Consistent Valuation Across Curves Using Pricing Kernels
aa r X i v : . [ q -f i n . M F ] F e b Consistent Valuation Across CurvesUsing Pricing Kernels
Andrea Macrina † ‡ ∗ , Obeid Mahomed §† Department of Mathematics, University College LondonLondon WC1E 6BT, United Kingdom ‡ Department of Actuarial Science, University of Cape TownRondebosch 7701, South Africa § African Institute of Financial Markets and Risk ManagementUniversity of Cape Town, Rondebosch 7701, South AfricaFebruary 19, 2018
Abstract
The general problem of asset pricing when the discount rate differs from the rate atwhich an asset’s cash flows accrue is considered. A pricing kernel framework is used tomodel an economy that is segmented into distinct markets, each identified by a yieldcurve having its own market, credit and liquidity risk characteristics. The proposedframework precludes arbitrage within each market, while the definition of a curve-conversion factor process links all markets in a consistent arbitrage-free manner. Apricing formula is then derived, referred to as the across-curve pricing formula, whichenables consistent valuation and hedging of financial instruments across curves (andmarkets). As a natural application, a consistent multi-curve framework is formulatedfor emerging and developed inter-bank swap markets, which highlights an importantdual feature of the curve-conversion factor process. Given this multi-curve framework,existing multi-curve approaches based on HJM and rational pricing kernel models arerecovered, reviewed and generalised, and single-curve models extended. In anotherapplication, inflation-linked, currency-based, and fixed-income hybrid securities areshown to be consistently valued using the across-curve valuation method.
Keywords:
Pricing kernel approach; rational pricing models; multi-curve term struc-tures; OIS and LIBOR; spread models; HJM; multi-curve potential model; linear-rationalterm structure models; inflation-linked and foreign-exchanged securities; valuation inemerging markets. ∗ Corresponding author: [email protected] Introduction
The fundamental problem that we consider is the valuation of a financial instrument us-ing a discounting rate which differs from the rate at which the instrument’s future cashflows accrue. Since such financial instruments are synonymous with fixed income assets,we will focus thereon. Nonetheless, we have no reason to believe that the framework thatwe develop cannot be extended to the valuation of a generic financial asset. The financialcrisis brought this valuation problem to the foreground when substantial spreads emergedbetween inter-bank interest rates that were previously bound by single yield curve consis-tencies, culminating in a new valuation paradigm of multiple yield curves—one used fordiscounting (the overnight indexed swap (OIS) yield curve) and others used for forecastingof cash flows (the y -month inter-bank offered rate (IBOR) curve, y =
1, 3, 6, 12). However,this problem was also prevalent pre-crisis when an economy is considered, which has aninter-bank swap market, a government bond market and trades in the global economy viathe foreign exchange market, resulting in three different curves: the nominal swap curve,the government bond curve and the foreign exchange basis curve. The valuation of anyfinancial instrument that is issued in one of these markets, but has cash flows that are de-termined by any of the other markets, once again manifests the fundamental problem.In this paper, we directly address the fundamental problem, articulated above, in a gen-eral sense. Considering the aftermath of the financial crisis however, academic literature onmulti-curve interest rate modelling (in the context of the developed inter-bank swap mar-ket) has evolved rapidly. Here we classify this literature into four categories or modellingapproaches and provide a non-exhaustive list of references and a brief summary of the maincontributions therein.The first category is short-rate models . Kijima et al. [ ] propose a three-yield curvemodel (discount, swap and government bond curve) for an economy with the respectiveshort-rates governed by Gaussian, exponentially quadratic models. Kenyon [ ] and Morini& Runggaldier [ ] consider Vasicek, Hull-White (HW) and Cox-Ingersoll-Ross (CIR) short-rate models for the OIS, IBOR and / or OIS-IBOR spread curves. Filipovi´c & Trolle [ ] propose a Vasicek process with stochastic long-term mean as the OIS short-rate model withexplicit models for default and liquidity risk. Alfeus et al. [ ] adopt a novel approachof modelling “roll-over risk” explicitly in a reduced-form setting and consider multi-factor2IR-type processes for this and the OIS short-rate. Heath-Jarrow-Morton (HJM) models constitute the second category. Pallavicini & Tarenghi [ ] , Fujii et al. [ ] , Moreni & Pallavicini [ ] , Crépey et al. [ ] and Miglietta [ ] allfocus on a hybrid HJM-LMM (LIBOR Market Model) approach where the OIS curve is mod-elled using the classical HJM model, while the IBOR forward rates are modelled in an adhoc manner. Crépey et al. [ ] pioneered the use of the HJM framework via a credit riskanalogy, while Miglietta [ ] and Grbac & Runggaldier [ ] do the same using a foreignexchange (FX) analogy. Pallavicini and Tarenghi [ ] focus on aspects of calibration, whileMoreni & Pallavicini [ ] propose a specific Markovian factor representation which expe-dites calibration. Crépey et al. [ ] consider Lévy driven models, while Cuchiero et al. [ ] consider a general semimartingale setup with multiplicative OIS-IBOR spreads.Category three is the class of LIBOR Market Models (LMMs) . Morini [ ] , Mercurio [ ] , [ ] , [ ] , [ ] and Bianchetti [ ] were the first to extend the LMM to a multi-curve setting,with the latter doing so via an FX analogy. Mercurio [ ] and Mercurio and Xie [ ] for-malised the first approach utilising an additive spread between OIS and IBOR forward ratesthat were modelled as martingales under the classical forward measure, while Ametranoand Bianchetti [ ] formalised the associated multi-curve bootstrapping process. Grbac etal. [ ] provide an alternative to the aforementioned approach using a class of affine LIBORmodels, first proposed by Keller-Ressel et al. [ ] .The fourth and final catergory are pricing kernel models . At the present time, we are onlyaware of Crépey et al. [ ] and Nguyen & Seifried [ ] who have formulated multi-curvesystems with pricing kernels. We highlight here that, in our opinion, both these papersadopt a hybrid pricing kernel-LMM approach since the OIS curve is modelled with a pricingkernel while the IBOR process is modelled in an ad hoc fashion—we will expand on this inSections 3 and 5. In this paper we develop a pure pricing-kernel based approach, which webelieve to be the first of such a modelling class.For a detailed review of the post-crisis multi-curve interest rate paradigm from both, atheoretical and practical perspective, we refer the reader to Bianchetti & Morini [ ] , Grbac& Runggaldier [ ] and Henrard [ ] .The solution that we propose rests upon a pricing formula, which we call the across-curve pricing formula . This formula has a pricing kernel-based model for the economy as itsfoundation. More specifically, the pricing kernel framework models the set of yield curvesassociated with the respective economy under consideration. This enables us to link theset of yield curves in a consistent arbitrage-free manner through the definition of a curve-conversion factor process . This conversion process plays an important dual role, giving rise3o the across-curve pricing formula that enables consistent valuation and hedging of financialinstruments across curves. It turns out that the curve-conversion factor process is consistentwith an FX process in multi-currency modelling in a pricing kernel framework, and there-fore our approach is also consistent with the FX analogy first proposed by Bianchetti [ ] forinterest rate derivatives (or the developed inter-bank swap market) in an LMM setting. Inour work we are interested in more than developed markets or the inter-bank swap market,and endeavour to build consistent relations among a wide variety of developed and emergingmarket fixed-income assets including inflation-linked notes, FX contracts, and hybrid prod-ucts such as inflation-linked FX instruments. Here, we mention Flesaker & Hughston [ ] and [ ] for an argitrage-free pricing kernel approach to the valuation of FX securities, andto Frey & Sommer [ ] if one were to consider extending classical short rate models, basedon diffusion processes with deterministic coefficients, for FX-rates. The approach by Jarrow& Yildirim [ ] , based on the HJM-framework, might be treated as in Section 4 and usedfor inflation-linked pricing as shown in Section 6, later in this paper. We note here the earlywork in 1998 by Hughston [ ] who produced a general arbitrage-free approach to the pric-ing of inflation derivatives, in which—to our knowledge—a foreign exchange analogy wasused in such a context, for the first time. In Hughston’s work, the CPI is treated like a foreignexchange rate that links the nominal and the real price systems as if they were domestic andforeign currencies, respectively. The work by Pilz & Schlögl [ ] on modelling commodityprices re-interprets a multi-currency LMM approach. Similarities can be seen when applyingour approach to multi-currency and multi-curve LIBOR models, as developed in Section 6.3,where an FX-LIBOR forward rate agreement is priced. In all that follows, we refer to thediscounting curve as the x -curve and the forecasting curve as the y -curve. Therefore, whendescribing our framework, we speak of the xy-formalism , while we refer to the applicationthereof as the xy-approach .With regard to the multi-curve system adopted by developed market practitioners for theirinter-bank swap market, we will show that there is a natural formulation of such a systemwithin our framework. Moreover, we will show that this natural formalism is not adopted bypractitioners, or the market in a strict sense in general. Rather, practitioners have adopted amore rigid version of the flexible multi-curve system we propose, the choice of which resultsand ensures simpler specifications for fundamental interest rate products, i.e. forward rateagreements (FRAs) and interest rate swaps (IRSs). We also formulate a multi-curve systemfor emerging markets , one that is remarkably consistent with the corresponding developedmarket system—this feature being entirely attributable to the critical dual role played bythe curve-conversion factor process . We will expand on this in Sections 2 and 3.4he remainder of this paper is structured as follows: Section 2 introduces the curve-conversion factor process and the across-curve pricing formula . Section 3 introduces consis-tent multi-curve interest rate systems for developed and emerging inter-bank swap markets.Section 4 reviews and reformulates existing HJM multi-curve modelling approaches withinthe context of the xy-approach, and introduces a new framework that we call the xy-HJMframework. Section 5 introduces a generic class of rational multi-curve models and revisitsrecent rational multi-curve approaches based on pricing kernels in light of the xy-framework.Moreover, the linear-rational term structure models are shown to belong to a more generalclass of pricing-kernel-based (rational) models and are extended to the multi-curve setup.In Section 6, the across-curve pricing approach is adopted to price inflation-linked and FXsecurities, including hybrid contracts. In Section 7 we draw various conclusions and takethe opportunity to summarise our findings. In this section we define the curve-conversion factor process and deduce what we term the across-curve pricing formula . At the basis of the curve-conversion factor process lies theassumption that, within a given economy, there is a distinct market associated with eachcurve. Each of these markets are characterised by its own set of market, liquidity and creditrisk factors. In turn, each set of market, liquidity and credit risk factors may be systematic oridiosyncratic in nature. The curve-conversion factor process plays a dual role: (i) it providesa mechanism—akin to a ladder—that enables one to transit consistently from one discountcurve system to another; and (ii) it facilitates the equivalent representation of cash flowsacross markets (or curves), no matter what financial instrument is implicitly being pricedor interest rate system being modelled. This feature enables consistent valuation acrossdifferent curves (or markets). The paradigm we shall adopt for the development of theacross-curve pricing approach is one based on pricing kernels. Previous works developingand applying the pricing kernel paradigm comprise, e.g., Constantinides [ ] , Flesaker &Hughston [ ] and [ ] , Rogers [ ] , Jin & Glasserman [ ] , Hughston & Rafailidis [ ] ,Akahori et al. [ ] , Macrina [ ] , and Filipovi´c et al. [ ] . Next, we introduce the stochasticbasis and the pricing kernel system.We consider a filtered probability space ( Ω , F , ( F t ) t ≥ , P ) , where ( F t ) t ≥ denotes thefiltration and P the real-world probability measure. We introduce an ( F t ) -adapted pricingkernel process ( h t ) t ≥ , which governs the inter-temporal relation between asset values atdifferent times in a financial market. It is a fundamental ingredient in the so-called standard5o-arbitrage pricing formula, for a non-dividend-paying financial asset H , given by H t = h t E [ h T H T | F t ] . (2.1)The no-arbitrage asset price process ( H t ) t ≥ is obtained by taking the conditional expec-tation of the random cash flow H T , occurring at the fixed future date T ≥ t ≥
0, that isdiscounted by the pricing kernel. Standard references, in which asset pricing using pricingkernels is discussed, include, e.g., Hunt & Kennedy [ ] , Duffie [ ] , Cochrane [ ] , andGrbac & Runggaldier [ ] .In order for us to deduce the across-curve pricing formula—seen as an extension to thepricing formula (2.1)—we assume the existence of a set of (continuous-time) ( F t ) -adaptedpricing kernel processes ( h yt ) t ≥ , where y =
0, 1, 2, . . . , n , each linked to a distinct y -market.The price H yt at t ∈ [ T ] of a non-dividend-paying financial asset H , with (random) cashflow H yT at the fixed future date T , is then given by H yt = h yt E (cid:2) h yT H yT | F t (cid:3) . (2.2)The superscript y emphasises that the pricing formula (2.2) holds for the valuation of assetsin the y -economy (or in the y -market). In fact, the pricing kernel process ( h yt ) governs theinter-temporal relation between the present value of financial assets and their future cashflows in the associated y -economy. It then follows in a straightforward manner, that theprice process ( P ytT ) ≤ t ≤ T of a zero coupon bond (ZCB), with payoff H yT = P yT T = T and quoted in the y -market, is given by P ytT = h yt E (cid:2) h yT | F t (cid:3) .The discount bond system—spanned in theory by a continuum, but in practice a finite num-ber of maturities T = T , T , . . . , T n —generates a term structure curve. Since this curve isindexed by the particular market y , we refer to it as the y-curve . In all that follows, we sin-gle out one of the set of the y -markets (and thereby its associated y -curve) and refer to itas the x -market (and its associated term structure curve as the x -curve); of course then thismarket also has an associated ( F t ) -adapted pricing kernel ( h xt ) . The x -market is the mar-ket within which pricing (or discounting) occurs, while the y -market will denote the marketwithin which the cash flows of the financial instruments are forecasted (or accrued).The fundamental pricing problem that is considered in this paper is one where a financial6nstrument accrues cash flows at a rate of interest that differs from that used for discounting.First we consider the problem of cash flow forecasting and equivalent representation underdifferent curves (or markets), before we tackle the problem of valuation (or discounting).An equivalent cash flow representation across curves (or markets) is justified in Appendix Ausing no-arbitrage portfolio-based strategies. These findings are formalised in the followingdefinition that introduces the curve-conversion factor process . Definition 2.1.
Consider an economy with n distinct markets characterised by a set of pricingkernel processes ( h yt ) and associated discount bond systems ( P ytT ) , where y =
0, 1, 2, . . . , n and ≤ t ≤ T . The converted value C xt in the x -market at time t of any spot cash flow C yt determined in the y-market is given by C xt = h yt h xt C yt , where x , y =
0, 1, . . . , n. The converted value C xt ( t , T ) at time t in the x -market of any forwardcash flow C yt (t,T), measurable at time t but payable at time T , determined in the y-market isgiven by C xt ( t , T ) = h yt P ytT h xt P xtT C yt ( t , T ) , where x , y =
0, 1, . . . , n. These two relations are combined by the definition of the ( F t ) -adaptedcurve-conversion factor process Q x ytT = E (cid:2) h yT | F t (cid:3) E (cid:2) h xT | F t (cid:3) = h yt P ytT h xt P xtT , (2.3) where t ∈ [ T ] is the time until which the cash flow being converted is measurable and T > is the payment date. We note that the cash flows C xt ( t , T ) and C yt ( t , T ) are linked by the identity C xt ( t , T ) = Q x ytT C yt ( t , T ) , for t ∈ [ T ] . With this definition at hand, we now have the necessary toolto resolve the fundamental pricing problem considered in this paper, i.e. valuing a genericfinancial instrument that accrues cash flows under one curve, the y -curve, but is pricedunder another curve, the x -curve. Our approach is consistent with the FX analogy proposedby Bianchetti [ ] , but formalised in an economy modelled by a set of pricing kernels—wedescribe our approach as the xy-formalism. At the heart of this formalism is the pricingformula presented next. We refer to this formula as the across-curve pricing formula . Therelation of this novel formula to the fundamental pricing formula (2.1) is shown in the proof7f the following proposition. Proposition 2.1.
Let ≤ s ≤ t ≤ T . Consider a generic financial asset H that has a single F t -measurable cash flow H yt ( t , T ) occurring at the fixed time T ≥ t and determined by the y-curve(or the y-market). It is noted that in the time interval [ t , T ] , the quantity H yt ( t , T ) is fixed atthe value observed at time t ≥ . Within the x y-approach, the price process ( H x ysT ) ≤ s ≤ T of afinancial instrument, determined by the x -curve (or x -market) and contingent on the asset H,is given by H x ysT = h xs E (cid:2) h xt P xtT Q x ytT H yt ( t , T ) | F s (cid:3) , 0 ≤ s < t , P xsT Q x ytT H yt ( t , T ) , t ≤ s ≤ T . (2.4) The curve-conversion factor process ( Q x ytT ) ≤ t ≤ T is introduced in Definition 2.1.Proof. For information we note that, by an application of the relation (2.2), the price process ( H yt ) ≤ t ≤ T of the financial asset H is deduced to be H yt = h yt E (cid:2) h yT H yt ( t , T ) | F t (cid:3) = H yt ( t , T ) P ytT , (2.5)since the cash flow H yt ( t , T ) is F t -measurable and it occurs at T . At time t ∈ [ T ] , we con-vert H yt ( t , T ) to the corresponding value H xt ( t , T ) in the x -market by use of the conversionfactor Q x ytT : H xt ( t , T ) = Q x ytT H yt ( t , T ) . (2.6)Now we insert the converted cash flow H xt ( t , T ) in the standard no-arbitrage formula (2.1)(or formula (2.2), where y = x -curve) where h t = h xt is assumed. Wehave, H xsT = h xs E (cid:2) h xt H xt ( t , T ) | F s (cid:3) . (2.7)Given that H xt ( t , T ) is F t -measurable, we deduce the following by the tower property ofconditional expectation: H xsT = h xs E (cid:2) E (cid:2) h xT | F t (cid:3) H xt ( t , T ) | F s (cid:3) = h xs E (cid:2) h xt P xtT H xt ( t , T ) | F s (cid:3) , (2.8)for 0 ≤ s < t . In addition, for t ≤ s ≤ T , we have H xsT = P xsT H xt ( t , T ) . Recalling that H xt ( t , T ) = Q x ytT H yt ( t , T ) , and by choosing to write H x ysT for H xsT in order to emphasise theinteraction between the x - and the y -curves, the proof is complete. We add that the one-8o-one across-curve extension to the standard pricing formula (2.1) is recovered by setting t = T in the relation (2.4). Remark 1.
When t = T and H yT T = , using Proposition 2.1, we may define the ZCBP x ysT = h xs E (cid:2) h yT | F s (cid:3) = Q x yss P ysT = P xsT Q x ysT , (2.9) for s ∈ [ T ] , which has two representations using the definition of the conversion factor (2.3). Given Proposition 2.1, we can now present the dual role played by the curve-conversionfactor process , within the xy-formalism, which is described in the following corollary.
Corollary 2.1.
Within the xy-formalism, if the cash flow H yt ( t , T ) is directly observable in theeconomy, then the curve-conversion factor process enables valuation by acting at the level of thediscounting curve as follows:H x ysT = h xs E (cid:2) h xt P xtT Q x ytT H yt ( t , T ) | F s (cid:3) = h xs E (cid:2) h yt P ytT H yt ( t , T ) | F s (cid:3) . (2.10) However, if the curve-converted cash flow H x ytT is directly observable in the economy, then thecurve-conversion factor process enables valuation by acting at the level of the cash flow asfollows: H x ysT = h xs E (cid:2) h xt P xtT Q x ytT H yt ( t , T ) | F s (cid:3) = h xs E (cid:2) h xt H x ytT | F s (cid:3) , (2.11) where ( H x ysT ) ≤ s ≤ t is the x -market value of H yt ( t , T ) , for s ≤ t ≤ T .Proof. If H yt ( t , T ) is determined in the y -market and directly observable (i.e. quoted) withinthe economy, then according to Proposition 2.1 the value of such a payoff within the x -market, at the future terminal time T , is given by H x yT T = Q x ytT H yt ( t , T ) , (2.12)which is model-implied, since the curve-conversion factor process Q x ytT is determined bythe specific forms of the pricing kernels ( h xt ) and ( h yt ) , respectively. Therefore, since H x yT T is not directly observable in the economy due to Q x ytT , the curve-conversion factor processis subsumed into the discounting process in Eq. (2.4) for 0 ≤ s < t , by observing that h xt P xtT Q x ytT = h yt P ytT , which yields Eq. (2.10).Conversely, if H yt ( t , T ) is determined in the y -market but the converted quantity H x yT T is9irectly observable within the economy, then H yt ( t , T ) = H x yT T Q x ytT , (2.13)is model-implied, which is subsumed into the cash flow process by observing that H x ysT = P xsT Q x ytT H yt ( t , T ) for t ≤ s ≤ T from Eq. (2.4), which yields Eq. (2.11). Remark 2.
Corollary 2.1 proves to be critical in Section 3, where consistent mutli-curve sys-tems are derived for both, developed and emerging inter-bank swap markets. With regard toFRAs (the fundamental inter-bank swap market derivative), which has an IBOR process as itsunderlying, it turns out that the y-market determined IBOR process is directly observable inthe emerging market, but its curve-converted equivalent is directly observable in the developedmarket. In this instance, the dual nature of the curve-conversion factor process caters for thisapparent cross-economy market inconsistency, resulting in one consistent modelling framework.
In Appendix B, we provide the consistent set of changes of numeraire assets and associ-ated equivalent probability measures, which ensure that no arbitrage is produced when theacross-pricing formula is applied using an equivalent martingale measure.
First we consider the definition of a spot IBOR, i.e. a deposit rate that is offered at a fixedtime t ≥ t + δ > t , so that the associated tenor is given by δ >
0. Thenwe may define (or represent) the spot IBOR process via ZCB instruments by L t ( t , t + δ ) = δ (cid:129) P t t + δ − ‹ , (3.1)where t ≥ δ >
0, and where P t t + δ is the price at time t of a ZCB, with tenor δ , thatmatures at time t + δ . In the classical single-curve framework, where IBORs are consid-ered an appropriate proxy for risk-free rates and where a tradable discount bond systemis assumed, one can then proceed to define the forward IBOR process via the canonicalno-arbitrage pricing relation L t ( T i − , T i ) = h t P tT i E (cid:2) h T i − P T i − T i L T i − ( T i − , T i ) (cid:12)(cid:12) F t (cid:3) , (3.2)10or 0 ≤ t ≤ T i − , and where δ i = T i − T i − is the IBOR tenor and ( h t ) t ≥ is the pricing kernelprocess. By use of the relation (3.1) with t = T i − and δ = δ i , and the ZCB pricing relation h t P tT i = E [ h T i − P T i − T i | F t ] , one obtains the forward IBOR process L t ( T i − , T i ) = δ i (cid:18) P tT i − P tT i − (cid:19) , (3.3)for 0 ≤ t ≤ T i − . We note that the product of the pricing kernel process and the discountedforward IBOR process ( h t P tT i L t ( T i − , T i )) ≤ t ≤ T i − is an (( F t ) , P ) -martingale, which is anal-ogous to the forward IBOR process being a martingale under the Q T i -forward measure inthe classical single-curve theory.The classical relation (3.3) states that the forward IBOR value at time t can be replicatedby a linear combination of zero-coupon bonds, i.e. by one maturing at the IBOR reset date T i − and another ZCB maturing at the IBOR settlement date T i . In a market where thespread between an overnight indexed swap (OIS) rate and the corresponding IBOR is non-zero, relation (3.3) is no longer acceptable. That is, the now risky IBOR can no longer bereplicated using risk-free
ZCBs. In oder words, the IBOR market is exposed to risk factorswhich are not necessarily affecting the risk-free
ZCB market, while the risk exposure alsovaries depending on the IBOR tenor δ i = T i − T i − one is investing in. Hence, one needsto assume that holding a financial contract written on a 3-month IBOR exposes an investorto a different risk profile than when holding an instrument written on a 6-month IBOR.It follows that assuming risk-free ZCBs can replicate the same risk exposures as contractswritten on an IBOR is wrong because: (a) an IBOR may be subject to more risk sourcesthan the risk-free
ZCBs; and (b) the number of risk factors affecting an IBOR contract maydepend on the IBOR tenor.We ask the following question: If one insisted on keeping the relation (3.3), albeit subjectto modifications, how would one need to adjust— in a consistent and arbitrage-free manner —the relation between an IBOR model and the associated ZCBs in a multi-curve setup? It turnsout that the answer is an extension based on the xy-formalism introduced above. Here ishow we do it.First, we consider a collection of interest rate curves indexed by x , y =
0, 1, 2, ..., n wherewe refer to the x -curve as the discounting curve and the y -curve as the forecasting curve . Anexample for a pair of curves ( x , y ) may be the pair (
0, 1 ) where the 0-curve is the OIScurve and the 1-curve is the 1-month IBOR curve. The case where x = y is the (classical)11ingle-curve economy. Next we make the relationship (3.3) curve-dependent and write L yt ( T i − , T i ) = δ i ‚ P ytT i − P ytT i − Œ . (3.4)Thus, the y -ZCB system P ytT i has an associated y -tenored IBOR, which is subject to thesame set of risk factors, i.e. the y -tenored IBOR defines the y -ZCB system. Moreover, the y -ZCB price process satisfies the martingale relation h yt P ytT i = E [ h yT i | F t ] , which is to saythat no-arbitrage is assumed within the self-consistent y -market.Next we detail the development of consistent multi-curve interest rate systems inspiredby the xy-formalism for both, emerging and developed markets. In this section we consider the simpler case of an emerging market , in particular one whereno OIS zero-coupon yield curve exists. To be precise, the spot overnight rate is observablebut there are no tradable and liquid overnight indexed swaps, i.e. there is no OIS derivativemarket to enable the construction of a yield curve. For more information on the specificnuances and issues relating to emerging inter-bank swap markets, we refer the reader toJakarasi et al. [ ] , and references therein, who consider the problem of estimating an OISzero-coupon yield curve in South Africa. In such a market, all forecasting and discountingof cash flows is done by one liquid, risky y -tenored IBOR zero-coupon yield curve, only.To derive the multi-curve discounting system within the xy-formalism, we first considerthe pricing of standard FRAs. FRAs are the fundamental primitive securities in any interestrate market, which facilitate price discovery for forward IBORs. The FRA considered herehas reset time T i − > T i > T i − , which is also assumed to be thesettlement time, and is therefore written on the future spot IBOR L yT i − ( T i − , T i ) . The valueat time t ∈ [ T i ] of this FRA is denoted by V y ytT i , with the first character of the superscriptindicating the discount curve, and the second character denoting the forecasting curve. Fora unit nominal, the FRA’s payoff at T i is given by V y yT i T i = δ i € L yT i − ( T i − , T i ) − K y Š , (3.5)where K y is an arbitrary strike rate expressed in the y -market. We emphasise that the FRA’spayoff is actually measurable at time T i − , however the actual cash flow is only paid at time12 i . As a consequence, we may also define the in-advance FRA payoff V y yT i − T i at T i − , whichis the value V y yT i T i discounted by P yT i − T i , by V y yT i − T i = P yT i − T i δ i € L yT i − ( T i − , T i ) − K y Š . (3.6)Using the pricing formula (2.4) with x = y , along with relations (3.2), (3.3) and (3.4), theFRA price process is derived as V y ytT i = δ i h yt E ” h yT i − V y yT i − T i (cid:12)(cid:12) F t — = δ i h yt E ” h yT i − P yT i − T i € L yT i − ( T i − , T i ) − K y Š (cid:12)(cid:12) F t — = δ i P ytT i (cid:0) L yt ( T i − , T i ) − K y (cid:1) = P ytT i − − ( + δ i K y ) P ytT i . (3.7)By setting V y ytT i =
0, the fair FRA rate process is recovered and is given by K y yt ( T i − , T i ) = L yt ( T i − , T i ) , (3.8)for t ∈ [ T i − ] . The notation K y yt ( T i − , T i ) emphasises that this fair FRA strike rate applieswhen the y -curve is used for both, discounting and forecasting.Next we consider a standard IRS with unit nominal, reset times { T , T , . . . , T n − } , pay-ment times { T , T , . . . , T n } , referencing the y -tenored IBOR and arbitrary fixed swap rateunder the y -market denoted by S y . Again applying pricing relation (2.4) with x = y , to-gether with relations (3.2), (3.3) and (3.4), the IRS price process is derived as V y ytT n = n X i = δ i h yt E ” h yT i € L yT i − ( T i − , T i ) − S y Š (cid:12)(cid:12) F t — = P ytT − P ytT n − S y n X i = δ i P ytT i , (3.9)for t ≤ T . Using the same notation convention as with the FRA, the fair IRS rate process isgiven by S y yt ( T , T n ) = P ytT − P ytT n P ni = δ i P ytT i , (3.10)for t ≤ T . For a brief treatment of bootstrapping in an emerging market, we here refer to The market convention is to understand the right-hand-side of (3.5) as the rate (or quote) observed at T i − and applied at T i on one unit of currency giving the payoff value V y yT i T i of the contract at T i . Since this value ispaid at T i , we denote it V y yT i T i , and use the subscripts T i . Next we consider the more complex case of a developed market where, in general, an OISmarket exists. In such a market, cash flows are forecast using the y -tenored IBOR zero-coupon yield curve but discounted using the OIS zero-coupon yield curve. Such a productfeature is also consistent with the notion of collateralisation. We consider the same FRA asin the emerging market case, however we now assume that discounting occurs under the x -curve (or the OIS curve, to be more specific). We now have to make use of relation (2.4)in order to define the FRA’s payoff. Proposition 3.1.
The developed market FRA with reset time T i − , expiry time T i and unitnominal has a terminal payoff, within the x -market, given byV x yT i T i = Q x yT i − T i δ i € L yT i − ( T i − , T i ) − K y Š = Q x yT i − T i V y yT i T i , (3.11) where, as before, δ i = T i − T i − and K y is the strike rate within the y-market. The in-advanceFRA payoff is then given by V x yT i − T i = P xT i − T i Q x yT i − T i V y yT i T i , (3.12) which is the discounted value of the terminal payoff within the x -market.Proof. A direct application of relation (2.4) leads to the result in Proposition 3.1. Like the emerging market
FRA, notice that the developed market
FRA’s payoff is also measurable at T i − with the actual cash flow occurring at T i .Before we consider the derivation of the value of the developed market FRA, the followinglemmas will prove to be useful in this regard.
Lemma 3.1.
The converted y-tenored forward IBOR processL x yt ( T i − , T i ) = Q x ytT i L yt ( T i − , T i ) , (3.13) for t ∈ [ T i − ] , satisfies the martingale relationL x ys ( T i − , T i ) = h xs P xsT i E ” h xT i L x yT i − ( T i − , T i ) (cid:12)(cid:12) F s — , (3.14) for ≤ s ≤ t ≤ T i − . roof. This statement follows from Eqs (2.3), (3.13) and (3.2).
Lemma 3.2.
The fair forward price K x of a forward contract initiated at time t to exchangea cash flow K y , determined in the y-market, for a cash flow of K x , in the x -market, with K y being converted at Q x yT i − T i but the final payoff occurring at expiry T i > T i − ≥ t is given byK x = h yt P ytT i h xt P xtT i K y = Q x ytT i K y . (3.15) Proof.
The value of such a forward contract is given by V x ytT i = h xt E ” h xT i € K y Q x yT i − T i − K x Š (cid:12)(cid:12) F t — = K y h yt h xt P ytT i − P xtT i K x , (3.16)which follows from Eq. (2.3) and the tower property of conditional expectations, whilesetting V x ytT i = K x yields the required result.We now have the necessary results to derive the value of the developed market FRA,which is presented in the following theorem.
Theorem 3.1.
The value of the developed market FRA with reset time T i − , expiry time T i andunit nominal, within the x -market, is given byV x ytT i = δ i P xtT i (cid:0) L x yt ( T i − , T i ) − K x (cid:1) , (3.17) for t ∈ [ T i − ] , where δ i = T i − T i − and K x is the strike rate within the x -market.Proof. Using Proposition 3.1, the value of the developed market
FRA, for t ∈ [ T i − ] , isgiven by V x ytT i = h xt E ” h xT i − P xT i − T i Q x yT i − T i δ i € L yT i − ( T i − , T i ) − K y Š (cid:12)(cid:12) F t — = δ i P xtT i L x yt ( T i − , T i ) − δ i P xtT i Q x ytT i K y , (3.18)with the first term following from Lemma 3.1 and the second term from Eq. (2.3). Eq.(3.17) follows from applying the result of Lemma 3.2 to the second term and factorisingaccordingly. 15he value of this FRA is commensurate with the value of a multi-curve (or basis) FRA ina developed market , i.e. the price dynamics are consistent with the standard FRA contracttraded in developed markets. The form of the developed market FRA’s value within thexy-framework leads to the following definition for the multi-curve forward IBOR process. Definition 3.1.
The multi-curve market-implied y-tenored forward IBOR process is given byL x yt ( T i − , T i ) = Q x ytT i L yt ( T i − , T i ) = P x ytT i δ i P xtT i ‚ P x ytT i − P x ytT i − Œ , (3.19) for t ∈ [ T i − ] , where ( P x ytT i ) is defined in Remark 1. Moreover, we may also derive the fair developed market
FRA rate given the value of theFRA provided by Theorem 3.1.
Corollary 3.1.
The fair FRA rate process K x yt ( T i − , T i ) at time t of a developed market FRAwritten on the market-implied y-tenored forward IBOR (3.13), with reset time T i − and settle-ment time T i , is given by K x yt ( T i − , T i ) = L x yt ( T i − , T i ) , (3.20) for t ∈ [ T i − ] .Proof. Setting the value of the developed market FRA, given by Eq. (3.17), equal to zero,we find that K x = L x yt ( T i − , T i ) at time t . Then for any time t ∈ [ T i − ] , the result for thefair FRA rate process, K x yt ( T i − , T i ) = L x yt ( T i − , T i ) , follows accordingly. Remark 3.
Relation (3.20) is the direct multi-curve analogy to the single-curve relation (3.8).In fact, for x = y one recovers the single-curve expressions (3.7) and (3.8). Remark 4.
Using Definition 3.1, one may re-state the value of the developed market FRA asV x ytT i = P x ytT i − − ( + δ i K y ) P x ytT i , (3.21) for t ∈ [ T i − ] , which is the direct multi-curve analogy to the emerging market FRA value(3.7) with the y-ZCBs replaced by the x y-ZCBs. Now that we have these results, it is also important to consider the relationship between L x yt ( T i − , T i ) and L xt ( T i − , T i ) . In particular, one would want L x yt ( T i − , T i ) ≥ L xt ( T i − , T i ) due to the greater degree of risk associated with the multi-curve y -tenored forward IBORprocess versus the corresponding x -tenored process. The following corollary reveals the16onditions under which this feature is achieved, by making use of the associated forwardcapitalisation factor (FCF) processes. Corollary 3.2.
The multi-curve market-implied y-tenored FCF process v x yt ( T i − , T i ) , observedat time t ≤ T i − and applying over the period [ T i − , T i ] , defined byv x yt ( T i − , T i ) : = + δ i L x yt ( T i − , T i ) , (3.22) is greater than or equal to the corresponding x -tenored FCF processv xt ( T i − , T i ) : = + δ i L xt ( T i − , T i ) , (3.23) if interest rates are non-negative and h yt ≤ h xt for all t ∈ [ T i − ] where T i − ≤ T i .Proof. Using Eq. (3.22) and Definition 3.1, we can show that v x yt ( T i − , T i ) = + δ i Q x ytT i L yt ( T i − , T i )= + Q x ytT i (cid:0) v yt ( T i − , T i ) − (cid:1) = − Q x ytT i + v x yt ( T i − , T i ) ,where v yt ( T i − , T i ) is the y -tenored FCF and v x yt ( T i − , T i ) : = Q x ytT i v yt ( T i − , T i ) is the y -tenored FCF represented equivalently in the x -market. Then, using Definition 2.1 and Eq.(2.9), we can show that v x yt ( T i − , T i ) = Q x ytT i P ytT i − P ytT i = Q x ytT i P x ytT i − P x ytT i = P xtT i − P xtT i Q x ytT i − = v xt ( T i − , T i ) Q x ytT i − .Now in order to have v xt ( T i − , T i ) ≤ v x yt ( T i − , T i ) , we must have that v xt ( T i − , T i ) ≤ − Q x ytT i + v xt ( T i − , T i ) Q x ytT i − v xt ( T i − , T i ) € − Q x ytT i − Š ≤ − Q x ytT i − Q x ytT i − ≤ − Q x ytT i ,where the last inequality holds if interest rates are non-negative, i.e. v xt ( T i − , T i ) ≥ Q x ytT i − ≥ Q x ytT i if interest rates are non-negative and h yt ≤ h xt for all t ∈ [ T i − ] where T i − ≤ T i . This may be easily evidenced by setting t = T i − and allowing T i to vary,while also using the linear and monotonic properties of conditional expectations.17his corollary proves that the xy-approach, applied to a developed market , yields a y -market interest rate system which is dominated by the x -market system, i.e. P ytT ≤ P xtT for 0 ≤ t ≤ T . Furthermore, this y -market system provides a forward IBOR process, L yt ( T i − , T i ) , and enables the construction of a conversion factor process Q x ytT i , which facili-tates the definition of the developed market y -tenored forward IBOR process L x yt ( T i − , T i ) .Therefore, while the y -market system is still fictitious, given that it cannot be directly ob-served, we still consider it to be a model-consistent system given our curve-conversionframework that is inspired by currency modelling. Remark 5.
Using the FCF, one may also express the terminal payoff of the developed marketFRA by V x yT i T i = Q x yT i − T i € v yT i − ( T i − , T i ) − v yK Š , (3.24) where v yK : = ( + δ i K y ) . Then, applying the same results as before, the value of the FRA, fort ∈ [ T i − ] , is V x ytT i = P xtT i (cid:0) v x yt ( T i − , T i ) − v xK (cid:1) , (3.25) where v xK = Q x ytT i v yK . If we define the multi-curve y-tenored forward IBOR process byL x yt ( T i − , T i ) : = δ i (cid:0) v x yt ( T i − , T i ) − (cid:1) , (3.26) and the multi-curve x -market equivalent FRA strike rate byK x : = δ i (cid:0) v xK − (cid:1) , (3.27) we then recover the developed market FRA price process: V x ytT i = P xtT i δ i (cid:0) L x yt ( T i − , T i ) − K x (cid:1) .We note that in this model v x yt ( T i − , T i ) = v xt ( T i − , T i ) Q x ytT i − so that + δ i L x yt ( T i − , T i ) + δ i L xt ( T i − , T i ) = h yt P ytT i − h xt P xtT i − , (3.28) and L x yt ( T i − , T i ) ≥ L xt ( T i − , T i ) if interest rates are non-negative and h xt ≤ h yt for all t ∈ [ T i − ] and for T i − ≤ T i . This is the approach adopted by Nguyen & Seifried [ ] and itshall be revisited in Section 5. Two comments on their multi-curve model, given the context ofthe xy-approach, follow:(i) The quantities L x yt ( T i − , T i ) and K x which determine the FRA’s floating and fixed cashflows are derived from the curve-converted quantities v x yt ( T i − , T i ) and v xK respectively. his is in contrast with L x yt ( T i − , T i ) and K x , the directly comparable curve-convertedquantities used in the xy-framework. Therefore, these derived quantities are no longerconsistent with a currency modelling analogy, with each differing from the correctly con-verted quantities by an additive factor of ( Q x ytT i − ) /δ i .(ii) Observation (i) is further supported by equation (3.28) which shows that the conver-sion factor process effectively models the spread between the multi-curve y-tenored FCFand the corresponding x -tenored FCF, as opposed to the classical forward exchange rate.Moreover, the derived y-market system has almost no relation to the developed markety-tenored interest rate system, that one seeks to model, since the model derived y-marketsystem dominates the x-market system, i.e. P xtT ≤ P ytT for ≤ t ≤ T .
Remark 6.
The mathematical quantity that directly models the y-tenored forward IBOR pro-cess is L x y · ( · , · ) and not L y · ( · , · ) . This is a consequence of industry standards in developed mar-kets, that the product of the x -pricing kernel and the x -curve discounted y-tenored forwardIBOR process is a martingale under the P -measure. In the x y-approach, this implies thath xs P xsT i L ys ( T i − , T i ) = E ” h xt P xtT i L yt ( T i − , T i ) | F s — , (3.29) for ≤ s ≤ t ≤ T i − . It is not possible to achieve this relationship within the xy-framework,given our representation of the y-tenored forward IBOR process (3.4). However this relation-ship is achieved if we replace L y · ( · , · ) with L x y · ( · , · ) . Our market-implied y-tenored forwardIBOR process, L x y · ( · , · ) , reveals the convolution of a conversion factor (which facilitates the mar-ket’s martingale assumption (3.29)) and the model y-tenored forward IBOR process, L y · ( · , · ) .This result questions the utility of the y-ZCB system in the developed market context. The y-ZCB system is a model construct, derived from the y-tenored model-consistent or model-impliedforward IBOR process, L y · ( · , · ) , which unravels the market’s martingale adjustment from the ob-served y-tenored market-implied IBOR process, L x y · ( · , · ) , via the conversion factor Q y x ·· . Remark 7.
The xy-framework advocates the following price process for a multi-curve FRAV x ytT i = δ i P x ytT i (cid:0) L yt ( T i − , T i ) − K y (cid:1) , (3.30) for t ∈ [ T i ] . We note that the conversion factor (or martingale adjustment) has been appliedto the discounting x-ZCB system and not to the model for the y-tenored forward IBOR process. owever, we note that the terminal FRA payoff would now beV x yT i T i = δ i h yT i h xT i € L yT i − ( T i − , T i ) − K y Š . This allows us to disentangle the y-ZCB system from the x-ZCB system, which enables us tomodel the y-curve discounting in a consistent, robust and rigorous fashion. From an economicsperspective, if one compares the return generated from an xy-FRA to a yy-FRA, one can showthat V y ytT i V y y T i > V x ytT i V x y T i = V x ytT i V y y T i , (3.31) as required, since discounting at the x -curve essentially represents a collateralised FRA whichshould therefore return the holder less than an equivalent investment in a non-collateralisedFRA, represented by the y-curve discounting. Next we consider the developed market
IRS, i.e. one which forecasts cash flows underthe y -curve but discounts under the x -curve, unlike the emerging market IRS.
Theorem 3.2.
The value of a developed market IRS, within the x -market, with reset times { T , T , . . . , T n − } , payment times { T , T , . . . , T n } and unit nominal, referencing the y-tenoredIBOR is given by V x ytT n = n X i = δ i P xtT i (cid:0) L x yt ( T i − , T i ) − S x (cid:1) , (3.32) for t ≤ T , where δ i = T i − T i − and where S x is the fixed swap rate within the x -market.Proof. Starting with the emerging market version of the IRS with fixed swap rate S y withinthe y -market and applying pricing relation (2.4), analagous to Proposition 3.1, the developedmarket IRS price process is given by V x ytT n = n X i = δ i h xt E ” h xT i − P xT i − T i Q x yT i − T i € L yT i − ( T i − , T i ) − S y Š (cid:12)(cid:12) F t — , (3.33)which, upon application of Lemma 3.1 and Equation (2.3), simplifies to V x ytT n = n X i = δ i P xtT i € L x yt ( T i − , T i ) − Q x ytT i S y Š , (3.34)for t ≤ T . The result follows by observing that the fixed IRS rate may be expressed in the x -market by S x = S y ( P ni = δ i P xtT i Q x ytT i ) / ( P ni = δ i P xtT i ) . This may be justified in an analogous20ashion to the fixed FRA rate, but this time making use of a fixed-for-fixed swap contract asopposed to a forward contract, as in Lemma 3.2. Remark 8.
Using Definition 3.19, one may re-state the value of the developed market IRS asV x ytT n = ” P x ytT − P x ytT n — − S y n X i = δ i P x ytT i , (3.35) for t ≤ T , which is the direct multi-curve analogy to the emerging market IRS value (3.9) withthe y-ZCBs replaced by the x y-ZCBs. Corollary 3.3.
The fair fixed swap rate process S x yt ( T , T n ) of a developed market IRS writtenon the market-implied y-tenored forward IBOR (3.13), with reset times { T , T , . . . , T n − } ,payment times { T , T , . . . , T n } and unit nominal, is given byS x yt ( T , T n ) = P x ytT − P x ytT n P ni = δ i P xtT i , (3.36) for t ≤ T .Proof. Setting the value of the developed market IRS equal to zero, given by Eq. (3.35), itfollows that the y -market fair fixed IRS rate is S y = € P x ytT − P x ytT n Š / P ni = δ i P x ytT i at time t .Using the proof of Theorem 3.2 and Remark 1, the x -market fair fixed IRS rate (convertingthe y -market rate) is given by S x = S y ( P ni = δ i P xtT i Q x ytT i ) / ( P ni = δ i P xtT i ) at time t . Then forany time t ≤ T , the result for the developed market fair IRS rate follows accordingly bysetting S x yt ( T , T n ) = S x .For a brief treatment of bootstrapping in a developed market, we here refer to AppendixC.2. Now that we have a good understanding of how the xy-formalism enables the modelling ofmulti-curve interest rate systems in developed markets , we may consider resolving the sameproblem for the case of an emerging market . Our first hurdle in moving from a developed toan emerging market setting is the non-existence of the OIS curve.Recall that we have assumed the existence of a collection of interest rate curves indexedby x , y =
0, 1, 2, . . . , n where we refer to the x -curve as the discounting curve and the y -curve as the forecasting curve . In a common developed market , n = ( h yt ) which are in turn calibratedusing liquid linear and non-linear interest rate market instruments. In a common emergingmarket , only one IBOR tenor is usually tradable and liquid, therefore it is not possible tocalibrate the entire set of pricing kernel processes ( h yt ) which span the common developedinterest rate market. This leads us to the following remark. Remark 9.
In the common emerging market, only one IBOR tenor, y ∗ , is tradable and liquidthereby enabling the specification and calibration of a well-defined pricing kernel process ( h y ∗ t ) .Pricing kernel processes for all other IBOR tenors ( b h yt ) are to be estimated statistically (orotherwise) as a suitable functional form of ( h y ∗ t ) , i.e. b h yt = f € h y ∗ t Š . (3.37) where f : R + → R + is measurable and adapted, such that the corresponding estimated y-ZCB(and y-curve) systems, ( b P ytT ) , may be constructed via b P ytT = b h yt E (cid:2)b h yT | F t (cid:3) = f (cid:0) h y ∗ t (cid:1) E ” f € h y ∗ T Š | F t — , (3.38) for ≤ t ≤ T .
In Remark 9, if the function f ( · ) is linear, then the estimated y -ZCB is given by b P ytT = f (cid:0) h y ∗ t (cid:1) f € P y ∗ tT Š , (3.39)which implies that it is possible to directly replicate the estimated y -ZCB through either astatic or dynamic replication strategy using the y ∗ -ZCB. However, this may not be possible,in general, if the function f ( · ) is convex (concave), as the estimated y -ZCB will be governedby the following inequality b P ytT ≥ ( ≤ ) f (cid:0) h y ∗ t (cid:1) f € P y ∗ tT Š , (3.40)which follows by the application of Jensen’s inequality. The xy-formalism may now be ap-plied in the emerging market setting, assuming the existence of a collection of interest ratecurves, indexed by x , y =
0, 1, . . . , y ∗ , . . . , n , that are modelled by the calibrated pricingkernel process ( h y ∗ t ) and the set of estimated pricing kernel processes ( b h yt ; y = y ∗ ) . First,we consider the developed market FRA within the emerging market context, i.e. one where22he payoff is forecasted by the y -curve and then discounted by the x -curve. The terminaland in-advance FRA payoffs remain unchanged and are identical to Eqs (3.11) and (3.12),respectively , with the FRA price process also assuming the familiar form V x ytT i = δ i P x ytT i (cid:0) L yt ( T i − , T i ) − K y (cid:1) = P x ytT i − − ( + δ i K y ) P x ytT i , (3.41)for t ∈ [ T i − ] , while L yt ( T i − , T i ) continues to be the correct forward IBOR process. Noticethat the derivation of equation (3.41) follows by a direct application of Corollary 2.1. Definition 3.2.
The multi-curve emerging market y-tenored forward IBOR process is given byL yt ( T i − , T i ) = δ i ‚ P ytT i − P ytT i − Œ , (3.42) for t ∈ [ T i − ] , unlike the developed market which required the definition of the market-implied y-tenored forward IBOR process L x yt ( T i − , T i ) : = Q x ytT i L yt ( T i − , T i ) for t ∈ [ T i − ] . This is due to the fact that there is currently no market standard for pricing an emergingmarket
FRA that is forecasted and discounted under different curves, with the only observ-able market quantity being the spot IBOR process L yt ( t , t + δ ) for t ≥
0. It is also possi-ble, as in the case of developed markets , to define a fair FRA rate process, K x yt ( T i − , T i ) = L yt ( T i − , T i ) , however one would not be able to observe this quantity in the market (sincethese FRAs are not traded, in general), therefore this would be a model-implied quantity .Similarly, we may consider the standard developed market IRS in the context of an emerg-ing market . The value of the IRS at some time t ≤ T , making use of the same relations asbefore, is again given by V x ytT n = n X i = δ i P x ytT i (cid:0) L yt ( T i − , T i ) − S y (cid:1) = P x ytT − P x ytT n − S y n X i = δ i P x ytT i , (3.43)where L yt ( T i − , T i ) , for t ∈ [ T i − ] , continues to be the correct forward IBOR process,analagous to the FRA result. As with the FRA, the fair IRS rate process is model-implied Assuming that one insists on maintaining measurability of the payoff at the IBOR reset time T i − . If the y -tenored IBOR corresponds to the most liquid and tradable tenor, i.e. y = y ∗ , then one will alsohave access to the set of forward IBOR processes L yt ( T i − , T i ) for 0 ≤ t ≤ T i − , from the standard and liquidlytradable set of single-curve emerging market FRAs, and K x yt ( T i − , T i ) = K y yt ( T i − , T i ) = L yt ( T i − , T i ) . y -tenored IBOR process is the tradable tenor and t = T =
0) and given by S x yt ( T , T n ) = ( P x ytT − P x ytT n ) / ( P ni = δ i P x ytT i ) for t ≤ T .In a multi-curve emerging market interest rate system, within the xy-framework, the ini-tial (estimated) y -ZCB systems may be constructed in a completely analogous fashion to thesingle-curve emerging market relations, see Appendix C, since K x y ( T i − , T i ) = L y ( T i − , T i ) and S x y ( T n ) = ( − P y T n ) / ( P ni = δ i P y T i ) . That is, all initial model-implied quantities areonly dependent on the y -curve or y -ZCB system.If we consider a FRA and an IRS within this context with payoffs forecasted by the y ∗ -curve and discounted by one of the other curves, denoted by the x -curve, then the pricingformulae are given by V x y ∗ tT i = P x y ∗ tT i − − ( + δ i K y ∗ ) P x y ∗ tT i , (3.44)and V x y ∗ tT n = ” P x y ∗ tT − P x y ∗ tT n — − S y n X i = δ i P x y ∗ tT i , (3.45)from Eqs (3.41) and (3.43), respectively. At this juncture, it is important to note that the x y ∗ -ZCB, ( P x y ∗ tT ) , plays the same role as the y ∗ -ZCB, ( P y ∗ tT ) , does in the single-curve emergingmarket setting in Section 3.1. This leads us to the following definition for the x y -ZCBsystem, in general. Definition 3.3.
In the multi-curve interest rate system derived within the xy-framework, thex y-ZCB system, ( P x ytT ) , defined byP x ytT = h xt E (cid:20) h xT (cid:18) h yT h xT (cid:19) ( ) (cid:12)(cid:12) F t (cid:21) = P xtT Q x ytT = Q x yt t P ytT , may be considered to be a quanto-bond assuming(i) the x -curve with varying notional defined by the forward conversion factor Q x ytT ; or(ii) the y-curve with varying notional defined by the spot conversion factor Q x yt t . Remark 10.
Within the developed market context—where the nominal OIS curve is consideredto be the distinct, single-curve tradable system, which we shall denote here as the x ∗ -curve—one may dynamically replicate y-ZCBs and x ∗ y-ZCBs, where y = x ∗ , via the following set ofx ∗ -curve quanto-bonds P ytT = Q x ∗ ytT Q x ∗ yt t P x ∗ tT and P x ∗ ytT = Q x ∗ ytT P x ∗ tT ,24 hereas, within the emerging market context where one nominal IBOR swap curve is consideredto be the distinct, single-curve tradable system, which we have denoted as the y ∗ -curve, one maydynamically replicate x -ZCBs and x y ∗ -ZCBs, where x = y ∗ , via the following set of y ∗ -curvequanto-bonds P xtT = Q x y ∗ t t Q x y ∗ tT P y ∗ tT and P x y ∗ tT = Q x y ∗ t t P y ∗ tT . In this section we develop Heath-Jarrow-Morton (HJM) multi-curve interest rate systemsbased on the xy-formalism introduced in this paper. The xy-HJM multi-curve system will bederived using results from Section 3.We consider the filtered probability space ( Ω , F , ( F t ) , P ) where ( F t ) ≤ t is the filtrationgenerated by two sets of independent multi-dimensional P -Brownian motions ( W t ) t ≥ and ( Z t ) t ≥ , respectively. Being synonymous with the xy-formalism, we consider an economywith two distinct markets, x and y , where x may be interpreted as a proxy default-free OIS-based market and y as a risky IBOR-based market. Furthermore, we assume that the x - and y -markets are driven by the multi-dimensional P -Brownian motions ( W xt ) t ≥ = ( W t ) t ≥ and ( W yt ) t ≥ = ( W t , Z t ) t ≥ respectively, where ( W t ) t ≥ is n -dimensional and ( Z t ) t ≥ is m -dimensional. This allows us to define the pricing kernel process associated with each market. Definition 4.1.
The ( F t ) -adapted x - and y-market pricing kernel processes ( h xt ) ≤ t and ( h yt ) ≤ t satisfy, respectively, d h xt h xt = − r xt d t − λ xt d W xt , d h yt h yt = − r yt d t − λ yt d W yt , (4.1) where ( r xt ) t ≥ and ( r yt ) t ≥ are the short rates of interest; and ( λ xt ) t ≥ and ( λ yt ) t ≥ = ( λ xt , λ zt ) t ≥ are the n- and ( n + m ) -dimensional market price of risk processes associated with the x- andy-markets, respectively. Next, let ( X tT ) ≤ t ≤ T and ( Y tT ) ≤ t ≤ T be (well-defined) processes, respectively satisfyingd X tT X tT = (cid:129) − A xtT + (cid:12)(cid:12) Σ xtT (cid:12)(cid:12) ‹ d t − (cid:0) Σ xtT + λ xt (cid:1) d W xt ,d Y tT Y tT = (cid:129) − A ytT + (cid:12)(cid:12) Σ ytT (cid:12)(cid:12) ‹ d t − (cid:0) Σ ytT + λ yt (cid:1) d W yt , (4.2)for 0 ≤ t ≤ T , where A ( · ) tT = R Tt a ( · ) tu d u is 1-dimensional, | · | denotes the Euclidean norm,25nd Σ ( · ) tT = R Tt σ ( · ) tu d u with Σ xtT and Σ ytT being n - and ( n + m ) -dimensional, respectively. Theprocesses ( σ xtT ) , ( σ ytT ) = ( σ wtT , σ ztT ) and ( a ( · ) tT ) are generic adapted processes satisfying theimplicit integrability conditions, with σ wtT and σ ztT being n - and m -dimensional, respectively.We may then define the respective ZCB prices as follows: Definition 4.2.
Setting P xtT : = X tT / h xt and P ytT : = Y tT / h yt , the ( F t ) -adapted x - and y-marketZCB-systems satisfy, respectively, the dynamical equations d P xtT P xtT = (cid:129) r xt − A xtT + (cid:12)(cid:12) Σ xtT (cid:12)(cid:12) − λ xt Σ xtT ‹ d t − Σ xtT d W xt ,d P ytT P ytT = (cid:129) r yt − A ytT + (cid:12)(cid:12) Σ ytT (cid:12)(cid:12) − λ yt Σ ytT ‹ d t − Σ ytT d W yt , (4.3) following the application of Ito’s Lemma. Invoking the classical HJM drift condition, A ( · ) tT = | Σ ( · ) tT | , results in ( h ( · ) t P ( · ) tT ) ≤ t ≤ T being a P -(local) martingale which is a requirement for thexy-HJM framework. Proposition 4.1.
Assuming that the x - and y-market ZCB-systems are differentiable in T ,the instantaneous forward rate processes ( f xtT ) ≤ t ≤ T and ( f ytT ) ≤ t ≤ T , respectively defined byf xtT = − ∂ T ln (cid:0) P xtT (cid:1) and f ytT = − ∂ T ln (cid:0) P ytT (cid:1) , satisfy d f xtT = (cid:0) a xtT + λ xt σ xtT (cid:1) d t + σ xtT d W xt ,d f ytT = (cid:0) a ytT + λ yt σ ytT (cid:1) d t + σ ytT d W yt , (4.4) which are consistent with the classical HJM instantaneous forward rate model.Proof. By direct application of Ito’s Lemma, the logarithm of the ZCB price process isln € P ( · ) tT Š = ln € P ( · ) T Š + Z t € r s − A ( · ) sT − λ ( · ) s Σ ( · ) sT Š d s − Z t Σ ( · ) sT d W ( · ) s , (4.5)and therefore taking the negative and differentiating with respect to T gives − ∂∂ T ln € P ( · ) tT Š = Z t € a ( · ) sT + λ ( · ) s σ ( · ) sT Š d s + Z t σ ( · ) sT d W ( · ) s , (4.6)which yields the required instantaneous forward rate result.Grbac & Runggaldier [ ] provide a thorough account of the approaches that have beenadopted in modeling a developed market multi-curve interest rate system with the HJM26ramework. Here we reprise the key results, given the economy that has already been in-troduced in this section, in order to contextualise the xy-HJM framework within the ex-isting body of literature. Grbac & Runggaldier [ ] note that all approaches that havebeen adopted model the x -market ZCB-system ( P xtT ) with the classical HJM model whilethe multi-curve market-implied y -tenored forward IBOR process, which we denote here by L x yt ( T i − , T i ) , is modeled in one of three ways:(i) L x yt ( T i − , T i ) is specified in an ad hoc fashion (usually) inspired by the LIBOR MarketModel (LMM) such that this approach is referred to as a hybrid HJM-LMM ;(ii) L x yt ( T i − , T i ) : = ( v yt ( T i − , T i ) − ) /δ i where, as before, v yt ( T i − , T i ) : = P ytT i − / P ytT i defines the FCF such that under certain parameter restrictions (see Proposition 4.2below) ( h xt P xtT i v yt ( T i − , T i )) ≤ t ≤ T i − is a P -(local) martingale; and(iii) L x yt ( T i − , T i ) : = E [ h xT i L yT i − ( T i − , T i ) (cid:12)(cid:12) F t ] / ( h xt P xtT i ) assuming the classical HJM driftcondition for the y -market ZCB system.In each approach ( h xt P xtT i L x yt ( T i − , T i )) ≤ t ≤ T i − is a P -(local) martingale, as required. Model(i) is inconsistent with our approach, since our focus is on modeling ZCB-systems directlyand implying simple spot and forward rate models therefrom, therefore we merely makenote of (i) for completeness. Models (ii) and (iii) are comparable to our approach, thereforewe expand upon them below. Proposition 4.2.
If the following parameter restrictions hold:A ytT i − A ytT i − = − (cid:12)(cid:12)(cid:12) Σ ytT i − Σ ytT i − (cid:12)(cid:12)(cid:12) + Σ xtT i € Σ wtT i − Σ wtT i − Š − λ zt € Σ ztT i − Σ ztT i − Š , (4.7) then the process ( h xt P xtT i v yt ( T i − , T i )) ≤ t ≤ T i − a P -(local) martingale, thereby enabling the useof Model (ii).Proof. Applying Ito’s Lemma to h xt P xtT i v yt ( T i − , T i ) , using Eqs (4.1) and (4.3), we haved € h xt P xtT i v yt ( T i − , T i ) Š h xt P xtT i v yt ( T i − , T i )= • A ytT i − A ytT i − + (cid:12)(cid:12)(cid:12) Σ ytT i − Σ ytT i − (cid:12)(cid:12)(cid:12) − Σ xtT i € Σ wtT i − Σ wtT i − Š + λ zt € Σ ztT i − Σ ztT i − Š˜ d t + € Σ wtT i − Σ wtT i − − Σ xtT i − λ xt Š d W xt + € Σ ztT i − Σ ztT i − Š d Z t , (4.8)from which it follows that the required martingale condition is achieved only if Eq. (4.7) isenforced. 27 emark 11. In Grbac & Runggaldier [ ] , both the x - and y-markets have the same sourcesof risk, i.e. are driven by the same set of Brownian motions, which resolves the parameterrestrictions to A ytT i − A ytT i − = − € Σ ytT i − Σ ytT i − Š + Σ xtT i € Σ ytT i − Σ ytT i − Š , (4.9) for ≤ t ≤ T i − . Model (iii) requires one to compute a conditional expectation, E [ h xT i L yT i − ( T i − , T i ) (cid:12)(cid:12) F t ] ,which is possible given our model choices, i.e. Eqs (4.1) and (4.3), along with the classicalHJM drift condition applied to the y -market ZCB-system. Note that in Grbac & Runggaldier [ ] , this model is justified by analogies to credit and foreign exchange modeling. Theirmodel setup leads to two different parameter restrictions, depending on which analogy isassumed. Our pricing kernel-based HJM setup leads to a unique parameter restriction (theclassical HJM drift condition for the y -market ZCB-system) which subsumes both analo-gies, since our setup does not require us to specify an exchange rate process in an ad hocexogenous manner. This may be seen in the following proposition, recalling the results fromProposition 2.1. Proposition 4.3.
The xy-HJM framework’s forward curve-conversion factor process ( Q x ytT ) ≤ t ≤ T satisfies d Q x ytT Q x ytT = ( Σ xtT − Σ wtT )( Σ xtT d t + λ xt d t + d W xt ) − ( Σ ztT + λ zt ) d Z t , (4.10) while the spot curve-conversion factor process ( Q x yt t ) t ≥ satisfies d Q x yt t Q x yt t = (cid:0) r xt − r yt (cid:1) d t + λ xt d W xt − λ yt d W yt , (4.11) where λ xt d W xt − λ yt d W yt = − λ zt d Z t .Proof. Using the definition of the conversion factor, Eq. (2.3), along with Definition 4.2, ob-serve that Q x ytT = Y tT / X tT while Q x yt t = h yt / h xt . The result then follows by a straightforwardapplication of Ito’s Lemma using Eqs. (4.2) and (4.1). Remark 12.
By the Girsanov Theorem, it is straightforward to show that there is a multi-dimensional Q x -Brownian motion ( W Q x t ) that satisfies d W Q x t = λ xt d t + d W xt upon changingmeasure from P to the x -market risk-neutral measure Q x . Moreover there is also a multi-dimensional Q Tx -Brownian motion ( W Q Tx t ) that satisfies d W Q Tx t = Σ xtT d t + d W Q x t upon changingmeasure from Q x to the x -market T -forward measure Q Tx .
28n this paper we have proposed the xy-formalism for multi-curve interest rate modeling,and in turn advocated model structures for both multi-curve emerging and developed mar-ket forward IBOR processes (see Definitions 3.2 and 3.1, respectively). We document thesemulti-curve forward IBOR processes within the xy-HJM context in the next definition.
Definition 4.3.
Within the xy-HJM framework, the multi-curve emerging market y-tenoredforward IBOR process is given byL yt ( T i − , T i ) = δ i (cid:0) v yt ( T i − , T i ) − (cid:1) , (4.12) with the FCF process, v yt ( T i − , T i ) , satisfying d v yt ( T i − , T i ) v yt ( T i − , T i ) = € Σ ytT i − Σ ytT i − Š € Σ ytT i d t + λ yt d t + d W yt Š , (4.13) for ≤ t ≤ T i − , such that the process ( h yt P ytT i v yt ( T i − , T i )) ≤ t ≤ T i − is a P -(local) martingale.The multi-curve developed market y-tenored forward IBOR process is given byL x yt ( T i − , T i ) = Q x ytT i L yt ( T i − , T i ) = δ i € v x yt ( T i − , T i ) − Q x ytT i Š , (4.14) with the converted FCF process, v x yt ( T i − , T i ) : = Q x ytT i v yt ( T i − , T i ) , satisfying d v x yt ( T i − , T i ) v x yt ( T i − , T i ) = € Σ xtT i − Σ wtT i − Š € Σ xtT i d t + λ xt d t + d W xt Š − ( Σ ztT i − + λ zt ) d Z t , (4.15) for ≤ t ≤ T i − , such that the process ( h xt P xtT i v x yt ( T i − , T i )) ≤ t ≤ T i − is a P -(local) martingale. We note that ( h xt P xtT i L x yt ( T i − , T i )) ≤ t ≤ T i − is also a P -(local) martingale, however themulti-curve developed market y -tenored forward IBOR process does not have an elegantdifferential representation as it is essentially the difference between two stochastic pro-cesses, these being the converted FCF process and the curve-conversion factor process. Remark 13.
The only parameter restrictions required by the xy-HJM framework are the clas-sical HJM drift conditions for both the x - and y-market ZCB systems. Therefore model (iii), aspresented in Grbac & Runggaldier [ ] , is also a viable model for the developed market forwardIBOR process, albeit an unnatural one given the incompatibility between the x -market pricingkernel (h xt ) and the y-market forward IBOR process (L yt ( T i − , T i ) ). Another viable modelwithin the xy-HJM framework is that of Nguyen & Seifried [ ] , given by equation (3.26),however recall the observations in Remark 5 regarding this model.
29n the next section, rational multi-curve models are introduced. Such models, and inparticular those produced in Section 5.2, provide a rich class of flexible and tractable spec-ifications for xy-HJM multi-curve models and associated spread dynamics.
As reported in Grbac & Runggaldier [ ] , multi-curve rational interest rate models based onthe pricing kernel approach have appeared in Crépey et al. [ ] and in Nguyen & Seifried [ ] . The multi-curve approach proposed by Crépey et al. [ ] assumes a discount bondsystem associated with an overnight-indexed swap (OIS) market and introduces a (forward)LIBOR process that has a built-in spread when compared to the OIS rate. The OIS-baseddiscount bond price system, which in our setup would correspond to the x -curve ZCB pricesystem, is generated by pricing kernel models driven by stochastic factors. The (forward)LIBOR process is derived by pricing a forward rate agreement (FRA) written on the LIBOR.The factor-based model of the multi-curve (forward) LIBOR process is then deduced fromthe no-arbitrage relation the FRA price process is required to satisfy. The LIBOR model turnsout to be a rational function(al) of stochastic drivers that is given in units of the OIS pricingkernel proxy. Thus, whenever the LIBOR dynamics depend on an idiosyncratic driving factor(not affecting the OIS pricing kernel proxy), an OIS-LIBOR spread is generated that dependson a spread-idiosyncratic stochastic factor. The source of the spread can be readily readoff from the expression of the LIBOR model owing to the transparency of the multi-curveapproach brought forward. Given that the OIS-LIBOR spread is obtained by focusing on howthe offer rate is modelled, we refer to Crépey et al. [ ] , and also the Nguyen & Seifried [ ] , as a rate-based modelling approach . A feature that is rather telling in understanding the structure of multi-curve models, andthus helps in their classification, is the nature of the discount and the forecasting curve,respectively. In the multi-curve models by Crépey et al. [ ] , the term-structure of thediscount (OIS-based) curve is constructed by a rational model. However, the LIBOR modelis postulated in a rather ad-hoc manner and ensues directly from modelling the payoff of theforward rate agreement written on it. Similar to the hybrid HJM-LMM models in Section 4,the forecasting curve (i.e. LIBOR-based term structure) is constructed akin to LIBOR marketmodels. This is why we refer to Crépey et al. [ ] , and to some extent also to Nguyen &30eifried [ ] , as rational-LMM hybrid models . Next, we establish the relations between thesemodels and the framework presented in this paper. Proposition 5.1.
Let K ( t ; T i − , T i ) be the value at time t ∈ [ T i − ] of the fair FRA rateobtained in Crépey et al. [ ] , Section 2.1. Then it holds that K ( t ; T i − , T i ) = K x yt ( T i − , T i ) ,where K x yt ( T i − , T i ) is determined by Eq. (3.20).Proof. By setting P tT i = P xtT i , it follows that L ( t ; T i − , T i ) = P xtT i L x yt ( T i − , T i ) = P x ytT i L yt ( T i − , T i ) , (5.1)where L ( t ; T i − , T i ) is the LIBOR specified in Crépey et al. [ ] , Eq. (2.6).Furthermore, in Section 2.2 of Crépey et al. [ ] , a particular class of rational LIBORmodels is presented that becomes the workhorse, later in the paper. Next we show howsuch class is obtained within the x y -framework. Remark 14.
From the relation (5.1) and by recalling that P ytT i = E [ h yT i |F t ] / h yt , we deducethat L ( t ; T i − , T i ) = δ i h xt € E ” h yT i − (cid:12)(cid:12) F t — − E ” h yT i (cid:12)(cid:12) F t —Š . (5.2) Next we specify the discounting and forecasting kernels as follows:h xt = P x t + b ( t ) A ( ) t , (5.3) h yt = P y t + ¯ b ( t ) A ( ) t + ¯ b ( t ) A ( ) t , (5.4) where, for i =
1, 2, 3 , the processes ( A ( i ) t ) are martingales. The quantities P x t , P y t , b ( t ) and ¯ b i ( t ) , i =
2, 3 , are suitably chosen deterministic functions. The correspondence to the rationalmulti-curve LIBOR models by Crépey et al. [ ] , Section 2.2, is found by settingL ( T i − , T i ) = δ i € P y T i − − P y T i Š , b ( T i , T i − ) = δ i (cid:2) ¯ b ( T i − ) − ¯ b ( T i ) (cid:3) , b ( T i , T i − ) = δ i (cid:2) ¯ b ( T i − ) − ¯ b ( T i ) (cid:3) . (5.5) The specifications (5.5) cause a slight loss of generality. However, whether in practical termssuch specifications are indeed restrictive can be decided once this model class is calibrated toactual market data.
We now turn our attention to the rational multi-curve models presented in Nguyen &Seifried [ ] . They propose to make use of the so-called FX-analogy to motivate pricing31ernel models for the spread observed between the OIS rate and LIBOR. In particular inSection 4, Theorem 4.1, a multiplicative spread is considered. The spread is given by theratio of a conditional expectation of the OIS-based pricing kernel (state-price deflator) anda conditional expectation of a hypothetical pricing kernel. The latter deflator may be asso-ciated with a foreign currency, although they distance themselves from such an interpreta-tion, c.f. Section 4.2 of Nguyen & Seifried [ ] . It is our view that, although the OIS-LIBORspread is interpreted as a kind of currency exchange rate in their work, the deduced rationalmulti-curve LIBOR models are of the kind that Crépey et al. [ ] derive. This is especiallyso because the rational LIBOR models developed in Nguyen & Seifried [ ] are rate-basedmodels —just as those produced by Crépey et al. [ ] —which relate the OIS forward rateand LIBOR, directly. As they seek to dissociate themselves from the work of Bianchetti [ ] ,who, among other authors, unequivocally sticks to the FX-analogy, we show that the ratio-nal models in Nguyen & Seifried [ ] do not need to rely on the FX-analogy in order to bederived within a pricing kernel setup. Proposition 5.2.
In Nguyen & Seifried [ ] , the multi-curve fair FRA rate L ∆ ( t ; T , T + ∆ ) isgiven by L ∆ ( t ; T , T + ∆ ) = ∆ ‚ p ( t , T ) p ( t , T + ∆ ) E (cid:2) D ∆ T | F t (cid:3) E [ D T | F t ] − Œ , (5.6) for t ∈ [ T ] . This model can be obtained by the following specification of the LIBOR process ( L ( t ; T i − , T i )) ≤ t ≤ T i − , for i =
1, 2, . . . , n, in Crépey et al. [ ] , Section 2.1, Eq. (2.7):L ( t ; T i − , T i ) = ∆ D t (cid:0) E (cid:2) D ∆ T | F t (cid:3) − E [ D T + ∆ | F t ] (cid:1) , (5.7) where T i − = T and T i = T + ∆ .Proof. Relation (5.7) is directly obtained by equating the fair FRA rate (4.2) in Nguyen &Seifried [ ] with the fair FRA rate (2.7) in Crépey et al. [ ] . This shows that the OIS-LIBOR spread models, given in Theorem 4.1 in Nguyen & Seifried [ ] , do not necessitatethe use of the FX-analogy in order to derive (rate-based) multi-curve discounting modelsin a pricing kernel approach. While ( D t ) corresponds to the OIS-associated pricing kernelprocess ( π t ) in Crépey et al. [ ] , there is indeed no reason to identify the process ( D ∆ t ) with a fictitious pricing kernel associated with a foreign currency / economy. It may just beviewed as an idiosyncratic component of the LIBOR process. Remark 15.
Comparing Eq. (5.7) with Eq. (5.2) we observe a discrepancy in the way thatthe conversion to a multi-curve setup is obtained in Nguyen & Seifried [ ] . The source of uch incongruence is discussed in Remark 5, (i). The difference is resolved by the followingadjustment in the multi-curve model (5.6):L ∆ ( t ; T , T + ∆ ) = ∆ ‚ p ( t , T ) p ( t , T + ∆ ) E (cid:2) D ∆ T | F t (cid:3) E [ D T | F t ] − · Q ( t , T + ∆ ) Œ , (5.8) where, based on to the x y-approach, the conversion factor Q ( t , T + ∆ ) , or spread process, isgiven by Q ( t , T + ∆ ) = E (cid:2) D ∆ T + ∆ | F t (cid:3) E [ D T + ∆ | F t ] . (5.9) The adjustment allows the model to be derived by a consistent application of the FX-analogy ina pricing kernel setup as produced in the x y-approach developed in this paper.
Unlike the preceding rational-LMM hybrid multi-curve models, we now consider rationalmodels for both, the discounting curve and the forecasting curve, that is for the ZCB priceprocess ( P xtT i ) ≤ t ≤ T i and ( P ytT i ) ≤ t ≤ T i , respectively, which feature as desirable properties (i)tractability, (ii) transparency of the dependence structure among the risk factors and thus(iii) a good understanding of the resulting model for the spread dynamics between the x - (discounting) and the y - (forecasting) curves. The rational price models considered byMacrina [ ] , and by Crépey et al. [ ] for multi-curve interest rate modelling in particular,offer the set of properties we require. For the x - and y -ZCB, we postulate the following: P xtT i = P x T i Q mk = Z xk ( t , T i ) P x t Q mk = Z xk ( t ) , P ytT i = P y T i Q n ℓ = Z y ℓ ( t , T i ) P y t Q n ℓ = Z y ℓ ( t ) , (5.10)where Z xk ( t , T i ) = ( + b xk ( T i ) A xt , k ) and Z y ℓ ( t , T i ) = ( + b y ℓ ( T i ) A yt , ℓ ) are taken to be positiveprocesses. The quantities P x t and P y T i are the initial term structures of the x and y ZCBs, b k and b ℓ are deterministic functions, and ( A xt , k ) and ( A yt , ℓ ) are martingales with respectto some ( P -equivalent) probability measure. For further (technical) details, we refer toMacrina [ ] and Crépey et al. [ ] . We take a closer look at ( P ytT i ) , although the structuralproperties of the model also apply to ( P xtT i ) . The return process of the forecasting ZCB isgiven by ln € P ytT i Š = ln ‚ P y T i P y t Œ + n X ℓ = ln ‚ + b y ℓ ( T i ) A yt , ℓ + b y ℓ ( t ) A yt , ℓ Œ . (5.11)33he associated short rate process ( r yt ) is given by r yt = − ‚ ∂ t P y t P y t + n X ℓ = θ yt , ℓ Œ , (5.12)where we define the ( A yt , ℓ ) -driven factor component ( θ yt , ℓ ) by θ yt , ℓ = ∂ t b y ℓ ( t ) A yt , ℓ + b y ℓ ( t ) A yt , ℓ . (5.13)Now let us assume, for the sake of the explanation, that the number n of factor components isgiven by the particular tenor y . So, let a =
1, 2, 3, . . ., y = a which is the 3-month, 6-month,9-month, 12-month, etc LIBOR tenor, and n = a + n =
2. Then, we have the following additive structurefor the short rate model associated with the corresponding forecasting ZCBs: y = ( P tT i ) : r t = − (cid:18) ∂ t P t P t + θ ( t ) + θ ( t ) (cid:19) , y = ( P tT i ) : r t = − (cid:18) ∂ t P t P t + θ ( t ) + θ ( t ) + θ ( t ) (cid:19) ,... ... y = a -month-tenored ZCB, ( P ytT i ) : r yt = − ‚ ∂ t P y t P y t + a + X ℓ = θ y ℓ ( t ) Œ , a =
1, 2, 3, . . . .(5.14)Depending on the specific interbank offer rate market, we could envisage the situation where θ i ℓ = θ j ℓ for all i , j = a . This would mean that the various y -curves only differed by thenumber of factor components driving the corresponding short rates (i.e, forecasting ZCBs).We would then have y -month-tenored ZCB, ( P ytT i ) : r yt = − ‚ ∂ t P y t P y t + a + X ℓ = θ ℓ ( t ) Œ , a =
1, 2, 3, . . . , (5.15)where the short rate model of the 1m-tenored ZCB is recovered by setting a =
0. From theFRA price process (3.21), one sees that the quantity responsible for the consistent transferfrom a single-curve to a multi-curve setting is the quanto-bond with price process ( P x ytT i ) .34ext we introduce the multiplicative class of rational models for the “multi-curve” quanto-bond price process ( P x ytT i ) ≤ t ≤ T i : P x ytT i = h xt E ” h yT i (cid:12)(cid:12) F t — = P y T i Q n ℓ = Z y ℓ ( t , T i ) P x t Q mk = Z xk ( t ) . (5.16)The model for the short rate of interest ( r x yt ) ≤ t , associated with the quanto-bond, is ob-tained by r x yt = − ∂ T i ln ( P x ytT i ) | T = t , assuming that the quanto-bond price function is differ-entiable in its maturity T i . It follows that r x yt = r yt . One could argue that it is somewhatartificial to introduce the x -discounting bond because, after all, the x -curve may be a specific y -curve. We wish however to allow for more generality: there is no reason why the typeof model ought to be the same for the x -ZCB and for the y -ZCB. It is only for conveniencethat we here decide to consider the same type of pricing model for both types of bonds. Inany case, the discounting curve—identified with the one-day deposit—can be viewed as the y = P xtT i = P tT i : r xt = r t = − (cid:18) ∂ t P t P t + θ ( t ) (cid:19) . (5.17)A byproduct of the multi-curve modelling approach based on bonds as considered in thispaper, is the implicit, or rather emerging, spread models. Within the rational models, theprocess for the spread between the y and y +
3m curves is given by s y , y + mtT i = P y + mtT i P ytT i = P y t P y + m t P y + m T i P y T i ∆ a + ( t , T i ) , a =
1, 2, 3, . . . , (5.18)where the stochastic spread process ( ∆ a + ( t , T i )) ≤ t ≤ T i is given by ∆ a + ( t , T i ) = + b ya + ( T i ) A yt , a + + b ya + ( t ) A yt , a + . (5.19)We note that the stochastic spread is positive assuming that the rates underlying the tenorsare non-negative, see Corollary 3.2. Filipovi´c et al. [ ] introduce the so-called linear-rational term structure (LRTS) models. Inthis section we show how the multi-curve extension to the LRTS is produced by showing that35he LRTS models belong to the more general class introduced in the previous section. Wethus prove that (a) the LRTS models belong to the class of models developed in Macrina [ ] when an infinite-time horizon is considered, and (b) that the pricing kernel generating theLRTS is a weighted heat kernel (WHK). Pricing kernels generated by WHKs in an infinite timehorizon setting are introduced in Akahori et al. [ ] and developed in Akahori & Macrina [ ] in the case tha the WHK is driven by a time-inhomogeneous Markov process. In particular,we shall show that the LRTS models produce bond price processes ( P tT ) ≤ t ≤ T of the form P tT = P T + b ( T ) A t P t + b ( t ) A t , ( ≤ t ≤ T ) (5.20)which are identified as a class of Markov functionals. The function b ( t ) , 0 ≤ t ≤ T , isdeterministic and ( A t ) ≤ t is a martingale process. The explicit construction of this class ofterm structure models is presented in Macrina [ ] . Definition 5.1.
Linear-Rational Term Structure Models, Filipovi´c et al. [ ] .Let ( Z t ) ≤ t denote the multivariate process with state space E ⊂ R m that satisfies the stochasticdifferential equation d Z t = κ ( θ − Z t ) d t + d M t , (5.21) where κ ∈ R m × m and θ ∈ R m , and where ( M t ) ≤ t is an m-dimensional martingale. Let ( ζ t ) ≤ t denote the pricing kernel process defined by ζ t = e − α t ( φ + ψ Z t ) , (5.22) where α ∈ R , φ ∈ R , and ψ ∈ R m such that φ + ψ z > for all z ∈ E. The linear-rationalterm structure, generated by the linear pricing kernel process ( ζ t ) ≤ t , have zero-coupon bondprice processes ( P tT ) ≤ t ≤ T given byP tT = e − α ( T − t ) φ + ψθ + ψ e ( T − t ) ( Z t − θ ) φ + ψ Z t , (5.23) where T is the bond maturity date. Proposition 5.3.
The stochastic differential equation (5.21) has the unique solution given byZ t = e − κ t (cid:18) Z + κ Z t e κ s d s θ (cid:19) + e − κ t A t . (5.24)36 he process ( A t ) ≤ t , defined by A t = Z t e κ s d M s , (5.25) is a martingale.Proof. That the mean-reverting process (5.24) is the unique solution to the SDE (5.21)follows from a straightforward application of Ito’s Lemma. To show that ( A t ) ≤ t is a martin-gale, one remarks that E [ | A t | ] < ∞ for all t ≥ E [ A u |F s ] = A s , for 0 ≤ s ≤ u . Thelatter follows by calculating E [ R us d [ φ ( t ) M t ] | F s ] , where 0 ≤ s ≤ t ≤ u , and by applyingFubini’s theorem. One then obtains E [ A u |F s ] − A s = E [ φ ( u ) M u − φ ( s ) M s | F s ] − E (cid:20)Z us M t ∂ t φ ( t ) d t (cid:12)(cid:12) F s (cid:21) =
0, (5.26)which completes the proof.
Theorem 5.1.
The pricing kernel process ( ζ t ) ≤ t that generates the linear-rational term struc-ture models, specified in Definition 5.1, is given by ζ t = ζ [ P t + b ( t ) A t ] , (5.27) where ζ = φ + ψ Z . The positive, deterministic function ( P t ) ≤ t ≤ T is the initial term structureof the associated T -maturity bond system with price processP tT = P T + b ( T ) A t P t + b ( t ) A t ( ≤ t ≤ T ) (5.28) where P t , the deterministic function b ( t ) and the martingale ( A t ) are determined byP t = e − α t φ + ψ Z (cid:20) φ + ψ e − κ t (cid:18) Z + κ Z t e κ s d s θ (cid:19)(cid:21) , 0 ≤ t ≤ T , (5.29) b ( t ) = e − α t φ + ψ Z ψ e − κ t , 0 ≤ t ≤ T , (5.30) A t = Z t e κ s d M s , t ≥
0. (5.31)
Proof.
One direction is straightforward: it suffices to insert (5.29), (5.30) and (5.31) in(5.28) to obtain (5.23). The other direction, i.e. beginning from Definition 5.1, goes as37ollows: The solution (5.24) is inserted in (5.23) to obtain P tT = e − α T ” φ + ψ e − κ T € Z + κ R t e κ s d s θ Š— + e − α T ψκ R t e − κ ( T − s ) d s θ + e − α T ψ e − κ T A t e − α t ” φ + ψ e − κ t € Z + κ R t e κ s d s θ Š— + e − α t ψ e − κ t A t .(5.32)Next, we define the functions γ ( t , T ) , λ ( t , T ) and ˜ b ( t ) by γ ( t , T ) = e − α T (cid:20) φ + ψ e − κ T (cid:18) Z + κ Z t e κ s d s θ (cid:19)(cid:21) , (5.33) λ ( t , T ) = e − α T ψκ Z t e − κ ( T − s ) d s θ , (5.34)˜ b ( t ) = e − α t ψ e − κ t (5.35)for t ∈ [ T ] , and therewith express the bond price process in the form P tT = γ ( t , T ) + λ ( t , T ) + ˜ b ( T ) A t γ ( t , t ) + ˜ b ( t ) A t . (5.36)The initial term structure P t , 0 ≤ t ≤ T , satisfies the relation γ (
0, 0 ) P t = γ ( t )+ λ ( t ) = γ ( t , t ) . Furthermore, γ ( t , T ) + λ ( t , T ) − [ γ ( T ) + λ ( T )] = P tT = γ ( t , T ) + λ ( t , T ) − [ γ ( T ) + λ ( T )] + γ (
0, 0 ) P T + ˜ b ( T ) A t γ (
0, 0 ) P t + ˜ b ( t ) A t , (5.37)and immediately obtain (5.28) by observing that b ( t ) = ˜ b ( t ) /γ (
0, 0 ) for 0 ≤ t ≤ T . Corollary 5.1.
The Linear-Rational Term Structure models can be expressed in the formP tT = P T (cid:2) + ¯ b ( T ) A t (cid:3) P t (cid:2) + ¯ b ( t ) A t (cid:3) , (5.38) for ≤ t ≤ T , where ¯ b ( t ) = b ( t ) / P t . This is the form (5.10) for m = , and thus thenecessary basis for the extension to the multi-curve linear-rational term structure models viaTheorem 3.1 and Definition 3.19, in a developed market, and via Definitions 3.2 and 3.3 inthe emerging market.Proof. This follows directly from (5.28).
Remark 16.
We emphasise that the form (5.28), or equivalently (5.38), shows that the Linear-Rational Term Structure has, by (5.29), a functionally fully specified initial term structure P t f bond prices for t ∈ [ T ] . Also, the models (5.28) specified by (5.29)-(5.31) produce anexample of the larger class (5.20), or equivalently (5.38), of term structure models that canaccommodate unspanned stochastic volatility as considered in Filipovi´c et al. [ ] , Section C. Next we consider weighted heat kernel processes over an infinite-time horizon, see Aka-hori et al. [ ] , and in particular the case where the propagator is a conditional expectation,as in Akahori & Macrina [ ] and Macrina [ ] . Such weighted heat kernels are used togenerate (explicit) pricing kernel processes. The definition that follows provides weightedheat kernels in a multivariate setting. Definition 5.2.
Let ( X t ) ≤ t be an m-dimensional ( F t ) -adapted Markov process, F ( t , x ) bea vector-valued and deterministic function in R m , and w ( t , u ) a matrix-valued deterministicfunction in R m × m . Furthermore, let the functions f ( t ) ∈ R and f ( t ) ∈ R m be deterministic.The process ( π t ) ≤ t is a weighted heat kernel defined by π t = f ( t ) + f ( t ) Z ∞ w ( t , u ) E [ F ( t + u , X t + u ) | F t ] d u , (5.39) where t ∧ u ≥ , and f ( t ) , f ( t ) , F ( t , x ) and w ( t , u ) are chosen such that ( π t ) is a positiveand finite (scalar-valued) process. The next statement asserts that the pricing kernel process ( ζ t ) in Filipovi´c et al. [ ] isa weighted heat kernel and it establishes the relation between ( ζ t ) and the class (5.39). Theorem 5.2.
The pricing kernel (5.22) that generates the linear-rational term structure mod-els by Filipovi´c et al. [ ] , is a special case of the process (5.39) where the following holds:1. Let ( X t ) be the Markov process ( Z t ) that satisfies (5.21).2. F ( t , X t ) = Z t , for all t ≥ .3. w ( t , u ) = e − β ( t + u ) , β ∈ R m × m invertible where β κ = κβ for κ ∈ R m × m invertible.4. The functions f ( t ) and f ( t ) are give byf ( t ) = f ( t ) (cid:2) ( β + κ ) − − β − (cid:3) e − β t θ + e − α t φ , (5.40) f ( t ) = e − α t ψ e β t ( β + κ ) , (5.41) where φ ∈ R , α ∈ R , θ ∈ R m , ψ ∈ R m and β ∈ R m × m with β κ = κβ for κ ∈ R m × m . Itis assumed that ( β + κ ) is invertible. roof. One direction is straightforward: It suffices to insert items 1- 4 into Equation (5.39)to obtain the pricing kernel process (5.22). In the other direction, that is starting from(5.39), one makes the initial assumptions that the first and second items shall hold. Thisleads to E [ Z t + u ) | F t ] = e − κ ( t + u ) (cid:18) Z + κ Z t + u e κ s d s θ (cid:19) + e − κ ( t + u ) Z t e κ s d M s . (5.42)Then, by choosing the ansatz given in the third item, one obtains Z ∞ w ( t , u ) E [ F ( t + u , X t + u ) | F t ] d u = ( β + κ ) − e − β t Z t + (cid:2) β − − ( β + κ ) − (cid:3) e − β t θ . (5.43)Thus, the functions f ( t ) and f ( t ) are selected such that the pricing kernel process (5.22)is obtained, that is (5.40) and (5.41). In this section, we show how the across-curve valuation approach developed in this paperextends to the pricing of other fixed-income financial instruments. The curve-conversionfactor process, developed in the present work, may conveniently be applied to the pric-ing and hedging of inflation-linked and foreign-exchange (FX) securities. In particular, thequanto-bond process ( P x ytT ) ≤ t ≤ T plays an important role in the pricing of hybrid securities,suchlike inflation-linked foreign-exchange products, where consistent asset valuation canstill be a challenge. It is customary in inflation-linked price modelling and hedging to consider two economies,the so-called nominal and real economies. Such a viewpoint matches the x y -concept somuch so that the curve-conversion factor associated with inflation-linked pricing is obtainedwith little effort. But this is the strength and appeal we see in this formalism. The nominal(N), cash-based economy is associated with the x -curve, and the real (R), goods / services-based economy is associated with the y -curve. So, we set x = N and y = R . Next we applythe scheme developed in Sections 2 and 3 of this paper.We assume positive pricing kernel processes ( h Nt ) ≤ t and ( h Rt ) ≤ t for the nominal and thereal economies, respectively. The process ( C t ) ≤ t of the consumer price index links prices40etween the nominal and the real economies by C t = C h Rt h Nt , (6.1)where C is the base price level at time 0 (not necessarily normalised to one). The price P NRtT at time t ≤ T of an inflation-linked discount bond with cash flow C T at maturity T isgiven by P NRtT = h Nt E • h NT C T C | F t ˜ = h Nt E (cid:2) h RT | F t (cid:3) . (6.2)In the x y -formalism, where we recall x = N and y = R , we may write the price process ( P NRtT ) in terms of the conversion formula P NRtT = P NtT Q NRtT , (6.3)where ( P NtT ) is the price process of the nominal discount bond, and where Q NRtT = E (cid:2) h RT | F t (cid:3) E (cid:2) h NT | F t (cid:3) ( ≤ t ≤ T ) (6.4)is the curve-conversion factor (spread process) linking discounting on the nominal N -curveand forecasting on the real R -curve. The expression for the quanto-bond (6.2) can be ob-tained in a straightforward fashion from Eq. (2.4) by setting t = T, thereafter replacing thepricing time s with t , and further by setting x = N , y = R and H RT =
1. The nominal curveserves as the base-curve; hence the curve-conversion factor process (6.4), in the relation(6.3), quantifies the number of positions in the nominal T -maturity discount bond neces-sary to replicate the no-arbitrage value at t ∈ [ T ] of the inflation-linked discount bondwith value P NRtT at time t . Given that the nominal discount bond P NtT and the inflation-linkeddiscount bond P NRtT are traded sufficiently on a market, one can imply from the market theinflation-linked conversion factor Q NRtT = P NRtT P NtT . (6.5)The pricing formulae for an inflation-linked forward rate agreement (or inflation-linkedzero-coupon swap) and for a year-on-year swap contract can be expressed in terms of theconversion factor. The derivations of such pricing formulae follow those for the forwardrate agreement and the swap contracts presented in Section 3. Price models for inflation-linked securities, which are based on explicit pricing kernel models—hence, on explicit curve41onversion factor processes—have been developed by Dam et al. [ ] . Such models featurea high degree of flexibility and good calibration properties. We consider two currencies i and j in the respective nominal (cash-based) economies N i and N j . Here we show that the forward foreign exchange rate , which converts an amountof domestic currency j into an amount of foreign currency i at a fixed future data, is givenby today’s spot exchange rate multiplied with the appropriate currency conversion factor.We set x = i and y = j in the xy-formalism, see Sections 2 and 3. In the following, weabbreviate “foreign exchange" with “FX".We denote by ( X i jt ) ≤ t the spot FX rate process, which converts, e.g., GBP to EUR. Weemphasise that the notation i j implies, in this example, EUR / GBP. By ( F i jtT ) ≤ t ≤ T we denotethe process of the forward FX rate. We conjecture the following relation: F i jtT = X i jt P jtT P itT . (6.6)Here, ( P itT ) ≤ t ≤ T and ( P jtT ) ≤ t ≤ T are assumed to be the nominal OIS discount bond priceprocesses denominated in the i (EUR) and j (GBP) currencies, respectively. We acknowledgethat the correct discount bond price processes, in practice, are those determined by therespective FX basis curves. While these may be easily incorporated into the frameworkvia pricing kernels and associated curve-conversion factor processes, we ignore this factthroughout this section for ease of exposition. We note that F i jt t = X i jt , t ∈ [ T ] . The i -and j -denominated economies are assumed to be equipped with the respective (nominal)pricing kernel processes ( h it ) and ( h jt ) . By recalling the price formula of a discount bond, itfollows from the conjecture (6.6) that F i jtT = X i jt h it E [ h jT | F t ] h jt E (cid:2) h iT | F t (cid:3) = X i jt h it h jt Q i jtT , (6.7)where the FX conversion factor ( Q i jtT ) for the currency pair ( i , j ) has the familiar form Q i jtT = E [ h jT | F t ] E (cid:2) h iT | F t (cid:3) . (6.8)Next we validate the conjecture (6.6) by pricing an FX forward contract in this setup.42e model the spot FX rate process ( X i jt ) by X i jt = X i j h jt h it , (6.9)and, by recalling (6.7), we obtain F i jtT = X i jt h it h jt Q i jtT = X i j Q i jtT . (6.10)This is the relation we would expect to emerge in the xy-approach for the forward FX process.The stochastic price dynamics of the forward FX contract are determined by the ratio of theforecasting curves in the two economies denominated in units of the respective currencies.We shall now see whether the expression (6.10) for the forward FX rate is indeed the fairrate obtained from pricing the FX forward contract. Proposition 6.1.
Let ( P itT ) ≤ t ≤ T and ( P jtT ) ≤ t ≤ T be the price processes of the discount bondsdenominated in the i and j currencies, respectively. Let ( X i jt ) t ≥ be the spot FX rate processexchanging j currency for i currency at time t ≥ . Then, for ≤ t ≤ T , the fair forward FXrate is given by F i jtT = X i j Q i jtT = X i jt P jtT P itT , (6.11) where ( Q i jtT ) ≤ t ≤ T is the curve-conversion process (6.8). Equation (6.11) confirms the expression given in conjecture (6.6). Furthermore, theFX curve-conversion factor process ( Q i jtT ) ≤ t ≤ T can be implied from the quoted forward FXrates and the spot rates on the market, that is, Q i jtT = F i jtT X i j . (6.12) Proof.
Consider the payoff V iT = X i jT / X i j − K i of an FX forward contract, with expiry date T > K i , denominated in i -currency. The price process ( V itT ) of the FXforward contract is given by V itT = h it E ” h iT € X i jT / X i j − K i Š (cid:12)(cid:12) F t — . (6.13)This follows as an application of the across-curve formula (2.4), where one sets x = i and43 = j , alongside t = T and where the pricing time s is replaced with t . Furthermore, H jT = − K i / Q i jT T shall hold, which is indeed a j -currency quantity. The relation H i jtT = V itT is obtained where we drop the j superscript in V itT to emphasise that the value V itT at time t ∈ [ T ] is given in units of the i -currency. Then, by recalling Eq. (6.9), it follows withease that V itT = P i jtT − K i P itT , (6.14)for t ∈ [ T ] . By setting V itT =
0, for all t ∈ [ T ] , we obtain the result stated in theproposition, where K itT = F i jtT / X i j = Q i jtT is the fair (strike) value for the forward currency-exchange process. We consider the situation whereby an investor wishes to enter a foreign-currency forwardcontract on LIBOR. An example might clarify this type of hybrid security. Suppose we con-sider GBP-based LIBOR, as quoted in the U.K. market. An investor wishes to enter a USD-denominated forward contract written on the GBP-based LIBOR quotes. This exposes theinvestor to the risk underlying the GBP-LIBOR market and the currency-exchange risk be-tween GBP & USD. We are interested in deducing the fair forward rate process of the USD-forward contract based on GBP-LIBOR. This is obtained with ease in the xy-approach bycombining the results in Sec. 3.2, on multi-curve systems in developed markets, and in Sec.6.2, above.
Proposition 6.2.
Consider ≤ t ≤ T i − < T i where T i − T i − is the tenor of the GBP-basedLIBOR and T i is the expiry date of the USD-denominated forward contract that is writtenon the GBP-based LIBOR. The fair forward rate process K x $ y £ t ( T i − , T i ) ≤ t ≤ T i − of the USD-denominated forward contract is given byK x $ y £ t ( T i − , T i ) = F $ £tT i X $ £ L x £ y £ t ( T i − , T i ) = L x $ y £ t ( T i − , T i ) , (6.15) where X $ £ is the spot USD / GBP exchange rate at t = , the fair forward USD / GBP exchangerate process ( F $ £tT i ) ≤ t ≤ T i is given by the relation (6.11) and the GBP-based LIBOR processL x £ y £ t ( T i − , T i ) ≤ t ≤ T i − is given by Eq. (3.13).Proof. The starting point is the y -tenored GBP-based LIBOR, given in Lemma 3.1, which we44onvert, at time T i − , into the x $ -market by the FX conversion factor (6.8). We obtain L x $ y £ T i − ( T i − , T i ) = Q $ £T i − T i L x £ y £ T i − ( T i − , T i ) = Q x $ y £ T i − T i L y £ T i − ( T i − , T i ) , (6.16)which is the USD-denominated GBP-based LIBOR. We note that the notation x £ and y £ standfor GBP-OIS and GBP-y-tenor, respectively. As mentioned earlier, we ignore the GBP-USD FXbasis curve, for simplicity. However, this curve may be easily incorporated into the valuationthrough an intermediate curve-conversion factor process for GBP-OIS to GBP-USD FX basis.Next, we write the price V x $ y £ tT i at time t ≥ T i ≥ t . That is, V x $ y £ tT i = X $ £ δ i h x $ t E ” h x $ T i € L x $ y £ T i − ( T i − , T i ) − K x $ Š | F t — , (6.17)where K x $ is the strike rate of the contract. By Eq. (6.16) and the tower property of condi-tional expectation, it then follows that V x $ y £ tT i = h y £ t h x $ t P x $ tT i L y £ t ( T i − , T i ) − K x $ P y £ tT i . (6.18)Setting V x $ y £ tT i = t ∈ [ T i ] gives the result (6.15), where Eqs (2.3), (3.2), (3.13)and (6.11) are used. Given that the relations for prices of inflation-linked and FX securities are available in thexy-approach, we can move on to the valuation of another hybrid financial instrument. Weconsider the price process of a contract that gives exposure to inflation in the domesticeconomy and is priced in a foreign currency. To answer this question, we take the exampleof a forward bet on inflation / deflation in the j -economy valued in units of the i -currency.This could be taking a bet at t ∈ [ T ) on the growth in the value of the U. K. price indexin EUR, X i jT C jT / C j , at the fixed future date T . Proposition 6.3.
Let ( C jt ) t ≥ be the j-economy price index process (6.1) and ( X i jt ) t ≥ the spotFX rate (6.9). Consider the random payoff V N i T = X i jT C jT / C j − K N i , where K N i is the nominali-currency strike value, and T is the fixed expiry date. The price process ( V N i tT ) ≤ t ≤ T of theinflation-linked FX forward with cash flow V N i T is given byV N i tT = X i j P N i tT Q N i R j tT − K N i P N i tT , (6.19)45 here ( P N i tT ) is the price process of the nominal discount bond in the i-economy, and where theconversion factor process ( Q N i R j tT ) ≤ t ≤ T is defined byQ N i R j tT = E ” h R j T | F t — E ” h N i T | F t — . (6.20) The fair forward inflation-linked FX rate process ( F N i R j tT ) ≤ t ≤ T is given byF N i R j tT = F i jtT P N j R j tT P N j tT . (6.21) Proof.
We begin with Proposition 2.1: Set x = N i and y = R j alongside t = T , and thereafterreplace the pricing time s with t . This gives, H N i R j tT = h N i t E ” h N i T Q N i R j T T H R j T | F t — . (6.22)For the real-economy random cash flow at time T > H R j T = X i j − K N i / Q N i R j T T , whichis a quantity denominated in units of the j -real-economy. Now we calculate the price V N i tT at time t ∈ [ T ] of the hybrid contract. We write H N i R j tT = V N i tT to emphasise that the value V N i tT at time t ∈ [ T ] is given in nominal units of the economy with currency i . We have: V N i tT = h N i t E ” h N i T € X i jT C jT / C j − K N i Š (cid:12)(cid:12) F t — (6.23) = X i j h N i t E ” h R j T | F t — − P N i tT K N i , (6.24)where Eqs (6.1) and (6.9) are used. This can be expressed in terms of the appropriateconversion factor process ( Q N i R j tT ) ≤ t ≤ T . That is, V N i tT = X i j P N i tT Q N i R j tT − P N i tT K N i , (6.25)where Q N i R j tT = E ” h R j T | F t — E ” h N i T | F t — . (6.26)46etting V N i tT =
0, for all t ∈ [ T ] , we obtain the fair inflation-linked FX forward process: F N i R j tT = X i j Q N i R j tT = F i jtT P N j R j tT P N j tT , (6.27)which concludes the proof.In summary, Equation (6.21) states that the fair rate of an inflation-linked FX forwardis given by F i jtT / P N j tT units of the bond P N j R j tT , which is linked to inflation in the (domestic) j -economy. In developed markets, the assets with price F i jtT , P N j tT and P N j R j tT , respectively,are (mostly) liquidly traded. The relation (6.21) determines the consistent hedge for the i -currency inflation-linked FX forward in terms of the j -economy FX forward, the inflation-linked bond and the zero-coupon bond in the j -market. We thus have (i) the consistentcurve-conversion formula Q N i R j tT = Q i jtT Q N j R j tT , linking inflation-indexed and FX securities, and(ii) the equivalent consistent relation (6.21) between the inflation-indexed and FX forwardrates. In this paper, a framework is developed that allows for the consistent pricing and hedgingof financial assets, which depend on a spread between the rates their values accrue and arediscounted at. Such a situation is manifest in fixed-income markets, in particular, wherethe return of instruments may accrue at one benchmark rate, e.g. LIBOR, and is discountedat another benchmark rate, e.g. the OIS rate. The paradigm for modelling the prices oftenor-based fixed-income products is the so-called multi-curve term structure framework.Although the approach we develop in this paper is applicable whenever spreads amongdifferent curves (term structures) need to be modelled, we consider fixed-income as themarket within which we develop what we term consistent valuation across curves . We choosethe modelling paradigm of pricing kernels to construct the consistent price systems thatgive rise to, and also rely on, the curve-conversion process that allows for no-arbitrageprice conversions from one curve to another, as e.g. required in multi-curve interest ratemodelling. This can be viewed as a kind of currency foreign-exchange analogy, and we drawseveral parallels with this view while we develop the xy-approach .After the introduction of the curve-dependent discounting systems, we produce thecurve-conversion factor process that links cash-flows associated with different curves andhence gives rise to consistent prices of assets, which accrue value according to the fore- asting curve and are discounted according to the discounting curve . The dual nature ofthe curve-conversion factor also allows for conversion of curves. The deduced across-curvepricing formula gives rise to the consistent set of numeraire assets and associated (risk-neutral) probability measures so as to avoid the introduction of arbitrage opportunities ina multi-curve market—or a ‘spread market’—see Section 2. The curve-conversion mecha-nism enables the introduction of tenor-based zero-coupon bonds without undermining theno-arbitrage requirement.An intriguing by-product of the across-curve pricing kernel approach we develop is thatit proposes consistent pricing relations for multi-curve systems in emerging markets wherea derivatives market on one of the benchmarks, say the OIS system, is absent and needsto be estimated. For example, the liquidly traded tenor may be used to calibrate the pric-ing kernel model underlying the zero-coupon bond price system associated with the liquidtenor. Although a multi-curve interest rate system is available in emerging markets, it hasan idiosyncratic and proprietary nature. Given an estimation methodology, one can applythe xy-approach as a standard to consistently price instruments in a multi-curve emergingmarket. We show that the across-curve valuation method is applicable in developed (liquid)markets as much as in emerging (less liquid) markets and that fixed-income products, suchas forward rate agreements, may be understood and priced with the same ease in both typesof markets.Recently, interest in so-called rational models has grown and the advantages of usingthis class of models to produce tractable interest-rate models, and extensions to the multi-curve setting, have been recognised. We develop generic pricing kernel models for across-curve valuation and show how rational multi-curve models, such as those of Crépey et al. [ ] and Nguyen & Seifried [ ] are recovered within our xy-approach, and furthermorethe linear-rational term structure models by Filipovi´c et al. [ ] may be generalised to amulti-curve environment. Moreover, important contributions have been made by severalauthors to produce multi-curve extensions of the Heath-Jarrow-Morton framework. We tryto contribute to this research area by investigating the HJM-framework from the perspectiveof our across-curve valuation scheme therewith suggesting multi-curve HJM-models.Finally we show how inflation-linked, currency-based, and fixed-income hybrid securi-ties can be priced by applying our consistent across-curve valuation method using pricingkernels. Acknowledgments . The authors are grateful to participants of the Avior QuantitativeFinance Research Seminar (Cape Town, March 2016), ACQuFRR & RMB Masterclass on48Catching up with Emerging Markets” (Johannesburg, July 2016), CFE 2016 (Sevilla, De-cember 2016), STM2016 Workshop, Institute of Statistical Mathematics, Japan (Tokyo, July2016), JAFEE 2016 (Tokyo, August 2016), and of the Seminar at the Graduate School of In-ternational Corporate Strategy, Hitotsubashi University (Tokyo, March 2017) for commentsand suggestions. Moreover, questions, comments and suggestions by Henrik Dam, MartinoGrasselli, Matheus Grasselli and Erik Schlögl have been very much appreciated and haveled us to understand and develop our work on an across-curve valuation approach betterand further. The authors thank two anonymous reviewers with comments which helpedimprove this paper.
A No-arbitrage strategy for conversion of cash flows and curves
Let us first consider a simple arbitrage relationship for an economy with default-free andcredit-risky interest rate curves, while assuming perfect market liquidity. Assume that the x -curve is the default-free curve while the y -curve is one of a potential set of credit-riskycurves. Consider the following simple strategy, at time 0:(i) Sell one unit of the numeraire asset in the x -market for 1 / h x ; and(ii) Buy one unit of the numeraire asset in the y -market which costs 1 / h y ,which costs zero to setup, i.e. V t =
0, since h x = h y =
1. Transaction (i) is equivalent toborrowing money via the x -market’s money market, while (ii) is equivalent to a deposit intothe y -market’s money market. Then at any time t >
0, the value of this strategy will be V t = h yt − h xt ,which will be greater than zero if the risky entity that holds the investment has not defaultedby that time. Therefore, this strategy does not allow for arbitrage, in general. Now, let usassume that one is able to mitigate all of the default-risk associated with the entity offeringthe y -market investment via appropriate collateralisation. In such a circumstance, the valueof the strategy at any time t > V t which would be greater than zerowith certainty at any time t >
0. No arbitrage may be achieved by adjusting the y -marketdeposit by the ratio h yt / h xt . At any time t >
0, this ratio is merely the realised multiplicativespread between the discount factors realised in the x - and y -markets respectively.49 emark 17. In currency modelling, the ratio h yt / h xt models the spot exchange rate between thex - and y-currencies. In particular, 1 unit of y-currency may be exchanged for h yt / h xt units ofx -currency at time t. Another relevant arbitrage relationship to consider involves a finite horizon loan and in-vestment strategy. Maintaining the same assumptions as before, consider the same strategyas before at time 0, and then do the following at some time t ∈ ( T ) :(i) Sell 1 / h xt units of the x -market T -maturity bond for P xtT ; and(ii) Buy 1 / h yt units of the y -market T -maturity bond for P ytT ,which again costs zero to setup at time 0, as before, and terminates at time t when themoney market loan and deposit is transferred to fixed horizon alternatives. Now, at anytime s ∈ [ t ) , the same arguments apply as before while at time t the value of this strategywill be V t = h yt P ytT − h xt P xtT ,which again does not permit an arbitrage opportunity, due to the credit risk associated withthe investment leg of the strategy. If we again invoke collateralisation of the investmentleg of the strategy, then arbitrage is precluded at: (a) all times s ∈ [ t ) by adjusting the y -market deposit by the ratio h yt / h xt ; and (b) at time t by adjusting the y -market fixed termdeposit by the ratio h yt P ytT / h xt P xtT . If these adjustments are not enforced post collateralisa-tion, then one would be ensured of a risk-free profit equal to V t for all times t ∈ ( T ] . Remark 18.
In currency modelling, the ratio h yt P ytT / h xt P xtT models the forward exchange ratebetween the x - and y-currencies. In particular, one can agree at time t to exchange 1 unit ofy-currency for h yt P ytT / h xt P xtT units of x -currency at time T ≥ t. B Consistent changes of numeraire and measure
Here we discuss changes-of-measure, numeraire assets, martingales and therefore no-arbitragewithin the xy-formalism. The curve-conversion factor process (2.3) induces the changes-of-measure between all introduced y -markets (or y -curves). In particular, it governs no-arbitrage across all distinct markets associated with the economy under consideration. Todemonstrate this, we consider Proposition 2.1, along with an asset with a spot-defined fu-ture cash flow H yT and deduce that the value of such an asset in the x -market is H x ytT = h xt E (cid:2) h yT H yT | F t (cid:3) , (B.1)50or t ∈ [ T ] . For each of the markets z = x , y , we introduce a change-of-measure den-sity martingale ( m zt ) ≤ t ≤ T which changes measure from the real-world measure P to theequivalent (risk-neutral) measure Q z , along with the z -discount factor ( D zt ) such that 1 / D zt is the natural numeraire under Q z . This also means that the z -market’s pricing kernel maybe written as h zt = D zt m zt . The price process ( H x ytT ) ≤ t ≤ T can now be expressed, equivalently,in terms of (a) the Q x risk-neutral measure and (b) the Q y risk-neutral measure: H x ytT = D xt E Q x (cid:2) D xT Q x yT T H yT | F t (cid:3) = m yt m xt D xt E Q y (cid:2) D yT H yT | F t (cid:3) = Q x yt t H yt , (B.2)where we emphasise that ( D xt H x ytT ) ≤ t ≤ T and ( D yt H yt ) ≤ t ≤ T are Q x - and Q y -martingalesrespectively, by construction. Moreover, we may change measure from Q z to the T -forwardmeasure Q Tz via the Radon-Nikodym derivatived Q Tz d Q z = D zT P zT T D zt P ztT , (B.3)which acts on F T given information F t up to time t , and therefore we may now expressthe price process ( H x ytT ) ≤ t ≤ T equivalently, in terms of (a) the x -market T -forward measureand (b) the y -market T -forward measure: H x ytT = P xtT E Q Tx (cid:2) Q x yT T H yT | F t (cid:3) = m yt D yt P ytT m xt D xt E Q Ty (cid:2) H yT | F t (cid:3) = P xtT Q x ytT E Q Ty (cid:2) H yT | F t (cid:3) = P xtT Q x ytT H yt P ytT ,(B.4)where ( H x ytT / P xtT ) ≤ t ≤ T and ( H yt / P ytT ) ≤ t ≤ T are Q Tx - and Q Ty -martingales respectively. Equa-tions (B.2) and (B.4) clearly demonstrate the role of the curve-conversion factor process inchanging measure within and across markets. Furthermore, the price process’ martingaleproperty is preserved across markets (and curves), with the curve-conversion factor pro-cess again enabling this property. The xy-approach precludes arbitrage within and acrossdifferent markets (and curves). C Bootstrapping of initial term structures
C.1 Emerging markets
In an emerging market, one would have the following initial data: (a) the y -tenored spotIBOR L y ( T ) ; (b) a set of fair FRA rates { K y y ( T , T ) , K y y ( T , T ) , . . . , K y y ( T n − , T n ) } ; and(c) a set of fair IRS rates { S y y ( T n + ) , S y y ( T n + ) , . . . , S y y ( T n + m ) } . Using this data, one51ay construct the initial y -ZCB system by the relations P y T = + L y ( T ) δ , P y T i = P y T i − + K y y ( T i − , T i ) δ i , P y T n + j = − S y y ( T n + j ) P n + j − k = δ k P y T k + δ n + j S y y ( T n + j ) , (C.1)for i ∈ {
2, 3, . . . , n } and j ∈ {
1, 2, . . . , m } . In general, one will have to make use of a suitablenumerical bootstrapping technique to extend the y -ZCB system from the longest FRA matu-rity to the set of IRS maturities. These results are all consistent with a classical single-curveinterest rate framework. C.2 Developed markets
In a developed market, one would have the following initial data: (a) the y -tenored spotIBOR L x y ( T ) ; (b) a set of fair FRA rates { K x y ( T , T ) , K x y ( T , T ) , . . . , K x y ( T n − , T n ) } ;and (c) a set of fair IRS rates { S x y ( T n + ) , S x y ( T n + ) , . . . , S x y ( T n + m ) } . Using this data,one may construct the initial y -ZCB system by the relations P y T = − δ i P x T L x y ( T ) , P y T i = P y T i − − δ i P x T i K x y ( T i − , T i ) , P y T n + j = − S x y ( T n + j ) n + j X k = δ k P x T k , (C.2)for i ∈ {
2, 3, . . . , n } and j ∈ {
1, 2, . . . , m } . In general, one will have to make use of a suitablenumerical bootstrapping technique to extend the y -ZCB system from the longest FRA ma-turity to the set of IRS maturities. Interestingly, but not surprisingly, since P x yst = ( h ys / h xs ) P yst for 0 ≤ s ≤ t , it follows that L yt ( T i − , T i ) = δ i ‚ P ytT i − P ytT i − Œ = δ i ‚ P x ytT i − P x ytT i − Œ , (C.3)and therefore P x y t = P y t for t ≥ Remark 19.
Market practitioners may choose to construct a market-implied y-ZCB system, hich we will denote by { P y t } t ≥ , as followsP y T = + L x y ( T ) δ , P y T i = P y T i − + L x y ( T i − , T i ) δ i , P y T n + j = − P n + j − k = δ k S x y ( T n + j ) P y T k + δ n + j S x y ( T n + j ) , (C.4) for i ∈ {
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