OpenGamma Quaniaive Research Muli-curves: Variaions on a Theme Marc Henrard marc@opengamma.com OpenGamma Quaniaive Research n. 6 Ocober 2012
Absrac The muli-curves framework is ofen implemened in a way o recycle o one curve formulas; here is no fundamenal reasons behind ha choice. Here we presen differen approaches o he mulicurves framework. They vary by he choice of building blocks insrumens (Ibor coupon or fuures) and he definiion of curve (pseudo-discoun facors or direc forward rae). The feaures of he differen approaches are described.
Conens 1 Inroducion 1 2 Discouning 2 3 Coupon discoun facor muli-curves framework 2 3.1 Exisence and arbirariness............................... 3 3.2 Ineres Rae Swap.................................... 4 3.3 Libor Fuures....................................... 5 3.4 Curve building...................................... 6 4 Coupon forward rae muli-curves framework 7 4.1 Libor Fuures....................................... 8 4.2 Curve building...................................... 9 5 Fuures discoun facor muli-curves framework 10 5.1 Zero rae collaeral.................................... 12 6 Conclusion 13
1 Inroducion Up o 2008, he sandard approach o pricing Ibor 1 -relaed derivaives was o use a unique ineres rae curve, supposed o be risk-free, o boh discoun he cash-flows and esimaed he Ibor-relaed paymens. Wih he recen crisis i became more apparen ha he hypohesis was no realisic and ha a differen approach was necessary. Tha necessiy was already indicaed well before he crisis in some lieraure and by some praciioners. Earlier developmens like Tuckman and Porfirio (2003) and Boenkos and Schmid (2004) had poined o he weakness of he hen framework bu wihou providing a heoreically sound alernaive. A firs sep, spliing (risk-free) discouning and Ibor fixing, was proposed in a simplified se-up in Henrard (2007). The framework was laer exended o a more flexible se-up and is now he base for mos of he muli-curves developmens. I is in paricular described in Henrard (2010). Furher developmens have been done in differen direcions, in paricular in Kijima e al. (2009), Amerano and Bianchei (2009), Chibane and Sheldon (2009), Mercurio (2009), Morini (2009), Bianchei (2010), Pierbarg (2010), Moreni and Pallavicini (2010), Pallavicini and Tarenghi (2010), and Mercurio (2010a). One of he saring poins of he muli-curves framework is he exisence of a se of asses which are no direcly relaed o he discoun bonds. The usual choice is he Ibor coupons and he coupons are priced using pseudo-discoun facors linked o forward curves creaed o reproduce marke insrumen prices. This is why we refer here o ha framework as he coupon discoun facor muli-curves framework. From he exisence hypohesis, one defines pseudo-discoun facors in such a way ha he usual projec-and-discoun formulas previously used in swap pricing are sill valid. This way o proceed is purely convenional. I is also quie pracical as all sandard insrumens can be priced wih formulas similar o he one we are used o. Neverheless he approach is purely based on ha definiion and seleced o used he good old formulas; here is no deeper fundamenal reason behind i. There are poenially muliple oher coheren approaches o muli-curves discouning / esimaion frameworks. In his noe we review several alernaive approaches. They are based on differen ways o implemen he hypohesis on exisence of Ibor relaed producs. In he presenaion we resric ourselves o a single currency descripion; he exension o he muli-currency case can be done like in he coupon discoun facor case. The second approach is o model he forward raes direcly, wihou he pseudo-discoun facor inermediary in he forward curve. We refer o ha framework as he coupon forward rae mulicurves framework. The advanages of such approach are ha i uses formula very similar o he curren one and a he same ime models direcly he forward raes on which one may have more inuiion. The saw-ooh effec ha appears wih linear inerpolaion of raes in he coupon discoun facor framework disappears (a leas is lessen) even if he same linear inerpolaion scheme is used 2. The hird approach proposed is based on he price of ineres rae fuures and no of coupons and was firs described in Henrard (2012). I is called here he fuures discoun facor mulicurves framework. The fundamenal hypohesis on he exisence of Ibor coupons is replaced by an 0 Firs version: 18 Augus 2012; his version: 19 Ocober 2012. 1 By Ibor we mean any reference rae which is fixed in a way similar o Libor and in paricular Euribor, Tibor, Cibor, BBSW, ec. The descripion of he differen indexes and heir convenions can be found in Quaniaive Research (2012). 2 To our knowledge, i is he firs ime his approach is formally described. Some people menioned o he auhor closely relaed approaches as poenially aracive, bu a he ime hey had no been implemened. 1
hypohesis on he exisence of STIR fuures linked o Ibor indexes. In he one curve approach, he pricing of ineres rae fuures, one of he mos liquid ineres rae producs, has araced a lo of aenion. As a small lis of relaed lieraure, we indicae: Kirikos and Novak (1997), Cakici and Zhu (2001), Pierbarg and Renedo (2004), Henrard (2005), Jäckel and Kawai (2005). Obviously one would like o price he fuures also in he muli-curves framework. This was done for he one-facor Gaussian HJM model in Henrard (2010) using a deerminisic hypoheses on he discouning/forward spread and in Mercurio (2010b) in he LMM wih sochasic basis. Here he problem is somehow reverse. From a given fuure price, we ry o obain he price of he (non-margined) coupons. The echniques used are similar. A firs reading, par of his noe may seems quie axiomaic and disan from pracical consideraions. I is our believe ha hose quesions are fundamenal and in some circumsances may simplify he implemenaion and modelling of ineres rae producs. Cee hisoire es vraie puisque je l ai invenée... Boris Vian I is he arisic freedom of he mahemaician o inven his own axioms, hypohesis and definiions 3. I have used ha freedom o propose several axiomaic approaches o muli-curves. 2 Discouning The saring poin of he muli-curves framework is he discouning of known derivaives cash-flows; his is he firs hypohesis: D The insrumen paying one uni in u is an asse for each u. Is value in is denoed P D (, u). The value is coninuous in. Wih his curve we are able o value fixed cash-flows. 3 Coupon discoun facor muli-curves framework The framework described in his secion is adaped from Henrard (2010). Our goal is o price Ibor-relaed derivaives, in paricular IRSs. We need an hypohesis saying ha hose insrumens exis in he framework we are describing. We call a j-ibor floaing coupon a financial insrumen which pays a he end of a period he Ibor rae se a he sar of he period. The deails of he insrumen are as follow. The rae is se or fixed a a dae 0 for he period [ 1, 2 ](0 0 1 < 2 ), a he end dae 2 he amoun paid is he Ibor fixing muliplied by he convenional accrual facor. The lag beween 0 and 1 is he spo lag. The difference beween 2 and 1 is a j period. All periods and accrual facors should be calculaed according o he day coun, business day convenion, calendar and end-of-monh rule appropriae o he relevan Ibor indexes. As he period addiion, + period j is used ofen we adop he noaion + j for ha dae, wihou clarifying in which uni he j is; i is usually clear from he conex. Our exisence hypohesis for he Ibor coupons reads as 3 Mahemaicians should be allowed as much legal conrol on heir axioms and definiions han ficion wriers have on heir characers. Mahemaics requires as much imaginaion han ficion wriing. 2
I CPN 4 The value of a j-ibor floaing coupon is an asse for each enor and each fixing dae. Is value is a coninuous funcion of ime. This hypohesis is implici in mos of he lieraure menioned in he inroducion. I is imporan o sae i explicily as his is no a consequence of he exisence of he discouning curve. Once we have assumed ha he insrumen is an asse, we can give is value a name. We do i indirecly hrough he curves P CDF,j. Definiion 1 (Curve) The forward curve P CDF,j 5 is he coninuous funcion such ha, P CDF,j (, ) = 1, P CDF,j (, s) is an arbirary sricly posiive funcion for s < Spo() + j, and for 0, 1 = Spo( 0 ) and 2 = 1 + j ( P P D CDF,j ) (, 1 ) (, 2 ) P CDF,j (, 2 ) 1 is he value in of he j-ibor floaing coupon wih fixing dae 0 on he period [ 1, 2 ]. (1) A his sage he only link beween he curves and marke raes is ha he Ibor rae fixing in 0 for he period j, denoed I j 0, is I j 0 = 1 δ ( P CDF,j ) ( 0, Spo( 0 )) P CDF,j ( 0, Spo( 0 ) + j) 1 (2) where δ is he fixing period year fracion. To obain his equaliy he value ime-coninuiy was used. 3.1 Exisence and arbirariness Noe ha Definiion 1, which conains an arbirary funcion, is iself arbirary. One could fix any j period (no only he firs one) and deduce he res of he curve from here. Or one could even ake an arbirary decomposiion of he period j inerval in sub-inervals and disribue hose sub-inervals arbirarily on he real axis in such a way ha, modulo j period, hey recompose he iniial j period. One could also change he value of P j (, ) o any value as only he raios are used, never a value on is own. We should also add a couple of remarks on he daes. We use he noaion + j as if he ime displacemens were a real addiion. This is no he case. There is no inverse because due o non-business daes, several daes 1 can lead o he same 1 + j. This is no an excepional case; hree days a week, in he sandard following rule, have a 1 + j ending on he same Monday. Moreover, we se our noaion wih he 2 used in P D and he one used in P j he same. Again due o non-business day adjusmens, his will no always be he case in FRAs and IRSs. The paymen dae ( 2 in P D ) can be several days before he end of fixing period dae ( 2 in P CDF,j ). The difference is ofen one or wo days bu can be up o six. We will no make ha disincion here. 4CPN sands for CouPoN. 5 CDF sands for Coupon Discoun Facor. 3
3.2 Ineres Rae Swap Wih hypohesis (I CPN ) and he relaed definiion, he compuaion of he presen value of vanilla ineres rae swaps is sraighforward. The definiion was seleced for ha reason. An IRS is described by a se of fixed coupons or cash flows c i a daes i (1 i ñ). For hose flows, he discouning curve is used. I also conains a se of floaing coupons over he periods [ i 1, i ] wih i = i 1 + j (1 i n). The accrual facors for he periods [ i 1, i ] are denoed δ i. The value of a (fixed rae) receiver IRS is ñ c i P D (, i ) i=1 n i=1 ( P P D CDF,j ) (, i 1 ) (, i ) P CDF,j 1. (3) (, i ) In he one curve pricing approach, he IRS are usually priced hrough eiher he Ibor forward approach or he cash flow equivalen approach. The Ibor forward approach consiss in esimaing he forward Ibor rae from he discoun facors and discouning he resul from paymen dae o oday. To keep ha inuiion, we define he Ibor forward rae in our framework as he figure we have o use o keep he same formula. Definiion 2 (Forward rae) The Ibor forward rae over he period [ 1, 2 ] is given a ime by F CDF,j ( 1, 2 ) = 1 ( P CDF,j ) (, 1 ) δ P CDF,j (, 2 ) 1. (4) Wih ha definiion he IRS presen value is ñ c i P D (, i ) i=1 n i=1 P D (, i )δ i F CDF,j ( i 1, i ). Noe he fundamenal difference beween I j 0 and F CDF,j. The objec I j is, by hypohesis I CPN, a fundamenal elemen of our economy; he F CDF,j is purely a definiion. The definiions of F and I coincide on he fixing dae 0 : I j 0 = F CDF,j 0 (Spo( 0 ), Spo( 0 ) + j). The cash flow equivalen approach in exbook formulas consiss in replacing he (receiving) floaing leg by receiving he noional a he period sar and paying he noional a he period end. We would like o have a similar resul in our new framework. To his end we define: Definiion 3 (Spread) The spread beween a forward curve and he discouning curve is β CDF,j (u, u + j) = P CDF,j (, u) P CDF,j (, u + j) P D (, u + j) P D. (5) (, u) Obviously he value of his variable is consan a 1 if P D = P CDF,j. Wih ha definiion, a floaing coupon price is ( P P D CDF,j ) ( (, i 1 ) (, i ) P CDF,j 1 = P D (, i ) β CDF,j ( i 1, i ) P D ) (, i 1 ) (, i ) P D 1 (, i ) = β CDF,j ( i 1, i )P D (, i 1 ) P D (, i ). 4
This las value is equal o he value of receiving β CDF,j noional a he period sar and paying he noional a he period end. A consequence of hypohesis I CPN and he definiion of β CDF,j is ha β. CDF,j ( i 1, i ) is a maringale in he P D (., i 1 ) numeraire. The Ibor coupon value is β j ( i 1, i )P D (, i 1 ) P D (, i ). The coupon is an asse due o I CPN and so is value divided by he numeraire P D (, i 1 ) is a maringale. The second erm, a zero-coupon, is also an asse, hence is rebased value is also a maringale. β CDF,j P D (, i 1 )/P D (, i 1 ) = β CDF,j he P D (., i 1 )-measure. 3.3 Libor Fuures The rebased firs erm is hus also a maringale and is value is. This proves ha β. CDF,j ( i 1, i ) is a maringale under A general pricing formula for ineres rae fuures in he one-facor Gaussian HJM model in he one curve framework was proposed in Henrard (2005). The formula exended a previous resul proposed in Kirikos and Novak (1997). The formula was exended o he muli-curves framework in Henrard (2007). The goal is o obain a relaively simple, coheren and pracical approach o Ibor derivaives pricing. To achieve he simpliciy, our nex hypoheses are relaed o he spreads beween he curves, as defined hrough he quaniies β CDF,j. S0 CDF The muliplicaive coefficiens beween discoun facor raios, β CDF,j (u, u + j), as defined in Equaion (5), are consan 6 hrough ime: β CDF,j (u, u + j) = β CDF,j 0 (u, u + j) for all and u. We describe he pricing of fuures under he hypoheses I CPN and S0 CDF in a muli-curves one-facor Gaussian HJM model. The pricing of he fuures in he LMM wih sochasic basis is proposed in Mercurio (2010b). The fuure fixing or las rading dae is denoed 0. The fixing is on he Ibor rae beween 1 = Spo( 0 ) and 2 = 1 + j. The fixing accrual facor for he period [ 1, 2 ] is δ. The fixing is linked o he yield curve by (2). The fuures price in is denoed Φ j ( 1 ). On he fixing dae, he relaion beween he price and he rae is Φ j 0 ( 1 ) = 1 I j 0. The fuures margining is done on he fuures price (muliplied by he noional and he fuures accrual facor). The exac noaion for he HJM one-facor model used here is ha in Henrard (2005). When he discoun curve P D (,.) is absoluely coninuous (which is somehing ha is always he case in pracice as he curve is consruced by some kind of inerpolaion) here exiss f(, u) such ha ( u ) P D (, u) = exp f(, s)ds. (6) The shor rae associaed wih he curve is (r ) 0 T wih r = f(, ). The cash-accoun numeraire is N = exp( 0 r sds). In he HJM framework, he equaions in he cash-accoun numeraire measure associaed wih N are df(, u) = σ(, u)ν(, u)d + σ(, u)dw. 6 In his framework, consan spread is equivalen o deerminisic spread due o he maringale propery of β. 5
where ν(, u) = u σ(, s)ds. The model is one-facor Gaussian if W is a one-facor Brownian moion and σ is a deerminisic funcion. Theorem 1 (Fuures price) Le 0 0 1 2. In he one-facor Gaussian HJM model on he discouning curve under he hypoheses D, I CPN and S0 CDF, he price of he fuures fixing in 0 for he period [ 1, 2 ] wih accrual facor δ is given by Φ j = 1 1 ( P CDF,j ) (, 1 ) δ P CDF,j (, 2 ) γ() 1 (7) where = 1 γ()f CDF,j + 1 (1 γ()) δ ( 0 ) γ() = exp ν(s, 2 )(ν(s, 2 ) ν(s, 1 ))ds. Proof: Using he generic pricing fuure price process heorem (Hun and Kennedy, 2004, Theorem 12.6), Φ j ( 1 ) = E N [ 1 I j 0 F ] where E N [] is he cash accoun numeraire expecaion. In I j 0, he only non-consan par is he raio of j-pseudo-discoun facors which is, up o β j 0, he raio of discoun facors. Using (Henrard, 2005, Lemma 1) wice, we obain P D ( 0, 1 ) P D ( 0, 2 ) = P D (, 1 ) P D (, 2 ) exp + 0 ( 1 2 0 ν(s, 1 ) ν(s, 2 )dw s ). ν 2 (s, 1 ) ν 2 (s, 2 )ds Only he second inegral conains a sochasic par. This inegral is normally disribued wih variance 0 (ν(s, 1 ) ν(s, 2 )) 2 ds. The expeced discoun facors raio is reduced o P D ( (, 1 ) P D exp 1 0 0 ) ν 2 (s, 1 ) ν 2 (s, 2 )ds + (ν(s, 1 ) ν(s, 2 )) 2 ds. (, 2 ) 2 By hypohesis SI CDF, he coefficien β CDF,j 0 resul. 3.4 Curve building is consan, and so we have obained he announced A relaive sandard way o calibrae he curves P D and P CDF,j is o selec a se of marke insrumens for which he presen value is known and an equal number of node poins. An inerpolaion scheme is seleced and he raes on he node poins are calibraes o reproduce he marke prices. The marke forward raes F CDF,j 0 ( 1, 1 + j) can be compued from ha curve. A ypical forward rae curve is displayed 7 in Figure 1. The swap daa used o build he curve are he one used in (Andersen and Pierbarg, 2010, Secion 6.2) and he inerpolaion scheme is linear on (coninuously compounded) raes. We suppose ha he swap raes are fixed versus hree monhs Ibor and ha he discouning curve has fla marke raes a 4%. 7 All he numbers in he figures of his noe have been produced using OpenGamma OG-Analyics library. The library is open source and available a hp://www.opengamma.com. 6
7.5 Forward raes wih pseudo discoun facors 7 6.5 Rae (in %) 6 5.5 5 4.5 4 0 2 4 6 8 10 Time o sar (in year) Figure 1: Forward raes compued using pseudo-discoun facors. The circles indicae he zerocoupon rae associaed o he differen nodes. The familiar sawooh paern can be seen. There is wo angles in he curve for each node poin. One when he fixing period end dae is on one node and one when he sar dae is on he node. One of he reasons of his unpleasan shape is probably ha we have an inuiion on a marke quaniy (forward rae) bu model i indirecly hrough a raio of discoun facors where our inuiion is dilued. 4 Coupon forward rae muli-curves framework We inroduce a differen framework sill based on coupons. The forward raes are modelled direcly and no hrough pseudo-discoun facors. For his reason we refer o i as he coupon forward rae muli-curves framework. From a pure heoreical poin of view i is equivalen o he previous framework as here is a bijecion beween he pseudo-discoun facors (once he arbirary par is seleced) and he forward rae. From a pracical poin of view hey are differen as he descripion and inerpolaion schemes will be applied on he discoun facors or direcly on he forward raes and give differen resuls. This is in some sense similar o he HJM/LMM dualiy. One is echnically easier bu he oher refers o marke quaniies. For his framework, he same exisence hypohesis I CPN is used. The associaed definiion is now: Definiion 4 (Forward rae) The forward curve F CFWD,j 8 is he coninuous funcion such ha, 8 CFWD sands for Coupon ForWarD rae. 7
P D (, 2 )δf CFWD,j ( 1 ) (8) is he price in of he j-ibor coupon wih sar dae 1 and mauriy dae 2 ( 0 1 = Spo( 0 ) < 2 ). Noe ha in his framework, he Ibor discouning is impossible as here is no discoun facor associaed o he Ibor curves. The link beween he curves and marke raes is I j 0 = F CFWD,j 0. (9) There is no arbirary par anymore o he curve. The curve is defined unambiguously (as long as he corresponding marke insrumens exis) for all 1 Spo(0). Wih hypohesis (I CPN ) and Definiion 4, he compuaion of he presen value of vanilla ineres rae swaps is sraighforward. The definiion was seleced for ha reason. The IRS descripion is he same as in he previous secion. The value of a (fixed rae) receiver IRS is ñ c i P D (, i ) i=1 n i=1 P D (, i )δ i F CFWD,j ( i 1 ). (10) Definiion 5 (Spread) The spread beween a forward curve and he discouning curve is β CFWD,j (u, u + j) = (1 + δfu CFWD,j ) P D (, u + j) P D. (11) (, u) Wih ha definiion, a floaing coupon price is ( P D (, i )δf CFWD,j ( 1 ) = P D (, i ) β CFWD,j ( i 1, i ) P D ) (, i 1 ) P D 1 (, i ) = β CFWD,j ( i 1, i )P D (, i 1 ) P D (, i ). This las value is equal o he value of receiving β CFWD,j he noional a he period end. noional a he period sar and paying 4.1 Libor Fuures The goal is o obain a relaively simple, coheren and pracical approach o Ibor derivaives pricing. To achieve he simpliciy, our nex hypohesis is relaed o he spreads beween he curves, as defined hrough he quaniies β CFWD,j S0 CFWD β CFWD,j The spreads β CFWD,j (u, u + j) = β CFWD,j. (u, u + j), as defined in Equaion (11), are consan hrough ime: 0 (u, u + j) for all and u. We describe he pricing of fuures under he hypoheses I CPN and S0 CFWD in a muli-curves one-facor Gaussian HJM model. The noaion is he same as in he previous secion. 8
Theorem 2 Le 0 0 1 2. In he one-facor Gaussian HJM model on he discouning curve under he hypoheses D, I CPN and S0 CFWD, he price of he fuures fixing in 0 for he period [ 1, 2 ] wih accrual facor δ is given by Φ j = 1 γ()f CFWD,j + 1 (1 γ()) δ where ( 0 ) γ() = exp ν(s, 2 )(ν(s, 2 ) ν(s, 1 ))ds. Proof: Using he generic pricing fuure price process heorem (Hun and Kennedy, 2004, Theorem 12.6), Φ j = E N [ 1 I j 0 F ]. The value I j 0 when wrien in erm of β j 0 depends on he raio of discoun facors. Using (Henrard, 2005, Lemma 1) wice, we obain P D ( 0, 1 ) P D ( 0, 2 ) = P D ( (, 1 ) P D (, 2 ) exp 1 2 + 0 0 ν(s, 1 ) ν(s, 2 )dw s ). ν 2 (s, 1 ) ν 2 (s, 2 )ds Only he second inegral conains a sochasic par. This inegral is normally disribued wih variance 0 (ν(s, 1 ) ν(s, 2 )) 2 ds. The expeced discoun facors raio is reduced o P D ( (, 1 ) P D exp 1 0 0 ) ν 2 (s, 1 ) ν 2 (s, 2 )ds + (ν(s, 1 ) ν(s, 2 )) 2 ds. (, 2 ) 2 is consan, and so we have obained he an- By hypohesis S0 CFWD, he coefficien β CFWD,j 0 nounced resul. 4.2 Curve building The advanages of he approach is ha he marke raes on which we have some inuiion are modelled direcly. In some sense, and borrowing a well known name, i could be called he Libor Marke Model of curve descripion (no of curve dynamic as is namesake). There is no requiremen anymore of an arbirary par like in Definiion 1 of he discoun facor approach. The inerpolaion and consrains can be imposed direcly on he marke quaniies. Figure 2 presens he forward rae using he same daa as Figure 1 and he same linear inerpolaion scheme (even if applied o a differen quaniy). The comparison beween he wo approaches is done in Figure 3(a). I is o each marke maker or risk manager o decide which one he prefers. Wih he repored daa, hey marke rae curves display less zig-zag wih he direc rae approach. Wih some oher marke raes, he picure can be differen. In Figure 3(b), we zoomed on a par of he curve. Beyond he angles a dae for which here is no daa in he pseudo-discoun facor framework, one can also see he waves due o he week-end effecs which varies wih he monhs lenghs. 9
7.5 Forward raes wih forward curves 7 6.5 Rae (in %) 6 5.5 5 4.5 4 0 2 4 6 8 10 Time o sar (in year) Figure 2: Forward raes compued using direc forward rae curve. 5 Fuures discoun facor muli-curves framework The framework described his secion was firs presened in Henrard (2012). Our exisence hypohesis replaces he hypohesis I CPN. I links Ibor fuures o maringale fuures price processes in he sense of (Hun and Kennedy, 2004, Secion 12.4). I FUT The prices of he (j-ibor) fuures are maringale fuures price processes for each fixing dae. Once we have assumed ha he insrumen exiss in our economy, we can give is price a name. We do i indirecly hrough he curves P FDF,j 9. The noaions concerning fuures are he same as in he previous secions. Definiion 6 (Fuures pseudo-discoun curves) The forward curve P FDF,j is he coninuous funcion such ha, P FDF,j (, ) = 1, P FDF,j (, s) is an arbirary funcion for s < Spo() + j, and for 0, 1 = Spo( 0 ) and 2 = 1 + j Φ j ( 1 ) = 1 1 ( P FDF,j ) (, 1 ) δ P FDF,j (, 2 ) 1. (12) The fuures price is obained direcly from he pseudo-discoun facor curves (or more exacly he curve is obained direcly from he fuures prices) wihou convexiy adjusmen. Wih ha definiion, he link beween I j 0 and P FDF,j is ( ) I j 0 = 1 P FDF,j ( 0, 1 ) δ P FDF,j ( 0, f 2 ) 1. 9 FDF sands for Fuures Discoun Facor. 10
7.5 Forward raes wih forward curves Forward raes wih pseudo discoun facors 4.25 Forward raes wih forward curves Forward raes wih pseudo discoun facors 7 6.5 4.2 Rae (in %) 6 5.5 Rae (in %) 4.15 5 4.1 4.5 4 0 2 4 6 8 10 Time o sar (in year) 4.05 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 Time o sar (in year) (a) Full picure (b) Zomm picure Figure 3: A comparison of forward raes compued using pseudo-discoun facors and using direc forward rae curve. We will also use he following definiion of fuures rae: Definiion 7 (Fuures rae) The fuures rae is given by F FUT,j ( 1 ) = 1 Φ j ( 1 ). (13) In hypohesis I FUT and Definiion 6, we suppose he exisence of a coninuum of fuures wih all possible fixing dae 0. Obviously finance is discree in paymen daes, wih a mos one paymen by day, and any real number exiss in pracice only on discree daily poins. For fuures here is he exra consrain ha fuures are raded wih selemen dae only every monhs on he shor par of he curve and quarerly up o 10 years. This may appear as a lo less poins han he usual FRAs and swaps. This is no really he case as he FRA are no specially liquid and mainly raded only on he shor par wih a-bes monhly mauriies and above wo year, here are only swap wih annual mauriies. The fuures curve conains more poins in he 2 o 10 years range. The coupon curves conains more poins only in heory, no in pracice. Like in he coupon framework, we define a new variable: Definiion 8 (Spread) The variable β FDF,j ( 1, 2 ) is defined as a raio of discoun facors raios β FDF,j ( 1, 2 ) = P FDF,j (, 1 ) P D p (, 2 ) P FDF,j (, f 2 ) P D (, 1 ) = (1 + δf FUT,j ( 0 )) P D (, f 2 ) P D (, 1 ). (14) In he coupon framework an hypohesis ofen used is ha he raios β FDF,j are consan hrough ime. We propose o use he nex bes hing: a deerminisic spread hypohesis SD FDF The muliplicaive coefficiens beween discoun facor raios, β FDF,j ( 1, 2 ), defined in Equaion (14), are deerminisic for all 1. This is he equivalen o he consan spread hypohesis S0 CDF used in he coupon muli-curves framework. An Ibor coupon pays he amoun δ p I j 0 in 2. Is oday s value is given by he following heorem. 11
Theorem 3 (Coupon value) In he fuures muli-curves framework, under he hypohesis I FUT and SD FDF, he price of he Ibor coupon fixing in 0 for he period [ 1, 2 ] is given by P D (0, 1 )δ p 1 δ βfdf,j 0 ( 1, 2 ) P D (0, 2 ). Proof: The proof is immediae. I is enough o use he link beween I j 0 and P FDF,j, use he definiion of β FDF,j 0 and ake he expecaion wih P D (., 2 ) as numeraire. The formula is very closed o he one for Ibor coupon in he coupon muli-curves framework. The difference is ha here he β FDF,j is aken in 0, no in 0. We menioned above ha he quaniy is no consan, so he wo formulas are differen and his is o be expeced as he definiions of P j are differen. I was proved in Henrard (2010) ha he quaniy γ()p D (, 1 )/P D (, 2 ) is a N-maringale in he one-facor Gaussian HJM model for ( 0 ) γ() = exp ν(s, 2 )(ν(s, 2 ) ν(s, 1 ))ds and ν he bond volailiy in he one-facor Gaussian HJM model. This is he base of he pricing of fuures in he coupon framework. Our nex hypohesis is coheren wih ha observaion HJM1 The quaniies β FDF,j ( 1, 2 ) are such ha β FDF,j = β FDF,j 0 γ() γ(0). Under HJM1 hypohesis, we have he following equaliies P FDF,j (, 1 ) P FDF,j (, 2 ) = P D (, 1 ) P D (, 2 ) βfdf,j = P D (, 1 ) γ()βfdf,j 0 P D (, 2 ) γ(0) wih he firs and he las variables N-maringales in he one facor Gaussian HJM model. I may seem a very srong hypohesis o impose a specific model. This is equivalen o he HJM hypohesis o price fuures ofen done in curve consrucion. Theorem 4 In he fuures muli-curves framework, under he hypohesis I FUT and HJM1, in he one-facor Gaussian HJM model he price of he Ibor coupon fixing in 0 for he period [ 1, 2 ] is given by P D (0, 1 ) βfdf,j 0 ( 1, 2 ) γ(0) P D (0, 2 ) = P D (0, p 2 )δ 1 p γ(0) ( F FDF,j 0 + 1 δ (1 γ(0)) ). The convexiy adjusmen is now done on he Ibor coupon, no on he fuures anymore. Noe ha he adjusmen is obained by dividing by he coefficien γ(0) and no muliplying by i. Should he adjusmen be called a concaviy adjusmen? 5.1 Zero rae collaeral In his framework meaningful in pracice? There is a leas one scenario where i could become he sandard framework. There is a push for more sandardisaion of he producs and of he legal 12
erms (CSA in paricular). One paricular discussion is around he changes of he CSA erms and he relaed collaeral renumeraion. In he curren sandard erms an overnigh rae (Fed Fund; Eonia, ec.) is paid. One poenial soluion o simplify he erm of he CSAs, which has been proposed by several marke paricipans, is o pay zero ineres on he collaeral. This is equivalen o a fuures margining. If ha proposal, which simplifies a cerain number of pracical problems, is pu in place, his framework would be he naural one also for he swaps wih zero rae collaeral. Suppose ha here is a coninuous (daily) margining for swaps and he rae paid on he collaeral is 0. This is he similar as he margining on fuures: he difference of value wih he previous valuaion is paid and no ineres is paid on ha amoun. The general fuures price process heory, as described in (Hun and Kennedy, 2004, Secion 12.3), applies in ha case. Wha is he value of he new Ibor coupon wih CSA a rae 0? The coupon pays I j 0 in 2 and is a fuures price process up o ha dae. According o he general fuures price heorem is value in 0 is [ ] [ ] E N I j 0 = 1 E N 1 I j 0 = 1 Φ j 0 ( 1). Noe ha he fac ha he coupon pays in 2 has no impac on he valuaion. The value is known a he fixing dae 0 and from ha dae on he require collaeral is paid. Noe ha paymens in advance or in arrear have he same value. Maybe here is would be a legal disincion beween he amoun paid as collaeral and he amoun paid as coupon, bu if we ignore ha disincion, from a cash-flow exchange, everyhing is exchanged as soon as he cash-flow is known. The general collaeral principle is sill valid: a promise o pay omorrow is fulfilled by paying oday he (discouned) expeced value and adaping he amoun up o he final paymen. The difference here is he discouning a a zero rae and he fac ha no adapaion is required afer he las fixing. The no-adapaion afer las fixing is also he case for he collaeral wih deerminisic ineres. 6 Conclusion We presened several muli-curves frameworks. They are differeniaed by he fundamenal marke insrumens (coupons or fuures) and by he way he forward curves are represened (pseudodiscoun facors or direc marke forward raes). We described hree of he four combinaions; he exension o he fourh is immediae. The coupons pseudo-discoun facor framework is he sandardly used combinaion, mos for hisorical reasons han for deep fundamenal reasons. In some circumsances, oher combinaions can be more efficien. References Amerano, F. and Bianchei, M. (2009). Boosrapping he illiquidiy: muliple yield curves consrucion for marke coheren forward raes esimaion. Working paper, Banca IMI/Banca InesaSanpaolo. Available a SSRN: hp://ssrn.com/absrac=1371311. 1 Andersen, L. and Pierbarg, V. (2010). Ineres Rae Modeling Volume I: Foundaions and Vanilla Models. Alanic Financial Press. 6 Bianchei, M. (2010). Two curves, one price. Risk, pages 74 80. 1 13
Boenkos, W. and Schmid, W. (2004). Cross currency swap valuaion. Working Paper 2, HfB - Business School of Finance & Managemen. 1 Cakici, N. and Zhu, J. (2001). Pricing eurodollar fuures opions wih he Heah-Jarrow-Moron models. The Journal of Fuures Markes, 21(7):655 680. 2 Chibane, M. and Sheldon, G. (2009). Building curves on a good basis. Technical repor, Shinsei Bank. Available a SSRN: hp://ssrn.com/absrac=1394267. 1 Henrard, M. (2005). Eurodollar fuures and opions: Convexiy adjusmen in HJM one-facor model. Working paper 682343, SSRN. Available a SSRN: hp://ssrn.com/absrac=682343. 2, 5, 6, 9 Henrard, M. (2007). The irony in he derivaives discouning. Wilmo Magazine, pages 92 98. 1, 5 Henrard, M. (2010). The irony in he derivaives discouning - Par II: he crisis. Wilmo Journal, 2(6):301 316. 1, 2, 12 Henrard, M. (2012). My fuure is no convex. Working paper, OpenGamma. Available a??? 1, 10 Hun, P. J. and Kennedy, J. E. (2004). Financial Derivaives in Theory and Pracice. Wiley series in probabiliy and saisics. Wiley, second ediion. 6, 9, 10, 13 Jäckel, P. and Kawai, A. (2005). The fuure is convex. Wilmo Magazine, pages 1 13. 2 Kijima, M., Tanaka, K., and Wong, T. (2009). A muli-qualiy model of ineres raes. Quaniaive Finance, pages 133 145. 1 Kirikos, G. and Novak, D. (1997). Convexiy conundrums. Risk, pages 60 61. 2, 5 Mercurio, F. (2009). Ineres raes and he credi crunch: new formulas and marke models. Technical repor, QFR, Bloomberg. 1 Mercurio, F. (2010a). A LIBOR marke model wih sochasic basis. Risk, 23(12):84 89. 1 Mercurio, F. (2010b). A LIBOR marke model wih sochasic basis. Working paper., Bloomberg L.P. Available a: hp://ssrn.com/absrac=1583081. 2, 5 Moreni, N. and Pallavicini, A. (2010). Parsimonious HJM modelling for muliple yield-curve dynamics. Working paper, SSRN. Available a hp://ssrn.com/absrac=1699300. 1 Morini, M. (2009). Solving he puzzle in he ineres rae marke. Working paper, IMI Bank Inesa San Paolo. Available a SSRN: hp://ssrn.com/absrac=1506046. 1 Pallavicini, A. and Tarenghi, M. (2010). Ineres-rae modeling wih muliple yield curves. Working paper 1629688, SSRN. Available a hp://ssrn.com/absrac=1629688. 1 Pierbarg, V. (2010). Funding beyond discouning: collaeral agreemens and derivaives pricing. Risk, 23(2):97 102. 1 14
Pierbarg, V. and Renedo, M. (2004). Eurodollar fuures convexiy adjusmens in sochasic volailiy models. Working Paper 610223, SSRN. Available a SSRN: hp://ssrn.com/absrac=610223. 2 Quaniaive Research (2012). Ineres Rae Insrumens and Marke Convenions Guide. OpenGamma. Available a hp://docs.opengamma.com/display/doc/analyics. 1 Tuckman, B. and Porfirio, P. (2003). Ineres rae pariy, money marke basis swaps and crosscurrency basis swaps. Fixed income liquid markes research, Lehman Brohers. 1 15
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OpenGamma Quaniaive Research 1. Marc Henrard. Adjoin Algorihmic Differeniaion: Calibraion and implici funcion heorem. November 2011. 2. Richard Whie. Local Volailiy. January 2012. 3. Marc Henrard. My fuure is no convex. May 2012. 4. Richard Whie. Equiy Variance Swap wih Dividends. May 2012. 5. Marc Henrard. Deliverable Ineres Rae Swap Fuures: Pricing in Gaussian HJM Model. Sepember 2012. 6. Marc Henrard. Muli-Curves: Variaions on a Theme. Ocober 2012.
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