Consistent Valuation Across Curves Using Pricing Kernels

Transcription

1 Consisen Valuaion Across Curves Using Pricing Kernels Andrea Macrina, Obeid Mahomed Deparmen of Mahemaics, Universiy College London arxiv: v2 [q-fin.mf] 16 Feb 2018 London WC1E 6BT, Unied Kingdom Deparmen of Acuarial Science, Universiy of Cape Town Rondebosch 7701, Souh Africa African Insiue of Financial Markes and Risk Managemen Universiy of Cape Town, Rondebosch 7701, Souh Africa July 7, 2018 Absrac The general problem of asse pricing when he discoun rae differs from he rae a which an asse s cash flows accrue is considered. A pricing kernel framework is used o model an economy ha is segmened ino disinc markes, each idenified by a yield curve having is own marke, credi and liquidiy risk characerisics. The proposed framework precludes arbirage wihin each marke, while he definiion of a curveconversion facor process links all markes in a consisen arbirage-free manner. A pricing formula is hen derived, referred o as he across-curve pricing formula, which enables consisen valuaion and hedging of financial insrumens across curves (and markes). As a naural applicaion, a consisen muli-curve framework is formulaed for emerging and developed iner-bank swap markes, which highlighs an imporan dual feaure of he curve-conversion facor process. Given his muli-curve framework, exising muli-curve approaches based on HJM and raional pricing kernel models are recovered, reviewed and generalised, and single-curve models exended. In anoher applicaion, inflaion-linked, currency-based, and fixed-income hybrid securiies are shown o be consisenly valued using he across-curve valuaion mehod. Keywords: Pricing kernel approach; raional pricing models; muli-curve erm srucures; OIS and LIBOR; spread models; HJM; muli-curve poenial model; linear-raional erm srucure models; inflaion-linked and foreign-exchanged securiies; valuaion in emerging markes. Corresponding auhor: a.macrina@ucl.ac.uk 1

2 1 Inroducion The fundamenal problem ha we consider is he valuaion of a financial insrumen using a discouning rae which differs from he rae a which he insrumen s fuure cash flows accrue. Since such financial insrumens are synonymous wih fixed income asses, we will focus hereon. Noneheless, we have no reason o believe ha he framework ha we develop canno be exended o he valuaion of a generic financial asse. The financial crisis brough his valuaion problem o he foreground when subsanial spreads emerged beween iner-bank ineres raes ha were previously bound by single yield curve consisencies, culminaing in a new valuaion paradigm of muliple yield curves one used for discouning (he overnigh indexed swap (OIS) yield curve) and ohers used for forecasing of cash flows (he y-monh iner-bank offered rae (IBOR) curve, y= 1,3,6,12). However, his problem was also prevalen pre-crisis when an economy is considered, which has an iner-bank swap marke, a governmen bond marke and rades in he global economy via he foreign exchange marke, resuling in hree differen curves: he nominal swap curve, he governmen bond curve and he foreign exchange basis curve. The valuaion of any financial insrumen ha is issued in one of hese markes, bu has cash flows ha are deermined by any of he oher markes, once again manifess he fundamenal problem. In his paper, we direcly address he fundamenal problem, ariculaed above, in a general sense. Considering he afermah of he financial crisis however, academic lieraure on muli-curve ineres rae modelling (in he conex of he developed iner-bank swap marke) has evolved rapidly. Here we classify his lieraure ino four caegories or modelling approaches and provide a non-exhausive lis of references and a brief summary of he main conribuions herein. The firs caegory is shor-rae models. Kijima e al. [33] propose a hree-yield curve model (discoun, swap and governmen bond curve) for an economy wih he respecive shor-raes governed by Gaussian, exponenially quadraic models. Kenyon[32] and Morini & Runggaldier[42] consider Vasicek, Hull-Whie (HW) and Cox-Ingersoll-Ross (CIR) shorrae models for he OIS, IBOR and/or OIS-IBOR spread curves. Filipović & Trolle[16] propose a Vasicek process wih sochasic long-erm mean as he OIS shor-rae model wih explici models for defaul and liquidiy risk. Alfeus e al. [3] adop a novel approach of modelling roll-over risk explicily in a reduced-form seing and consider muli-facor 2

3 CIR-ype processes for his and he OIS shor-rae. Heah-Jarrow-Moron (HJM) models consiue he second caegory. Pallavicini & Tarenghi [45], Fujii e al.[21], Moreni & Pallavicini[41], Crépey e al.[10] and Migliea[39] all focus on a hybrid HJM-LMM (LIBOR Marke Model) approach where he OIS curve is modelled using he classical HJM model, while he IBOR forward raes are modelled in an ad hoc manner. Crépey e al. [9] pioneered he use of he HJM framework via a credi risk analogy, while Migliea[39] and Grbac & Runggaldier[23] do he same using a foreign exchange (FX) analogy. Pallavicini and Tarenghi[45] focus on aspecs of calibraion, while Moreni & Pallavicini[41] propose a specific Markovian facor represenaion which expedies calibraion. Crépey e al.[10] consider Lévy driven models, while Cuchiero e al.[12] consider a general semimaringale seup wih muliplicaive OIS-IBOR spreads. Caegory hree is he class of LIBOR Marke Models (LMMs). Morini[40], Mercurio[35], [36],[37],[38] and Bianchei[5] were he firs o exend he LMM o a muli-curve seing, wih he laer doing so via an FX analogy. Mercurio[36] and Mercurio and Xie[43] formalised he firs approach uilising an addiive spread beween OIS and IBOR forward raes ha were modelled as maringales under he classical forward measure, while Amerano and Bianchei[4] formalised he associaed muli-curve boosrapping process. Grbac e al.[22] provide an alernaive o he aforemenioned approach using a class of affine LIBOR models, firs proposed by Keller-Ressel e al.[31]. The fourh and final caergory are pricing kernel models. A he presen ime, we are only aware of Crépey e al.[11] and Nguyen & Seifried[44] who have formulaed muli-curve sysems wih pricing kernels. We highligh here ha, in our opinion, boh hese papers adop a hybrid pricing kernel-lmm approach since he OIS curve is modelled wih a pricing kernel while he IBOR process is modelled in an ad hoc fashion we will expand on his in Secions 3 and 5. In his paper we develop a pure pricing-kernel based approach, which we believe o be he firs of such a modelling class. For a deailed review of he pos-crisis muli-curve ineres rae paradigm from boh, a heoreical and pracical perspecive, we refer he reader o Bianchei & Morini[6], Grbac & Runggaldier[23] and Henrard[24]. The soluion ha we propose ress upon a pricing formula, which we call he acrosscurve pricing formula. This formula has a pricing kernel-based model for he economy as is foundaion. More specifically, he pricing kernel framework models he se of yield curves associaed wih he respecive economy under consideraion. This enables us o link he se of yield curves in a consisen arbirage-free manner hrough he definiion of a curveconversion facor process. This conversion process plays an imporan dual role, giving rise 3

4 o he across-curve pricing formula ha enables consisen valuaion and hedging of financial insrumens across curves. I urns ou ha he curve-conversion facor process is consisen wih an FX process in muli-currency modelling in a pricing kernel framework, and herefore our approach is also consisen wih he FX analogy firs proposed by Bianchei[5] for ineres rae derivaives (or he developed iner-bank swap marke) in an LMM seing. In our work we are ineresed in more han developed markes or he iner-bank swap marke, and endeavour o build consisen relaions among a wide variey of developed and emerging marke fixed-income asses including inflaion-linked noes, FX conracs, and hybrid producs such as inflaion-linked FX insrumens. Here, we menion Flesaker & Hughson[18] and[19] for an argirage-free pricing kernel approach o he valuaion of FX securiies, and o Frey & Sommer[20] if one were o consider exending classical shor rae models, based on diffusion processes wih deerminisic coefficiens, for FX-raes. The approach by Jarrow & Yildirim[29], based on he HJM-framework, migh be reaed as in Secion 4 and used for inflaion-linked pricing as shown in Secion 6, laer in his paper. We noe here he early work in 1998 by Hughson[25] who produced a general arbirage-free approach o he pricing of inflaion derivaives, in which o our knowledge a foreign exchange analogy was used in such a conex, for he firs ime. In Hughson s work, he CPI is reaed like a foreign exchange rae ha links he nominal and he real price sysems as if hey were domesic and foreign currencies, respecively. The work by Pilz & Schlögl[46] on modelling commodiy prices re-inerpres a muli-currency LMM approach. Similariies can be seen when applying our approach o muli-currency and muli-curve LIBOR models, as developed in Secion 6.3, where an FX-LIBOR forward rae agreemen is priced. In all ha follows, we refer o he discouning curve as he x-curve and he forecasing curve as he y-curve. Therefore, when describing our framework, we speak of he xy-formalism, while we refer o he applicaion hereof as he xy-approach. Wih regard o he muli-curve sysem adoped by developed marke praciioners for heir iner-bank swap marke, we will show ha here is a naural formulaion of such a sysem wihin our framework. Moreover, we will show ha his naural formalism is no adoped by praciioners, or he marke in a sric sense in general. Raher, praciioners have adoped a more rigid version of he flexible muli-curve sysem we propose, he choice of which resuls and ensures simpler specificaions for fundamenal ineres rae producs, i.e. forward rae agreemens (FRAs) and ineres rae swaps (IRSs). We also formulae a muli-curve sysem for emerging markes, one ha is remarkably consisen wih he corresponding developed marke sysem his feaure being enirely aribuable o he criical dual role played by he curve-conversion facor process. We will expand on his in Secions 2 and 3. 4

5 The remainder of his paper is srucured as follows: Secion 2 inroduces he curveconversion facor process and he across-curve pricing formula. Secion 3 inroduces consisen muli-curve ineres rae sysems for developed and emerging iner-bank swap markes. Secion 4 reviews and reformulaes exising HJM muli-curve modelling approaches wihin he conex of he xy-approach, and inroduces a new framework ha we call he xy-hjm framework. Secion 5 inroduces a generic class of raional muli-curve models and revisis recen raional muli-curve approaches based on pricing kernels in ligh of he xy-framework. Moreover, he linear-raional erm srucure models are shown o belong o a more general class of pricing-kernel-based (raional) models and are exended o he muli-curve seup. In Secion 6, he across-curve pricing approach is adoped o price inflaion-linked and FX securiies, including hybrid conracs. In Secion 7 we draw various conclusions and ake he opporuniy o summarise our findings. 2 Across-curve pricing formula In his secion we define he curve-conversion facor process and deduce wha we erm he across-curve pricing formula. A he basis of he curve-conversion facor process lies he assumpion ha, wihin a given economy, here is a disinc marke associaed wih each curve. Each of hese markes are characerised by is own se of marke, liquidiy and credi risk facors. In urn, each se of marke, liquidiy and credi risk facors may be sysemaic or idiosyncraic in naure. The curve-conversion facor process plays a dual role: (i) i provides a mechanism akin o a ladder ha enables one o ransi consisenly from one discoun curve sysem o anoher; and (ii) i faciliaes he equivalen represenaion of cash flows across markes (or curves), no maer wha financial insrumen is implicily being priced or ineres rae sysem being modelled. This feaure enables consisen valuaion across differen curves (or markes). The paradigm we shall adop for he developmen of he across-curve pricing approach is one based on pricing kernels. Previous works developing and applying he pricing kernel paradigm comprise, e.g., Consaninides[8], Flesaker & Hughson[17] and[19], Rogers[47], Jin & Glasserman[30], Hughson & Rafailidis[26], Akahori e al.[1], Macrina[34], and Filipović e al.[15]. Nex, we inroduce he sochasic basis and he pricing kernel sysem. We consider a filered probabiliy space(ω,,( ) 0,), where( ) 0 denoes he filraion andhe real-world probabiliy measure. We inroduce an( )-adaped pricing kernel process(h ) 0, which governs he iner-emporal relaion beween asse values a differen imes in a financial marke. I is a fundamenal ingredien in he so-called sandard 5

6 no-arbirage pricing formula, for a non-dividend-paying financial asse H, given by H = 1 h [h T H T ]. (2.1) The no-arbirage asse price process(h ) 0 is obained by aking he condiional expecaion of he random cash flow H T, occurring a he fixed fuure dae T 0, ha is discouned by he pricing kernel. Sandard references, in which asse pricing using pricing kernels is discussed, include, e.g., Hun & Kennedy[27], Duffie[14], Cochrane[7], and Grbac & Runggaldier[23]. In order for us o deduce he across-curve pricing formula seen as an exension o he pricing formula (2.1) we assume he exisence of a se of (coninuous-ime)( )-adaped pricing kernel processes(h y ) 0, where y= 0,1,2,...,n, each linked o a disinc y-marke. The price H y a [0, T] of a non-dividend-paying financial asse H, wih (random) cash flow H y T a he fixed fuure dae T, is hen given by H y = 1 h y h y T H y T. (2.2) The superscrip y emphasises ha he pricing formula (2.2) holds for he valuaion of asses in he y-economy (or in he y-marke). In fac, he pricing kernel process(h y ) governs he iner-emporal relaion beween he presen value of financial asses and heir fuure cash flows in he associaed y-economy. I hen follows in a sraighforward manner, ha he price process(p y ) 0 T of a zero coupon bond (ZCB), wih payoff H y T = P y T T = 1 a he fixed mauriy T and quoed in he y-marke, is given by P y = 1 h y h y T. The discoun bond sysem spanned in heory by a coninuum, bu in pracice a finie number of mauriies T= T 1, T 2,..., T n generaes a erm srucure curve. Since his curve is indexed by he paricular marke y, we refer o i as he y-curve. In all ha follows, we single ou one of he se of he y-markes (and hereby is associaed y-curve) and refer o i as he x-marke (and is associaed erm srucure curve as he x-curve); of course hen his marke also has an associaed( )-adaped pricing kernel(h x ). The x-marke is he marke wihin which pricing (or discouning) occurs, while he y-marke will denoe he marke wihin which he cash flows of he financial insrumens are forecased (or accrued). The fundamenal pricing problem ha is considered in his paper is one where a financial 6

7 insrumen accrues cash flows a a rae of ineres ha differs from ha used for discouning. Firs we consider he problem of cash flow forecasing and equivalen represenaion under differen curves (or markes), before we ackle he problem of valuaion (or discouning). An equivalen cash flow represenaion across curves (or markes) is jusified in Appendix A using no-arbirage porfolio-based sraegies. These findings are formalised in he following definiion ha inroduces he curve-conversion facor process. Definiion 2.1. Consider an economy wih n disinc markes characerised by a se of pricing kernel processes(h y y ) and associaed discoun bond sysems(p ), where y= 0,1,2,...,n and 0 T. The convered value C x in he x-marke a ime of any spo cash flow C y deermined in he y-marke is given by C x = hy h x C y, where x, y= 0,1,...,n. The convered value C x (, T) a ime in he x-marke of any forward cash flow C y (,T), measurable a ime bu payable a ime T, deermined in he y-marke is given by C x (, T)= hy P y h x P x C y (, T), where x, y= 0,1,...,n. These wo relaions are combined by he definiion of he( )-adaped curve-conversion facor process Q x y = h y T h x T = hy P y h x P x, (2.3) where [0, T] is he ime unil which he cash flow being convered is measurable and T> 0 is he paymen dae. We noe ha he cash flows C x y (, T) and C (, T) are linked by he ideniy C x (, T)= (, T), for [0, T]. Wih his definiion a hand, we now have he necessary ool Q x y C y o resolve he fundamenal pricing problem considered in his paper, i.e. valuing a generic financial insrumen ha accrues cash flows under one curve, he y-curve, bu is priced under anoher curve, he x-curve. Our approach is consisen wih he FX analogy proposed by Bianchei[5], bu formalised in an economy modelled by a se of pricing kernels we describe our approach as he xy-formalism. A he hear of his formalism is he pricing formula presened nex. We refer o his formula as he across-curve pricing formula. The relaion of his novel formula o he fundamenal pricing formula (2.1) is shown in he proof 7

8 of he following proposiion. Proposiion 2.1. Le 0 s T. Consider a generic financial asse H ha has a single - measurable cash flow H y (, T) occurring a he fixed ime T and deermined by he y-curve (or he y-marke). I is noed ha in he ime inerval[, T], he quaniy H y (, T) is fixed a he value observed a ime 0. Wihin he x y-approach, he price process(h x y st ) 0 s T of a financial insrumen, deermined by he x-curve (or x-marke) and coningen on he asse H, is given by H x y st = 1 h x s h x P x Qx y H y (, T) s, 0 s<, P x st Qx y H y (, T), s T. The curve-conversion facor process(q x y ) 0 T is inroduced in Definiion 2.1. (2.4) Proof. For informaion we noe ha, by an applicaion of he relaion (2.2), he price process (H y ) 0 T of he financial asse H is deduced o be H y = 1 h y h y T H y (, T) y y = H (, T)P, (2.5) since he cash flow H y (, T) is -measurable and i occurs a T. A ime [0, T], we conver H y (, T) o he corresponding value H x (, T) in he x-marke by use of he conversion facor Q x y : H x y (, T)= Qx H y (, T). (2.6) Now we inser he convered cash flow H x (, T) in he sandard no-arbirage formula (2.1) (or formula (2.2), where y=0 is aken o be he x-curve) where h = h x is assumed. We have, H x st = 1 h x hs x H x (, T) s. (2.7) Given ha H x (, T) is -measurable, we deduce he following by he ower propery of condiional expecaion: H x st = 1 h x s h x T H x (, T) 1 s = h x hs x P x H x (, T) s, (2.8) for 0 s<. In addiion, for s T, we have H x st = P x st H x (, T). Recalling ha H x y (, T)= Qx H y x y (, T), and by choosing o wrie HsT for H x st in order o emphasise he ineracion beween he x- and he y-curves, he proof is complee. We add ha he one- 8

9 o-one across-curve exension o he sandard pricing formula (2.1) is recovered by seing =T in he relaion (2.4). Remark 1. When =T and H y T T = 1, using Proposiion 2.1, we may define he ZCB P x y st = 1 h x s h y T s = Q x y ss P y st = P x st Qx y st, (2.9) for s [0, T], which has wo represenaions using he definiion of he conversion facor (2.3). Given Proposiion 2.1, we can now presen he dual role played by he curve-conversion facor process, wihin he xy-formalism, which is described in he following corollary. Corollary 2.1. Wihin he xy-formalism, if he cash flow H y (, T) is direcly observable in he economy, hen he curve-conversion facor process enables valuaion by acing a he level of he discouning curve as follows: H x y st = 1 h x hs x P x Qx y H y (, T) 1 s = h y hs x P y H y (, T) s. (2.10) However, if he curve-convered cash flow H x y is direcly observable in he economy, hen he curve-conversion facor process enables valuaion by acing a he level of he cash flow as follows: H x y st = 1 h x hs x P x Qx y H y (, T) 1 s = h x hs x H x y s, (2.11) where(h x y st ) 0 s is he x-marke value of H y (, T), for s T. Proof. If H y (, T) is deermined in he y-marke and direcly observable (i.e. quoed) wihin he economy, hen according o Proposiion 2.1 he value of such a payoff wihin he x- marke, a he fuure erminal ime T, is given by H x y T T = Qx y H y (, T), (2.12) which is model-implied, since he curve-conversion facor process Q x y he specific forms of he pricing kernels(h x ) and(hy is deermined by x y ), respecively. Therefore, since HT T, he curve-conversion facor process is no direcly observable in he economy due o Q x y is subsumed ino he discouning process in Eq. (2.4) for 0 s<, by observing ha h x P x Qx y = hy P y, which yields Eq. (2.10). Conversely, if H y x y (, T) is deermined in he y-marke bu he convered quaniy HT T is 9

10 direcly observable wihin he economy, hen H y x y HT T (, T)= Q x y, (2.13) is model-implied, which is subsumed ino he cash flow process by observing ha H x y P x st Qx y H y (, T) for s T from Eq. (2.4), which yields Eq. (2.11). st = Remark 2. Corollary 2.1 proves o be criical in Secion 3, where consisen muli-curve sysems are derived for boh, developed and emerging iner-bank swap markes. Wih regard o FRAs (he fundamenal iner-bank swap marke derivaive), which has an IBOR process as is underlying, i urns ou ha he y-marke deermined IBOR process is direcly observable in he emerging marke, bu is curve-convered equivalen is direcly observable in he developed marke. In his insance, he dual naure of he curve-conversion facor process caers for his apparen cross-economy marke inconsisency, resuling in one consisen modelling framework. In Appendix B, we provide he consisen se of changes of numeraire asses and associaed equivalen probabiliy measures, which ensure ha no arbirage is produced when he across-pricing formula is applied using an equivalen maringale measure. 3 Pricing kernel approach o muli-curve sysems Firs we consider he definiion of a spo IBOR, i.e. a deposi rae ha is offered a a fixed ime 0 by a se of suiably credi-raed banks wihin a given economy. We assume ha he mauriy of said IBOR is +δ>, so ha he associaed enor is given byδ>0. Then we may define (or represen) he spo IBOR process via ZCB insrumens by L (, +δ)= 1 δ 1 1, (3.1) P +δ where 0,δ>0, and where P +δ is he price a ime of a ZCB, wih enorδ, ha maures a ime + δ. In he classical single-curve framework, where IBORs are considered an appropriae proxy for risk-free raes and where a radable discoun bond sysem is assumed, one can hen proceed o define he forward IBOR process via he canonical no-arbirage pricing relaion L (T i 1, T i )= 1 h P i h Ti 1 P Ti 1 T i L Ti 1 (T i 1, T i ), (3.2) 10

11 for 0 T i 1, and whereδ i = T i T i 1 is he IBOR enor and(h ) 0 is he pricing kernel process. By use of he relaion (3.1) wih = T i 1 andδ=δ i, and he ZCB pricing relaion h P i =[h Ti 1 P Ti 1 T i ], one obains he forward IBOR process L (T i 1, T i )= 1 Pi 1 1, (3.3) δ i P i for 0 T i 1. We noe ha he produc of he pricing kernel process and he discouned forward IBOR process(h P i L (T i 1, T i )) 0 Ti 1 is an(( ),)-maringale, which is analogous o he forward IBOR process being a maringale under he T i-forward measure in he classical single-curve heory. The classical relaion (3.3) saes ha he forward IBOR value a ime can be replicaed by a linear combinaion of zero-coupon bonds, i.e. by one mauring a he IBOR rese dae T i 1 and anoher ZCB mauring a he IBOR selemen dae T i. In a marke where he spread beween an overnigh indexed swap (OIS) rae and he corresponding IBOR is nonzero, relaion (3.3) is no longer accepable. Tha is, he now risky IBOR can no longer be replicaed using risk-free ZCBs. In oder words, he IBOR marke is exposed o risk facors which are no necessarily affecing he risk-free ZCB marke, while he risk exposure also varies depending on he IBOR enorδ i = T i T i 1 one is invesing in. Hence, one needs o assume ha holding a financial conrac wrien on a 3-monh IBOR exposes an invesor o a differen risk profile han when holding an insrumen wrien on a 6-monh IBOR. I follows ha assuming risk-free ZCBs can replicae he same risk exposures as conracs wrien on an IBOR is wrong because: (a) an IBOR may be subjec o more risk sources han he risk-free ZCBs; and (b) he number of risk facors affecing an IBOR conrac may depend on he IBOR enor. We ask he following quesion: If one insised on keeping he relaion (3.3), albei subjec o modificaions, how would one need o adjus in a consisen and arbirage-free manner he relaion beween an IBOR model and he associaed ZCBs in a muli-curve seup? I urns ou ha he answer is an exension based on he xy-formalism inroduced above. Here is how we do i. Firs, we consider a collecion of ineres rae curves indexed by x, y= 0,1,2,..., n where we refer o he x-curve as he discouning curve and he y-curve as he forecasing curve. An example for a pair of curves(x, y) may be he pair(0,1) where he 0-curve is he OIS curve and he 1-curve is he 1-monh IBOR curve. The case where x= y is he (classical) 11

12 single-curve economy. Nex we make he relaionship (3.3) curve-dependen and wrie L y (T i 1, T i )= 1 y P i 1 δ i P y 1. (3.4) i Thus, he y-zcb sysem P y has an associaed y-enored IBOR, which is subjec o he i same se of risk facors, i.e. he y-enored IBOR defines he y-zcb sysem. Moreover, he y-zcb price process saisfies he maringale relaion h y P y =[h y i T i ], which is o say ha no-arbirage is assumed wihin he self-consisen y-marke. Nex we deail he developmen of consisen muli-curve ineres rae sysems inspired by he xy-formalism for boh, emerging and developed markes. 3.1 Discouning sysems in emerging markes In his secion we consider he simpler case of an emerging marke, in paricular one where no OIS zero-coupon yield curve exiss. To be precise, he spo overnigh rae is observable bu here are no radable and liquid overnigh indexed swaps, i.e. here is no OIS derivaive marke o enable he consrucion of a yield curve. For more informaion on he specific nuances and issues relaing o emerging iner-bank swap markes, we refer he reader o Jakarasi e al.[28], and references herein, who consider he problem of esimaing an OIS zero-coupon yield curve in Souh Africa. In such a marke, all forecasing and discouning of cash flows is done by one liquid, risky y-enored IBOR zero-coupon yield curve, only. To derive he muli-curve discouning sysem wihin he xy-formalism, we firs consider he pricing of sandard FRAs. FRAs are he fundamenal primiive securiies in any ineres rae marke, which faciliae price discovery for forward IBORs. The FRA considered here has rese ime T i 1 > 0 and mauriy ime T i > T i 1, which is also assumed o be he selemen ime, and is herefore wrien on he fuure spo IBOR L y T i 1 (T i 1, T i ). The value a ime [0, T i ] of his FRA is denoed by V y y i, wih he firs characer of he superscrip indicaing he discoun curve, and he second characer denoing he forecasing curve. For a uni nominal, he FRA s payoff a T i is given by V y y T i T i =δ i L y T i 1 (T i 1, T i ) K y, (3.5) where K y is an arbirary srike rae expressed in he y-marke. We emphasise ha he FRA s payoff is acually measurable a ime T i 1, however he acual cash flow is only paid a ime 12

13 T i. 1 As a consequence, we may also define he in-advance FRA payoff V y y T i 1 T i a T i 1, which is he value V y y T i T i discouned by P y T i 1 T i, by V y y T i 1 T i = P y T i 1 T i δ i L y T i 1 (T i 1, T i ) K y. (3.6) Using he pricing formula (2.4) wih x= y, along wih relaions (3.2), (3.3) and (3.4), he FRA price process is derived as V y y i = δ i h y h y T V y y i 1 T i 1 T i = δ i h y h y P L y y (T Ti 1 Ti 1 Ti Ti 1 i 1, T i ) K y = δ i P y y L i (T i 1, T i ) K y = P y i 1 (1+δ i K y )P y i. (3.7) By seing V y y i = 0, he fair FRA rae process is recovered and is given by K y y (T i 1, T i )= L y (T i 1, T i ), (3.8) for [0, T i 1 ]. The noaion K y y (T i 1, T i ) emphasises ha his fair FRA srike rae applies when he y-curve is used for boh, discouning and forecasing. Nex we consider a sandard IRS wih uni nominal, rese imes{t 0, T 1,..., T n 1 }, paymen imes{t 1, T 2,..., T n }, referencing he y-enored IBOR and arbirary fixed swap rae under he y-marke denoed by S y. Again applying pricing relaion (2.4) wih x= y, ogeher wih relaions (3.2), (3.3) and (3.4), he IRS price process is derived as V y y n = n δ i h y i=1 = P y 0 P y n S y h L y y (T Ti Ti 1 i 1, T i ) S y n i=1 δ i P y i, (3.9) for T 0. Using he same noaion convenion as wih he FRA, he fair IRS rae process is given by S y y (T 0, T n )= P y P y 0 n n i=1 δ ip y, (3.10) i for T 0. For a brief reamen of boosrapping in an emerging marke, we here refer o 1 The marke convenion is o undersand he righ-hand-side of (3.5) as he rae (or quoe) observed a T i 1 and applied a T i on one uni of currency giving he payoff value V y y T i T i of he conrac a T i. Since his value is paid a T i, we denoe i V y y T i T i, and use he subscrips T i. 13

14 Appendix C Discouning sysems in developed markes Nex we consider he more complex case of a developed marke where, in general, an OIS marke exiss. In such a marke, cash flows are forecas using he y-enored IBOR zerocoupon yield curve bu discouned using he OIS zero-coupon yield curve. Such a produc feaure is also consisen wih he noion of collaeralisaion. We consider he same FRA as in he emerging marke case, however we now assume ha discouning occurs under he x-curve (or he OIS curve, o be more specific). We now have o make use of relaion (2.4) in order o define he FRA s payoff. Proposiion 3.1. The developed marke FRA wih rese ime T i 1, expiry ime T i and uni nominal has a erminal payoff, wihin he x-marke, given by V x y T i T i = Q x y T i 1 T i δ i L y T i 1 (T i 1, T i ) K y = Q x y T i 1 T i V y y T i T i, (3.11) where, as before,δ i = T i T i 1 and K y is he srike rae wihin he y-marke. The in-advance FRA payoff is hen given by V x y T i 1 T i = P x T i 1 T i Q x y T i 1 T i V y y T i T i, (3.12) which is he discouned value of he erminal payoff wihin he x-marke. Proof. A direc applicaion of relaion (2.4) leads o he resul in Proposiion 3.1. Like he emerging marke FRA, noice ha he developed marke FRA s payoff is also measurable a T i 1 wih he acual cash flow occurring a T i. Before we consider he derivaion of he value of he developed marke FRA, he following lemmas will prove o be useful in his regard. Lemma 3.1. The convered y-enored forward IBOR process for [0, T i 1 ], saisfies he maringale relaion for 0 s T i 1. L x y (T i 1, T i )=Q x y L y i (T i 1, T i ), (3.13) L x y s (T i 1, T i )= 1 hs x P x h x T L x y i T (T i 1 i 1, T i ) s, (3.14) st i 14

15 Proof. This saemen follows from Eqs (2.3), (3.13) and (3.2). Lemma 3.2. The fair forward price K x of a forward conrac iniiaed a ime o exchange a cash flow K y, deermined in he y-marke, for a cash flow of K x, in he x-marke, wih K y being convered a Q x y T i 1 T i bu he final payoff occurring a expiry T i > T i 1 is given by K x = Proof. The value of such a forward conrac is given by h y P y i h x P x K y = Q x y K y. (3.15) i i V x y i = 1 h x h x K y Q x y Ti T i 1 T K x i = K y hy h x P y P x i K x, (3.16) i which follows from Eq. (2.3) and he ower propery of condiional expecaions, while seing V x y = 0 and solving for K x yields he required resul. i We now have he necessary resuls o derive he value of he developed marke FRA, which is presened in he following heorem. Theorem 3.1. The value of he developed marke FRA wih rese ime T i 1, expiry ime T i and uni nominal, wihin he x-marke, is given by V x y i =δ i P x i L x y (T i 1, T i ) K x, (3.17) for [0, T i 1 ], whereδ i = T i T i 1 and K x is he srike rae wihin he x-marke. Proof. Using Proposiion 3.1, he value of he developed marke FRA, for [0, T i 1 ], is given by V x y i = 1 h x = δ i P x h x T P x i 1 T i 1 T Q x y i T i 1 Tδ i i L y T (T i 1 i 1, T i ) K y L x y i (T i 1, T i ) δ i P x i Q x y i K y, (3.18) wih he firs erm following from Lemma 3.1 and he second erm from Eq. (2.3). Eq. (3.17) follows from applying he resul of Lemma 3.2 o he second erm and facorising accordingly. 15

16 The value of his FRA is commensurae wih he value of a muli-curve (or basis) FRA in a developed marke, i.e. he price dynamics are consisen wih he sandard FRA conrac raded in developed markes. The form of he developed marke FRA s value wihin he xy-framework leads o he following definiion for he muli-curve forward IBOR process. Definiion 3.1. The muli-curve marke-implied y-enored forward IBOR process is given by P x y x y L x y (T i 1, T i )= Q x y L y i (T P i i 1 i 1, T i )= δ i P x i P x y 1, (3.19) i for [0, T i 1 ], where(p x y i ) is defined in Remark 1. Moreover, we may also derive he fair developed marke FRA rae given he value of he FRA provided by Theorem 3.1. Corollary 3.1. The fair FRA rae process K x y (T i 1, T i ) a ime of a developed marke FRA wrien on he marke-implied y-enored forward IBOR (3.13), wih rese ime T i 1 and selemen ime T i, is given by for [0, T i 1 ]. K x y (T i 1, T i )= L x y (T i 1, T i ), (3.20) Proof. Seing he value of he developed marke FRA, given by Eq. (3.17), equal o zero, we find ha K x = L x y (T i 1, T i ) a ime. Then for any ime [0, T i 1 ], he resul for he fair FRA rae process, K x y (T i 1, T i )= L x y (T i 1, T i ), follows accordingly. Remark 3. Relaion (3.20) is he direc muli-curve analogy o he single-curve relaion (3.8). In fac, for x= y one recovers he single-curve expressions (3.7) and (3.8). Remark 4. Using Definiion 3.1, one may re-sae he value of he developed marke FRA as V x y i = P x y i 1 (1+δ i K y )P x y i, (3.21) for [0, T i 1 ], which is he direc muli-curve analogy o he emerging marke FRA value (3.7) wih he y-zcbs replaced by he x y-zcbs. Now ha we have hese resuls, i is also imporan o consider he relaionship beween L x y (T i 1, T i ) and L x (T i 1, T i ). In paricular, one would wan L x y (T i 1, T i ) L x (T i 1, T i ) due o he greaer degree of risk associaed wih he muli-curve y-enored forward IBOR process versus he corresponding x-enored process. The following corollary reveals he 16

17 condiions under which his feaure is achieved, by making use of he associaed forward capialisaion facor (FCF) processes. Corollary 3.2. The muli-curve marke-implied y-enored FCF process v x y (T i 1, T i ), observed a ime T i 1 and applying over he period[t i 1, T i ], defined by v x y (T i 1, T i ) := 1+δ i L x y (T i 1, T i ), (3.22) is greaer han or equal o he corresponding x-enored FCF process v x (T i 1, T i ) := 1+δ i L x (T i 1, T i ), (3.23) if ineres raes are non-negaive and h y h x for all [0, T i 1 ] where T i 1 T i. Proof. Using Eq. (3.22) and Definiion 3.1, we can show ha v x y (T i 1, T i ) = 1+δ i Q x y L y i (T i 1, T i ) = 1+ Q x y y v i (T i 1, T i ) 1 = 1 Q x y i + v x y (T i 1, T i ), where v y (T i 1, T i ) is he y-enored FCF and v x y (T i 1, T i ) := Q x y v y i (T i 1, T i ) is he y- enored FCF represened equivalenly in he x-marke. Then, using Definiion 2.1 and Eq. (2.9), we can show ha v x y (T i 1, T i )=Q x y P y i 1 i P y i = Q x y P x y i 1 i P x y i = P x i 1 P x i Q x y i 1 = v x (T i 1, T i )Q x y i 1. Now in order o have v x (T i 1, T i ) v x y (T i 1, T i ), we mus have ha v x (T i 1, T i ) 1 Q x y + v x i (T i 1, T i )Q x y v x (T i 1, T i ) 1 Q x y 1 Q x y i 1 i 1 Q x y i 1 1 Q x y i, where he las inequaliy holds if ineres raes are non-negaive, i.e. v x (T i 1, T i ) 1. Finally, Q x y Q x y i 1 if ineres raes are non-negaive and h y i h x for all [0, T i 1 ] where T i 1 T i. This may be easily evidenced by seing =T i 1 and allowing T i o vary, while also using he linear and monoonic properies of condiional expecaions. i 1 17

18 This corollary proves ha he xy-approach, applied o a developed marke, yields a y- marke ineres rae sysem which is dominaed by he x-marke sysem, i.e. P y P x for 0 T. Furhermore, his y-marke sysem provides a forward IBOR process, L y (T i 1, T i ), and enables he consrucion of a conversion facor process Q x y, which faciliaes he definiion of he developed marke y-enored forward IBOR process L x y (T i 1, T i ). i Therefore, while he y-marke sysem is sill ficiious, given ha i canno be direcly observed, we sill consider i o be a model-consisen sysem given our curve-conversion framework ha is inspired by currency modelling. Remark 5. Using he FCF, one may also express he erminal payoff of he developed marke FRA by V x y T i T = Q x y i T i 1 T v y i T (T i 1 i 1, T i ) v y K, (3.24) where v y K :=(1+δ ik y ). Then, applying he same resuls as before, he value of he FRA, for [0, T i 1 ], is V x y i = P x i v x y (T i 1, T i ) v x K, (3.25) where v x K = Qx y v y i K. If we define he muli-curve y-enored forward IBOR process by L x y (T i 1, T i ) := 1 x y v (T i 1, T i ) 1, (3.26) δ i and he muli-curve x-marke equivalen FRA srike rae by we hen recover he developed marke FRA price process: V x y i K x := 1 δ i v x K 1, (3.27) We noe ha in his model v x y (T i 1, T i )= v x (T i 1, T i )Q x y so ha i 1 = P x i δ i L x y (T i 1, T i ) K x. 1+δ i L x y y (T i 1, T i ) h 1+δ i L x(t i 1, T i ) = i 1 h x P x, (3.28) i 1 and L x y (T i 1, T i ) L x (T i 1, T i ) if ineres raes are non-negaive and h x h y for all [0, T i 1 ] and for T i 1 T i. This is he approach adoped by Nguyen & Seifried[44] and i shall be revisied in Secion 5. Two commens on heir muli-curve model, given he conex of he xy-approach, follow: (i) The quaniies L x y (T i 1, T i ) and K x which deermine he FRA s floaing and fixed cash flows are derived from he curve-convered quaniies v x y (T i 1, T i ) and v x K respecively. 18 P y

19 This is in conras wih L x y (T i 1, T i ) and K x, he direcly comparable curve-convered quaniies used in he xy-framework. Therefore, hese derived quaniies are no longer consisen wih a currency modelling analogy, wih each differing from he correcly convered quaniies by an addiive facor of(q x y i 1)/δ i. (ii) Observaion (i) is furher suppored by equaion (3.28) which shows ha he conversion facor process effecively models he spread beween he muli-curve y-enored FCF and he corresponding x-enored FCF, as opposed o he classical forward exchange rae. Moreover, he derived y-marke sysem has almos no relaion o he developed marke y-enored ineres rae sysem, ha one seeks o model, since he model derived y-marke sysem dominaes he x-marke sysem, i.e. P x P y for 0 T. Remark 6. The mahemaical quaniy ha direcly models he y-enored forward IBOR process is L x y (, ) and no L (, ). y This is a consequence of indusry sandards in developed markes, ha he produc of he x-pricing kernel and he x-curve discouned y-enored forward IBOR process is a maringale under he-measure. In he x y-approach, his implies ha h x s P x st L y i s (T i 1, T i )= h x P x L y i (T i 1, T i ) s, (3.29) for 0 s T i 1. I is no possible o achieve his relaionship wihin he xy-framework, given our represenaion of he y-enored forward IBOR process (3.4). However his relaionship is achieved if we replace L (, ) y wih L x y (, ). Our marke-implied y-enored forward IBOR process, L x y (, ), reveals he convoluion of a conversion facor (which faciliaes he marke s maringale assumpion (3.29)) and he model y-enored forward IBOR process, L (, ). y This resul quesions he uiliy of he y-zcb sysem in he developed marke conex. The y- ZCB sysem is a model consruc, derived from he y-enored model-consisen or model-implied forward IBOR process, L (, ), y which unravels he marke s maringale adjusmen from he observed y-enored marke-implied IBOR process, L x y (, ), via he conversion facor Q y x. Remark 7. The xy-framework advocaes he following price process for a muli-curve FRA V x y i =δ i P x y i L y (T i 1, T i ) K y, (3.30) for [0, T i ]. We noe ha he conversion facor (or maringale adjusmen) has been applied o he discouning x-zcb sysem and no o he model for he y-enored forward IBOR process. 19

20 However, we noe ha he erminal FRA payoff would now be V x y T i T i =δ i h y T i h x T i L y T i 1 (T i 1, T i ) K y. This allows us o disenangle he y-zcb sysem from he x-zcb sysem, which enables us o model he y-curve discouning in a consisen, robus and rigorous fashion. From an economics perspecive, if one compares he reurn generaed from an xy-fra o a yy-fra, one can show ha V y y i V y y 0T i > V x y i V x y 0T i = V x y i V y y 0T i, (3.31) as required, since discouning a he x-curve essenially represens a collaeralised FRA which should herefore reurn he holder less han an equivalen invesmen in a non-collaeralised FRA, represened by he y-curve discouning. Nex we consider he developed marke IRS, i.e. one which forecass cash flows under he y-curve bu discouns under he x-curve, unlike he emerging marke IRS. Theorem 3.2. The value of a developed marke IRS, wihin he x-marke, wih rese imes {T 0, T 1,..., T n 1 }, paymen imes{t 1, T 2,..., T n } and uni nominal, referencing he y-enored IBOR is given by V x y n = n i=1 δ i P x i L x y (T i 1, T i ) S x, (3.32) for T 0, whereδ i = T i T i 1 and where S x is he fixed swap rae wihin he x-marke. Proof. Saring wih he emerging marke version of he IRS wih fixed swap rae S y wihin he y-marke and applying pricing relaion (2.4), analagous o Proposiion 3.1, he developed marke IRS price process is given by V x y n = n δ i h x i=1 h x T P x i 1 T i 1 T Q x y i T i 1 T L y i T (T i 1 i 1, T i ) S y, (3.33) which, upon applicaion of Lemma 3.1 and Equaion (2.3), simplifies o V x y n = n i=1 δ i P x L x y i (T i 1, T i ) Q x y S y, (3.34) i for T 0. The resul follows by observing ha he fixed IRS rae may be expressed in he x-marke by S x = S y ( n i=1 δ ip x Q x y i )/( n i i=1 δ ip x ). This may be jusified in an analogous i 20

21 fashion o he fixed FRA rae, bu his ime making use of a fixed-for-fixed swap conrac as opposed o a forward conrac, as in Lemma 3.2. Remark 8. Using Definiion 3.19, one may re-sae he value of he developed marke IRS as V x y = P x y n P x y 0 S y n n i=1 δ i P x y i, (3.35) for T 0, which is he direc muli-curve analogy o he emerging marke IRS value (3.9) wih he y-zcbs replaced by he x y-zcbs. Corollary 3.3. The fair fixed swap rae process S x y (T 0, T n ) of a developed marke IRS wrien on he marke-implied y-enored forward IBOR (3.13), wih rese imes{t 0, T 1,..., T n 1 }, paymen imes{t 1, T 2,..., T n } and uni nominal, is given by for T 0. S x y (T 0, T n )= P x y P x y 0 n n i=1 δ ip x, (3.36) i Proof. Seing he value of he developed marke IRS equal o zero, given by Eq. (3.35), i follows ha he y-marke fair fixed IRS rae is S y = P x y P x y 0 / n n i=1 δ ip x y a ime. i Using he proof of Theorem 3.2 and Remark 1, he x-marke fair fixed IRS rae (convering he y-marke rae) is given by S x = S y ( n i=1 δ ip x Q x y i )/( n i i=1 δ ip x ) a ime. Then for i any ime T 0, he resul for he developed marke fair IRS rae follows accordingly by seing S x y (T 0, T n )=S x. C.2. For a brief reamen of boosrapping in a developed marke, we here refer o Appendix 3.3 Consisen muli-curve discouning in emerging markes Now ha we have a good undersanding of how he xy-formalism enables he modelling of muli-curve ineres rae sysems in developed markes, we may consider resolving he same problem for he case of an emerging marke. Our firs hurdle in moving from a developed o an emerging marke seing is he non-exisence of he OIS curve. Recall ha we have assumed he exisence of a collecion of ineres rae curves indexed by x, y= 0,1,2,...,n where we refer o he x-curve as he discouning curve and he y- curve as he forecasing curve. In a common developed marke, n=4wih 0 denoing he 21

22 nominal OIS curve, 1 he 1-monh IBOR curve, 2 he 3-monh IBOR curve, 3 he 6-monh IBOR curve and 4 he 12-monh IBOR curve. Moreover, he sochasic evoluion of each of hese curves are modelled via a pricing kernel process(h y ) which are in urn calibraed using liquid linear and non-linear ineres rae marke insrumens. In a common emerging marke, only one IBOR enor is usually radable and liquid, herefore i is no possible o calibrae he enire se of pricing kernel processes(h y ) which span he common developed ineres rae marke. This leads us o he following remark. Remark 9. In he common emerging marke, only one IBOR enor, y, is radable and liquid hereby enabling he specificaion and calibraion of a well-defined pricing kernel process(h y ). Pricing kernel processes for all oher IBOR enors(h y ) are o be esimaed saisically (or oherwise) as a suiable funcional form of(h y ), i.e. h y = f h y. (3.37) where f : + + is measurable and adaped, such ha he corresponding esimaed y-zcb (and y-curve) sysems,(p y ), may be consruced via for 0 T. P y = 1 y h y h T 1 = f h y f h y T, (3.38) In Remark 9, if he funcion f( ) is linear, hen he esimaed y-zcb is given by P y = 1 f f h y P y, (3.39) which implies ha i is possible o direcly replicae he esimaed y-zcb hrough eiher a saic or dynamic replicaion sraegy using he y -ZCB. However, his may no be possible, in general, if he funcion f( ) is convex (concave), as he esimaed y-zcb will be governed by he following inequaliy P y ( ) 1 f f h y P y, (3.40) which follows by he applicaion of Jensen s inequaliy. The xy-formalism may now be applied in he emerging marke seing, assuming he exisence of a collecion of ineres rae curves, indexed by x, y = 0,1,..., y,...,n, ha are modelled by he calibraed pricing kernel process(h y ) and he se of esimaed pricing kernel processes( h y ; y y ). Firs, we consider he developed marke FRA wihin he emerging marke conex, i.e. one where 22

23 he payoff is forecased by he y-curve and hen discouned by he x-curve. The erminal and in-advance FRA payoffs remain unchanged and are idenical o Eqs (3.11) and (3.12), respecively 2, wih he FRA price process also assuming he familiar form V x y i = δ i P x y i L y (T i 1, T i ) K y = P x y i 1 (1+δ i K y )P x y i, (3.41) for [0, T i 1 ], while L y (T i 1, T i ) coninues o be he correc forward IBOR process. Noice ha he derivaion of equaion (3.41) follows by a direc applicaion of Corollary 2.1. Definiion 3.2. The muli-curve emerging marke y-enored forward IBOR process is given by L y (T i 1, T i )= 1 y P i 1 δ i P y 1, (3.42) i for [0, T i 1 ], unlike he developed marke which required he definiion of he markeimplied y-enored forward IBOR process L x y (T i 1, T i ) := Q x y L y i (T i 1, T i ) for [0, T i 1 ]. This is due o he fac ha here is currenly no marke sandard for pricing an emerging marke FRA ha is forecased and discouned under differen curves, wih he only observable marke quaniy being he spo IBOR process L y (, +δ) for 0. I is also possible, as in he case of developed markes, o define a fair FRA rae process, K x y (T i 1, T i )= L y (T i 1, T i ), however one would no be able o observe his quaniy in he marke (since hese FRAs are no raded, in general), herefore his would be a model-implied quaniy 3. Similarly, we may consider he sandard developed marke IRS in he conex of an emerging marke. The value of he IRS a some ime T 0, making use of he same relaions as before, is again given by V x y n = n i=1 δ i P x y i L y (T i 1, T i ) S y = P x y 0 P x y n S y n i=1 δ i P x y i, (3.43) where L y (T i 1, T i ), for [0, T i 1 ], coninues o be he correc forward IBOR process, analagous o he FRA resul. As wih he FRA, he fair IRS rae process is model-implied 2 Assuming ha one insiss on mainaining measurabiliy of he payoff a he IBOR rese ime T i 1. 3 If he y-enored IBOR corresponds o he mos liquid and radable enor, i.e. y=y, hen one will also have access o he se of forward IBOR processes L(T y i 1, T i ) for 0 T i 1, from he sandard and liquidly radable se of single-curve emerging marke FRAs, and K x y (T i 1, T i )= K y y (T i 1, T i )= L y (T i 1, T i ). 23

24 (unless he y-enored IBOR process is he radable enor and =T 0 = 0) and given by S x y (T 0, T n )=(P x y P x y 0 )/( n n i=1 δ ip x y ) for T i 0. In a muli-curve emerging marke ineres rae sysem, wihin he xy-framework, he iniial (esimaed) y-zcb sysems may be consruced in a compleely analogous fashion o he single-curve emerging marke relaions, see Appendix C, since K x y 0 (T i 1, T i )= L y 0 (T i 1, T i ) and S x y 0 (0, T n)=(1 P y 0T)/( n n i=1 δ ip y 0T). Tha is, all iniial model-implied quaniies are i only dependen on he y-curve or y-zcb sysem. If we consider a FRA and an IRS wihin his conex wih payoffs forecased by he y - curve and discouned by one of he oher curves, denoed by he x-curve, hen he pricing formulae are given by and V x y i = P x y i 1 (1+δ i K y )P x y i, (3.44) V x y = P x y n P x y 0 S y n n i=1 δ i P x y i, (3.45) from Eqs (3.41) and (3.43), respecively. A his juncure, i is imporan o noe ha he x y -ZCB,(P x y ), plays he same role as he y -ZCB,(P y ), does in he single-curve emerging marke seing in Secion 3.1. This leads us o he following definiion for he x y-zcb sysem, in general. Definiion 3.3. In he muli-curve ineres rae sysem derived wihin he xy-framework, he x y-zcb sysem,(p x y ), defined by P x y = 1 y h h x h x T T (1) h x = P x Qx y = Qx y P y, T may be considered o be a quano-bond assuming (i) he x-curve wih varying noional defined by he forward conversion facor Q x y ; or (ii) he y-curve wih varying noional defined by he spo conversion facor Q x y. Remark 10. Wihin he developed marke conex where he nominal OIS curve is considered o be he disinc, single-curve radable sysem, which we shall denoe here as he x -curve one may dynamically replicae y-zcbs and x y-zcbs, where y x, via he following se of x -curve quano-bonds P y = Qx y P x Q x y and P x y = Q x y P x, 24

25 whereas, wihin he emerging marke conex where one nominal IBOR swap curve is considered o be he disinc, single-curve radable sysem, which we have denoed as he y -curve, one may dynamically replicae x-zcbs and x y -ZCBs, where x y, via he following se of y -curve quano-bonds P x y Qx = Q x P y y 4 xy-hjm muli-curve models x y and P = Q x y P y. In his secion we develop Heah-Jarrow-Moron (HJM) muli-curve ineres rae sysems based on he xy-formalism inroduced in his paper. The xy-hjm muli-curve sysem will be derived using resuls from Secion 3. We consider he filered probabiliy space(ω,,( ),) where( ) 0 is he filraion generaed by wo ses of independen muli-dimensional-brownian moions(w ) 0 and (Z ) 0, respecively. Being synonymous wih he xy-formalism, we consider an economy wih wo disinc markes, x and y, where x may be inerpreed as a proxy defaul-free OISbased marke and y as a risky IBOR-based marke. Furhermore, we assume ha he x- and y-markes are driven by he muli-dimensional-brownian moions(w x ) 0=(W ) 0 and(w y ) 0=(W, Z ) 0 respecively, where(w ) 0 is n-dimensional and(z ) 0 is m- dimensional. This allows us o define he pricing kernel process associaed wih each marke. Definiion 4.1. The( )-adaped x- and y-marke pricing kernel processes(h x ) 0 and(h y ) 0 saisfy, respecively, dh x h x = r x d λx dw x, dh y h y = r y d λy dw y, (4.1) where(r x ) 0 and(r y ) 0 are he shor raes of ineres; and(λ x ) 0 and(λ y ) 0=(λ x,λz ) 0 are he n- and(n+ m)-dimensional marke price of risk processes associaed wih he x- and y-markes, respecively. Nex, le(x ) 0 T and(y ) 0 T be (well-defined) processes, respecively saisfying dx = X dy = Y A x + 1 Σ x 2 A y + 1 y Σ 2 for 0 T, where A ( ) = T 2 d Σ x +λx 2 d Σ y +λy dw x, dw y, (4.2) a ( ) udu is 1-dimensional, denoes he Euclidean norm, 25