On multicurve models for the term structure

Transcription

1 On mulicurve models for he erm srucure Laura Morino Diparimeno di Maemaica Pura ed Applicaa Universià di Padova, Via Triese 63, I-3511-Padova Wolfgang J. Runggaldier Diparimeno di Maemaica Pura ed Applicaa Universià di Padova, Via Triese 63, I-3511-Padova Absrac In he conex of muli-curve modeling we consider a wo-curve seup, wih one curve for discouning (OIS swap curve) and one for generaing fuure cash flows (LIBOR for a give enor). Wihin his conex we presen an approach for he clean-valuaion pricing of FRAs and CAPs (linear and nonlinear derivaives) wih one of he main goals being also ha of exhibiing an adjusmen facor when passing from he one-curve o he wo-curve seing. The model iself corresponds o shor rae modeling where he shor rae and a shor rae spread are driven by affine facors; his allows for correlaion beween shor rae and shor rae spread as well as o exploi he convenien affine srucure mehodology. We briefly commen also on he calibraion of he model parameers, including he correlaion facor. Mahemaics Subjec Classificaion : Primary 91G30; Secondary 91G0, 60H30. Keywords : Mulicurve models, affine facor models, ineres rae derivaives, clean valuaion, adjusmen facors. 1 Inroducion In he wake of he big crisis one has winessed a significan increase in he spreads beween LIBORs of differen enors as well as he spread beween a LIBOR and he discoun curve (LIBOR-OIS). This has led o he consrucion of mulicurve models where, ypically, fuure cash flows are generaed Presen affiliaion: Deloie Consuling Srl., Milano. 1

2 hrough curves associaed o he underlying raes, bu are discouned by anoher curve. The majoriy of he models ha have been considered reflecs he usual classical disincion beween i) shor rae models; ii) HJM seup; iii) BGM or LIBOR marke models. By analogy o credi risk we may call he firs wo caegories of models as boom-up models, while he hird one could be classified as op-down. In addiion, mehodologies have appeared ha are relaed o foreign exchange. Here we consider only he firs wo seups. We begin by discussing some issues arising wih he HJM mehodology and concenrae hen on shor rae models. The hird seup (op-down) is mainly presen in work by F. Mercurio and co-auhors (see e.g. 19, 0), bu also in oher recen work such as 16. There are advanages and disadvanages wih each seup. Among he possible advanages of shor rae models is he fac ha hey lead more easily o a Markovian seing, which is convenien for various calculaions (see 8). On he oher hand, one of he major advanages of HJM over a direc shor rae modeling is ha he model is auomaically calibraed o he iniial erm srucure. Shor rae models in a muli-curve seup have already appeared in he lieraure, e.g. 18, 17, 11. To presen he basic ideas in a simple way, here we consider a wocurve model, namely wih a curve for discouning and one for generaing fuure cash flows. The choice of he discoun curve is no unique; we follow he common choice of considering he OIS swap curve. For he risky cash flows wihou collaeral we consider a single LIBOR (i.e. for a given enor srucure). We presen an approach for he pricing of some basic LIBOR-relaed derivaives, namely FRAs and CAPs (linear/nonlinear) and consider only clean valuaion formulas, namely wihou counerpary risk. Alhough real pricing problems require a more global approach (see e.g. he discussions in 13, 14,, 6 as well as in recen work by D.Brigo and co-auhors such as 1, ), clean valuaion formulas are neverheless useful for various reasons: as poined ou in 8, marke quoes ypically reflec prices of fully collaeralized ransacions so ha clean price formulas may urn ou o be sufficien for calibraion also when using he model o compue possible value adjusmens; furhermore (see 8), TVA adjusmens are ofen compued on op of clean prices. Concerning mehodology, since our approach is of he boom-up ype ha considers shor rae modeling, we heavily exploi he advanages of an affine erm srucure. This is in conras wih opdown approaches, where (see 19, 0) log-normal models are common (see

3 however 16 and 15 for affine LIBOR models wih general disribuions in a mulicurve conex). Tradiionally, ineres raes are defined o be coheren wih he bond prices p(, T ), which represen he expecaion of he marke concerning he fuure value of money. For he discree compounding forward LIBORs, which we denoe here by L(; T, S), his leads o ( < T < S) L(; T, S) = 1 S T ( ) p(, T ) p(, S) 1 (1.1) which can also be jusified as represening he fair value of he fixed rae in a FRA on he LIBOR. Since we consider only a single LIBOR ha corresponds o a given enor srucure, we assume S = T + (for enor ). In his way one obains a single curve for he erm srucure. The acual LIBOR raes, which in wha follows we shall denoe by L(; T, T + ), are deermined by he LIBOR panel ha akes ino accoun various facors such as credi risk, liquidiy, ec. (see he discussion in 11). Following some of he recen lieraure, in paricular 7 (see also 18), we keep he formal relaionship (1.1) beween LIBOR raes and bond prices, bu replace he risk-free bond prices p(, T ) by ficiious risky bond prices p(, T ) ha are supposed o be affeced by he same facors as he acual LIBORs and ha, analogously o he risk-free bond prices, we define hen as p(, T ) = E {exp Q T (r u + s u )du F (1.) where r is he classical shor rae, whereas s represens he shor rae spread (hazard rae in case of only defaul risk). Noice ha in his way he spread is inroduced from he ouse. Noice also ha he ficiious bond prices p(, T ) are no acual prices. Since in wha follows we are ineresed in FRAs and CAPs ha are based on he T spo LIBOR L(; T, T + ), we acually posulae he relaionship (1.1) only a he incepion ime = T. Our saring poin is hus he following relaionship L(T ; T, T + ) = 1 ( ) 1 p(t, T + ) 1 (1.3) where we have aken ino accoun he fac ha also for he risky bonds we have p(t, T ) = 1. In addiion o he pricing of FRAs and CAPs in our wo-curve seup, our major goal here is o derive a relaionship beween heoreically risk-free and acual FRAs (possibly also CAPs) hereby exhibiing an adjusmen facor which plays a role analogous o ha of he quano adjusmens in he pricing of cross-currency derivaives or he muliplicaive forward basis in 1. 3

4 The model.1 Preliminary consideraions We sar wih some commens concerning HJM-like approaches o beer moivae our shor rae approach. Given he bond price processes p(, T ) and p(, T ), in order o apply an HJM-approach, we need o inroduce corresponding forward rae processes f T () and f T () ha lead o a forward rae spread expressed as g T () := f T () f T (). One hen also obains corresponding shor raes and a shor rae spread, namely r = f (), r = f (), s = g () = r r. Noice ha a consisen model should lead o p(, T ) p(, T ), which implies f T () f T () or, equivalenly g T () 0 < T T, where T is a given maximal mauriy. An exensive sudy wihin he mulicurve HJM approach has appeared in 7. The driving random process is a Levy and a corresponding HJM drif condiion is derived. Condiions are given for he non-negaiviy of raes and spreads; explici formulas are obained for various ineres rae derivaives. Wha may no be fully saisfacory in 7 is ha: i) some difficulies arise when dealing no only wih credi risk, bu also oher risks such as liquidiy. In paricular, when looking for a condiion ha corresponds o he defaulable HJM drif condiion; ii) a ficiious defaul has o be considered explicily (wih pre defaul bond prices). The sudy in 7 is coninued in he recen paper 8 wih he main purpose of aking ino accoun also counerpary risk and funding coss and of deermining various valuaion adjusmens on op of he clean prices. The mehodology in 8 is again based on an HJM approach, bu wih explici ingrediens for he induced shor rae models in order o obain a Markovian srucure and o be able o acually perform he value adjusmen calculaions. In paricular, he auhors in 8 use a Levy Hull & Whie exended Vasicek model for r and inroduce an addiional facor ha can be inerpreed as represening a shor rae spread. In his laer sense i becomes analogous o he approach o be presened here. Anoher HJM-based approach, limied o defaul risk, appears in 3 wih emphasis on obaining Markovian models wih sae dependen volailiies. The driving processes are of he jump-diffusion ype. The difficulies here appear o be given by he fac ha, for convenien specificaions of he volailiies, one obains deerminisic shor rae spreads. For more general, sochasic volailiies he auhors obain only approximae Markovianiy. These difficulies have been overcome in he subsequen paper 4, where he auhors obain finie-dimensional Markovian realizaions also wih sochasic spreads and, in addiion, obain a correlaion srucure beween 4

5 credi spread, ineres rae and he sochasic volailiy. When rying o exend heir approach o a muli curve seing, beyond ha implied by credi risk alone, here appear hough some compuaional difficulies due o he sochasic volailiy. Before coming now o describing our shor rae model, we recall some basics concerning FRAs. We sar from he Definiion.1. A FRA (forward rae agreemen) is an OTC derivaive ha allows he holder o lock in a < T he ineres rae beween he incepion dae T and he mauriy T + a a fixed value K. A mauriy T +, a paymen based on K is made and one based on L(T ; T, T + ) is received. We shall denoe he value of he FRA a < T by F RA T (, K). In our wo-curve risky seup, he fair price of a FRA in < T wih fixed rae K and noional N is F RA T (, K) = N p(, T + )E T + L(T ; T, T + ) K F = Np(, T + )E T + 1 p(t,t + ) (1 + K) F (.1) where E T + denoes expecaion under he (T + ) forward measure Q T +. Noice ha he simulaneous presence of p(, T + ) and p(, T + ) does no allow for he convenien reducion of he formula o a simpler form as in he one-curve seup.. Descripion of he model iself For he shor-rae model approach we shall have o sar by modeling direcly he shor rae r and he shor rae spread s and we do i under he sandard maringale measure Q (o be calibraed o he marke) for he risk-free money marke accoun as numeraire. In order o accoun for a possible (negaive) correlaion beween r and s we inroduce a facor model: given hree independen affine facor processes Ψ i, i = 1,, 3 le { r = Ψ Ψ 1 s = κψ 1 + Ψ 3 (.) where κ is a consan ha measures he insananeous correlaion beween r and s (negaive correlaion for κ > 0). This seup could be generalized in various ways, in paricular by using more facors o drive s. In view of he exising lieraure one could, insead of using an affine model srucure as we do i here, consider e.g. ambi-ype processes as presened in 5. Such a model, which is no of he semimaringale ype, allows also for analyical compuaions and gives he possibiliy o ake ino accoun long-range dependence. Remaining wihin he pure credi risk seing where, see he commen afer (1.), he spread is given by he defaul inensiy, some of he facors affecing he spread could be given a specific meaning as in 9 5

6 where, using an HJM-ype approach, he auhors consider a spread field process wih one of he variables represening he raing of he issuer. The approach in 9 could possibly be generalized also io he presen seing. A common approach o modeling he facors in an affine conex is o assume hem of he ype of a square roo diffusion. This guaranees posiiviy of he spread, bu he negaive correlaion comes a he expense of possibly negaive ineres raes (even if only wih small probabiliy). Wih such a model, by passing o he (T + ) forward measure, one can compue he value of a FRA and of he fair fixed rae. For various reasons, in paricular in view of our main goal o obain an adjusmen facor, i is convenien o be able o have he same facor model for FRAs wih differen mauriies. We herefore aim a performing he calculaions under a single reference measure, namely he sandard maringale measure Q. More precisely, for he facor processes we assume he following affine diffusions under Q ha are of he Vasicek ype, namely dψ 1 = (a 1 b 1 Ψ 1 )d + σ 1 dw 1 dψ i = (a i b i Ψ i )d + σ i (.3) Ψ i dwi, i =, 3 where a i, b i, σ i are posiive consans wih a i (σ i ) / for i =, 3, and w i independen Wiener processes. We have chosen a Vasicek-ype model for simpliciy, bu he resuls below can be easily exended o he Hull & Whie version of he Vasicek model. Noice ha he facor Ψ 1 may ake negaive values implying ha, no only r, bu also s may become negaive (see however laer under commens on he main resul ). Resuls compleely analogous o hose ha we shall obain here for he above pure diffusion model may be derived also for affine jump-diffusions a he sole expense of more complicaed noaion. 3 Main resul (FRAs) 3.1 Preliminary noions and resuls Recalling he expression for a FRA under he forward measure, namely F RA T (, K) = Np(, T + )E T + 1 p(t, T + ) (1 + K) F, (3.1) one has ha he crucial quaniy o compue is ν,t := E T + 1 p(t, T + ) F (3.) 6

7 and ha he fixed rae o make he FRA a fair conrac a ime is K := 1 ( ν,t 1) (3.3) In he classical single curve case we have insead ν,t := E T + 1 p(t, T + ) F = p(, T ) p(, T + ) (3.4) p(,t ) being p(,t + ) an F maringale under he (T + ) forward measure. The fair fixed rae in he single curve case is hen K = 1 (ν,t 1) = 1 ( ) p(, T ) p(, T + ) 1 (3.5) and noice ha, in order o compue K, no ineres rae model is needed (conrary o K ). Due o he affine dynamics of Ψ i (i = 1,, 3) under Q, we have for he risk-free bond { p(, T ) = E Q exp { T r u du F = E Q T exp (Ψ1 u Ψ u)du F = exp A(, T ) B 1 (, T )Ψ 1 B (, T )Ψ The coefficiens saisfy B 1 b 1 B 1 1 = 0, B 1 (T, T ) = 0 B b B (σ ) (B ) + 1 = 0, B (T, T ) = 0 A = a 1 B 1 (σ1 ) (B 1 ) + a B, A(T, T ) = 0 leading, in paricular, o (3.6) (3.7) B 1 (, T ) = 1 ) (e b1 (T ) b 1 1. (3.8) For he risky bond we have insead { p(, T ) = E Q exp T (r u + s u )du F { = E Q exp T ((κ 1)Ψ1 u + Ψ u + Ψ 3 u)du F = exp Ā(, T ) B 1 (, T )Ψ 1 B (, T )Ψ B 3 (, T )Ψ 3 (3.9) 7

8 This ime he coefficiens saisfy B 1 b 1 B1 + (κ 1) = 0, B1 (T, T ) = 0 B b B (σ ) ( B ) + 1 = 0, B (T, T ) = 0 B 3 b 3 B3 (σ3 ) ( B 3 ) + 1 = 0, B3 (T, T ) = 0 Ā = a 1 B1 (σ1 ) ( B 1 ) + a B + a 3 B3, Ā(T, T ) = 0 (3.10) leading, in paricular, o B 1 (, T ) = 1 κ ( ) b 1 e b1 (T ) 1 = (1 κ) B 1 (, T ) (3.11) From he above 1 s order equaions i follows ha B 1 (, T ) = (1 κ) B 1 (, T ) B (, T ) = B (, T ) Ā(, T ) = A(, T ) a 1 κ T B 1 (u, T )du Leing hen we obain + (σ1 ) κ T (B1 (u, T )) du (σ 1 ) κ T B 1 (u, T )du a 3 T B 3 (u, T )du (3.1) Ã(, T ) := Ā(, T ) A(, T ) (3.13) p(, T ) = exp Ā(, T ) B 1 (, T )Ψ 1 B (, T )Ψ B 3 (, T )Ψ 3 + κb 1 (, T )Ψ 1 = p(, T ) exp Ã(, T ) + κb 1 (, T )Ψ 1 B 3 (, T )Ψ 3 so ha, puing for simpliciy B 1 := B 1 (T, T + ), one may wrie (3.14) p(t, T + ) p(t, T + ) = exp Ã(T, T + ) κ B 1 Ψ 1 T + B 3 (T, T + )Ψ 3 T. (3.15) 3. The resul iself We inroduce he Definiion 3.1. We call adjusmen facor he process { Ad T, p(t, T + ) := E Q p(t, T + ) F, (3.16) 8

9 and shall prove he following Proposiion 3.1. We have ν,t = ν,t Ad T, exp κ (σ1 ) (b 1 ) 3 ( 1 e b1 ) ( 1 e b1 (T ) ) (3.17) wih wo adjusmen facors on he righ, of which he firs one can be expressed as { Ad T, = e Ã(T,T + ) E Q e κ B 1 Ψ 1 T + B 3 (T,T + )Ψ 3 T F (3.18) := A(θ, κ, Ψ 1, Ψ 3 ) wih θ := (a i, b i, σ i, i = 1,, 3). One may noice he analogy here wih he muliplicaive forward basis in 1. As a consequence of he previous proposiion we have he following relaion beween he fair value K of he fixed rae in an acual FRA and he fair value K in a corresponding riskless one: Corollary 3.1. The following relaionship holds ( K = K + 1 ) Ad T, exp κ (σ1 ) (1 ) ( ) (b 1 ) 3 e b1 1 e b1 (T ) 1 (3.19) Noice ha he facor given by he exponenial is equal o 1 for zero correlaion, i.e. for (κ = 0). 3.3 Commens on he main resul Commens concerning he adjusmen facors An easy inuiive inerpreaion of he main resul can be obained in he case of κ = 0 (independence of r and s ): in his case we have r + s > r implying p(t, T + ) < p(t, T + ) so ha Ad T, 1 (he exponenial adjusmen facor is equal o 1). As expeced, from Proposiion 3.1 and Corollary 3.1 i hen follows ha ν,t ν,t, K K (3.0) To gain some inuiion for he cases when κ 0, le p κ (, T ), ν κ,t, AdT,,κ denoe he given quaniies by sressing ha he correlaion parameer has value κ. Noice ha p(, T ) and hus also ν,t do no depend on κ. Consider hen he case κ > 0, which is he sandard case implying negaive correlaion beween r and s. (The case κ < 0 is analogous/dual). For illusraive purposes we disinguish beween he wo evens {Ψ 1 > 0, T, T + 9

10 , {Ψ 1 < 0, T, T + where he laer occurs only wih small probabiliy (in realiy, Ψ 1 will be posiive for cerain values of and negaive for he remaining ones). On {Ψ 1 > 0, T, T + we now have p κ (T, T + ) < p 0 (T, T + ) ν κ,t > ν0,t ν κ,t /ν,t > ν 0,T /ν,t (3.1) Recalling hen ν κ,t = ν,t Ad T,,κ exp κ (σ1 ) ( ) ( ) (b 1 ) 3 1 e b1 1 e b1 (T ) (3.) he las inequaliy in (3.1) can be seen o be in line wih he fac ha, in his case, in (3.) he exponenial facor is > 1 and Ad T,,κ > Ad T,,0 (recall Definiion 3.1). On he oher hand, on {Ψ 1 < 0, T, T +, we have p κ (T, T + ) > p 0 (T, T + ) ν κ,t /ν,t < ν 0,T /ν,t (3.3) This inequaliy can be seen o be in line wih he fac ha, here, Ad T,,κ < Ad T,,0, bu he exponenial facor is sill > 1. This can neverheless be explained by noicing ha, in his case, r is relaively large and r + s is closer o r (may be even < r ). This implies a push of ν,t κ /ν,t owards smaller values han in he previous case Commens concerning he use of he resuls for calibraion For wha concerns calibraion of our model o FRA and oher available marke daa, noice ha he coefficiens a 1, a, b 1, b, σ 1, σ can be calibraed in he usual way on he basis of he observaions of defaul-free bonds p(, T ) (if we had a Hull& Whie exension of our Vasicek-ype model (.3) hen also for his model he calibraion could be performed as in he sandard case). To calibrae a 3, b 3, σ 3, noice ha, conrary o p(, T ), he risky bonds p(, T ) are no observable (relaion (1.3) does no imply a unique inverse relaionship o deermine p(, ( T ) from observaions ) of he LIBORs). One can however observe K = 1 p(,t ) p(,t + ) 1 as well as he risky FRA rae K. Recalling hen Corollary 3.1 and he fac ha Ad T, = A(θ, κ, Ψ 1, Ψ 3 ), noice ha, having calibraed a i, b i, σ i (i = 1, ), from he observaions of K and K one could hus calibrae a 3, b 3, σ 3 as well as κ. If here is a way o deermine direcly Ad T, (e.g. by observing he FRA raes for uncorrelaed r and s ), hen he relaionship beween K and K as expressed in Corollary 3.1 would allow o calibrae separaely κ. We furhermore recall ha, as poined ou in 8, calibraion of clean prices is sufficien also when using he model o compue possible value adjusmens. 10

11 3.4 Proof of he main resul Since he quaniies of ineres, namely ν,t and ν,t were defined under he forward measure (see (3.) and (3.4)), as a firs sep we perform a change from he forward measure Q T + o he sandard maringale measure Q. To his effec, puing b := exp 0 r udu, he densiy process for changing p(,t + ) from Q o Q T + is L = p(0,t + )b. We can hus wrie ν,t = E T + { 1 p(t,t + ) F { = L 1 E Q LT + p(t,t + ) F { 1 = p(,t + ) EQ exp T p(t,t + ) r u du p(t,t + ) F (3.4) Recalling he expression for p(t, T + )/ p(t, T + ) (see (3.15)) his becomes { 1 ν,t = p(,t + ) EQ e R T r udu exp Ã(T, T + ) κ B 1 Ψ 1 T + B 3 (T, T + )Ψ 3 T F = 1 p(,t + ) exp Ã(T, T + ) E Q { e B 3 (T,T + )Ψ 3 T F E Q { e R T ( Ψ1 u+ψ u)du e κ B 1 Ψ 1 T F (3.5) To proceed, consider he process F given by he las facor in (3.5), namely { F := E Q e R T ( Ψ1 u +Ψ u )du e κ B 1 Ψ 1 T F (3.6) Due o he affine dynamics of Ψ i, i = 1,, and he independence of Ψ 1 and Ψ, we may wrie { F := E Q e R { T Ψ 1 u du e κ B 1 Ψ 1 T F E Q e R T Ψ u du F = exp α 1 (, T ) β 1 (, T )Ψ 1 exp α (, T ) β (, T )Ψ where he coefficiens saisfy β 1 b 1 β 1 1 = 0, β 1 (T, T ) = κ B 1 (3.7) β b β (σ ) (β ) + 1 = 0, β (T, T ) = 0 α 1 = (σ1 ) (β 1 ) + a 1 β 1, α 1 (T, T ) = 0 α = a β, α (T, T ) = 0 11 (3.8)

12 Recalling also (3.6)-(3.8), he soluions of he sysem (3.8) can be expressed as β 1 (, T ) = (b 1 1 κ B 1 + 1)e b 1 b1 (T ) 1 = B 1 (, T ) + κ B 1 e b1 (T ) β (, T ) = B (, T ) α 1 (, T ) = (σ1 ) = (σ1 ) T (β1 (u, T )) du a 1 T β 1 (u, T )du (B1 (u, T )) du a 1 T B 1 (u, T )du e b1 (T u) du T α (, T ) = a T Consequenly F = exp (σ 1 ) a T + (σ1 ) (κ B 1 ) T +κ B 1 (σ 1 ) T B (u, T )du B 1 (u, T )e b1 (T u) du a 1 κ B 1 T T (B1 (u, T )) du a 1 T B 1 (u, T )du B (u, T )du B 1 (, T )Ψ 1 B (, T )Ψ exp (σ 1 ) (κ B 1 ) T e b1 (T u) du a 1 κ B 1 T κ B 1 e b1 (T ) Ψ 1 exp κ B 1 (σ 1 ) T B 1 (u, T )e b1 (T u) du = p(, T ) exp (σ 1 ) (κ B 1 ) T exp κ B 1 (σ 1 ) T B 1 (u, T )e b1 (T u) du On he oher hand, recalling (3.15), one obains { E Q p(t,t + ) p(t,t + ) F e b1 (T u) du e b1 (T u) du e b1 (T u) du a 1 κ B 1 T e b1 (T u) du κ B 1 e b1 (T ) Ψ 1 = e Ã(T,T + ) E Q { e B 3 (T,T + )Ψ 3 T F E Q { e κ B 1 Ψ 1 T F (3.9) (3.30) (3.31) where, due o he affine dynamics of Ψ 1 {, we may wrie E Q e κ B 1 Ψ 1 T F = exp ᾱ(, T ) β(, T )Ψ 1 wih ᾱ( ) and β( ) saisfying { β b 1 β = 0, β(t, T ) = κ B1 ᾱ = a 1 β (σ 1 ) ( β), ᾱ(t, T ) = 0 (3.3) (3.33) 1

13 so ha β(, T ) = κ B 1 e b1 (T ) ᾱ(, T ) = a 1 κ B 1 T e b1 (T u) du + (σ1 ) (κ B 1 ) T e b1 (T u) du (3.34) and,consequenly, { E Q e κ B 1 Ψ 1 T F = exp exp κ B 1 e b1 (T ) Ψ 1 a 1 κ B 1 T e b1 (T u) du + (σ1 ) (κ B 1 ) T e b1 (T u) du (3.35) Combining (3.5) wih (3.30) as well as wih (3.31) ogeher wih (3.35), we obain { 1 ν,t = p(,t + ) exp Ã(T, T + ) E Q e B 3 (T,T + )Ψ 3 T F F { = p(,t ) p(t,t + ) p(,t + ) EQ The resul hen follows noicing ha B 1 T p(t,t + ) F exp κ(σ 1 ) T B1 B 1 (u, T )e b1 (T u) du. B 1 (u, T )e b1 (T u) du = 1 (b 1 ) 3 ( 1 e b1 ) ( 1 e b1 (T ) ). 4 Aspecs of CAP pricing 4.1 Preliminary commens (3.36) (3.37) This par is relaed o work in progress, bu we wan neverheless o presen some ideas on how our resuls obained for FRAs (linear derivaives) can be exended o nonlinear derivaives. To discuss a specific case, we concenrae here on he pricing of a single Caple, wih srike K, mauriy T on he spo LIBOR for he period T, T +. Using he forward measure Q T +, is price in < T is hen given by Capl T, () = p(, T + )E T + { ( L(T ; T, T + ) K ) + F { ( = p(, T + )E T + 1 p(t,t + ) K ) (4.1) + F wih K := 1 + K. As model, we may use he same risky shor rae model as for he FRAs ha we may consider as already calibraed (for he sandard maringale measure Q). I may hus suffice o derive jus a pricing algorihm ha 13

14 need no also be used for calibraion. The mos convenien way o price a Caple is, as in (4.1) and as we do i below, o compue he expecaions under he forward measure. Noice however ha expecaions wih respec o a forward measure can be easily compued by performing a change o he sandard maringale measure (see e.g. (3.4)), namely he one for which we may already have calibraed he model. Besides pricing, i may be desirable o obain also here an adjusmen facor. 4. A possible pricing mehodology For he pricing, in he forward measure, we may use Fourier ransform mehods as in 7 and 8 hereby represening he claim as ( e K ) + wih := log p(t, T + ) (4.) We hen need only o compue he momen generaing funcion of, which is a linear combinaion of he facors (his compuaion is feasible hanks o he affine srucure) and use he Fourier ransform of f(x) = ( e x K ) +, which is well-known. Noice ha one could possibly also apply a Gram-Charlier expansion as in 18. Wih he Fourier ransform mehod he price in = 0 of he Caple can hen be obained in he form (see 8) Capl(0, T, T + ) = p(0, T + ) π K1 iv R T + M (R + iv) dv (4.3) (R + iv) (R + iv 1) M T + where ( ) is he momen generaing funcion of under he (T + T + ) forward measure and R is such ha M (R + iv) is finie. This momen generaing funcion can be compued for each of he various forward measures in erms of he Q characerisics of he facors, analogously o he compuaions in secion 3.4 (see, in paricular, (3.4)). From hese compuaions one can also see ha he Radon-Nikodym-derivaive o change from Q o Q T + can in fac be expressed in explici form and i preserves he affine srucure, see Corollary 10. in 10 (For a recen accoun on condiions for an absoluely coninuous measure ransformaion o preserve he affine srucure see 1). If M T + (z) is he momen generaing funcion of wih p(t, T + ) insead of p(t, T + ), hen M T + (z) = M T + (z)a(z; θ, κ, Ψ 1 0, Ψ 0, Ψ 3 0) (4.4) { ( ) 1 where A(z; θ, κ, Ψ 1 0, Ψ 0, Ψ3 0 ) = ET + M T + (z) e z. Now, from he 14

15 expression for p(, T ) in (3.6) we obain (z) = E T + { e z = E T + { e z log p(t,t + ) = E T + { exp za(t, T + ) + zb 1 (T, T + )Ψ 1 T + zb (T, T + )Ψ T M T + (4.5) On he oher hand, from he expression for p(, T ) in (3.9) (see also he varian in (3.14), where he parameer κ appears explicily) we obain ( ) 1 A(z; θ, κ, Ψ 1 0, Ψ 0, Ψ3 0 ) = M T + (z) E T + { e z log p(t,t + ) = E T + { exp zā( ) + z B 1 ( )Ψ 1 T + z B ( )Ψ T + z B 3 ( )Ψ 3 (4.6) T where ( ) sands for (T, T + ). Given he affine naure of he facors, boh expressions in (4.5) and (4.6) can be explicily compued as a funcion of he parameers of he model and he iniial values Ψ 1 0, Ψ 0, Ψ3 0 of he facors, as expressed by he symbol A(z; θ, κ, Ψ 1 0, Ψ 0, Ψ3 0 ). We may now consider A(z; θ, κ, Ψ 1 0, Ψ 0, Ψ3 0 ) as adjusmen facor for his nonlinear example given by he Caples. I is no as explici as he adjusmen facor for he FRAs in (3.17) and (3.19) and we are presenly working on obaining a more explici form also in his case. Acknowledgemens: We are graeful o Giulio Migliea as well as o Claudio Fonana and Zorana Grbac for very consrucive commens. References 1 M. Bianchei, Two Curves, One Price: Pricing & Hedging Ineres Rae Derivaives, Decoupling Forwarding and Discouning Yield Curves, ariv: v4 (Aug. 01) D. Brigo, M. Morini, A. Pallavicini, Counerpary Credi Risk, Collaeral and Funding wih pricing cases for all asse classes. Wiley, Forhcoming, C. Chiarella, C. Nikiopoulos Sklibosios, E. Schloegl, A Markovian defaulable erm srucure model wih sae dependen volailiies, Inernaional Journal of Theoreical and Applied Finance 10 (007), pp C. Chiarella, S.C. Maina, C. Nikiopoulos Sklibosios, Markovian defaulable HJM erm srucure models wih unspanned sochasic volailiy. Quaniaive Finance Research Cenre Research Paper no. 83 (010), Universiy of Technology, Sydney. 5 J.M. Corcuera, G. Farkas, W. Schouens, E. Valkeila. A shor rae model using ambi processes. In: Malliavin Calculus and Sochasic Analysis. A Fesschrif in Honor of David Nualar. Springer Proceedings in 15

16 Mahemaics & Saisics Volume 34 (013), pp Springer Science+Business Media, New York. 6 S. Crépey, R. Gerboud, Z. Grbac, N. Ngor, Counerpary risk and funding: The four wings of he TVA. Inernaional Journal of Theoreical and Applied Finance (online 30 April 013). 7 S. Crépey, Z. Grbac and H.-N. Nguyen, A muliple-curve HJM model of inerbank risk. Mahemaics and Financial Economics 6(3) (01), pp S. Crépey, Z. Grbac, N. Ngor, D. Skovmand, A Levy HJM muliple-curve model wih applicaion o CVA compuaion. Preprin R. Douady, M. Jeanblanc, A raing-based model for credi derivaives, European Invesmen Review 1 (00), pp D. Filipović, Term Srucure Models A Graduae Course, Springer Verlag (009). 11 D. Filipović, A.B. Trolle, The erm srucure of inerbank risk. Journal of Financial Economerics 109 (013), pp C. Fonana, J.M. Mones, A unified approach o pricing and risk managemen of equiy and credi risk, Journal of Compuaional and Applied Mahemaics, 59(B) (014), pp M. Fuji, Y. Shimada, A.Takahashi, A noe on he consrucion of muliple swap curves wih and wihou collaeral. CARF Working Paper Series F-154, M. Fuji, Y. Shimada, A.Takahashi, A marke model of ineres raes wih dynamic basis spreads in he presence of collaeral and muliple currencies. Wilmo Magazine 54 (011), pp Z. Grbac, A. Papapanoleon, J. Schoenmakers, D. Skovmand, Affine LIBOR models wih muliple curves: heory, examples and calibraion. In preparaion. 16 M. Keller-Ressel, A. Papapanoleon, J. Teichmann, The Affine Libor Models, Mahemaical Finance 3 (013), pp C. Kenyon, Shor-Rae Pricing Afer he Liquidiy and Credi Shocks: Including he Basis (Augus 18, 010). Available a SSRN: hp://ssrn.com/absrac= or hp://dx.doi.org/10.139/ssrn M. Kijima, K.Tanaka, T. Wong, A muli-qualiy model of ineres raes, Quaniaive Finance, 9() (009), pp

17 19 F. Mercurio, Ineres Raes and The Credi Crunch: New Formulas and Marke Models. Bloomberg Porfolio Research Paper F. Mercurio, LIBOR Marke Models wih Sochasic Basis. Bloomberg Educaion and Quaniaive Research Paper A. Pallavicini, D. Brigo, Ineres-Rae Modelling in Collaeralized Markes: Muliple curves, credi-liquidiy effecs, CCPs, (013) ariv: v1 V. Pierbarg, Funding beyond discouning: collaeral agreemens and derivaives pricing. Risk Magazine, 4 (010), pp