis-swaps are quoed wih mauriies ranging from one year up o hiry years, so ha each leg conains a srip 8

9 of many Euribor raes. The averaging effec weakens he liquidiy impac, bu does no cancel i ou. Indeed, in figure 3 we see ha he basis-swap spread beween one year and six-monh Euribor raes was ofen negaive in he las rimeser of However, we find ha in mos cases quoes of basis-swap spreads are posiive. In he lieraure only Fujii e al. (2010) and Mercurio (2010) force consrains on basis-swap spreads by direc modelling basis-swap spreads wih respec o he EONIA raes wih non-negaive processes, bu his condiion does no ensure ha all quoed basis-swap spreads remain posiive. Wihin our modeling framework, i will be quie difficul o impose consrains on basis-swap spreads posiiviy, since we will no model hem direcly. 3 Exending he HJM framework Our goal is o exend he classical HJM framework o include curves associaed o differen enors by modelling forward Libor raes by means of a common family of (Markov) processes. In he lieraure oher auhors proposed generalizaions of he HJM framework, see for insance Chiarella (2010) or Carmona (2004). In paricular, in recen papers Marìnez (2009) and Fujii (2010) exended he HJM framework o incorporae muliple-yield curves and o deal wih foreign currencies. Our approach differs from he previous ones mainly on wo relevan poins. Firs, we model only observed raes as in Libor marke model approaches, avoiding he inroducion of quaniies such as forecasing curve bonds or insananeous raes. Second, we consider a common family of processes for all he yield curves of a given currency, so ha we are able o build parsimonious ye flexible models. As a consequence of he discussion of previous secions, and in order o keep he model as simple as possible, le us summarize he basic requiremens he model mus fulfill: i) exisence of a risk free curve, wih insananeous forward raes f (T ) ii) exisence of Libor raes, ypical underlying of raded derivaives, wih associaed forwards F (T, x) iii) no arbirage dynamics of he f (T ) and he F (T, x) (boh being T -forward measure maringales) ensuring he limi case of Eq.(4) iv) possibiliy of wriing boh he f (T ) and he F (T, x) as funcion of a common family of Markov processes. While he firs wo requisies are relaed o he se of financial quaniies we are abou o model, he las wo are condiions we impose on heir dynamics, and will be graned by a befiing choice of model volailiies. 9

10 3.1 Generalized dynamics According o requiremens i) and ii) we model risk-free forward insananeous raes f (T ) and (risky) forward Libor raes F (T, x), for which we choose, under he T forward measure, he following SDE 3. df (T ) = σ (T ) dw d(k(t, x) + F (T, x)) (k(t, x) + F (T, x)) = Σ (T, x) dw σ (T ) := σ (T ; T, 0) Σ (T, x) := T T x du σ (u; T, x), (5) where we inroduced he family of volailiy (row) vecor processes σ (u; T, x), he (row) vecor of independen Brownian moions W, and he se of shifs k(t, x) ha are required o saisfy 4 lim x 0 x k(t, x) = 1, ha is, k(t, x) 1/x if x 0. The idenificaion of he volailiy of risk free insananeous forward raes f (T ) wih σ (T ; T, 0) is easily jusified if we explicily inegrae he SDE for F (T, x) (( F (T, x) = k(t, x) 1 + F ) { 0(T, x) exp 1 k(t, x) 2 0 Σ s (T, x) 2 ds + 0 } ) Σ s(t, x) dw s 1 and ake he limi x 0, such ha Σ s (T, x) xσ s (T ; T, 0) + O(x 2 ) and he exponenial may be expanded in series of x. The paricular choice of a shifed forward Libor dynamics ensures he limi of Eq.(4) and is formally equivalen o he evoluion of risk-free simple raes E (T, x), which are for insance shifed lognormal when sandard HJM volailiies leads o an Hull and Whie model. In lieraure, direc modelling of shifed forward raes is also considered in Eberlein and Kluge (2007) (see also references herein), and in Papapanaleon (2010). By means of he change of numeraire echnique we have ha dw (T ) = dw (rn) d W (rn), log P (T ) ( T ) (rn) = dw + duσ (u; u, 0) d where W (T ) and W (rn) are sandard Brownian moions under T forward and risk-neural measure, respecively. I is hen sraighforward o wrie he dynamics of forward Libor 3 See appendix A for vecor and marix noaion. 4 This assumpion may be generalized asking ha k(t, x) φ(t, x)/x for a funcion φ such ha lim x 0 φ(t, x) = 1. 10

11 raes and insananeous risk-free raes under he risk neural measure as [ d(k(t, x) + F (T, x)) T ] (k(t, x) + F (T, x)) = Σ (T, x) du σ (u; u, 0)d + dw, [ T ] (6) df (T ) = σ (T ) du σ (u; u, 0)d + dw W being a risk-neural measure mulidimensional sandard Brownian moion. 3.2 Consrains on he volailiy process Le us analyse more in deail he dynamics of he shifed forward Libors under risk-neural measure. By inegraing he SDE over he ime period [0, ] we ge ln ( ) k(t, x) + F (T, x) k(t, x) + F 0 (T, x) = 0 Σ s(t, x) [ dw s 1 2 Σ s(t, x)ds + T s ] duσs(u; u, 0)ds. To ensure he racabiliy and a Markovian specificaion of he model, we exend he single-curve HJM approach of Richken and Sankarasubramanian (1995), by seing σ (u; T, x) := h q(u; T, x)g(, u) { u } g(, u) := exp dy λ(y) q(u; u, 0) = 1, (7) where h is a marix adaped process, q is a diagonal marix deerminisic funcion (i.e. q ij = q i 1 i=j ) and λ is a deerminisic array funcion. The condiion on q when T = u is needed o ensure ha in he limi case x 0 we recover he sandard Richen-Sankarasubramanian separabiliy condiion. By plugging he expression for he volailiy ino Eq.(6), i is possible o work ou he expression ending up wih he represenaion ln ( ) k(t, x) + F (T, x) k(t, x) + F 0 (T, x) = G (, T x, T ; T, x) ( X + Y (G 0 (,, T ) 12 )) G(, T x, T ; T, x), (8) where we have defined he Iô sochasic process X N X i := g i (s, ) (h ik,sdw k,s + (h sh ) s ) ik dy g k (s, y)ds, i = 1,..., N s k=1 and he auxiliary marix process Y 0 Y ik := 0 ds g i (s, )(h sh s ) ik g k (s, ) i, k = 1,..., N 11

12 wih X0 i = 0 and Y0 ik = 0, as well as he vecorial deerminisic funcions G 0 (, T 0, T 1 ) := T1 G(, T 0, T 1 ; T, x) := T 0 dy g(, y) T1 T 0 dy q(y; T, x)g(, y). The limi case x 0, as previously deailed for general σ (u; T, x) sill holds and we may check ha f (T ) = lim x 0 F (T, x) Dynamics of sae variables Equaion (8) is he analogous of sandard HJM reconsrucion formula and is he main resul of our paper. Le us noice ha i reurns a reconsrucion formula for forward Libor raes, while sandard HJM one is based on bonds. This imporan feaure is consisen wih he requiremen of a model capable o direcly describe marke relevan quaniies. Thanks o our assumpion we are fully able o describe insananeous forward raes (i.e. discouning curve bonds) and forward Libor raes once we know he sae variables {X, Y }, which saisfy, under he risk neural measure, he following coupled (S)DE dx i = dy ik N k=1 ( Y ik λ i ()X i ) d + h dw = [ (h h ) ik (λ i () + λ k ())Y ik Le us noice ha forward Libor diffusion pre-facors 5 G(, T x, T ; T, x) depend on he q(u; T, x). This flexibiliy is a desirable feaure, as i allows for a locally uned dynamics for forward Libor raes, as we show in he nex secion Exac calibraion and sensiiviies Our approach focuses on marke quaniies and leaves us he freedom of choosing he q(u; T, x) and he κ(t, x) such as o exacly calibrae a selecion of marke daa. These free parameers are independen from he skew/smile paerns ha endogenously come wih he risk free HJM dynamics of Eonia single-period raes. This is a relevan advanage over oher microscopic muli-curve models where he dynamics of microscopic quaniies uniquely deermines implied volailiy paerns for raes of any enor and mauriy. Thus, as relevan enors and mauriies form a discree se (for insance x {1, 3, 6, 12} monhs), we may reasonably se κ(t, x) o be a piece-wise-consan deerminisic funcion ] d. 5 Acually, saring from (8), and swiching o he erminal Q T measure, we have df (T, x) = [κ(t, x) + F (T, x)]g (, T x, T ; T, x) h dw. 12

13 o be exacly calibraed o a subse of caple or swapion skews. Furher, we have he same possibiliy for a subse of a-he-money caple or swapion volailiies by properly defining he q(u; T, x) process. For insance, i is possible o se q(u; T, x) := ˆq(T, x)p(u), wih ˆq a scalar funcion and p an array funcion. Wih his prescripion ˆq may be used o exacly calibrae a subse of a-he-money quoes, while he array p allows o selec he subse of he X ha is relevan for he diffusion of F (T, x). In his way we may associae he dynamics of a chosen rae o a selecion of relevan diffusion modes. The possibiliy of an exac calibraion o a subse of a-he-money caple or swapion volailiies and skews easily allows for sensiiviy compuaion, and is similar o wha happens in sochasic local volailiy models, see for insance Torrealba (2010), where, afer having calibraed he parameers of he volailiy process, he local erm allows for an exac calibraion o some relevan marke quoes. 3.3 Eonia simple raes For sake of compleeness we may also compue Eonia simple raes E (T, x) by plugging he separable volailiy form wihin he relaionship { T } 1 + xe (T, x) = exp dy f (y) T x such ha ( ) 1 + xe (T, x) ln = 1 + xe 0 (T, x) G 0(, T x, T ) ( ( X + Y G 0 (,, T ) 1 )) 2 G 0(, T x, T ). (9) Le us noice ha if we se q(u; T, x) 1, hen G 0 (, T x, T ) G(, T x, T, x, T ) such ha F (T, x) and E (T, x) would differ only in heir shifs and iniial values. If we moreover choose κ(t, x) = 1/x, we would obain a model wih perfec insananeous correlaion beween Libors and Eonia simple raes in which 1 + xf (T, x) 1 + xe (T, x) = 1 + xf 0(T, x) 1 + xe 0 (T, x), hence showing ha he saic correcion model of Henrard (2009) is a paricular case of his exended HJM framework. 13

14 3.4 Swap raes Our framework also allows us o derive an (approximaed) expression for swap raes dynamics. Le us consider a swap wih a x enor floaing leg and a x enor fixed one paying a imes {T a+1,..., T b } and {Tā+1,..., T b}, respecively. The swap par rae equaing he wo legs is S ab (x, x) := b k=a+1 τ kp (T k )F (T k, x) b k=ā+1 τ kp (T k ) where he quaniies wih a bar refer o he fix leg. We inroduce he weighs w as w ab k ()(x, x) := τ k P (T k ) b k=ā+1 τ kp (T k ) and perform he usual freezing (see Errais and Mercurio (2005)) echnique o obain, under he swap measure Q ab, ds ab (x, x) = b k=a+1 b k=a+1 w ab k ()df (T k, x) w ab k () [κ(t k, x) + F (T k, x)] Σ (T k, x) dw ( S ab (x, x) + ψ ab (x, x) ) b k=a+1 δ ab k Σ (T k, x) dw, where δ ab k (x) := ψ ab (x, x) := b k=a+1 τ kp 0 (T k )κ(t k, x) b j=ā+1 τ jp 0 (T j ) τ k P (T k )(κ(t k, x) + F 0 (T k, x)) b j=a+1 τ jp (T j )(κ(x, T j ) + F 0 (x, T j )). Wih similar argumens we ge an expression also for basis swap spreads, since we have B ab 3.5 Volailiy dynamics (x, x ) =: S ab (x, x ) S ab (x, x ). As in he single-curve HJM framework we can add a sochasic volailiy process o our model by exending he filraion o include also he informaion generaed by he volailiy process. A popular choice is o model he marix process h by means of a square-roo process (see for insance Trolle and Schwarz (2009) and reference herein). 14

15 We sar by replacing he h process by h := v R where R is a lower riangular marix, while he variance v dynamics under risk neural measure is given by is a vecor process whose dv = κ (θ v ) d + ν v dz, v 0 = v where κ,θ,ν, v are consan deerminisic vecors, and Z is a vecor of independen Brownian moions correlaed o he W processes as given by ρ ii d := d Z, W i, and ρ ij = 0 for i j where ρ is a diagonal deerminisic correlaion marix. Wih his choice we ge shifed Heson dynamics for marke raes, so ha we can calculae opion pricing wih usual Fourier ransform echniques (see Lewis (2001)). 4 Model calibraion and numerical examples As shown in Pallavicini and Tarenghi (2010) here are evidences ha he money marke for Euro area has moved o a muli-curve seing for wha concerns he pricing of plainvanilla insrumens like ineres-rae swaps, bu he siuaion is no so clear for derivaive conracs, where he calibraion of volailiy and correlaion parameers may hide he impac of which yield curve is used in pricing. In paricular, his holds for CMS swaps and CMS opions, while he swapion marke has evidences of pricing in he old singlecurve approach, alhough some conribuors sar quoing in muli-curve framework from Sepember On he oher hand, he money marke for Euro area does no quoe opions on all rae enors. In Euro area only opions on he six-monhs enor are widely lised, while he hreemonhs enor is presen only in few quoes (swapions wih one-year enor and cap/floors wih mauriies up o wo years), and opions on he oher rae enors are missing. Thus, any model which requires a differen dynamics for each erm-srucure, has he problem ha marke quoes canno be found o fix all is degrees of freedom. Here, we selec a simple bu realisic volailiy specificaion for he muli-curve HJM framework, which we calibrae o a-he-money swapion prices quoed by ICAP R on Bloomberg R plaform on 12 of Augus We limi ourselves o a very simple calibraion daa-se since we are ineresed only in highlighing a relevan propery of our framework, namely he possibiliy o build all volailiy erm-srucures saring from few marke quoes. We address o Pallavicini and Tarenghi (2010) for furher calibraion deails for a simpler model specificaion (which consider independenly each yield curve). In paricular, we consider a simple exension of a wo-facor Gaussian model (see G2++ model in Brigo and Mercurio (2006)), ha we call Weighed Gaussian model (WG2++ model), since he erms depending on Libor enors appear as muliplicaive weighs of he 15

16 X processes. Noice ha we could use more han wo facors (WGn++ model), or we could add sochasic volailiy o calibrae also he swapion volailiy smile, leading o a Weighed Heson model (WHn++ model). 4.1 The Weighed Gaussian model As an example we inroduce a simple specificaion of our generic HJM muli-curve approach, wih differen dynamics for each forward Libor rae. In pracice i is a generalizaion of a shifed n-facor Hull and Whie model associaed o risk-free raes. Le us se he volailiy process h o be in he form h := ε()hr, where h is a diagonal consan marix h ij = h i δ i=j, R is a lower riangular marix represening he pseudo-square roo of a correlaion marix ρ, and we allow for a ime varying common volailiy shape in he form ε() := 1 + (β β 1 )e β 2, where β 0, β 1, β 2 are hree posiive consans. Microscopical Markov facors X and Y evolve under he risk free measure, as ( n ) dx i = Y ij λ i X i d + ε()h i dŵ i dy ij j=1 = ( ε 2 ()h i h j ρ ij + (λ i + λ j )Y ij d Ŵ i Ŵ j = ρ ij d ) d where he λ i are non negaive consans, and dŵ := R dw. The risk free shor rae is given as usual by n r := f 0 () + and he shif erm f 0 () allows o recover = 0 risk free yield curve 6. As for he enor-mauriy facors q, we chose a mauriy independen form of he ype k=1 X i q i (u; T, x) := e xη i. Numerical ess are done wih n = 2, hence leading o en free parameers {λ 1, λ 2, h 1, h 2, η 1, η 2, ρ 12, β 0, β 1, β 2 }, and for sake of simpliciy we se κ(t, x) = 1/x. By consrucion his model suppors differen forecasing curves and we boosrapped iniial forward Libors F 0 (T, x) by means of he Eonia erm srucure (discouning) and differen enors Euribor erm srucures. 6 As he Y are deerminisic, his model is ofen wrien by explicily compuing he Y -relaed quaniies such as he drif of he X. Those quaniies are hen incorporaed ino a generic shif. 16 (10)

17 Eonia Euribor Dae zero rae 3M rae 6M rae 13-Aug % 16-Aug % % % 17-Aug % % % 23-Aug % % % 30-Aug % % % 06-Sep % % % 16-Sep % % % 18-Oc % % % 16-Nov % % % 16-Dec % % % 17-Jan % % % 16-Feb % % % 16-May % % % 16-Aug % % % 16-Aug % % % Eonia Euribor Dae zero rae 3M rae 6M rae 16-Aug % % % 18-Aug % % % 17-Aug % % % 16-Aug % % % 16-Aug % % % 16-Aug % % % 16-Aug % % % 17-Aug % % % 16-Aug % % % 18-Aug % % % 16-Aug % % % 16-Aug % % % 16-Aug % % % 16-Aug % % % 16-Aug % % % Table 1: Eonia erm-srucure expressed in erm of ACT/360 zero-raes and Euribor erm srucures for hree- six- monh enors expressed in erms of ACT/360 forward raes. Boosrapping deails can be found on Pallavicini and Tarenghi (2010). Daa boosrapped from marke quoes observed on 12 of Augus Benchmark models In our numerical examples we compare he resuls of he WG2++ model wih respec o oher wo HJM-like models, all wih wo driving facors and ime-dependen volailiies. We discoun flows by means of he Eonia erm srucure and use for forecasing purposes he Euribor erm srucures. The G2++ model of Brigo and Mercurio (2006). This is a single-curve (old-syle) model which we exend o incorporae ime-dependen volailiies via he common ime-dependen facor ε(). I is obained by seing η j 0, (i.e. q 1,) and F 0 (T, x) E 0 (T, x). For his model we use, as discouning and forwarding curve, a erm srucure obained wih old-syle sandard echniques from deposis, fuures and swap raes. The MMG model of Pallavicini and Tarenghi (2010). This is an uncerain parameer muli-curve model which we resric o have only one scenario. I is obained by seing η j 0, uses separae forwarding and discouning curves and, as discussed in Sec.3.3, reduces o Henrard saic correcion model. I has eigh free parameers (λ 1,h 1,λ 2,h 2,ρ 12,β 0,β 1,β 2 ) and uses he same curves as he Weighed Gaussian. 17

18 1Y 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y 10Y 15Y 20Y 1Y 0.27% 0.56% 0.85% 1.12% 1.37% 1.61% 1.85% 2.08% 2.30% 2.50% 3.44% 4.24% 2Y 0.43% 0.83% 1.22% 1.59% 1.95% 2.30% 2.63% 2.95% 3.26% 3.56% 4.83% 5.95% 3Y 0.51% 0.98% 1.44% 1.87% 2.31% 2.71% 3.11% 3.48% 3.84% 4.20% 5.61% 6.90% 4Y 0.55% 1.07% 1.56% 2.02% 2.47% 2.92% 3.35% 3.77% 4.17% 4.55% 6.05% 7.42% 5Y 0.58% 1.11% 1.63% 2.12% 2.59% 3.06% 3.51% 3.94% 4.37% 4.77% 6.37% 7.80% 6Y 0.59% 1.15% 1.69% 2.20% 2.68% 3.15% 3.60% 4.04% 4.48% 4.89% 6.53% 8.02% 7Y 0.60% 1.16% 1.72% 2.24% 2.74% 3.21% 3.67% 4.12% 4.57% 5.00% 6.69% 8.20% 8Y 0.61% 1.18% 1.74% 2.26% 2.77% 3.25% 3.71% 4.17% 4.63% 5.07% 6.80% 8.33% 9Y 0.62% 1.19% 1.75% 2.28% 2.79% 3.28% 3.75% 4.22% 4.68% 5.13% 6.88% 8.45% 10Y 0.62% 1.19% 1.76% 2.29% 2.81% 3.31% 3.79% 4.24% 4.70% 5.14% 6.90% 8.49% 15Y 0.59% 1.14% 1.69% 2.21% 2.70% 3.20% 3.68% 4.14% 4.59% 5.04% 6.72% 8.27% 20Y 0.55% 1.07% 1.58% 2.08% 2.57% 3.05% 3.51% 3.95% 4.37% 4.75% 6.39% 7.81% Table 2: A-he-money swapion prices quoed by ICAP R on Bloomberg R plaform on 12 of Augus Underlying swap s enor on columns, is saring ime on rows. In Euro area swapions wih one-year enor are claims o ener a swap whose floaing leg is indexed wih he hree-monh Euribor rae, while all he oher refer o he six-monh Euribor rae. 4.3 Iniial forwarding and discouning curves The iniial yield curves can be boosrapped from he money marke quoes. We refer again o Pallavicini and Tarenghi (2010) and references herein for a complee discussion. Here, we adop heir mehodology. In paricular we use he Eonia erm-srucure o discoun cash flows, as a proxy for he risk-free yield curve (see also Fujii e al. (2010) and Mercurio (2010)). In able 1 we show he Eonia erm-srucure expressed in erm of ACT/360 zero-raes, and he Euribor erm srucures for hree- six- monh enors expressed in erms of ACT/360 forward raes. 4.4 Swapion pricing formula Swapion prices are given by he following expecaion value under risk-neural measure [ { Ta } ] Π ab := E exp du r u A ab (T a ; x)(s ab (T a ; x, x) K) + Under he approximaion of Secion 3.4, which is prey good for ATM swapions, we ge a log-shifed dynamics for swap raes such ha Π ab = A ab (; x) Bl(K + ψ ab (x, x), S ab (; x, x) + ψ ab (x, x), Γ ab (x)) 18

19 where he annuiy A ab is given by b A ab (; x) := τ k P (T k ) k=a+1 and Bl( ) is he usual Black formula wih given srike, forward rae and volailiy. paricular he volailiy Γ ab is given by In wih he deerminisic vecor ab defined as Γ ab (x) := ( ab (x)) Σ ab ab (x) ab (x) := he ηx b k=a+1 and he deerminisic marix Σ ab defined as Σ ab := Ta δ ab k (x) e λt k 1 e λt k λ du (e λu ) ρ e λu ε 2 (u). For our calibraion examples we consider a-he-money swapion prices quoed by ICAP R on Bloomberg R plaform on 12 of Augus 2010, as given in able 2. Noice ha in he Euro area swapions wih one-year enor are claims o ener a swap whose floaing leg is indexed wih he hree-monh Euribor rae, while all he oher refer o he six-monh Euribor rae. Swapions referring o oher Euribor enors or o Eonia are no acively quoed. 4.5 Calibraion examples In able 3 we lis he model parameers obained from he calibraion procedure. Noice ha he wo driving processes X 1 and X 2 operae on wo differen ime scales. Indeed, by consrucion he firs process has always a speed of mean reversion smaller han he one of he second process. This consrain is enforced while calibraing o avoid a degenerae problem. In he figure on he righ side of able 2 he volailiy backbones of each driving facor, namely he produc ε()h k wih k {1, 2} ploed wih respec o ime in years. We can see ha, allowing for more degrees of freedom along he Euribor enor space as we increase he complexiy of he model, he volailiies of he wo driving processes X 1 and X 2 spli apar: a higher volailiy for he process wih higher speed of mean reversion (process acing on a shorer ime scale). Calibraion errors in erm of implied swapion volailiies are shown in figure 4. We can see ha he calibraion error for swapions wih a enor of one year is less as long as he model allows for incorporaing muliple yield curves (MMG) and differeniaing heir 19

20 Model Pars G2++ MMG WG2++ λ η h λ η h ρ β β β χ 2 100% 22.08% 16.99% Table 3: Model parameers obained from he calibraion procedure o a-he-money swapion prices quoed by ICAP R on Bloomberg R plaform on 12 of Augus The Las row shows he calibraion error normalized o he one obained wih he G2++ model. On he righ panel he volailiy backbone of each driving facor, namely he produc ε()h k wih k {1, 2} wih respec o ime in years. dynamics (WG2++). Indeed, as previously saed in he Euro area swapions wih oneyear enor are claims o ener a swap whose floaing leg is indexed wih he hree-monh Euribor rae, while all he oher refer o he six-monh Euribor rae. We show in figure 5 he implied volailiy for swapions of differen enors and expiries as prediced by he WG2++ model and by he benchmark models. We show boh claims o ener a swap whose floaing leg is indexed wih he hree-monh Euribor rae and he ones referring o he six-monh Euribor rae. Noice ha only he WG2++ model and he MMG model are able o differeniae beween he wo ypes of swapions. In paricular, we observe ha only he WG2++ model is able o preserve such difference also for longer swapion enors. Indeed, he MMG model produces he spli because of differen iniial yield curves, while he WG2++ model relies also on a dynamical mechanism. 5 Conclusions and furher developmens Ineres-rae modelling requires a framework able o incorporae many iniial yield curves, one for each Libor rae enor plus one for discouning. Classical models may be exended in many ways, bu, unforunaely, he marke is oo young o quoe opions on all enors: only limied number of quoes are available and hey are concenraed only in few enors. Thus, a model, which allows a minimal exension of classical frameworks and, a he same ime, allows for more complex dynamics when quoes will be available, is a relevan ool 20

21 Figure 4: Differences in basis poins beween marke and model-implied volailiies, namely calibraion error in erm of implied swapion volailiies. Upper panel is G2++ model, lower-lef panel is MMG model, and lower-righ panel is WG2++ model. Each panel shows on he lef axis he underlying swap s enor, while on he righ axis is saring ime. for boh quans and praciioners. In his paper we presened an exension of he HJM model which is able o deduce he dynamics of he discouning yield curve and of marke Libor raes of any enor saring from a single family of Markov processes. Furher, we calibrae a simplified version of he model, he Weighed Gaussian model, wih wo driving facors and deerminisic volailiy, o a-he-money swapion prices o size he effec of he new degree of freedom inroduced o model differen enors. We also made a comparison wih wo benchmark models, already published in he lieraure, which urn ou o be special cases of our model. Our nex sep will be he calibraion of a sochasic volailiy version of he model, he Weighed Heson model, o he whole cube of swapion prices, and a he same ime we hope he marke evolves by quoing opions on more enors. 21

22 Figure 5: Each panel shows he implied volailiies by changing he underlying swap s saring ime. Top-lef panel is one-year underlying swap s enor, op-righ wo-year enor, boom-lef five-year enor, boom-righ en-year enor. A Appendix: vecor and marix noaion When we consider a vecor quaniy v, we hink i as a marix wih only one row, if a column vecor is needed we use he ransposiion operaor, namely v. Furher, we inroduce also he vecor whose enries are all of ones and we name i 1. Le us consider wo marix quaniies a and b, whose elemens are respecively a ij and b ij wih 1 i n and 1 j m. We define elemen-wise muliplicaion as he marix ab wih elemens: (ab) ij := a ij b ij and, in he same fashion, also muliplicaion by a vecor v, whose elemens are v i wih 1 i n, or a scalar κ as (va) ij := v i a ij, (κa) ij := κa ij 22

23 while index conracion as he marix a b wih elemens: (a b) jk := n a ij b ik i=1 References [1] M. Bianchei (2009). Two Curves, One Price: Pricing ad Hedging Ineres Rae Derivaives Using Differen Yield Curves for Discouning and Forwarding. Available a hp://ssrn.com/absrac= [2] M. Bianchei (2010). Muliple Curves, One Price: The Pos Credi-Crunch Ineres Rae Marke. Talk kep a Risk and modelling fixed income ineres raes, Marcus Evans conference, London, April. [3] W. Boenkos and W.M. Schmid (2005). Cross currency swap valuaion. Available a hp://ssrn.com/absrac= [4] D. Brigo, and F. Mercurio (2006). Ineres Rae Models: Theory and Pracice wih Smile, Inflaion and Credi, Second Ediion, Springer Verlag. [5] R. Carmona (2004). HJM: a Unified Approach o Dynamic Models for Fixed Income, Credi and Equiy Markes. in Paris-Princeon Lecures on Mahemaical Finance 2004, Springer. [6] C. Chiarella, S.C. Maina and C. Nikiipoulos Sklibosios, (2010). Markovian Defaulable HJM Term Srucure Models wih Unspanned Sochasic Volailiy. Available a hp://ssrn.com/absrac= [7] M. Chibane and G. Sheldon (2009). Building Curves on a Good Basis. Available a hp://ssrn.com/absrac= [8] E. Eberlein, and W. Kluge (2007). Calibraion of Lévy erm srucure models. In M. Fu, R. A. Jarrow, J.-Y. Yen, and R. J. Ellio (Eds.), Advances in Mahemaical Finance: In Honor of Dilip B. Madan, pg Birkhäuser. [9] J. Eisenschimd and J. Tapking (2009). Liquidiy Risk Premia in Unsecured Inerbank Money Markes. ECB Working Paper Series, 1025, 3. [10] E. Errais and F. Mercurio (2005). Yes, Libor Models can Capure Ineres Rae Derivaives Skew: A Simple Modelling Approach. Available a hp://ssrn.com/absrac= [11] C. Fries (2010). Discouning Revisied: valuaion under funding, counerpary risk and collaeralizaion (2010). Available a hp://ssrn.com/absrac=

24 [12] E. Fruchard, C. Zammouri and E. Willems (1995). Basis for change, Risk, Vol. 8, No.10, 70-75, Ocober. [13] M. Fujii, Y. Shimada and A. Takahashi (2010). On he Term Srucure of Ineres Raes wih Basis Spreads, Collaeral and Muliple Currencies. Available a hp://ssrn.com/absrac= [14] P. Hun, J. Kennedy, and A. Pelsser (2000) Markov-funcional ineres rae models, Finance and Sochasics, Springer, vol. 4(4), pages [15] D. Heah, R. Jarrow, A. Moron (1992). Bond Pricing and he Term Srucure of Ineres Raes: a new mehodology. Economerica 60, [16] M. Henrard, M. (2007). The Irony in he Derivaives Discouning. Wilmo Magazine, July 2007, [17] M. Henrard (2009). The Irony in he Derivaives Discouning Par II: The Crisis. Preprin, Dexia Bank, Brussels. [18] C. Kenyon (2010). Shor-Rae Pricing afer he Liquidiy and Credi Shocks: Including he Basis. Available a hp://ssrn.com/absrac= [19] M. Kijima, K. Tanaka and T. Wong (2009). A Muli-Qualiy Model of Ineres Raes, Quaniaive Finance 9(2), [20] A. Lewis (2001). A simple opion formula for general jump-diffusion and oher exponenial Lévy processes. Available a hp://ssrn.com/absrac= [21] T. Marìnez (2009). Drif condiions on a HJM model wih sochasic basis spreads. Available a hp:// [22] F. Mercurio (2009). Ineres Raes and The Credi Crunch: New Formulas and Marke Models. Bloomberg Porfolio Research Paper No FRONTIERS. Available a hp://ssrn.com/absrac= [23] F. Mercurio (2010). LIBOR Marke Models wih Sochasic Basis. Available a hp://ssrn.com/absrac= [24] N.Moreni and M. Morini (2010). A noe on muliple curve erm srucure modeling wih credi/liquidiy spread, Banca IMI Inernal Repor. [25] M. Morini (2009). Solving he Puzzle in he Ineres Rae Marke. Available a hp://ssrn.com/absrac= [26] M. Morini and A. Prampolini (2010). Risky Funding: A unified framework for counerpary and liquidiy charges. Available a hp:// 24

25 [27] A. Papapanaleon (2010).Old and New Approaches o Libor Modeling.Saisica Neerlandica 64, 3. [28] A. Pallavicini and M. Tarenghi (2010). Ineres-Rae Modeling wih Muliple Yield Curves. Available a hp://ssrn.com/absrac= [29] P. Richken and L. Sankarasubramanian (1995). Volailiy Srucures of Forward Raes and he Dynamics of he Term Srucure. Mahemaical Finance 7, [30] M. Torrealba (2010). Modelling he Spread and Applicaions o Callable Spread Opions. Talk presened a 6h WBS Fixed Income Conference, Madrid [31] A. Trolle and E. Schwarz (2009). A General Sochasic Volailiy Model for he Pricing of Ineres Rae Derivaives. Review of Financial Sudies, vol. 22(5), pages