Rational Multi-Curve Models with Counterparty-Risk Valuation Adjustments

Transcription

1 Raional Muli-Curve Models wih Counerpary-Risk Valuaion Adjusmens Séphane Crépey 1, Andrea Macrina 2,3, Tuye Mai Nguyen 1, David Skovmand 4 1 Laboraoire de Mahémaiques e Modélisaion d Évry, France 2 Deparmen of Mahemaics, Universiy College London, Unied Kingdom 3 Deparmen of Acuarial Science, Universiy of Cape Town, Souh Africa 4 Deparmen of Finance, Copenhagen Business School, Denmark Sepember 21, 215 Absrac We develop a muli-curve erm srucure seup in which he modelling ingrediens are expressed by raional funcionals of Markov processes. We calibrae o LIBOR swapions daa and show ha a raional wo-facor lognormal muli-curve model is sufficien o mach marke daa wih accuracy. We elucidae he relaionship beween he models developed and calibraed under a risk-neural measure Q and heir consisen equivalence class under he real-world probabiliy measure P. The consisen P-pricing models are applied o compue he risk exposures which may be required o comply wih regulaory obligaions. In order o compue counerpary-risk valuaion adjusmens, such as CVA, we show how defaul inensiy processes wih raional form can be derived. We flesh ou our sudy by applying he resuls o a basis swap conrac. Keywords: Muli-curve ineres rae erm srucure models, LIBOR, raional asse pricing models, calibraion, counerpary-risk, risk managemen, Markov funcionals, basis swap. 1

2 1 Inroducion In his work we endeavour o develop muli-curve ineres rae models which exend o counerpary risk models in a consisen fashion. The aim is he pricing and risk managemen of financial insrumens wih price models capable of discouning a muliple raes (e.g. OIS and LIBOR) and which allow for correcions in he asse s valuaion scheme so o adjus for counerpary-risk inclusive of credi, deb, and liquidiy risk. We hus propose facor-models for (i) he Overnigh Index Swap (OIS) rae, (ii) he London Inerbank Offer Rae (LIBOR), and (iii) he defaul inensiies of wo counerparies involved in bilaeral OTC derivaive ransacions. The hree ingrediens are characerised by a feaure hey share in common: he rae and inensiy models are all raional funcions of he underlying facor processes. Since we have in mind he pricing of asses as well as he managemen of risk exposures, we also need o work wihin a seup ha mainains price consisency under various probabiliy measures. We will for insance wan o price derivaives by making use of a risk-neural measure Q while analysing he saisics of risk exposures under he real-world measure P. This poin is paricularly imporan when we calibrae he ineres rae models o derivaives daa, such as implied volailiies, and hen apply he calibraed models o compue counerpary-risk valuaion adjusmens o comply wih regulaory requiremens. The presened raional models allow us o develop a comprehensive framework ha begins wih an OIS model, evolves o an approach for consrucing he LIBOR process, includes he pricing of fixed-income asses and model calibraion, analyses risk exposures, and concludes wih a credi risk model ha is applied for he analysis of counerpary-risk valuaion adjusmens (XVA). The issue of how o model muli-curve ineres raes and incorporae counerpary-risk valuaion adjusmens in a pricing framework has moivaed much research. For insance, research on muli-curve ineres rae modelling is presened in Henrard (27, 21, 214), Kijima, Tanaka, and Wong (29), Kenyon (21), Bianchei (21), Mercurio (21b, 21a, 21c), Fujii, Shimada, and Takahashi (21, 211), Bianchei and Morini (213), Filipović and Trolle (213), Moreni and Pallavicini (214) or Crépey, Grbac, Ngor, and Skovmand (215). On counerpary-risk valuaion adjusmen, we menion wo recen books by Brigo, Morini, and Pallavicini (213) and Crépey, Bielecki, and Brigo (214); more references are given as we go along. Pricing models wih raional form have appeared before. Flesaker and Hughson (1996) pioneered such pricing models and in paricular inroduced he so-called raional log-normal model for discoun bond prices. Furher relaed sudies include Rukowski (1997), Döberlein and Schweizer (21) and Hun and Kennedy (24), Brody and Hughson (24), Hughson and Rafailidis (25), Brody, Hughson, and Mackie (212), Akahori, Hishida, Teichmann, and Tsuchiya (214), Filipović, Larsson, and Trolle (214), Macrina and Parbhoo (214) or Nguyen and Seifried (214). However, as far as we know, he presen paper is he firs o apply raional pricing models in a muli-curve seup, along wih Nguyen and Seifried (214) who develop a raional mulicurve model in he spiri of Rogers (1997) based on a muliplicaive spread, and i is he only raional pricing paper dealing wih XVA compuaions. We shall see ha, despie he simpliciy of hese models, hey perform surprisingly well when comparing o oher, 2

3 in principle more elaborae, proposals such as Crépey, Grbac, Ngor, and Skovmand (215) or Moreni and Pallavicini (213, 214). Oher recen relaed research includes Filipović, Larsson, and Trolle (214), for he sudy of unspanned volailiy and is regulaory implicaions, Cuchiero, Keller-Ressel, and Teichmann (212), for momen compuaions in financial applicaions, and Cheng and Tehranchi (214), moivaed by sochasic volailiy modelling. This paper consiss of hree main pars: (a) Developmen of he novel raional mulicurve ineres rae approach along wih he raional credi-inensiy models necessary for he compuaion of he counerpary-risk valuaion adjusmens. This maerial is presened in Secion 2. (b) Clean valuaion of ineres rae securiies and model calibraion (Secions 3 and 4) where we consider specific raional facor models for he muli-curve ineres raes and defaul inensiy models wih he goal in mind of singling ou he mos parsimonious model ha bes calibraes o available derivaives daa. (c) Counerpary-risk valuaion adjusmens (Secion 5), which are compued for basis swaps priced wih he raional rae models. The basis swap case sudy gives also he opporuniy o highligh he imporance of consisen pricing and hedging under P and Q. The main novel research conribuions are lised as follows: (i) The raional models for muli-curve erm srucures whereby we derive he LIBOR process by pricing a forward rae agreemen under he real-world probabiliy measure. In doing so we apply a pricing kernel model. The shor rae model arising from he pricing kernel process is aken as a proxy model for he OIS rae. In view of derivaive pricing in subsequen secions, we also provide an alernaive derivaion of he raional muli-curve ineres rae models by saring wih he risk-neural measure. We call his mehod boom-up risk-neural approach. (ii) We explain he advanages one gains from he resuling codebook for he LIBOR process, which we model as a raional funcion where he denominaor is he sochasic discoun facor associaed wih he uilised probabiliy measure. We calibrae hree specificaions of our muli-curve framework and assess hem for he qualiy of fi and on posiiviy of raes and spread. We show ha a one-facor raional model is oo rigid o be able o calibrae o given daa, a shorcoming ha one canno ge rid of even when he driving facor feaures jumps in is dynamics. We conclude by emphasising a wo-facor lognormal OIS-LIBOR model as he mos parsimonious raional model wih good racabiliy and calibraion properies. To our knowledge, his is he firs ime such an in-deph calibraion analysis has been performed on raional ineres rae models. (iii) We show he explici relaionship in our seup beween pricing under an equivalen measure and he real-world probabiliy measure. We compue he risk exposure associaed wih holding a basis swap and plo he quaniles under boh probabiliy measures for comparison. As an example, we apply Lévy random bridges o describe he dynamics of he facor processes under P. This enables us o inerpre he re-weighing of he risk exposure under P as an effec ha could be relaed o, e.g., forward guidance provided by a cenral bank. (iv) We propose new credi defaul inensiy models wih raional form, which can be guaraneed o ake posiive values a all imes and have he same appealing mahemaical racabiliy as he raional (muli-curve) ineres rae models. (v) We compue XVA, ha is, he counerpary-risk valuaion adjusmens due o credi, deb, and liquidiy risk, based on raional muli-curve 3

4 ineres rae and raional credi-inensiy models. 2 Raional muli-curve erm srucures We model a financial marke by a filered probabiliy space (Ω, F, P, {F } ), where P denoes he real probabiliy measure and {F } is he marke filraion. The noarbirage pricing formula for a generic (non-dividend-paying) financial asse wih price process {S T } T, which is characerised by a cash flow S T T a he fixed dae T, is given by S T = 1 π E P [π T S T T F, (2.1) where {π } U is he pricing kernel embodying he iner-emporal discouning and riskadjusmens, see e.g. Hun and Kennedy (24). Once he model for he pricing kernel is specified, he OIS discoun bond price process {P T } T is deermined as a special case of formula (2.1) by The associaed OIS shor rae of ineres is obained by P T = 1 π E P [π T F. (2.2) r = ( T ln P T ) T =, (2.3) where i is assumed ha he discoun bond sysem is differeniable in is mauriy parameer T. The rae {r } is non-negaive if he pricing kernel {π } is a supermaringale and vice versa. We nex go on o infer a pricing formula for financial derivaives wrien on LIBOR. In doing so, we also derive a price process (2.6) ha we idenify as deermining he dynamics of he FRA rae (2.7). I is his formula ha, in our work, reveals he naure of he so-called muli-curve erm srucure whereby he OIS rae and he LIBOR raes of differen enors are reaed as disinc discoun raes. 2.1 Generic muli-curve ineres rae models We propose a new mehod for consrucing muli-curve pricing models for securiies wrien on LIBOR by saring wih he valuaion of a forward rae agreemen (FRA). We consider T T 2 T i T n, where T, T i,..., T n are fixed daes, and le N be a noional, K a srike rae and δ i = T i T i 1. The fixed leg of he FRA conrac is given by NKδ i and he floaing leg payable in arrears a ime T i is modelled by Nδ i L(T i ; T i 1, T i ), where he random rae L(T i ; T i 1, T i ) is F Ti 1 -measurable. The ne cash flow a he mauriy dae T i of he FRA conrac reads H Ti = Nδ i [K L(T i ; T i 1, T i ). (2.4) The FRA price process is hen given by an applicaion of (2.1), ha is, for T i 1, by H Ti = 1 π E P [ π Ti H Ti F = Nδ i [KP Ti L(, T i 1, T i ), (2.5) 4

5 where we define L(; T i 1, T i ) := 1 π E P [ π Ti L(T i ; T i 1, T i ) F. (2.6) We call he above process he LIBOR process. We noe here ha even hough LIBOR is no as such a raded asse, he process (2.6) may be inerpreed as a price process of a radable asse wih cash flow L(T i ; T i 1, T i ) a ime T i. The fair spread of he FRA a ime (he value of K a ime such ha H Ti = ), called he FRA rae, is hen expressed in erms of L(; T i 1, T i ) by K = L(; T i 1, T i ) P Ti. (2.7) For imes up o and including T i 1, he LIBOR process (2.6) can be wrien in erms of a condiional expecaion of an F Ti 1 -measurable random variable. In fac, for T i 1, E P [ π Ti L(T i ; T i 1, T i ) F = [ E P E P [ π Ti L(T i ; T i 1, T i ) F Ti 1 F (2.8) = [ E P E P [ π Ti F Ti 1 L(Ti ; T i 1, T i ) F, (2.9) and hus L(, T i 1, T i ) = 1 π E P [ E P [ π Ti F Ti 1 L(Ti ; T i 1, T i ) F. (2.1) The (pre-crisis) classical approach o LIBOR modelling defines he price process {H Ti } of a FRA by H Ti = N [ (1 + δ i K)P Ti P Ti 1, (2.11) see, e.g., Hun and Kennedy (24). By equaing wih (2.5), we see ha he classical single-curve LIBOR model is obained in he special case where L(; T i 1, T i ) = 1 δ i ( PTi 1 P Ti ). (2.12) Remark 2.1. Unless markes feaure invered yield curves, one expecs he posiive-spread relaion L(; T, T + δ i ) < L(; T, T + δ j ), for enors δ j > δ i, o hold. We shall reurn o his relaionship in Secion 4 where various model specificaions are calibraed and he posiiviy of he spread is checked. For recen work on muli-curve modelling wih focus on spread modelling, we refer o Cuchiero, Fonana and Gnoao (214). 2.2 Muli-curve models wih raional form In order o consruc explici LIBOR processes, he pricing kernel {π } and he random variable L(T i ; T i 1, T i ) need o be specified in he definiion (2.6). For reasons ha will become apparen as we move forward in his paper including mahemaical racabiliy, good calibraion properies, and parsimonious modelling we op o apply he raional pricing models for a generic financial asse proposed in Macrina (214). These models besow a raional form on he price processes, here inended as a quoien of summands (slighly abusing he erminology ha usually refers o a quoien of polynomials ). The 5

6 basic pricing model wih raional form for a generic financial asse (for shor raional pricing model ) ha we borrow here is given by S T = S T + b 2 (T )A (2) + b 3 (T )A (3) P + b 1 ()A (1), (2.13) where S T is he value of he asse a =. There may be more ba-erms in he numeraor, bu wo (a mos) will be enough for all our purposes in his work. For T and i = 1, 2, 3, b i () are deerminisic funcions and A (i) = A i (, X (i) ) are maringale processes, no necessarily under P bu under an equivalen maringale measure M, which are driven by M-Markov processes {X (i) }. The deails of how he expression (2.13) is derived from he formula (2.1), and in paricular how explici examples for {A (i) } can be consruced, are shown in Macrina (214). Having oped for he paricular raional pricing model (2.13), i follows from he relaion (2.1) ha he pricing kernel model associaed wih he price process (2.13) necessarily has he form π = π [ P + b 1 ()A (1) M, (2.14) M where {M } is he P-maringale ha induces he change of measure from P o an auxiliary measure M under which he {A (i) } are maringales. The deerminisic funcions P and b 1 () are defined such ha P + b 1 ()A (1) is a non-negaive M-supermaringale (see e.g. Example 2.1), and hus in such a way ha {π } is a non-negaive P-supermaringale. By he equaions (2.2) and (2.3), i is sraighforward o see ha P T = P T + b 1 (T )A (1) P P + b 1 ()A (1), r = + b 1 ()A (1) P + b 1 ()A (1), (2.15) where he do-noaion means differeniaion wih respec o ime. Le us reurn o he modelling of raional muli-curve erm srucures and in paricular o he definiion of he (forward) LIBOR process. Puing equaions (2.6) and (2.1) in relaion, we see ha he class of models (2.13) naurally lends iself for he modelling of he LIBOR process (2.6) in he considered seup. Since (2.13) saisfies (2.1) by consrucion, so does he LIBOR model L(; T i 1, T i ) = L(; T i 1, T i ) + b 2 (T i 1, T i )A (2) + b 3 (T i 1, T i )A (3) P + b 1 ()A (1) (2.16) saisfy he maringale equaion (2.6) and in paricular (2.1) for T i 1. Based on our knowledge, his is he firs ime ha LIBOR is modelled in his way. In Macrina (214) a mehod based on he use of weighed hea kernels is provided for he explici consrucion of he M-maringales {A (i) } i=1,2. This furher applicaion allows for he developmen of (explici) LIBOR processes, which, if circumsances in financial markes require i, by consrucion ake posiive values a all imes. 2.3 Boom-up risk-neural approach Since we also deal wih counerpary-risk valuaion adjusmens, we presen anoher scheme for he consrucion of he LIBOR models, which we call boom-up risk-neural approach. As he name suggess, we model he muli-curve erm srucure by making use 6

7 of he risk-neural measure (via he auxiliary measure M) while he connecion o he P-dynamics of prices can be reinroduced a a laer sage, which is imporan for he calculaion of risk exposures and heir managemen. Boom-up refers o he fac ha he shor ineres rae will be modelled firs, hen followed by he discoun bond price and LIBOR processes. Similarly, in Secion 2.4, he defaul inensiy processes will be modelled firs, and hereafer he price processes of counerpary-risky asses will be derived hereof. We uilise he noaion E[... F = E [.... In he boom-up seing, we direcly model he shor risk-free rae {r } in he manner of he righ-hand side in (2.15), i.e. r = c 1() + b 1 ()A (1) c 1 () + b 1 ()A (1), (2.17) by posulaing (i) non-increasing deerminisic funcions b 1 () and c 1 () wih c 1 () = 1 (laer c 1 () will be seen o coincide wih P ), and (ii) an ({F }, M)-maringale {A (1) } wih A (1) = such ha h = c 1 () + b 1 ()A (1) (2.18) is a posiive ({F }, M)-supermaringale for all >. Example 2.1. Le A (1) = S (1) 1, where {S (1) } is a posiive M-maringale wih S (1) = 1, for example a uni-iniialised exponenial Lévy maringale. Then he supermaringale (2.18) is posiive for any given if < b 1 () c 1 (). Associaed wih he supermaringale (2.18), we characerise he (risk-neural) pricing measure Q by he M-densiy process {µ } T, given by ( µ = dq ) b 1 (s)da (1) s = E dm F c 1 (s) + b 1 (s)a (1), (2.19) s which is aken o be a posiive ({F }, M)-maringale. We noe ha, in principle, we allow for jumps in his seup, and hus we denoe by {A (1) } he lef-limi process of {A(1) }, where all semi-maringales ( are assumed righ-coninuous wih lef limis. Furhermore, we denoe by D = exp ) r s ds he discoun facor associaed wih he risk-neural measure Q. Lemma 2.1. h = D µ. Proof. The Io semi-maringale formula applied o ϕ(, A (1) ) = ln(c 1 ()+b 1 ()A (1) ) = ln(h ) and o ln(d µ ) gives he following relaions: ( ) d ln c 1 () + b 1 ()A (1) = r d + + d s ( b 1 ()da (1) c 1 () + b 1 ()A (1) b2 1 ()d[a(1), A (1) c 2(c 1 () + b 1 ()A (1) )2 ln ( c 1 () + b 1 ()A (1) ) b 1 () A (1) c 1 () + b 1 ()A (1) ), (2.2) 7

8 where (2.17) was used in he firs line, and where d ln(d µ ) = d ln D + d lnµ ln (µ ) = ln = r d + dµ µ d[µ, µc 2(µ ) 2 + d s = r d + ( µ µ ( = ln c 1 () + b 1 ()A (1) b 1 ()da (1) c 1 () + b 1 ()A (1) ( +d s ( ln(µ ) µ ) µ b2 1 ()d[a(1), A (1) c 2(c 1 () + b 1 ()A (1) ) ( ) ( = ln 1 + b 1() A (1) c 1 () + b 1 ()A (1) = ln ). )2 ln(µ ) b 1() A (1) c 1 () + b 1 ()A (1) c 1 () + b 1 ()A (1) c 1 () + b 1 ()A (1) ) ), (2.21) Therefore, d ln(h ) = d ln(d µ ). Moreover, h = D µ = 1. Hence h = D µ. I hen follows ha he price process of he OIS discoun bond wih mauriy T can be expressed, for T, by P T = E Q [ DT D = 1 D µ E M [D T µ T F = E M [ ht h = c 1(T ) + b 1 (T )A (1) c 1 () + b 1 ()A (1). (2.22) Thus, he process {h } plays he role of he pricing kernel associaed wih he OIS marke under he measure M. In paricular, we noe ha c 1 () = P for [, T and r = ( T ln P T ) T =. A consrucion inspired by he above formula for he OIS bond leads o he raional model for he LIBOR prevailing over he inerval [T i 1, T i ). The F Ti 1 - measurable spo LIBOR rae L(T i ; T i 1, T i ) is modelled in erms of {A (1) } and, in his paper, a mos wo oher M-maringales {A (2) } and {A (3) } evaluaed a T i 1 : L(T i ; T i 1, T i ) = L(; T i 1, T i ) + b 2 (T i 1, T i )A (2) T i 1 + b 3 (T i 1, T i )A (3) T i 1 P Ti + b 1 (T i )A (1) T i 1. (2.23) The (forward) LIBOR process is hen defined by an applicaion of he risk-neural valuaion formula (which is equivalen o he pricing formula (2.1) under P) as follows. For T i 1 we le L(; T i 1, T i ) = 1 [ E Q D [D T i L(T i ; T i 1, T i ) = E M DTi µ Ti L(T i ; T i 1, T i ) (2.24) D µ [ E M = E M [h Ti 1 T i L(T i ; T i 1, T i ), (2.25) and hus, by applying (2.18) and (2.23), h L(; T i 1, T i ) = L(; T i 1, T i ) + b 2 (T i 1, T i )A (2) + b 3 (T i 1, T i )A (3) P + b 1 ()A (1). (2.26) 8

9 Hence, we recover he same model and expression as in (2.16). The LIBOR models (2.26) (or (2.16)) are compaible wih an HJM muli-curve seup where, in he spiri of Heah, Jarrow and Moron (1992), he iniial erm srucures P Ti and L(; T i 1, T i ) are fied by consrucion. Example 2.2. Le A (i) = S (i) 1, where S (i) is a posiive M-maringale wih S (i) = 1. For example, one could consider a uni-iniialised exponenial Lévy maringale defined in erms of a funcion of an M-Lévy process {X (i) }, for i = 2, 3. Such a consrucion produces non-negaive LIBOR raes if b 2 (T i 1, T i ) + b 3 (T i 1, T i ) L(; T i 1, T i ). (2.27) If his condiion is no saisfied, hen he LIBOR model may be viewed as a shifed model, in which he LIBOR raes may become negaive wih posiive probabiliy. For differen kinds of shifs used in he muli-curve erm srucure lieraure we refer o, e.g., Mercurio (21a) or Moreni and Pallavicini (214). 2.4 Raional credi model For he counerpary-risk valuaion adjusmens (XVA) produced laer in his paper, we require credi-inensiy models, which we consruc in he same fashion as he raional muli-curve ineres rae models. The following novel raional defaul-inensiy models are developed by use of he boom-up risk-neural approach presened in he previous secion. We consider {X (i) } i=1,2,...,n, which are assumed o be ({F }, M)-Markov processes. For any muli-index (i 1,..., i d ), we wrie F (i 1,...,i d ) = l=1,...,d F X(i l ). The (marke) filraion {F } is given by {F (1,...,n) }. For he applicaion in he presen secion, we fix n = 6. Markov processes {X (1) } = {X (3) } and {X (2) } are uilised o drive he OIS and LIBOR models as described in Secion 2.3, in paricular he zero-iniialised ({F }, M)-maringales {A (i) } i=1,2,3. The Markov processes {X (i) }, i = 4, 5, 6, which are assumed o be muually M-independen as well as M-independen of he Markov processes i = 1, 2, 3, are applied o model {F }-adaped processes {γ (i) } i=4,5,6 defined by γ (i) (i) = ċi() + ḃi()a c i () + b i ()A (i), (2.28) where b i () and c i (), wih c i () = 1, are non-increasing deerminisic funcions, and where {A (i) } i=4,5,6 are zero-iniialised ({F }, M)-maringales of he form A(, X (i) ). Comparing wih (2.17), we see ha (2.28) is modelled in he same way as he OIS rae (2.17), nonnegaive in paricular, as an inensiy should be (see Remark 2.2). In line wih he boom-up consrucion in Secion (2.3), we now inroduce a densiy ({F }, M)-maringale {µ ν } T ha induces a measure change from M o he risk-neural measure Q: dq = µ ν, T, dm F 9

10 where {µ } is defined as in Secion 2.3. Here, we furhermore define ν = i 4 ν(i) where he processes ν (i) = E ( ) ḃ i ()da (i) (i) ċ i () + ḃi()a are assumed o be posiive rue ({F }, M)-maringales. Lemma 2.2. Le ξ denoe any non-negaive F (1,2,3) T -measurable random variable and le χ = j 4 χ i where, for j = 4, 5, 6, χ j is F (j) -measurable. Then T E R [ξ χ = E R [ξ j 4 E R [χ i, (2.29) for R = M or Q and for T. Proof. Since F (4,5,6) T is independen of F (1,2,3) and of ξ, E M [ ξ F (1,2,3) F (4,5,6) T = E M [ ξ F (1,2,3) Therefore, [ [ E M [ξ χ = E M E M ξ χ (1,2,3) F F (4,5,6) (1,2,3) T F F (4,5,6) [ [ = E M E M ξ F (1,2,3) F (4,5,6) T χ F (1,2,3) F (4,5,6) [ [ = E M E M ξ (1,2,3) F χ (1,2,3) F F (4,5,6) [ = E M ξ [ F (1,2,3) E M χ F (1,2,3) F (4,5,6) = E M [ξ E M [χ. Nex, he Girsanov formula in combinaion wih he resul for M-condiional expecaion yields: [ [ [ E Q [ξχ = µt ν T ξχ EM = E M µt ξ E M νt χ µ ν µ ν [ [ = E M νt µ T ξ E M µt ν T χ = E Q ν µ µ ν [ξ EQ [χ. The resul remains o be proven for he case ξ = 1, which is done similarly. For he XVA compuaions, we shall use a reduced-form counerpary-risk approach where he defaul imes of a bank b (we adop is poin of view) and of is counerpary c are modeled in erms of hree Cox imes τ i defined by τ i = inf { >. γ (i) s ds E i }. (2.3) Under Q, he random variables E i (i = 4, 5, 6) are independen and exponenially disribued. Furhermore, τ c = τ 4 τ 6, τ b = τ 5 τ 6, hence τ = τ b τ c = τ 4 τ 5 τ 6. We wrie γ c = γ (4) + γ (6), γ b = γ (5) + γ (6), γ = γ (4) + γ (5) + γ (6), 1

11 which are he so-called ({F }, Q)-hazard inensiy processes of he {G } sopping imes τ c, τ b and τ, where he full model filraion {G } is given as he marke filraion {F }- progressively enlarged by τ c and τ b (see, e.g., Bielecki, Jeanblanc, and Rukowski (29), Chaper 5). Wriing as before D = exp( r s ds), we noe ha Lemma 2.1 sill holds in he presen seup. Tha is, h = c 1 () + b 1 ()A (1) = D µ, an ({F }, M)-supermaringale, assumed o be posiive (e.g. under an exponenial Lévy maringale specificaion for {A (1) } as in Example 2.2). Furher, we inroduce Z (i) = exp( ds), for i = 4, 5, 6, and obain analogously ha γ(i) s k (i) := c i () + b i ()A (i) = Z i ν (i). (2.31) Wih hese observaions a hand, he following resuls follow from Lemma 2.2. We wrie k = i 4 k(i) and Z = i 4 Z(i). Proposiion 2.1. The ideniies (2.22) and (2.26) sill hold in he presen seup, ha is P T = E Q [e [ [ T r s ds = E Q DT = E M ht = c 1(T ) + b 1 (T )A (1) D h c 1 () + b 1 ()A (1) (2.32) and, for T i 1, Likewise, E Q E Q L(; T i 1, T i ) = L(; T i 1, T i ) + b 2 (T i 1, T i )A (2) + b 3 (T i 1, T i )A (3) P + b 1 ()A (1). (2.33) [e T E Q γ s ds = E Q [e T γ s ds γ c T [ ZT Z [e T (rs+γc)ds s = E Q = E M Proof. Using Lemma 2.2, we compue E Q [e T r sds = E Q = E M [ kt k = i=4,5,6 [ [ = E Q Z (5) T Z (5) T E Q = E Q [e T γ s ds [ D T Z (4) T Z(6) T D Z (4) Z (6) [ DT D [ ht h = E M = i=4,6 i=1,4,6 c i (T ) + b i (T )A (i) c i () + b i ()A (i), (2.34) Z (4) T Z(6) T Z (4) Z (6) ċ i (T ) + ḃi(t )A (i) c i () + b i ()A (i), (2.35) c i (T ) + b i (T )A (i) c i () + b i ()A (i). (2.36) [ [ [ ht ν T = E M ht E M νt h ν h ν = c 1(T ) + b 1 (T )A (1) c 1 () + b 1 ()A (1), (2.37) where he las equaliy holds by Lemma 2.1. This proves (2.32). The oher ideniies are proven similarly. 11

12 Remark 2.2. Equaions (2.32) and (2.34) are similar in naure and appearance. As i is he case for he resuling OIS rae {r } (2.17), he fac ha (2.31) is designed o be a supermaringale has as a consequence ha he associaed inensiy (2.28) is a non-negaive process. This is readily seen by observing ha {ν (i) } is a maringale and hus he drif of he supermaringale (2.31) is given by he necessarily non-negaive process {γ (i) } ha drives {Z (i) }. A ime =, we have A (i) =, hence only he erms c i (T ) remain in hese formulas. Since he formulae (2.32) and (2.33) are no affeced by he inclusion of he credi componen in his approach, he valuaion of he basis swap of Secion 5.2 remains unchanged. By making use of he so-called Key Lemma of credi risk, see for insance Bielecki, Jeanblanc, and Rukowski (29), he ideniy (2.36) is he main building block for he pre-defaul price process of a clean CDS on he counerpary (respecively he bank, subsiuing τ b for τ c in (2.36)). In paricular, he ideniies a = E Q [ e T (rs+γc s)ds = c 1 (T )c 4 (T )c 6 (T ), E Q [ e T (rs+γb s)ds = c 1 (T )c 5 (T )c 6 (T ), (2.38) for T, can be applied o calibrae he funcions c i (T ), i = 4, 5, 6, o CDS curves of he counerpary and he bank, once he dependence on he respecive credi risk facors has been specified. The calibraion of he noisy credi model componens b i (T )A (i), i = 4, 5, 6, would require CDS opion daa or views on CDS opion volailiies. If he enire model is judged underdeermined, more parsimonious specificaions may be obained by removing he common defaul componen τ 6 (jus leing τ c = τ 4, τ b = τ 5 ) and/or resricing oneself o deerminisic defaul inensiies by seing some of he sochasic erms equal o zero, i.e. b i (T )A (i) =, i = 4, 5 and/or 6 (as is he case for he one-facor ineres rae models in Secion 3). The core building blocks of our muli-curve LIBOR model wih counerparyrisk are he couerpary-risk kernels {k (i) }, i = 4, 5, 6, he OIS kernel {h }, and he LIBOR kernel given by he numeraor of he LIBOR process (2.26). We may view all kernels as defined under he M-measure, a priori. The respecive kernels under he P-measure, e.g. he pricing kernel {π }, are obained as explained a he end of Secion Clean valuaion The nex quesions we address are cenred around he pricing of LIBOR derivaives and heir calibraion o marke daa, especially LIBOR swapions, which are he mos liquidly raded (nonlinear) ineres rae derivaives. Since marke daa ypically reflec prices of fully collaerallised ransacions, which are funded a a remuneraion rae of he collaeral ha is bes proxied by he OIS rae, we consider in his secion clean valuaion ignoring counerpary-risk for he purpose of model calibraion in he nex secion and hus assume funding a he rae r. Tha is, in his par we do no make use of he credi-inensiy models proposed in Secion 2.4, bu will apply hem in Secion 5 for he compuaion of counerpary-risk and funding valuaion adjusmens. An ineres rae swap, see for insance Brigo and Mercurio (26), is an agreemen beween wo counerparies, where one sream of fuure ineres paymens is exchanged 12

13 for anoher based on a specified nominal amoun N. A popular ineres rae swap is he exchange of a fixed rae (conracual swap spread) agains he LIBOR a he end of successive ime inervals [T i 1, T i of lengh δ. Such a swap can also be viewed as a collecion of n forward rae agreemens. The swap price Sw a ime T is given by he following model-independen formula: Sw = Nδ n [L(; T i 1, T i ) KP Ti. i=1 Remark 3.3. In he marke, a floaing leg has ypically higher frequency han a fixed leg. For simpliciy we consider he case when he imings of he fixed and floaing paymens are he same. Of course, aking differen frequencies can be accommodaed. A swapion is an opion beween wo paries o ener a swap a he expiry dae T k (he mauriy dae of he opion). Is price a ime T k is given by he following M-pricing formula: Swn Tk = Nδ E M [h Tk (Sw Tk ) + F h [ ( n ) + = Nδ E M F h Tk [L(T k ; T i 1, T i ) KP Tk T h i = Nδ P + b 1 A (1) i=k+1 E M [ ( n i=k+1 [ L(; Ti 1, T i ) + b 2 (T i 1, T i )A (2) T k + b 3 (T i 1, T i )A (3) T k K(P Ti + b 1 (T i )A (1) T k ) ) + F, (3.39) using he formulae (2.22) and (2.26) for P Tk T i and L(T k ; T i 1, T i ). In paricular, he swapion prices a ime = can be rewrien by use of A (i) = S (i) 1 so ha [ ( ) Swn Tk = Nδ E M c 2 A (2) T k + c 3 A (3) T k c 1 A (1) + T k + c = Nδ E M [ ( c 2 S (2) T k + c 3 S (3) T k c 1 S (1) T k + c ) +, (3.4) where n n n c 2 = b 2 (T i 1, T i ), c 3 = b 3 (T i 1, T i ), c 1 = K b 1 (T i ), c = i=k+1 n i=k+1 i=k+1 i=k+1 [L(; T i 1, T i ) KP Ti, c = c + c 1 c 2 c 3. As we will see in several insances of ineres, hese expecaions can be compued efficienly wih high accuracy by various numerical schemes. Remark 3.4. The advanages of modelling he LIBOR process {L(; T i 1, T i )} by a raional funcion of which denominaor is he discoun facor (pricing kernel) associaed wih he 13

14 employed pricing measure (in his case M) are: (i) The raional form of {L(; T i 1, T i )} and also of {P Ti } produces, when muliplied wih he discoun facor {h }, a linear expression in he M-maringale drivers {A (i) }. This is in conras o oher akin pricing formulae in which he facors appear as sums of exponenials, see e.g. Crépey e al. (215), Equaion (33). (ii) The dependence srucure beween he LIBOR process and he OIS discoun facor {h } or he pricing kernel {π } under he P-measure is clear-cu. The numeraor of {L(; T i 1, T i )} is driven only by idiosyncraic sochasic facors ha influence he dynamics of he LIBOR process. We may call such drivers he LIBOR risk facors. Dependence on he OIS risk facors, in our model example {A (1) }, is produced solely by he denominaor of he LIBOR process. (iii) Usually, he FRA process K = L(; T i 1, T i )/P Ti is modelled direcly and more commonly applied o develop muli-curve frameworks. Wih such models, however, i is no guaraneed ha simple pricing formulae like (3.39) can be derived. We hink ha he codebook (2.6), and (2.26) in he considered example, is more suiable for he developmen of consisen, flexible and racable muli-curve models. We nex consider one-facor and wo-facor models in preparaion for calibraion o marke daa. The ulimae goal is o find a simple, parsimonious and racable ineres rae model ha can be accuraely calibraed o swapions wrien on LIBOR. 3.1 Univariae Fourier pricing We begin wih a much simplified model ha is driven by a single marke facor. We will evenually see ha such a choice is unrealisic and in paricular he resuling model is oo rigid o allow for saisfacory model calibraion. One migh jusify such a model by saying ha since in curren markes here are no liquidly-raded OIS derivaives and hence no useful daa is available, a pragmaic simplificaion is o assume deerminisic OIS raes {r }. However, his assumpion produces unrealisic fuure scenarios for he basis spreads beween OIS and LIBOR raes, which are no guaraneed o be posiive. Hence, we shall relax such an assumpion laer when we consider a wo-facor model, and hus OIS reurns o be a sochasic process. Le us assume A (1) =, and hence b 1 () plays no role eiher, so ha i can be assumed equal o zero. Furhermore, for a sar, we assume A (3) = and b 3 () =, and (3.4) simplifies o [ ( ) [ Swn Tk = Nδ E M c 2 A (2) + ( ) T k + c = Nδ E M c 2 S (2) + T k + c, where here c = c c 2. For c > he price is simply Swn Tk = Nδc. For c <, and in he case of an exponenial-lévy maringale model wih S (2) = e X(2) ψ 2 (1), where {X (2) } is a Lévy process wih cumulan ψ 2 such ha [ E M e zx(2) = exp [ψ 2 (z), (3.41) 14

15 we have where Swn Tk = Nδ 2π R c 1 iv R M (2) T k (R + iv) dv, (3.42) (R + iv)(r + iv 1) ( ) M (2) T k (z) = e T kψ 2 (z)+z ln(c 2 ) ψ 2 (1) and R is an arbirary consan ensuring finieness of M (2) T k (R + iv) for v R. For deails concerning (3.42), we refer o Eberlein, Glau and Papapanoleon (21). 3.2 One-facor lognormal model In he even ha {A (1) } = {A (3) } = and {A (2) A (2) = exp ( a 2 X (2) 1 2 a2 2 } is of he form ) 1, (3.43) where {X (2) } is a sandard Brownian moion and a 2 is a real consan, i follows from simple calculaions ha he swapion price is given, for c = c c 2, by [ ( ) Swn Tk =Nδ E M c 2 A (2) + T k +c (3.44) ( ( 1 2 =Nδ c 2 Φ a2 2 T ) ( k ln( c /c 2 ) 1 + c Φ a 2 Tk 2 a2 2 T k ln( c /c 2 ) a 2 Tk )), (3.45) where Φ(x) is he sandard normal disribuion funcion. In Secion 4, where we focus on model calibraion, we also consider a one-facor model ha is driven by an NIG-process, as an example of a model wih jumps. 3.3 Two-facor lognormal model We reurn o he price formula (3.4) and consider he case where he maringales {A (i) } are given, for i = 1, 2, 3, by A (i) = exp ( a i X (i) 1 ) 2 a2 i 1, (3.46) for real consans a i and sandard Brownian moions {X (1) } = {X (3) } and {X (2) } wih correlaion ρ. Then i follows ha [ ( Swn Tk = E M c 2 e X T k a a2 2 T k + c 3 e Y T k a a2 3 T k c 1 e Y ) T k a a2 1 T k + c, (3.47) where X N (, 1), Y N (, 1), (X Y ) = y N (ρy, (1 ρ 2 )). Hence, Swn Tk = = K(y)> (c 2 e x T k a a2 2 T k K(y)) + f(x y)f(y)dxdy ( ) (c 2 e x T k a a2 2 T k K(y)) + f(x y)dx f(y)dy ( + K(y)< (c 2 e x T k a a2 2 T k K(y)) + f(x y)dx ) f(y)dy, 15

16 where K(y) = c 1 (e a 1 Tk y 1 2 a2 1 T k 1) c 3 (e a 3 Tk y 1 2 a2 3 T k 1) c, f(y) = 1 e y2 2, 2π f(x y) = 2 2(1 ρ 2 ). 1 2π(1 ρ 2 ) e (x ρy) This expression can be simplified furher o obain Swn Tk = K(y)> [ ( c 2 e a 2 Tk ρy+ 1 2 a2 2 T k(1 ρ 2) ρy + a2 Tk (1 ρ 2 ) + ln(c 2 ) 1 2 Φ a2 2 T ) k K(y) 1 ρ 2 ( ρy + ln(c2 ) 1 2 K(y)Φ a2 2 T ) k K(y) f(y)dy 1 ρ 2 ( + c 2 e a 2 Tk ρ(y 1 2 a 2 Tk ρ K(y)) f(y)dy. K(y)< The calculaion of he swapion price is hen reduced o calculaing wo one-dimensional inegrals. Since he regions of inegraion are no explicily known, one has o numerically solve for he roos of K(y), which may have up o wo roos. Neverheless a full swapion smile can be calculaed in a small fracion of a second by means of his formula. 4 Calibraion The counerpary-risk valuaion adjusmens, abbreviaed by XVAs (CVA, DVA, ec.), can be viewed as long-erm opions on he underlying conracs. For heir compuaion, he effecs by he volailiy smile and erm srucure maer. Furhermore, for he planned XVA compuaions of muli-curve producs (e.g. basis swaps), which we shall consider in he nex secion, i is necessary o calibrae he proposed pricing model o financial insrumens wih underlying enors of δ = 3m and δ = 6m. Similar o Crépey e al. (215), we make use of he following EUR marke Bloomberg daa of January 4, 211 o calibrae our model: EONIA, hree-monh EURIBOR and six-monh EURIBOR iniial erm srucures on he one hand, and hree-monh and six-monh enor swapions on he oher. As in he HJM framework of Crépey e al. (215), o which he reader is referred for more deails in his regard, he iniial erm srucures are fied by consrucion in our seup. Wih regard o he calibraion o swapions, a firs, we calibrae he non-mauriy/enor-dependen parameers o he swapion smile for he 9Y1Y swapion wih a hree-monh enor underlying. The marke smile corresponds o a vecor of srike bps [ 2, 1, 5, 25,, 25, 5, 1, 2 around he underlying forward swap spread. Then, we make use of ahe-money swapions on hree and six-monh enor swaps all erminaing a exacly en years, bu wih mauriies from one o nine years. This co-erminal procedure is chosen wih a view owards he XVA applicaion in Secion 5, where a basis swap wih a en-year erminal dae is considered. 16

17 The general idea leading hrough Secion 4 is ha we calibrae he simples sochasic ineres rae model, driven by one facor, where wo cases are considered: (i) log-normal facor, and (ii) log-nig facor. I urns ou ha neiher of hese models calibrae o swapion daa saisfacorily, hus suggesing ha a wo-facor raional model is he nex class ha needs o be aken under consideraion. As we will find ou, he wo-facor raional log-normal muli-curve ineres rae model provides good calibraion properies and is deemed o be he winning model. Anoher conclusion is ha we do no need o include jump drivers in he specificaion of he wo-facor raional model. Model calibraion in a one-facor seup where, say, {A (2) } is he single sochasic facor, involves he following seps: 1. We calibrae he parameers of he driving maringale {A (2) } o he smile of he 9Y1Y swapion wih enor δ = 3m. This par of he calibraion procedure gives us also he values of b 2 (9, 9.25), b 2 (9.25, 9.5), b 2 (9.5, 9.75) and b 2 (9.75, 1), which we assume o be equal. 2. Nex, we consider he co-erminal, Y(1 )Y ATM swapions wih = 1, 2,..., 9 years. These are available wrien on he hree and six-monh raes. We calibrae he remaining values of b 2 one mauriy a a ime, going backwards and saring wih he 8Y2Y for he hree-monh enor and wih he 9Y1Y for he six-monh enor. This is done assuming ha he parameers are piecewise consan such ha b 2 (T, T +.25) = b 2 (T +.25, T +.5) = b 2 (T +.5, T +.75) = b 2 (T +.75, T +1) for each T =, 1,..., One-facor lognormal model In he one-facor lognormal specificaion of Secion 3.2, we calibrae he parameer a 2 and b = b 2 (9, 9.25) = b 2 (9.25, 9.5) = b 2 (9.5, 9.75) = b 2 (9.75, 1) wih Malab uilising he procedure lsqnonlin based on he pricing formula (3.44) (if c <, oherwise Swn Tk = Nδc ). This calibraion yields: a 2 =.537, b =.117. Forcing posiiviy of he underlying LIBOR raes means, in his paricular case, resricing b L(; 9.75, 1) =.328, c.f. (2.27). The consrained calibraion yields: a 2 =.1864, b =.328. The wo resuling smiles can be found in Figure 1, where we can see ha he unconsrained model achieves a reasonably good calibraion. However, enforcing posiiviy is highly resricive since he Gaussian model, in his seing, canno produce a downward sloping smile. 17

18 Implied volailiy Swapion 9Y1Y 3m Tenor Marke vol Lognormal 1d Lognormal 1d - Posiiviy consrained Srike Figure 1: Lognormal one-facor calibraion Nex we calibrae he b 2 parameers o he ATM swapion erm srucures of 3 monhs and 6 monhs enors. The resuls are shown in Figure 2. When posiiviy is no enforced he model can be calibraed wih no error o he marke quoes of he ATM co-erminal swapions. However, one can see from he figure ha he posiiviy consrain does no allow he b 2 funcion o ake he necessary values, and hus a very poor fi o he daa is obained, in paricular for shorer mauriies. Wih his in mind he naural quesion is wheher he posiiviy consrain is oo resricive. Informal discussions wih marke paricipans reveal ha posiive probabiliy for negaive raes is no such a criical issue for a model. As long as he probabiliy mass for negaive values is no subsanial, i is a feaure ha can be lived wih. Indeed assigning a small probabiliy o his even may even be realisic. 1 In order o invesigae he significance of he negaive raes and spreads issue menioned in Remark 2.1, we calculae lower quaniles for spo raes as well as he spo spread for he model calibraed wihou he posiiviy consrain. As Figure 3 shows, he lower quaniles for he raes are of no concern. Indeed i can hardly be considered pahological ha raes will be below -14 basis poins wih 1% probabiliy on a hree year ime horizon. Similarly, wih regard o he spo spread, he lower quanile is in fac posiive for all ime horizons. Furher calculaions reveal ha he probabiliies of he eigh year spo spread being negaive is and he nine year is.8 which again can hardly be deemed pahologically high. We find ha he model performs surprisingly well despie he parsimony of a one-facor lognormal seup. While posiiviy of raes and spreads are no achieved, he model assigns only small probabiliies o negaive values. However, he abiliy of fiing he smile wih such a parsimonious model is no saisfacory (cf. Fig. 1), which is our moivaion for he nex specificaion. 1 A broad panel of money marke raes are currenly negaive, including DKK (CIBOR), shor erm EURIBOR and CHF LIBOR. 18

19 ATM-swapion implied volailiy Value for b2 ATM-swapion implied volailiy Value for b2.27 Calibraion o 3m ATM-swapions - implied volailiy.25 Calibraion o 3m ATM-swapions - b2-parameer Marke vol Calibraed vol Calibraed vol - Posiiviy Consrained.2 Calibraed b 2 (T,T+.25) Calibraed b 2 (T,T+.25)- Posiiviy Consrained Expiry in years T.26 Calibraion o 6m ATM-swapions - implied volailiy.25 Calibraion o 6m ATM-swapions - b2-parameer Marke vol Calibraed vol Calibraed vol - Posiiviy Consrained.2 Calibraed b 2 (T,T+.5) Calibraed b 2 (T,T+.5) - Posiiviy Consrained Expiry in years T Figure 2: One-Facor Lognormal calibraion. (Lef) Fi o ATM swapion implied volailiy erm srucures. (Righ) Calibraed values of he b 2 parameers. (Top) δ = 3m. (Boom) δ = 6m. # % Lower Quaniles L(T;T;T+.5)-L(T;T;T+.25) L(T;T;T+.25) L(T;T;T+.5) T Figure 3: One-Facor Lognormal calibraion. 1% lower quaniles 19

20 Implied volailiy 4.2 Exponenial normal inverse Gaussian model The one-facor model, which is driven by a Gaussian facor {A (2) }, is able o capure he level of he volailiy smile. Neverheless, he model implied skew is slighly differen from he marke skew. To overcome his issue, we now consider a one-facor model driven by a richer family of Lévy processes. The process {A (2) } is now assumed o be he exponenial normal inverse Gaussian (NIG) M-maringale A (2) = exp ( ) X (2) ψ(1) 1, (4.48) where {X (2) } is an M-NIG-process wih cumulan ψ(z), see (3.41), expressed in erms of he paramerisaion 2 (ν, θ, σ) from Con and Tankov (23) as ( ) ψ(z) = ν ν 2 2zθ z 2 σ 2 ν, (4.49) where ν, σ > and θ R. The parameers ha need o be calibraed a firs are ν, θ, σ and b = b 2 (9, 9.25) = b 2 (9.25, 9.5) = b 2 (9.5, 9.75) = b 2 (9.75, 1). Afer he calibraion, we obain b =.431, ν =.2498, θ =.242, σ = Imposing b L(; 9.75, 1) =.328 o ge posiive raes we obain insead b =.291, ν =.1354, θ =.82, σ =.348. The wo fis are ploed in Figure 4. Here, imposing posiiviy comes a a much smaller cos when compared o he one facor Gaussian case. The NIG process has a richer srucure (more parameric freedom) and herefore is able o compensae for an imposed smaller level of he parameer b Swapion 9Y1Y 3m Tenor Marke vol exp-nig exp-nig - Posiiviy consrained Srike Figure 4: Exponenial-NIG calibraion 2 The Barndorff-Nielsen (1997) paramerisaion is recovered by seing µ =, α = 1 σ and δ = σν. θ 2 i σ 2 i + ν 2 i, β = θi σ 2 i 2

21 ATM-swapion implied volailiy Value for b2 ATM-swapion implied volailiy Value for b2 We coninue wih he second par of he calibraion of which resuls are found in Figure 5. Here we see ha enforcing posiiviy may have a small effec on he smile bu i means ha he volailiy srucure canno be made o mach swapions wih mauriy smaller han 7 years. Thus, enforcing posiiviy in his model produces limiaions which we wish o avoid. In Figure 6, we plo lower quaniles for he raes and spreads as for he one-facor.28 Calibraion o 3m ATM-swapions - implied volailiy.2 Calibraion o 3m ATM-swapions - b 2 -parameer Marke vol Calibraed vol Calibraed vol - Posiiviy Consrained Calibraed b 2 (T,T+.25) Calibraed b 2 (T,T+.25)- Posiiviy Consrained Expiry in years T.26 Calibraion o 6m ATM-swapions - implied volailiy.2 Calibraion o 6m ATM-swapions - b2-parameer.24 Marke vol Calibraed vol Calibraed vol - Posiiviy Consrained Calibraed b 2 (T,T+.5) Calibraed b 2 (T,T+.5) - Posiiviy Consrained Expiry in years T Figure 5: Exponenial-NIG calibraion. (Lef) Fi o ATM swapion implied volailiy erm srucures. (Righ) Calibraed values of he b 2 parameers. (Top) δ = 3m. (Boom) δ = 6m. lognormal model. While spo spreads remain posiive, he levels do no, and, as shown, he model assigns an unrealisically high probabiliy mass o negaive values. In fac he model assigns a 1% probabiliy o raes falling below -12% wihin 2 years! Thus, he one-facor exponenial-nig model loses much of is appeal because i canno fi long-erm smiles and shorer-erm ATM volailiies while mainaining realisic values for he ineres rae. 4.3 Calibraion of a wo-facor lognormal model The necessiy o produce a beer fi o he smile han wha can be achieved wih he one-facor Gaussian model, while mainaining realisically posiive raes and spreads, leads us o proposing he wo-facor specificaion presened in Secion 3.3. This model is heavily 21

22 .2 1% Lower Quaniles L(T;T;T+.5)-L(T;T;T+.25) L(T;T;T+.25) L(T;T;T+.5) T Figure 6: Exponenial-NIG calibraion calibraion. 1% lower quaniles. paramerised and he parameers a hand are no all idenified by he considered daa. We herefore fix he following parameers: a 1 = 1, a 3 = 1.6, (4.5) b 3 (T, T +.25) =.15L(; T ; T +.25), T [9, 9.75, (4.51) b 2 (T, T +.25) =.55L(, T ; T +.25), T [, (4.52) We assume ha b 1 is consan, i.e. b 1 = b 1 (T ) for T [, 1, and ha b 3, ouside of he region defined above, is piecewise consan such ha b 3 (T, T +.25) = b 3 (T +.25, T +.5) = b 3 (T +.5, T +.75) = b 3 (T +.75, T + 1) for each T =, 1..., 8 and b 3 (T, T +.5) = b 3 (T +.5, T + 1) holds for each T =, 1..., 9. We furhermore assume ha b 2 (T, T +.5) = b 2 (T, T +.25), T [, 9.5. These somewha ad hoc choices are made wih a view owards b 2 and b 3 being fairly smooh funcions of ime. We herewih apply a slighly alered procedure o calibrae he remaining parameers if compared o he scheme uilised for he one-facor models. 1. We firs calibrae o he smile of he 9Y1Y years swapion which gives us he parameers a 2, ρ, he assumed consan value of b 1, and b 2 (9, 9.25) o b 2 (9.75, 1) which are assumed equal o a consan b. Similar o he exponenial-nig model, we make use of four parameers in oal o fi he smile. 2. The remaining b 2 parameers are deermined a priori, so wha remains is o calibrae he values of b 3. The hree-monh enor values b 3 (T, T +.25) for T [, 8.75 are calibraed o ATM, co-erminal swapions saring from he 8Y2Y years and hen coninuing backwards o he 1Y9Y insrumens. For he six-monh enor producs, we calibrae b 3 (T, T +.5) for T [, 9.5 saring wih 9Y1Y and proceed backwards. These are he values we obain from he firs calibraion phase: b 1 =.2434, b =.2, a 2 =.1888, ρ =.953. The corresponding fi is ploed in he upper lef quadran of Figure 7. In order o check he robusness of he calibraed fi hrough ime, we also calibrae o hree alernaive daes. The qualiy of he fi appears quie saisfacory 22

23 Implied volailiy Implied volailiy Implied volailiy Implied volailiy.24 Swapion 9Y1Y 3m Tenor Swapion 9Y1Y 3m Tenor Marke vol Lognormal 2d b 1 = b=.217 a 2 = ;= Marke vol Lognormal 2d b 1 =.2772 b= a 2 =.1763 ;= Srike Srike.165 Swapion 9Y1Y 3m Tenor Swapion 9Y1Y 3m Tenor Marke vol Lognormal 2d b 1 = b= a 2 =.1937 ;= Marke vol Lognormal 2d b 1 = b= a 2 = ;= Srike Srike Figure 7: Lognormal wo-facor calibraion. 23

24 Value for b2 and comparable o he exponenial-nig model. For all four daes he calibraion is done enforcing he posiiviy condiion b 2 (T, T +.25) + b 3 (T, T +.25) L(; T, T +.25). However, he procedure yields he exac same parameers even if he consrain is relaxed. We hus conclude ha a beer calibraion appears no o be possible for hese daases by allowing negaive raes. Noe ha i is only for our firs daa se ha he calibraed correlaion ρ is as high as.953. In he oher hree cases we have ρ =.4118, ρ =.3964, and ρ = Figure 8 shows he parameers b 2 and b 3 obained a he second phase of he calibraion o he daa of 4 January 211. As wih he previous model (cf. he lef graphs of Figures 2 and 5), he volailiies are mached o marke daa wihou any error..22 3m ATM Swapion Calibraed parameer values.22 6m ATM Swapion Calibraed parameer values.2 b 2 (T,T+.25) b 3 (T,T+.25).2 b 2 (T,T+.5) b 3 (T,T+.5) Years Years Figure 8: Two-facor lognormal calibraion. (Lef) Parameer values fied o hree-monh ATM swapion implied volailiy erm srucures. (Righ) Parameer values fied o sixmonh ATM swapion implied volailiy erm srucures. We add here ha, alhough no visible from he graphs, he calibraed parameers saisfy he LIBOR spread posiiviy discussed in Remark 2.1. In conclusion, we find ha he wo-facor log-normal has he abiliy o fi he swapion smile very well, i can be conrolled o generae posiive raes and posiive spreads, and i is racable wih numerically-efficien closed-form expressions for he swapion prices. Given hese desirable properies, we discard he one-facor models and reain he wo-facor lognormal model for all he analyses in he remaining par of he paper. 5 XVA Analysis So far we have focused on so-called clean compuaions, i.e. ignoring counerpary-risk and assuming ha funding is obained a he risk-free OIS rae. In realiy, conracually specified counerparies a he end of a financial agreemen may defaul, and funding o ener or honour a financial agreemen may come a a higher cos han a OIS rae. Thus, various valuaion adjusmens need o be included in he pricing of a financial posiion. The price of a counerpary-risky financial conrac is compued as he difference beween he clean price, as in previous secions, and adjusmens accouning for counerpary-risk and 24