On multicurve models for the term structure.

Transcription

1 On mulicurve models for he erm srucure. Wolfgang Runggaldier Diparimeno di Maemaica, Universià di Padova WQMIF, Zagreb 2014

2 Inroducion and preliminary remarks Preliminary remarks 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 hrough curves associaed o he underlying raes, bu are discouned by anoher curve.

3 Inroducion and preliminary remarks Preliminary remarks The majoriy of he models ha have been considered reflec he usual classical disincion beween i) shor rae models (boom-up); ii) HJM seup; iii) BGM or LIBOR marke models (op-down). In addiion, mehodologies relaed o foreign exchange. Concerning i) and ii), shor rae models lead more easily o a Markovian srucure, while HJM allows for a direc calibraion o he iniial erm srucure.

4 Inroducion and preliminary remarks Preliminary remarks Here we concenrae on shor rae models. [Kenyon, Kijima-Tanaka-Wong, Filipovic-Trolle] A major goal wih his modeling choice will be o derive an easy relaionship beween risk-free and risky FRAs hereby exhibiing an adjusmen facor ha plays a role analogous o quano adjusmens in cross-currency derivaives or o he muliplicaive forward basis in [Bianchei].

5 Inroducion and preliminary remarks Preliminary remarks 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 S a a fixed value K. A mauriy S, a paymen based on K is made and one based on he relevan floaing rae (generally he spo Libor rae L(T ; T, S)) is received. Considering laer on a single enor, we le he mauriy be S = T + and denoe he value of he FRA a < T by FRA T (, K ).

6 Inroducion and preliminary remarks Preliminary remarks To presen he basic ideas in a simple way, here we consider a wo-curve model, namely wih a curve for discouning and one for generaing fuure cash flows: i) The choice of he discoun curve is no unique; we follow he common choice of considering he OIS swap curve. ii) For he risky cash flows wihou collaeral we consider a single LIBOR (i.e. for a given enor).

7 Inroducion and preliminary remarks Preliminary remarks We describe an approach ha we presen here for he case of pricing of FRAs (linear derivaives). We consider only clean valuaion formulas, namely wihou counerpary risk. To accoun for counerpar risk and funding issues, various value adjusmens are generally compued on op of he clean prices. As poined ou in [Crepey, Grbac, Ngor, Skovmand], marke quoes ypically reflec prices of fully collaeralized ransacions. The clean price formulas hus urn ou o be sufficien also for calibraion.

8 Inroducion and preliminary remarks Preliminary remarks 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 discree compounding forward raes his leads o ( < T < S) F(; T, S) = 1 ( ) p(, T ) S T p(, S) 1 The formula can also be jusified as represening he fair fixed rae a ime of a FRA, where he floaing rae received a S is F(T ; T, S) = 1 S T ( ) 1 p(t, S) 1

9 Inroducion and preliminary remarks Preliminary remarks In fac, he arbirage-free price in of such a FRA is (using he forward maringale measure Q S ) FRA T (, K ) = p(, S) E QS {(F(T ; T, S) K ) F } which is zero for K = E QS {(F(T ; T, S) F } = E QS { 1 S T ( p(t,t ) p(t,s) 1 ) F } = 1 S T ( p(,t ) p(,s) 1 )

10 Inroducion and preliminary remarks Preliminary remarks Since he discoun curve is considered o be given by he OIS zero-coupon curve (p(, T ) = p OIS (, T )), one uses also he noaion L D (; T, S) for F (; T, S) and calls i OIS forward rae. The pre-crisis (risk-free) forward Libor rae L(; T, S) was supposed o coincide wih he OIS forward rae, namely he following equaliy was supposed o hold L(; T, S) = L D (; T, S) = F(; T, S)

11 Inroducion and preliminary remarks Preliminary remarks Puing now S = T + (enor ), recall ha he risky LIBOR raes L(; T, T + ) are deermined by he LIBOR panel ha akes ino accoun various facors such as credi risk, liquidiy, ec. and his implies ha in general L(; T, S) F(; T, S) hus leading o a LIBOR-OIS spread. Following some of he recen lieraure, in paricular [Crepey- Grbac-Nguyen] (see also [Kijima-Tanaka-Wong]), we keep he formal relaionship beween discree compounding forward raes and bond prices also for he LIBORs, bu replace he risk-free bond prices p(, T ) by ficiious ones p(, T ) ha are supposed o be affeced by he same facors as he LIBORs.

12 Inroducion and preliminary remarks Preliminary remarks Since FRAs are based on he T spo LIBOR L(T ; T, T + ), we acually posulae he classical relaionship only a he incepion ime = T. Our saring poin is hus L(T ; T, S) = 1 ( 1 p(t, T + ) 1 Noice ha also for our risky bonds we have p(t, T ) = 1. )

13 FRA pricing FRAs In our wo-curve risky seup, he fair price of a FRA in < T wih S = T +, fixed rae K and noional N is hen [ ] FRA T (, K ) = N p(, T + )E T + L(T ; T, T + ) K F [ ] = Np(, T + )E T + 1 p(t,t + ) (1 + K ) F where E T + denoes expecaion under he (T + ) forward measure. The 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.

14 FRA pricing FRAs The crucial quaniy o compue in he FRA T (, K ) expression is [ ] ν,t := E T + 1 p(t, T + ) F. The fixed rae o make he FRA a fair conrac a ime is hen K := 1 ( ν,t 1)

15 FRA pricing FRAs In he classical single curve case we have insead [ ] ν,t := E T + 1 p(t, T + ) F = p(, T ) p(, T + ) being p(,t ) 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 To compue K no ineres rae model is needed (conrary o K ).

16 The risky shor rae model The model To compue he expecaion E T + we need a model for p(, T ). For his purpose recall firs he classical bond price formula (r is he shor rae) [ ] } T p(, T ) = E {exp Q r u du F wih Q he sandard maringale measure,.

17 The risky shor rae model The model We now define he risky bond prices as [ p(, T ) = E Q {exp T (r u + s u )du ] F } wih s represening he shor rae spread (hazard rae in case of only defaul risk). The spread is inroduced from he ouse. p(, T ) is no an acual price.

18 The risky shor rae model The model Nex we need a dynamical model for r and s and for his purpose we shall inroduce a facor model. 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 Q. We shall firs recall wo basic facor models for he shor rae.

19 The risky shor rae model The model Model A. The square-roo, exponenially affine model (CIR) model where r = I i=1 γ iψ i wih, under Q, (w i independen Q Wiener) dψ i = (a i b i Ψ i )d + σ i c i Ψ i + d i dw i I implies [ p(, T ) = E {exp Q = exp T ] } r u du F [ A(, T ) I i=1 Bi (, T )Ψ i ] For c i = 0 he square-roo model becomes a Gaussian mean revering (Hull-Whie) model.

20 The risky shor rae model The model The above model class includes various specific models ha have appeared in he lieraure such as e.g. he following wo-facor Gaussian shor rae model from [Filipovic-Trolle] (analogous models for he spreads) { dr = κ r (γ r )d + σ r dw r dγ = κ γ (θ γ γ ) + σ γ (ρ dw r + ) 1 ρ 2 dw γ

21 The risky shor rae model The model I suffices in fac o consider wo Gaussian facors dψ i = (a i b i Ψ i )d + σ i dw i, i = 1, 2 and pu { r = λ 1 Ψ 1 + λ 2 Ψ 2 γ = λ1 a 1 +λ 2 a 2 + b1 b 2 b 1 b 1 λ 2 Ψ 2 The given model class can also be easily generalized o affine jump-diffusion models (see e.g.[ Bjoerk, Kabanov, R.]); only he noaion becomes hen more involved.

22 The risky shor rae model The model Model B. The Gaussian, exponenially quadraic model [Pelsser, Kijima-Tanaka-Wong] (dual o square-roo exponenially affine) r = I 1 i=1 γ i Ψ i + I 2 i=i1 +1 γ i(ψ i )2 dψ i = b i Ψ i d + σi dw i I implies [ p(, T ) = E {exp Q ] } T r u du F [ = exp A(, T ) I 1 i=1 B i (, T )Ψ i ] I 2 i=i1 +1 Ci (, T )(Ψ i )2 Advanage of his model in derivaive pricing: he disribuion of Ψ i remains always Gaussian; in a square-roo model i is a χ 2 disribuion.

23 The risky shor rae model The model In presening join models for r and s we wan o allow for non-zero correlaion beween r and s. I is obained by considering common facors, he remaining ones being idiosyncraic facors. To obain an adjusmen facor, a leas one of he common facors has o saisfy a Gaussian model (Vasiček/Hull-Whie). By analogy o he pure shor rae case, also here we consider wo model classes.

24 The risky shor rae model The model Model A. (based on Morino-R. 2013) Given hree independen affine facor processes Ψ i, i = 1, 2, 3 le { r = Ψ 2 Ψ 1 s = κψ 1 + Ψ 3 where he common facor Ψ 1 allows for insananeous correlaion beween r and s wih correlaion inensiy κ (negaive correlaion for κ > 0). Oher facors may be added o drive s.

25 The risky shor rae model The model (Model A. cond.) Le, under Q, dψ 1 = (a 1 b 1 )Ψ 1 d + σ1 dw 1 dψ i = (a i b i )Ψ i d + σi Ψ i dw i, i = 2, 3 where a i, b i, σ i are posiive consans wih a i (σ i ) 2 /2 for i = 2, 3, and w i independen Q Wiener processes. Ψ 1 may ake negaive values implying ha, no only r, bu also s may become negaive (see laer).

26 The risky shor rae model The model Model B. (analogous o above) Given again hree independen affine facor processes Ψ i, i = 1, 2, 3 le { r = Ψ 1 + (Ψ 2 )2 s = κψ 1 + (Ψ 3 )2 where he common facor Ψ 1 allows again for insananeous correlaion beween r and s wih correlaion inensiy κ. Oher facors may be added o drive s. Under Q, dψ i = b i Ψ i d + σ i dw i, i = 1, 2, 3 where w i independen Q Wiener processes. Ψ 1 migh ake negaive values so ha also here r and s may become negaive.

27 Resuls Bond price relaions For case A. we have p(, T ) = exp [ A(, T ) B 1 (, T )Ψ 1 B 2 (, T )Ψ 2 ] p(, T ) = exp [ Ā(, T ) B 1 (, T )Ψ 1 B 2 (, T )Ψ 2 B 3 (, T )Ψ 3 ] wih B 1 (, T ) = (1 κ) B 1 (, T ), B2 (, T ) = B 2 (, T ). I follows ha p(, T ) = p(, T ) exp [Ã(, T ) + κb 1 (, T )Ψ 1 B 3 (, T )Ψ 3 where Ã(, T ) := Ā(, T ) A(, T ). ]

28 Resuls Bond price relaions Puing for simpliciy B 1 := B 1 (T, T + ), i follows ha p(t, T + ) [ Ã(T p(t, T + ) = exp, T + ) κ B 1 Ψ 1 T + B ] 3 (T, T + )Ψ 3 T and, defining an adjusmen facor as { } Ad T, p(t, T + ) := E Q p(t, T + ) F his facor 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 = A(θ, κ, Ψ 1, Ψ3 ) wih θ := (a i, b i, σ i, i = 1, 2, 3).

29 Resuls Main resul Proposiion: We have ν,t = ν,t [ Ad T, exp κ (σ1 ) 2 2(b 1 ) 3 ( 1 e b1 ) ( ) ] 2 1 e b1 (T ) The fair value K of he fixed rae in a risky FRA is hen relaed o K in a corresponding riskless FRA as follows: K = ( K + ) 1 Ad T, [ ) ( ) ] exp κ (σ1 ) e b1 1 e b1 (T ) 1 2(b 1 ) 3 ( The facor given by he exponenial is equal o 1 for zero correlaion (κ = 0).

30 Resuls Commens on he main resul: 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, since s = Ψ 3 > 0, 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 we hen have ν,t ν,t, K K

31 Resuls Commens on he main resul: calibraion The coefficiens a 1, a 2, b 1, b 2, σ 1, σ 2 can be calibraed in he usual way on he basis of he observaions of defaul-free bonds p(, T ). To calibrae a 3, b 3, σ 3, noice ha, conrary o p(, T ), he risky bonds p(, T ) are no observable (here is no 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.

32 Resuls Commens on he main resul: calibraion Recalling hen he Corollary, namely K = ( K + ) 1 Ad T, [ ) ( ) ] exp κ (σ1 ) 2 2 e b1 1 1 e b1 (T ) 1 (b 1 ) 3 ( and he fac ha Ad T, = A(θ, κ, Ψ 1, Ψ2 ), his allows o calibrae a 3, b 3, σ 3 as well as κ.

33 Resuls Bond price relaions For case B. we have analogously (( ) sands for (, T )) p(, T ) = exp [ A( ) B 1 ( )Ψ 1 C 2 ( )(Ψ 2 )2] p(, T ) = exp [ Ā( ) B 1 ( )Ψ 1 C 2 ( )(Ψ 2 )2 C 3 ( )(Ψ 3 )2] wih B 1 (, T ) = (1 + κ) B 1 (, T ), C2 (, T ) = C 2 (, T ). I follows ha p(, T ) = p(, T ) exp [Ã(, T ) + κb 1 (, T )Ψ 1 C 3 (, T )(Ψ 3 )2] where, again, Ã(, T ) := Ā(, T ) A(, T ).

34 Resuls Bond price relaions Puing again B 1 := B 1 (T, T + ), i follows ha p(t, T + ) p(t, T + ) =exp[ Ã(T,T + ) κ B 1 Ψ 1 T +C3 (T,T + )(Ψ 3 T )2 ] Inroducing he same adjusmen facor { } Ad T, p(t, T + ) := E Q p(t, T + ) F ha can again 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 = A(θ, κ, Ψ 1, Ψ3 ) where θ := (a i, b i, σ i, i = 1, 2, 3), one obains compleely analogous resuls as for case A.

35 Thank you for your aenion

36 Appendix Preliminary resuls for deermining ν,t Due o he affine dynamics of Ψ i (i = 1, 2, 3) under Q we have for he risk-free bond { [ p(, T ) = E Q exp ] } T r u d u F { [ ] } = E Q T exp (Ψ 1 u Ψ 2 u)d u F = exp [ A(, T ) B 1 (, T )Ψ 1 B 2 (, T )Ψ 2 ]

37 Appendix Preliminary resuls for deermining ν,t The coefficiens saisfy B 1 b 1 B 1 1 = 0, B 1 (T, T ) = 0 B 2 b 2 B 2 (σ2 ) 2 2 ) = 0, B 2 (T, T ) = 0 A = a 1 B 1 (σ1 ) 2 2 (B1 ) 2 + a 2 B 2, A(T, T ) = 0 in paricular B 1 (, T ) = 1 ) (e b1 (T ) b 1 1

38 Appendix Preliminary resuls for deermining ν,t 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)Ψ 1 u Ψ 2 u Ψ 3 u)du F = exp [ Ā(, T ) B 1 (, T )Ψ 1 B 2 (, T )Ψ 2 B 3 (, T )Ψ 3 ]

39 Appendix Preliminary resuls for deermining ν,t The coefficiens saisfy B 1 b 1 B1 + (κ 1) = 0, B1 (T, T ) = 0 B 2 b 2 B2 (σ2 ) 2 2 ( B 2 ) = 0, B2 (T, T ) = 0 B 3 b 3 B3 (σ3 ) 2 2 ( B 3 ) = 0, B3 (T, T ) = 0 Ā = a 1 B1 (σ1 ) 2 2 ( B 1 ) 2 + a 2 B2 + a 3 B3, Ā(T, T ) = 0 in paricular B 1 (, T ) = 1 κ ( ) b 1 e b1 (T ) 1 = (1 κ) B 1 (, T )

40 Appendix Preliminary resuls for deermining ν,t >From he 1 s order equaions i follows ha B 1 (, T ) = (1 κ) B 1 (, T ) B 2 (, T ) = B 2 (, T ) Ā(, T ) = A(, T ) a 1 κ T + (σ1 ) 2 2 κ2 T B 1 (u, T )du (B 1 (u, T )) 2 du + (σ 1 ) 2 κ T B 1 (u, T )du a 3 T B 3 (u, T )du Le Ã(, T ) := Ā(, T ) A(, T )

41 Aspecs of CAP pricing Nonlinear derivaives CAPs/Caples We concenrae on he pricing of a single Caple, wih srike K, mauriy T on he forward LIBOR for he period [T, T + ]. Using he forward measure, 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 ) } + F wih K := 1 + K.

42 Aspecs of CAP pricing Nonlinear derivaives CAPs/Caples 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). The aim, pursued in he case of he FRAs, of performing he calculaions under he same measure Q leads here o some difficulies and so we sick o forward measures. Depending on he pricing mehodology, one may hen need o change he dynamics of he facors o be valid under he various forward measures. The R.N.-derivaive o change from Q o he various forward measures can be expresses in explici form and i preserves he affine srucure.

43 Aspecs of CAP pricing Nonlinear derivaives CAPs/Caples I may hus suffice o derive jus a pricing algorihm ha need no also be used for calibraion. I remains however desirable o obain also here an adjusmen facor.

44 Aspecs of CAP pricing Nonlinear derivaives CAPs/Caples For he pricing, in he forward measure, we may use Fourier ransform mehods as in [CGN] and [CGNS] hereby represening he claim as ( e X K ) + wih X := log p(t, T + ) (possibly also a Gram-Charlier expansion as in [KTW]). Need only o compue he momen generaing funcion of X ha is a linear combinaion of he facors (compuaion is feasible hanks o he affine srucure) ( and use he Fourier ransform of f (x) = e x K ) +.

45 Aspecs of CAP pricing Nonlinear derivaives CAPs/Caples The price in = 0 can hen be obained in he form Capl(0, T, T + ) = p(0, T + ) 2π K 1 iv R + MT X (R + iv) (R + iv) (R + iv 1) dv M T + where X ( ) is he momen generaing funcion of X under he (T + ) forward measure.

46 Aspecs of CAP pricing Nonlinear derivaives CAPs/Caples If M T + X ( ) is he momen generaing funcion of X wih p(t, T + ) insead of p(t, T + ) hen M T + X ( ) = M T + X ( )A( ; θ, κ, Ψ 1 0, Ψ2 0, Ψ3 0 ) where, given he affine naure of he facors, A( ; θ, κ, Ψ 1 0, Ψ2 0, Ψ3 0 ) can be explicily compued. Since, for he above facorizaion o hold, A( ; θ, κ, Ψ 1 0, Ψ2 0, Ψ3 + 0 ) conains also (MT X ( )) 1, his may however no suffice o derive a saisfacory adjusmen facor as for FRAs.