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1 Ciy Research Online Ciy, Universiy of Lonon Insiuional Reposiory Ciaion: Karouzakis, N., Hagioannies, J. & Anriosopoulos, C Convexiy Ajusmen for Consan mauriy Swaps in a Muli-Curve Framework. Annals of Operaions Research, oi: 1.17/s This is he publishe version of he paper. This version of he publicaion may iffer from he final publishe version. Permanen reposiory link: hp://openaccess.ciy.ac.uk/17896/ Link o publishe version: hp://x.oi.org/1.17/s Copyrigh an reuse: Ciy Research Online aims o make research oupus of Ciy, Universiy of Lonon available o a wier auience. Copyrigh an Moral Righs remain wih he auhors an/or copyrigh holers. URLs from Ciy Research Online may be freely isribue an linke o. Ciy Research Online: hp://openaccess.ciy.ac.uk/ publicaions@ciy.ac.uk

2 DOI 1.17/s ANALYTICAL MODELS FOR FINANCIAL MODELING AND RISK MANAGEMENT Convexiy ajusmen for consan mauriy swaps in a muli-curve framework Nikolaos Karouzakis 1 John Hagioannies 2 Kosas Anriosopoulos 3 The Auhors 217. This aricle is publishe wih open access a Springerlink.com Absrac In his paper we propose a ouble curving seup wih isinc forwar an iscoun curves o price consan mauriy swaps CMS. Using separae curves for iscouning an forwaring, we evelop a new convexiy ajusmen, by eparing from he resricive assumpion of a fla erm srucure, an expan our seing o incorporae he more realisic an even challenging case of erm srucure ils. We calibrae CMS spreas o marke aa an numerically compare our ajusmens agains he Black an SABR sochasic alpha bea rho CMS ajusmens wiely use in he marke. Our analysis suggess ha he propose convexiy ajusmen is significanly larger compare o he Black an SABR ajusmens an offers a consisen an robus valuaion of CMS spreas across ifferen marke coniions. Keywors Convexiy ajusmen Consan mauriy swaps Muli-curve framework Yiel curve moelling Money marke insrumens 1 Inroucion The recen financial crisis has le, among ohers, o unpreceene behavior in he money markes, which has creae imporan iscrepancies on he valuaion of ineres rae financial insrumens. Imporan reference raes ha use o be highly correlae an moving ogeher for a long perio of ime, sare o iverge from one anoher. A characerisic example, B Nikolaos Karouzakis N.Karouzakis@sussex.ac.uk John Hagioannies j.hagioannies@ciy.ac.uk Kosas Anriosopoulos kanriosopoulos@escpeurope.eu 1 School of Business, Managemen an Economics, Universiy of Sussex, Brighon BN1 9SL, UK 2 Cass Business School, Ciy Universiy Lonon, 16 Bunhill Row, Lonon EC1Y 8TZ, UK 3 ESCP Europe Business School, 527 Finchley Roa, Lonon NW3 7BG, UK

3 ha has been wiely suie recenly, is he wiening of he sprea beween eposi raes Libor/Euribor an overnigh inex swap OIS raes of he same mauriy. A he same ime, he marke sare observing non-zero spreas beween swap raes of he same mauriy, bu base on ifferen frequencies of he unerlying Libor rae, or beween forwar rae agreemen FRA raes an forwar raes implie by consecuive eposis. These examples inicae ha financial players consier each enor as a separae marke, incorporaing ifferen crei an liquiiy premia, an as such, each one of hem is riven by is own ynamics. Such iscrepancies have, above all, quesione he mehoology use o boosrap he yiel curve, which has creae a layer of uncerainy on he mehos use o price an hege ineres rae financial insrumens. There are hree main issues associae wih he pre-crisis approach, which make i inconsisen. Firs, he informaion incorporae ino he basis spreas is no aken ino accoun. Secon, using a single yiel curve oes no allow us o consier he ifferen ynamics inrouce by each unerlying rae enor, making heging an pricing of ineres rae erivaives less sable. Finally, he no-arbirage assumpion inicaes ha a unique iscouning curve nees o be use, regarless of he number of he unerlying enors. In orer for marke paricipans o comply wih he menione marke feaures, hey sare builing a separae forwar curve for each given enor, so ha fuure cash flows are generae using he appropriae curve associae wih he unerlying rae enor. A he same ime, a single an unique iscouning curve ha o be use, in orer o calculae he presen value of conrac s fuure paymens. This le financial players o sar using he OIS swap curve, raher han he Libor curve, for he consrucion of a riskless erm srucure. The reason behin heir choice was mainly wofol. Firs, OIS is believe o conain very lile crei an liquiiy risk premia compare o Libor raes. Secon, he fac ha mos raes in he ineres rae marke are mainly cash collaeralize makes he funing cos for a financial insiuion no longer equal o he Libor rae, bu o he collaeral rae insea. For ha reason, he Libor rae ha was wiely use as a proxy for he risk-free iscouning rae, is now replace by he collaeral rae, which is assume o coincie wih he overnigh rae i.e. fe fun rae for USD, Eonia for EUR, ec.. The lieraure on he valuaion of ineres rae erivaives base on separae curves, for generaing fuure raes an for iscouning, is growing rapily. Previous conribuions focus on he valuaion of cross currency basis swaps see, Boenkos an Schmi 25; Kijima e al. 29; Fujii e al. 21; Henrar 21. Henrar 27b, is he firs o apply his mehoology o he single currency case, whereas Bianchei 21 is he firs o eal wih he pos-crisis siuaion. Furhermore, Amerano an Bianchei 29, Chibane an Shelon 29 an Morini 29, evelop new mehoologies for boosrapping muliple ineres rae yiel curves. On he oher han, many conribuions focus on exening pricing moels uner he muli-curve framework. Kijima e al. 29, apply he mehoology o suy wo shor rae moels, he Vasicek moel an he quaraic Gaussian moel, an use hem for he valuaion of bon opions an swapions. Mercurio 29, 21anGrbac e al. 215 exen he libor marke moel LMM o be compaible wih he muli-curve pracice an price caples an swapions, while more recenly, Pallavicini an Tarenghi 21, Crépey e al. 212, Moreni an Pallavicini 214 ancuchiero e al. 216 exen he classical Heah Jarrow Moron HJM framework o incorporae muliple curves in orer o price ineres rae proucs such as forwar saring ineres rae swaps IRS, plain vanilla European swapions an CMS sprea opions. Finally, imporan conribuions inclue Crépey e al. 215 who evelop a Levy-base HJM moel for crei value ajusmen CVA an Fanelli 216who evelop a efaulable HJM moel for pricing basis swaps in a muli-curve seup. In his paper, we follow he approach escribe in Mercurio 21 an Pallavicini an Tarenghi 21 o price a CMS. A CMS exchanges a swap rae wih a fixe ime o mauriy

4 agains fixe or floaing. In a common CMS, one woul swap a quarerly e.g. 3-monh Libor or semi-annual rae agains a 5 or 1-year swap rae. Whereas a regular floaing rae e.g. 6-monh Libor conains informaion abou shor-erm ineres raes, a CMS rae e.g. 1-year swap rae conains informaion abou he overall level of he yiel curve. This makes CMS a popular insrumen among invesors an porfolio managers. I gives invesors he abiliy o place bes on he shape of he yiel curve over ime. Generally, a consan mauriy payer will benefi from a flaening or inversion of he yiel curve an is expose o he risk of he yiel curve seepening. I also helps porfolio managers o hege a floaing rae eb wihou inroucing uraion risk from he heging insrumen. The mix of shor an long erm raes in he srucure of he CMS makes is value epen on he shape of he yiel curve. Sanar approaches for is valuaion involve he calculaion of a convexiy ajusmen. Such convexiy ajusmen canno be compue exacly, so previous lieraure uses eiher ahoc approximaions or uilising unrealisic assumpions. A common assumpion use in he relevan lieraure is ha he erm srucure of ineres raes is fla an only parallel shifs are allowe. There are wo main avenues owars pricing a CMS. In he firs, one ses up a erm srucure moel an uses some approximaion meho o compue he expece swap rae, uner he forwar measure. More specifically, Lu an Nefci 23 follow his irecion an work wih wo or more forwar raes joinly. Using he forwar libor moel, hey price a CMS swap an compare is empirical performance wih he sanar convexiy ajusmen propose by Hagan 25. They fin ha he convexiy ajusmen overesimaes CMS swap raes. Similarly, Henrar 27a uses one-facor LMM an HJM moels o approximae CMS swaps, while Brigo an Mercurio 26 use a wo-facor Gaussian shor rae moel G2++ moel o moel bon prices associae wih CMS proucs. Finally, in a recen work, Wu an Chen 21 price ifferen CMS-ype ineres rae erivaives wihin he LMM framework. They presen a new approach for fining he approximae isribuion of a CMS uner he forwar maringale measure. In he secon irecion, one uses replicaion argumens an he problem is formulae uner he swap measure. The price is base on he implie swapion volailiies which play he role of he isribuion of swap raes. For he replicaion proceure, he change from he forwar o he swap measure is neee an he Raon Nikoym erivaives nee o be approximae. Pelsser 23 is he firs o show ha he convexiy ajusmen can be inerpree as he sie effec of a change of numeraire. He approximaes he measure change by proposing a linearizaion of he swap rae an obains analyical soluions o he CMS price. Hagan 25 obains close-form formulae for he pricing of CMS swaps an opions by relaing hem o he swapion marke via a saic replicaion approach. Finally, Mercurio an Pallavicini 26 use a srike exrapolaion o saically replicae CMS swap/opions by moelling implie volailiies of European swapions using he SABR moel of Hagan e al. 22. Finally, in a recen work, Zheng an Kuen Kwok 211 propose a generalise saic replicaion approach o hege exoic swap conracs an annuiy opions using ifferen swapions. The main problem wih previous conribuions is ha he yiel curve is assume o be fla an only parallel shifs are allowe. However, in a swap where one pays Libor plus sprea an receives a 1-year CMS rae, he srucure is mainly sensiive o he slope of he ineres rae yiel curve an is almos immunise agains any parallel shif. In his paper, following Hagan 25, we apply he commonly use convexiy ajusmen in a new framework of ouble curving. We hen evelop a new convexiy ajusmen, by eparing from he resricing assumpion ha he erm srucure is fla, an we allow for a il. Using marke aa for Euro money marke insrumens Eonia, Euribor, CMS spreas an swapion volailiies, we fin

5 ou ha he new convexiy ajusmen is significanly larger han he one commonly mae in he lieraure. We finally compare our approach wih he SABR CMS ajusmen, inrouce by Mercurio an Pallavicini 25, 26, which is wiely use in he marke, an we fin ha our approach provies a beer fi o he marke s CMS sprea prices. The remaining of his paper is organise as follows. Secion 2 presens he valuaion framework for he main insrumens FRA, IRS, CMS consiere. Secion 3 shows he main resul of our work, ha is a new convexiy ajusmen ha akes ino accoun he il in he erm srucure, uner a ouble curving framework. Secion 4 briefly epics he smile-consisen convexiy ajusmen using SABR moel. Secions 5 an 6 presen he marke aa ha have been use an escribe numerical calculaions. Finally, Sec. 7 conclues. 2 The valuaion framework This secion inrouces he efiniions of basic insrumens uner he mui-curve environmen. I mosly follows he works of Brigo an Mercurio 26anMercurio 29. We inrouce wo curves, one for he iscouning process, say curve, an one for he forwaring, say curve f. Forwar raes can be efine for boh curves. Le us ake oay being ime zero an consier a enor srucure {T i } i=,...,n, wih T i < T i+1.leδ i = T i+1 T i be he accrual facor for he ime inerval ] T i, T i+1. Wihin his srucure, for each curve, he ime- wih T value of forwar raes is efine by, F ;, T i = 1 δ i P, P, T i ] 1 where P, T i, wih i = 1,...,n, enoing he ime- price of he T i -mauriy iscoun bon. Furhermore, we enoe by Q T i he T i -forwar probabiliy measure associae wih he relae expecaion. We assume a given single iscoun curve for use in he calculaion of all ne presen values NPVs, i.e., for iscouning all fuure cash flows. This curve is assume o be he OIS zero-coupon curve, srippe from marke OIS swap raes an is efine for every possible mauriy T i. All pricing measures we will consier are hose associae wih he OIS iscoun curve. Following Mercurio 21, we aop he sanar efiniion for he FRA rae. he numeraire P, T i, an by E QT i Definiion 1 Consier imes, T 1, T 2, wih T 1 < T 2. The ime- FRA rae FRA; T 1, T 2 is efine as he fixe rae o be exchange a ime T 2 for he Libor rae LT 1, T 2,soha he swap has zero value a ime. By no-arbirage pricing we ge, FRA; T 1, T 2 = E QT 2 LT 1, T 2 F ] 2 where Q T 2 enoing he T 2-forwar measure associae wih he numeraire P, T 2, E QT 2 he relae expecaion an F he informaion available in he marke a ime. Proposiion 1 Any simple compoune forwar rae spanning a ime inerval ening in T i, is a maringale uner he T i -forwar measure, Fu;, T i = E QT i ] F; Ti 1, T i F u, for u < T i. Following Bianchei 21 an Mercurio 21, working uner he single-curve framework, where he forwar an iscoun curves coincie f, from proposiion 1, he 1

6 forwar rae F f ; T 1, T 2 is by consrucion a maringale uner Q T 2, ] 1 P, T 1 F f ; T 1, T 2 = T 2 T 1 P, T 2 1 = E QT 2 LT 1, T 2 F ] 3 where LT 1, T 2 is he spo Libor rae efine by he usual no-arbirage relaionship beween Libor raes an zero coupon bon prices, which hols for non-efaulable counerparies an insrumens wih no liquiiy risk, ] 1 1 LT 1, T 2 = T 2 T 1 P T 1, T 2 1 = F f T 1 ; T 1, T 2 4 Base on ha, we can conclue ha he FRA rae FRA; T 1, T 2 coincies wih he forwar Libor rae, FRA; T 1, T 2 = F f ; T 1, T 2. In he muli-curve framework, however, Eq. 4 oes no hol. The forwar rae F f ; T 1, T 2 is no a maringale uner he forwar measure Q T 2, an he FRA rae is ifferen from he forwar rae, FRA; T 1, T 2 = F f ; T 1, T 2. Therefore, he presen value of a fuure Libor rae is no longer obaine by iscouning he corresponing forwar rae, bu by iscouning he corresponing FRA rae. Accoring o Mercurio 21, he FRA rae is he naural generalizaion of a forwar rae o he muli-curve case. This has a sraighforwar implicaion, when i comes o he valuaion of Ineres Rae Swaps. 2.1 Ineres rae swap We show how o evaluae an IRS uner he muli-curve framework. For simpliciy, we assume ha IRS enors for fixe an floaing legs are he same. The ime- value wih T of he floaing leg payoff is calculae by aking he iscoune expecaion uner he forwar measure Q T i+1, δ i P, T i+1 E QT i+1 ] LTi, T i+1 F 5 Using Eq. 2, he presen value of he swap s floaing leg is given as, n 1 δ i P, T i+1 FRA; T i, T i+1 6 i= Similarly, he value of he swap s fixe leg is given by he presen value of he fixe coupon paymens, K, pai on he fixe legs aes as, n 1 K δ i P, T i+1 7 i= Thus, he ime- value of he IRS o he fixe rae payer is given by, n 1 n 1 IRS, K ; T i = δ i P, T i+1 FRA; T i, T i+1 K δ i P, T i+1 8 i= I follows ha he fair forwar swap rae ha equaes he wo legs a ime T is, n 1 i= S; T, T n = δ i P, T i+1 FRA; T i, T i+1 n 1 i= δ 9 i P, T i+1 This is he forwar swap rae of an IRS, where cash flows are generae hrough curve f an iscoune wih curve. i=

7 2.2 Consan mauriy swap A consan mauriy swap conrac, is a swap where one of he legs pays receives perioically a swap rae wih a fixe ime o mauriy, c, while he oher leg receives pays eiher fixe or floaing. More commonly, one erm is se o a shor erm floaing inex such as he 3-monh Libor rae, while he oher leg is se o a long erm fixe rae such as he 1-year swap rae. Le { i, j } j=,...,c, be a se of rese aes, associae wih imes T i, i =,...,n, wih i, T i, T i+1 ] an i, j i, j 1 =, wih j = 1,...,c. We suppose ha = 6 monhs an assume for simpliciy ha i, = T i, an we will se Δ = δ/, wih Δ nee no be inegral. The forwar swap rae of he i h IRS a ime eermine by i, j, a ime i, enoe as, S i, j S, i,, i,c,isgivenby, S i, j = c 1 j= P, i, j+1 FRA; i, j, i, j+1 c 1 j= P, i, j+1 where P, i, j enoing he ime- price of he i, j -mauriy iscoun bon. Furhermore, we enoe by Q i, j he i, j -forwar swap probabiliy measure associae wih he numeraire P i, j = c j=1 P, i, j, an by E Q i, j he relae expecaion. Consier now, a c-year CMS saring a T wih paymen aes T i, i = 1,...,n. A each paymen ae T i, one pary pays receives δl Ti 1, T i + δr ] o is counerpary an receives pays δs Ti 1 1, ] from is counerpary, where, L Ti 1, T i enoes he δ-monh e.g. 3-monh spo Libor rae reseing a anapplieoaδ perio, T i ], S Ti 1 1,, wih j = 1,...,c, ishec-year spo swap rae, reseing every = 6-monhs a = i 1, an applie o a δ perio, T i ] an R,ishe CMS premium sprea, a consan chosen so ha he cos of he insrumen a ime,when he conrac is iniiae, is zero. For simpliciy, we wrie S for S Ti 1 1, an L T i for L Ti 1, T i. A ime T i, he CMS pays cashflow c i as, c i = δ S L T i R ] 11 where we suppose ha he counerpary pays floaing i.e. Libor + sprea an receives fixe i.e. he swap rae. The ime- value, wih, of he CMS can be obaine by aking he iscoune expecaion E QT i uner he forwar measure Q T i corresponing o he numeraire P T i = P, T i,as, V T i = P T i = P T i δ δe QT i E QT i S L T i R ] S ] FRA;, T i R where, for he FRA, we follow Eq. 2. A his poin i is imporan o emphasize he fac ha, naurally, he expecaion use o calculae he above payoff, is associae wih he paymen aes T i. However, uner he forwar measure, Q T i,heswaprae,s, is no a maringale. The convexiy ajusmen arises since he expece payoff is calculae in a worl which is forwar risk neural wih respec o a zero coupon bon. In ha worl, he expece unerlying swap rae upon which he payoff is base, oes no equal he forwar swap rae. The convexiy is jus he ifference beween he expece swap rae an he forwar swap rae. 1 12

8 When we consier pricing CMS-ype erivaives, i is convenien o compue he expecaion of he fuure CMS raes uner he forwar measure, ha is associae wih he paymen aes. However, he naural maringale measure of he CMS rae is he unerlying forwar swap measure. Convexiy correcion arises when one compues he expece value of he CMS rae uner he forwar measure ha iffers from he naural swap measure wih he unerlying forwar swap measure as numeraire. 3 Convexiy ajusmen Following Pelsser 23, we efine he convexiy ajusmen as he ifference in expecaion of some quaniy i.e., swap rae when he expecaions are compue uner wo ifferen measures i.e., forwar an swap measures. Therefore, expecaion E QT i S ] can be wrien as an expecaion which is a maringale uner is measure plus an ajusmen. This means ha he convexiy ajusmen is given as he ifference in expecaion uner he forwar measure an he forwar swap measure of he forwar swap rae, as, CA = E QT i S ] E Q i, j S ] 13 where uner he forwar swap measure Q i, j, corresponing o he numeraire P i, j = cj=1 P, i, j, he forwar swap rae, S, is a maringale. Assuming ha he convexiy ajusmen CA is known, he curren value of he CMS is given from Eqs. 12an13by, V T i = P T i δ E Q i, j = P T i δ S ] + CA FRA;, T i R S + CA FRA;, T i R 14 wherewehaveusehas is a maringale uner he forwar swap measure Q i, j an he presen value of he CMS is given by, n n PV = V T i = P T i δ S + CA FRA;, T i R ] 15 i=1 i=1 Given ha he cos of he CMS a ime- is zero, we have, ni=1 P T i R δ S + CA ] ni=1 δ P T i FRA;, T i = ni=1 δ P T i ni=1 δ P T 16 i where he secon par of he equaion is he forwar swap rae efine in Eq. 9. The convexiy ajusmen CA is eermine by changing numeraire in he firs erm of Eq. 13as, ] CA = E Q i, j G i Ti S where for i {1,...,n} we have efine he funcion G i = P T i P i, j. The convexiy ajusmen is approximae by approximaing he G i erm. G i

9 3.1 Fla erm srucure wih parallel shifs Following Hagan 25 anbrigo an Mercurio 26, we iniially erive an expression for he convexiy ajusmen when he erm srucure is fla an can only evolve wih parallel shifs. We enoe by r, he ime- value enor spo rae. For, he wo numeraires are given as, P T i = r Δ P 18 an, c ] P i, j 1 = 1 + r j=1 j P = P r 1 + r c 19 Using Eqs. 18an19, funcion G i is given as, G i = P T i P = P 1 + r Δ r 1 ] = i, j P 1 1 r r 1+r c Δ r c := Gr 2 An imporan assumpion we make when we work uner muliple curves, is he fac ha he swap rae is no a risk-free rae anymore. More specifically, following Liu e al. 26 an Filipović an Trolle 213 among ohers, we assume i o be equal o he risk-free rae r assume o be he OIS rae plus a sprea X, wrien as, S i, j = R = r + X 21 where, X is he sprea incorporaing he crei an liquiiy risk premia of he counerpary an R is he risky forwar swap rae efine in Eq. 1. Wha is worh menioning a his poin, is he fac ha uner he assumpion of a fla erm srucure, he forwar rae FRA; i, j, i, j+1, oes no epen on j i.e. assume o be consan. We approximae G using a firs-orer Taylor expansion as, Gr Ti 1 Gr 1 G r G S Gr r X r = GS S T X i 1 X Ti 1 S ] + X 22 Using Eqs. 17an22, he convexiy ajusmen can be approximae as, CA G S X i, j = GS X EQ S 2 S S ] + X Ti 1 X 2 1 = S E Q i, j 2 G S X GS X E Q i, j S X Ti 1 X 23 S 2 S S S In he above expression, here are wo expecaions we nee o calculae, E Q i, j an E Q i, j S ] X Ti 1. We assume ha uner Q i, j S ] 2 he wo processes i.e. swap rae an sprea, which are log-normal wih consan volailiy, are maringales an are of he form,

10 S = σ,s S W,S,forheswapraeanX = σ,x X W,X, for he sprea, where W,S an W,X are wo correlae wiener processes wih correlaion ρ s,x an σ,s an σ,x are eerminisic volailiies. Applying Io s Lemma, he wo expecaions are given as see Appenix 1, E Q i, j E Q i, j S ] 2 = S 2 exp σ,s 2 Ti 1 24 S X Ti 1 ] = S X exp ρ s,x σ,s σ,x 25 Using he wo expecaions, he convexiy ajusmen is given as, CA = K r e σ,s 2 1 X S e ρ s,x σ,s σ,x 1 ] 26 wih, K r = S 2 r r 1 + δr cr 1 + r c 1 27 Assuming ha here is no sprea in he marke i.e. if X =, we en up wih he well-known Black-like ajusmen formula propose by Hagan A erm srucure wih ils In his secion, we epar from he resricive an unrealisic assumpion of a fla erm srucure an we exen our analysis by allowing for a il. Since we no longer assume a fla erm srucure, he spo rae r T is now given by some eerminisic funcion f as follows, r T = f r,, T a 28 where r is he shor rae an a = a 1,...,a k is some vecor of parameers. The G funcion we nee o approximae is now given by, G i = P T i P i, j = cj=1 1 + r T i 1 + r i, j T i i, j = cj=1 T 1 + f T i i r 1 + f i, j i, j r 29 where, for simpliciy, we have wrien f T i r for r T i = f r,, T i an f i, j r for r i, j = f r,, i, j. As in he fla erm srucure case, we approximae G using a firs-orer Taylor expansion a r, as, Gr Ti 1, Gr, 1 G r r, Gr, r r + G r, Gr, 3 where G r an G enoe he parial erivaives of G wih respec o r an. In he previous caseweassumehas i, j = R. In he curren case, where we allow for a il in he erm srucure, we assume he following approximaions, S i, j R an S i, j R Ti 1.

11 Using Eqs. 17an3, he convexiy ajusmen can be approximae as, 2 CA = S 2 Gr r, E Q i, j S 1 Gr, S 2 Gr r, Gr, E Q i, j + G r, Gr, S 2 Gr r, Gr, S S X Ti 1 S 2 S X S As before, we assume ha uner Q i, j he wo log-normal wih consan volailiy processes i.e. swap rae an sprea are maringales an he expecaions are given by Eqs. 24 an 25. So, he convexiy ajusmen is now given by, CA = S e σ,s 2 1 X ] e ρ s,x σ,s σ,x 1 + G r, Gr, S 31 We also nee o calculae he wo erms G r r, Gr, an G r, Gr, ha incorporae he parial erivaives. Analyical expressions are given in Appenix 2. Finally, for funcion f we use he following parameric funcional form, base on he well-known Nelson an Siegel moel. f r,, T a, b, k = r + a + bt e kt a 32 wih, f T i r r = f i, j r r = 1 f T i r = ka + bt i be kt i r = ka + b i, j be k i, j f i, j 33 4 Smile-consisen convexiy ajusmen In orer o es he propose CMS convexiy ajusmens, we compare hem wih he smileconsisen convexiy ajusmen, which is wiely use in he marke. In he presence of a marke smile, when he erm srucure is no fla, bu may il, he ajusmen is necessarily more involve, if we aim o incorporae consisenly he informaion coming from he quoe implie volailiies. The proceure o erive a smile consisen convexiy ajusmen is escribe in Mercurio an Pallavicini 26 anpallavicini an Tarenghi 21, an is he one we will use here. For he consisen erivaion of CMS convexiy ajusmen, volailiy moelling is require. We use he SABR moel a popular marke choice for swapion smile analysis for he swap rae in orer o infer from i he volailiy smile surface. The SABR moel assumes ha S evolves uner he associae forwar swap measure Q i, j accoring o,

12 S = V S V = ɛv W V = α β Z 34 where, Z an W are Q i, j -sanar Brownian moions wih, Z W = ρ 35 an where β, 1], ɛ an α are posiive consans an ρ -1, 1]. The CMS convexiy ajusmen is given in Mercurio an Pallavicini 26as, CA SABR S ; δ = θ S 2 2 Black K, S,v imp K, S K 1 36 S where, lns/k + v 2 /2 lns/k v 2 /2 BlackK, S,v:= SΦ K Φ 37 v v v imp K, S := σ imp K, S Ti 1 38 An approximaion for he implie volailiy of he swapion wih mauriy is erive in Hagan e al. 22as, σ imp K, S α S 1 β ] z 2 K β2 24 ln 2 S K + 1 β4 192 ln4 S xz K β 2 α 2 ρβɛα 24 S 1 β + K 4 S + ɛ 2 2 3ρ2 1 β 2 24 K 39 where an z := ɛ α S 1 β 2 S K ln K { } 1 + 2ρz + z xz := ln 2 + z ρ 1 ρ 4 41 The above formula provies us wih an efficien approximaion for he SABR implie volailiy for each srike K. We consier a ifferen SABR moel for each swap rae conaine in he CMS payoff an we perform a calibraion of all he SABR parameers four parameers α,β,ρ,ɛ for each swap rae o swapion volailiy smile an CMS sprea quoe in he marke. See Mercurio an Pallavicini 26 an Pallavicini an Tarenghi 21 for a eaile escripion of he calibraion proceure.

13 5 Marke aa We use hree aa ses for his suy, one conaining Euro money marke insrumens for he consrucion of he yiel curves, a secon one conaining CMS swap spreas wih a mauriy of 5-years, where he associae unerlying swaps have a 1-year mauriy i.e. X 5,1, an a hir one conaining swapion volailiies for ifferen srikes, as well as implie black a-he-money ATM swapion volailiies. All marke aa was collece from Bloomberg. Daa ses are presene in eail below: For he iscouning curve, we use Eonia Fixing an OIS raes from 3-monhs o 3-years. For he 3-monh curve, we use Euribor 6-monhs fixing, FRA raes up o 15 monhs, an swaps from 2 o 3 years, paying an annual fix rae in exchange for he Euribor 3-monh rae. For he 6-monh curve, we use Euribor 6-monhs fixing, FRA raes up o 2 years, an swaps from 2 o 3 years, paying an annual fix rae in exchange for he Euribor 6-monh rae. The marke quoes a value for he CMS sprea which makes he CMS swap fair. However, i quoes he sprea only for a few CMS swap mauriies an enors usually 5, 1, 15, 2 an 3 years. In he Euro marke, he CMS enor is equal o 3 monhs, while he c-year IRS which is use as inexaion in he CMS has Libor paymens of 6-monhs or 1-year frequency. Thus, CMS spreas epen on hree ifferen curves in our framework; firs, he funing curve use o iscoun he cash flows of he CMS swap, which we consier o be he risk-free curve i.e. OIS curve; secon, he 3-monh forwaring curve for he Euribor raes pai in he secon leg of he CMS; an hir, he 6-monh or 1-year forwaring curve for he Euribor raes pai by he inexaion IRS. 6 Empirical resuls In his secion, we compare numerically he accuracy of he approximaions for he CMS convexiy ajusmens agains he Black an SABR moels convexiy ajusmens presene in Sec An empirical illusraion Our firs numerical example is base on Euro aa as of 3 February 26. We es a CMS wih mauriy of 5 years i.e. nδ = 5, where he associae unerlying swaps have mauriy of 1 years i.e. c = 1. The closing price for he CMS sprea is X 5,1 = 64.9 basis poins bps. The ATM swapion volailiy is σ5,1 AT M =.15, an swapion volailiies for ifferen srikes are given in Table 1. For he parameers of he erm srucure in case 2, we choose he values: a, b, k =.1,.2,.1. Finally, when we apply he case wih he sprea, we assume ha he sprea is consan a X = 1 bps, while is volailiy is σ,x =.1, an he correlaion is ρ s,x =.9. The calibraion proceure is performe by minimising he square ifference beween CMS spreas an swapion volailiies an he marke aa. Our resuls are summarize in Tables 2, 3 an 4. We enoe by case 1, he Black-like fla erm srucure convexiy ajusmen of Eq. 26 an by case 2, he il erm srucure convexiy ajusmen of Eq. 31. For he Black-like convexiy ajusmen, we se X =, in Eq. 26, while for he SABR moel, we use Eq. 36.

14 Table 1 This able repors marke quoes of swapion volailiies in % for ifferen srikes K Expiry Tenor years 1 years 6.54% 2.3%.93%.41%.3%.51%.68%.39% Each srike inicaes he ifference from he ATM volailiy Table 2 This able presens he marke price of he CMS sprea agains he calibrae price of he sprea using he case 1 Black-like convexiy ajusmen parallel shif, he case 2 secon convexiy ajusmen allows for il an he SABR moel Marke Case 1 Case 2 SABR Price Difference in bps The ifferences beween he marke price an he hree ifferen moels are provie in basis poins bps. In his case he sprea, X of he swap rae is no aken ino accoun Table 3 This able presens he marke price of he CMS sprea agains he calibrae price of he sprea using he case 1 Black-like convexiy ajusmen parallel shif, he case 2 secon convexiy ajusmen il an he SABR moel Marke Case 1 Case 2 SABR Price Difference in bps The ifferences beween he marke price an he hree ifferen moels are provie in basis poins bps. We incorporae he swap sprea, X, by assuming ha he swap rae is equal o he risk-free rae plus he sprea Our numerical resuls sugges ha in all cases marke aa are well reprouce. As Table 2 repors, he SABR moel performs slighly beer han our new convexiy ajusmen case 2, wih.89 bps compare o.83 bps, when he sprea is no aken ino accoun, an much beer compare o he Black-like formula case 1,.83 bps agains 2.53 bps. However, his is no he case when we ake ino accoun he swap sprea. The absolue ifference in bps beween our new convexiy ajusmen moel an he marke is significanly smaller compare o he SABR case, i.e..1 bps compare o.83 bps. Furhermore, as expece, Black s moel calibraion resuls, alhough beer han he non-sprea case 1.38 bps agains 2.53 bps, sill fail o fi he aa compare o he oher wo cases. In aiion, in Table 4,we repor he convexiy ajusmens for all four cases. We observe ha he convexiy ajusmen in he il case is significanly larger han he fla case, especially, when he swap sprea is incorporae. Furhermore, in he non-fla case, convexiy ajusmen presens a curvy shape compare o he earlier Black-like case, where he shape behaves in a more saic way. 6.2 Numerical examples In orer o furher es he accuracy of he approximaions for he CMS convexiy ajusmens agains he Black an SABR moels convexiy ajusmens, we calibrae he moels o ifferen aes spanning he perio from 27 o 212. This perio covers he mos ineresing phases of he unfoling of he global financial crisis an, as such, we can erive safer conclusions of how he propose convexiy ajusmens perform uner ifferen marke coniions i.e. perios of sabiliy an marke urmoil. In Table 5, we presen marke aa for CMS

15 Table 4 This able presens he convexiy ajusmen for all four ifferen cases, i.e. he Black-like fla erm srucure wih an wihou sprea, an he il erm srucure wih an wihou sprea i CA case 1 CA case 2 CA case 1 wih sprea CA case 2 wih sprea The whole srucure from year 1 o year 2 is given Table 5 This able repors marke CMS swap spreas in bps an marke ATM swapion volailiies aa for specific aes CMS swap spreas have cms mauriy of 5 years an associae unerlying swaps wih mauriy of 1 years Dae Marke CMS Marke volailiy % 1/8/ /1/ /5/ /6/ /3/ swap spreas, where he mauriy of he CMS is 5 years, an he associae unerlying swaps have a 1-year mauriy. Furhermore, he marke a-he-money swapion volailiies, for all ifferen aes, are repore, while Table 6 repors marke volailiy smiles across ifferen srikes an for ifferen aes. We can observe ha marke aa clearly show levels of urmoil in he marke uring he perio of he financial crisis. In Table 7, we repor all calibrae parameers. Our resuls for ifferen aes an marke aa are summarize in Table 8, where all prices are given in basis poins. Our numerical resuls sugges ha in each case an in each perio, he convexiy ajusmen in he case of he il erm srucure wih swap sprea incorporae, gives beer resuls in erms of fiing marke CMS spreas. Our new convexiy ajusmen il, gives sufficienly accurae an robus resuls across all marke scenarios sabiliy an urmoil an sprea levels.

16 Table 6 This able repors marke swapion volailiy smiles for ifferen aes Dae 2 % 1 % 5 % 25 % 25 % 5 % 1 % 2 % 1/8/ /1/ /5/ /6/ /3/ Srikes are expresse as absolue ifferences in basis poins w.r.. he ATM values Table 7 This able presens he moel parameers, obaine from he calibrae proceure, of he funcions escribe in Secs. 3 an 4 for he ifferen aes in our sample Dae 1/8/27 31/1/28 28/5/21 3/6/211 9/3/212 SABR parameers alpha bea rho epsilon f Nelson Siegel a b k Table 8 This able compares marke CMS sprea prices agains each ifferen case Dae 1/8/27 31/1/28 28/5/21 3/6/211 9/3/212 Marke Case 1 Fla Case 2 Til SABR Mk SABR Mk Til Absolue ifferences in bps beween marke CMS swap spreas an he moels are given Even in he perio of , where marke was experiencing an unpreceene urbulence, our convexiy ajusmen performs well, since he ifference beween he marke aa an he SABR moel is sufficienly higher han he case of he il erm srucure, wih a ifference of aroun 3 5 basis poins for he il compare o 7 9 basis poins for he SABR. Furhermore, in every case he resuls are in beween he limis of he bi-ask sprea of aroun 1 basis poins, inicaing ha he marke aa are well recovere across all perios. Finally, convexiy ajusmens for all ifferen cases, he Black-like fla erm srucure in re colour, he il erm srucure in green colour an he SABR moel in blue colour, are presene in Figs. 1, 2, 3, 4, 5 an 6. All cases ake ino accoun he sprea

17 Fig. 1 This figure shows he convexiy ajusmens for he fla erm srucure re, he il erm srucure green an he SABR moel blue. In all cases he swap sprea is aken ino accoun. In he lower panel,he calibrae volailiy smile is isplaye. Resuls are from a specific ae, 3/2/26. Colour figure online Fig. 2 This figure shows he convexiy ajusmens for he fla erm srucure re, he il erm srucure green an he SABR moel blue. In all cases he swap sprea is aken ino accoun. In he lower panel,he calibrae volailiy smile is isplaye. Resuls are from a specific ae, 1/8/27. Colour figure online on he swap rae. Furhermore, he oucome of he calibraion proceure uner he SABR moel i.e. he whole volailiy smile agains ifferen srikes is presene in he lower panel of he figures, where we observe ha he SABR moel is perfecly calibrae across ifferen aes. The only excepion is Ocober of 28, i.e. he peak of he financial crisis, where markes were uner severe pressures, ha he SABR moel sruggles o fi he volailiy smile. Regaring he convexiy ajusmens, in all cases an across ifferen perios, we epic similar characerisics. We observe ha convexiy ajusmen wih il erm

18 Fig. 3 This figure shows he convexiy ajusmens for he fla erm srucure re, he il erm srucure green an he SABR moel blue. In all cases he swap sprea is aken ino accoun. In he lower panel,he calibrae volailiy smile is isplaye. Resuls are from a specific ae, 31/1/28. Colour figure online Fig. 4 This figure shows he convexiy ajusmens for he fla erm srucure re, he il erm srucure green an he SABR moel blue. In all cases he swap sprea is aken ino accoun. In he lower panel,he calibrae volailiy smile is isplaye. Resuls are from a specific ae, 28/5/21. Colour figure online srucure is significanly larger han in he oher wo cases. Furhermore, he shape of he non-fla case presens a slope compare o he Black-like case where he convexiy ajusmens are fla an saic. This helps he moel perform well, especially in perios of marke urmoil.

19 Fig. 5 This figure shows he convexiy ajusmens for he fla erm srucure re, he il erm srucure green an he SABR moel blue. In all cases he swap sprea is aken ino accoun. In he lower panel,he calibrae volailiy smile is isplaye. Resuls are from a specific ae, 3/6/211. Colour figure online Fig. 6 This figure shows he convexiy ajusmens for he fla erm srucure re, he il erm srucure green an he SABR moel blue. In all cases he swap sprea is aken ino accoun. In he lower panel,he calibrae volailiy smile is isplaye. Resuls are from a specific ae, 9/3/212. Colour figure online 7 Conclusion In his paper we have evelope a new CMS convexiy ajusmen in a ouble-curve framework, ha separaes he iscouning an forwaring erm srucures. The moivaion of our suy comes from he unpreceene increase in he Libor-OIS sprea ha was experience uring he financial crisis, which has quesione he legiimacy of consiering boh Libor an OIS quoes as risk-free, an has raise vali issues in he consrucion of zero-coupon

20 curves, which clearly, can no longer be base on raiional boosrapping proceures. In ha vein, our work fills he gap of he shorcomings of single yiel curve moel ajusmens, wiely use in he lieraure, when one eals wih he issue of convexiy in money marke insrumens. In he ouble-curving environmen ha we escribe, we have erive he convexiy facor requiremen in he convenional case ha he erm srucure of ineres raes is fla, an is ynamic evoluion allows only for parallel shifs, an we have expane our seing o incorporae he more realisic an challenging case of erm srucure ils. The new erm appears o be approximaely linear in his parameer. In all compuaions, our resuls conclue ha he convexiy ajusmen of he il erm srucure case is significanly larger han he convexiy ajusmens implie by he Black an SABR moels. As an empirical illusraion, we have calibrae boh convexiy ajusmens o real marke aa, by using swapion volailiies, an calculae he ifferences beween marke quoes an our moel implie CMS spreas. We furher compare our resuls wih he wiely use by marke praciioners smile-consisen CMS ajusmen, using he SABR moel. We consiere a ifferen SABR moel for each swap rae conaine in he CMS payoff, an we performe a calibraion of all he SABR parameers o swapion volailiy smile an CMS spreas quoe by he marke. In all cases he swapion volailiy smiles are very well recovere by he calibrae SABR moels. Furhermore, our resuls emonsrae ha he propose convexiy ajusmens offer a marke consisen an robus valuaion of CMS spreas, an sugges ha CMS-ype of proucs shoul be price uner a muli-curve framework. Open Access This aricle is isribue uner he erms of he Creaive Commons Aribuion 4. Inernaional License hp://creaivecommons.org/licenses/by/4./, which permis unresrice use, isribuion, an reproucion in any meium, provie you give appropriae crei o he original auhors an he source, provie a link o he Creaive Commons license, an inicae if changes were mae. Appenix 1: Expecaion of swap rae an sprea To calculae E Q i, j S ] X Ti 1,weleY = lns Y = Y S S + Y X + 1 X 2 X, an we apply Io s Lemma. 2 Y S S Y 2 X 2 X 2 2 Y + S S X X = σ,s W,S + σ,x W,X 1 σ,s 2 + σ,x 2] So, ln S X = σ,s W,S + σ,x W,X 1 σ,s 2 + σ 2 ],X 43 2 which means ha, S X Ti 1 = S X e 2 1 ] σ,s 2 +σ,x ]T 2 i 1 σ e,s W Ti 1,S +σ,x W Ti 1,X 44

21 So, E Q i, j S ] ] ] X Ti 1 = E Q i, j S X e 2 1 σ,s 2 +σ,x 2 E Q i, j ] e σ,s W Ti 1,S+σ,X W Ti 1,X 45 In general, if: X N,σ 2 X, Y N,σ2 Y an EeX+Y ],henwehave,ee 1 2 varx+y ] wih varx + Y = varx + vary + 2ρ varxvary So, in our case, So, E Q i, j X = σ,s W Ti 1,S N,σ,S 2 46 Y = σ,x W Ti 1,X N,σ,X 2 47 varx + Y = σ,s 2 + σ,x 2 + 2ρ s,x σ,s 2 σ,x 2 = σ,s 2 + σ,x 2 + 2ρ s,x σ,s σ,x 48 ] ] e σ,s W Ti 1,S+σ,X W Ti 1,X = E Q i, j 2 e 1 σ,s 2 +σ,x 2 +2ρ s,x σ,s σ,x 49 So, finally, E Q i, j S X Ti 1 ] = S X e ρ s,x σ,s σ,x 5 Appenix 2: Parial erivaives Calculaion of parial erivaives: G r r, Gr, an G r, Gr, Gr, = cj=1 1 + f T i 1 + f i, j. We sar wih our funcion, T i r i, j r 51 an le, So, u = v = u r = T i v c r = j=1 1 + f T i c j=1 1 + f T i i, j T i r 1 + f i, j r T i 1 + f i, j i, j r 1 f T i r r r i, j 1 f i, j r r

22 So, T i 1 + f T T i i 1 T f i r r r G r r, = cj=1 1 + f i, j i, j r + cj=1 1 + f T i 1 + f i, j T i r i, j r an if we ivie each erm wih Gr,, wehave, cj=1 i, j 1 + f i, j i, j 1 i, j f r r r cj=1 1 + f i, j i, j r 56 G r r, Gr, = T i 1 + f T i r f T i r r + cj=1 i, j 1 + f i, j r i, j cj=1 1 + f i, j r i, j 1 f i, j r r 57 So: For he parial erivaive wih respec o, G r,, we procee as before, u T = 1 + f T i i T i f T i r 1 + f T i r v c = 1 + f i, j i, j 1 r ln 1 + f i, j r i, j 1 + f i, j r G r, = j=1 1 + f T i cj=1 cj=1 T i T r i 1 + f T i cj=1 1 + f i, j 1 + f i, j 1+ f T i r T i r f T i 1 + f i, j i, j r r + 1 ln 1 + f T i r ] 58 ] r + 1 ln1 + f T i r i, j r i, j 1 r ln1 + f i, j cj=1 So, if we ivie each erm wih Gr,, wehave, 1 + f i, j r i, j i, j r f i, j 1+ f i, j r f i, j r ] 59 ] r 6 G r, Gr, = 1 ln 1 + f T i r T i 1 + f T i r f T i r

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