domain: qfin math

text:
Claudio Fontana
email: fontana@math.unipd.it

AND Zorana Grbac
email: grbac@lpsm.paris

Thorsten Schmidt
email: thorsten.schmidt@stochastik.uni-freiburg.de

Term structure modelling with overnight rates beyond stochastic continuity

Keywords: April 15, 2022 Libor reform, alternative risk-free rate, SOFR, SONIA, âĆňSTR, affine processes, semimartingales, stochastic discontinuities, BSDE, local risk-minimization. JEL classification: C02, C60, E43, G12, G13. 2020 Mathematics Subject Classification: 60G15, 60G44, 60G57, 91G15, 91G20, 91G30

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Introduction

The ceasing publication of Libor rates by January 1, 2022 1 , with the exception of some tenors of the USD-indexed Libor, marks a major transition for interest rate markets. In this reform of interest rate benchmarks, a central role is played by overnight rates, such as SOFR (secured overnight financing rate) in the US, SONIA (Sterling overnight index average) in the UK and âĆňSTR (Euro short-term rate) in the Euro zone, generically referred to as risk-free rates (RFRs). At the same time, the problem of defining and constructing pertinent forward term rates which would replace the Libor rates of various tenors is currently under discussion, motivated by the needs of market participants.

A distinctive feature of RFRs is the presence of stochastic discontinuities in their dynamics, i.e., jumps or spikes occurring at predetermined dates or at regular intervals of time, as a result of regulatory and liquidity constraints. This is well illustrated by Figure 1. In particular, let us consider the spike observed on September 17, 2019. According to [START_REF] Anbil | What happened in money markets in september 2019?[END_REF], "Strains in money markets in September seem to have originated from routine market events, including a corporate tax payment date and Treasury coupon settlement. The outsized and unexpected moves in money market rates were likely amplified by a number of factors". The analysis of [START_REF] Anbil | What happened in money markets in september 2019?[END_REF] suggests that the date of this spike was largely known in advance (a corporate tax payment date coinciding with a Treasury coupon settlement), while the size of the jump was obviously not predictable. In addition, overnight rates tend to exhibit jumps in correspondence to meetings of the monetary policy authority and those meetings usually follow a predetermined calendar. In a recent work, [START_REF] Backwell | Expected and unexpected jumps in the overnight rate: consistent management of the Libor transition[END_REF] document that most of the variation in the SONIA rate over the years 2016-2020 occurs in correspondence to the meeting dates of the Monetary Policy Committee of the Bank of England. In this work, we develop a general framework for interest rate markets described by RFRs and forward term rates, explicitly allowing for stochastic discontinuities in their dynamics. At the time of writing, there is no consensus on the choice of forward term rates that will replace Libor rates. Therefore, we postulate general dynamics for an arbitrary family of forward term rates and consider them jointly with RFRs. Inspired by the extended Heath-Jarrow-Morton setup of [START_REF] Fontana | Term structure modelling for multiple curves with stochastic discontinuities[END_REF], we characterize absence of arbitrage by means of generalized drift conditions, with specific no-arbitrage restrictions related to the stochastic discontinuity dates. Moreover, we address the issue of compatibility between a forward term rate and the Libor fallback represented by the RFR compounded over the corresponding tenor. This leads to the study of a linear BSDE with stochastic discontinuities, whose driver is determined by the no-arbitrage restrictions. In general, such a BSDE may admit multiple solutions and we provide sufficient conditions ensuring uniqueness of the solution.

Towards practical applications, we provide a tractable specification based on affine semimartingales, which generalize affine processes beyond stochastic continuity (see [START_REF] Keller-Ressel | Affine processes beyond stochastic continuity[END_REF]). We introduce affine semimartingale models for an overnight rate, showing that they provide a natural extension of classical short-rate models based on affine processes to the case of stochastic discontinuities. As illustrated by a simple example, this class of models allows reproducing several stylized features of overnight rates, in particular spikes and jumps at fixed times. Moreover, we derive explicit pricing formulas for bonds and caplets in an extended Hull-White model with discontinuities.

Finally, we study the hedging of derivatives related to RFRs (or, more generally, derivatives written on Libor fallbacks determined by RFRs). The presence of stochastic discontinuities generates market incompleteness and, therefore, we resort to local risk-minimization. We show that the locally risk-minimizing strategy admits a decomposition into two components: a dynamic continuous-time strategy representing the Delta-hedging strategy, and an additional component that optimally rebalances the portfolio in correspondence to the stochastic discontinuity dates. We exemplify this result by considering the problem of hedging a SOFR-caplet by trading in a SOFR futures contract, which is the most liquidly traded contract written on SOFR at the time of writing.

Related literature. The reform of benchmark interest rates is receiving considerable attention from the financial community and, therefore, we limit this overview of the literature to some contributions that are specifically related to our work, referring to [START_REF] Henrard | LIBOR fallback and quantitative finance[END_REF], [START_REF] Klingler | Life after LIBOR[END_REF] and [START_REF] Piterbarg | Interest rates benchmark reform and options markets[END_REF] for a general analysis of the challenges of the Libor reform. From the perspective of RFR modelling, one of the first and most influential contributions is [START_REF] Lyashenko | Looking forward to backward-looking rates: a modeling framework for term rates replacing LIBOR[END_REF], where the classical Libor market model is extended to backward-looking rates (i.e., compounded RFRs). Several authors have extended classical short-rate models to RFRs. In this direction, one of the first contributions is [START_REF] Mercurio | A simple multi-curve model for pricing SOFR futures and other derivatives[END_REF], who develops a short rate model for SOFR by adding a deterministic spread to the OIS rate. The Hull-White model has been applied to RFRs in [START_REF] Hofman | Implied volatilities for options on backward-looking term rates[END_REF] and [START_REF] Turfus | Caplet pricing with backward-looking rates[END_REF]. More recently, [START_REF] Skov | Dynamic term structure models for SOFR futures[END_REF] have proposed a multi-factor Gaussian short-rate model in order to describe the SOFR futures market, while [START_REF] Fontana | Caplet pricing in affine models for risk-free rates[END_REF] formulates a short-rate model for RFRs based on a general affine process in view of pricing applications. Always in a short-rate perspective, [START_REF] Rutkowski | Pricing and hedging of SOFR derivatives under differential funding costs and collateralization[END_REF] propose a Vasiček joint model for SOFR and other reference rates and study the hedging of SOFR-based derivatives, also in the presence of funding costs and collateralization. A different approach is taken by [START_REF] Macrina | Rational savings account models for backward-looking interest rate benchmarks[END_REF], who adopt a linear rational model for the savings account and derive several explicit pricing formulas. We also mention [START_REF] Willems | SABR smiles for RFR caplets[END_REF], where an extended SABR model has been applied to the pricing of caplets in the post-Libor universe.

The papers mentioned in the previous paragraph do not consider the possible presence of stochastic discontinuities in the RFR dynamics. This phenomenon is however playing an increasingly important role in several recent works. In particular, [START_REF] Andersen | Spike modeling for interest rate derivatives with an application to SOFR caplets[END_REF] develop a model that can generate spikes in the SOFR dynamics, both at totally inaccessible times and at anticipated times. [START_REF] Gellert | Short rate dynamics: A fed funds and SOFR perspective[END_REF] show that a diffusive HJM model for instantaneous forward rates is compatible with the presence of jumps/spikes at fixed times in the short rate, consistently with the empirical evidence on SOFR. In the recent work [START_REF] Backwell | Expected and unexpected jumps in the overnight rate: consistent management of the Libor transition[END_REF], the SONIA rate is modelled via a short-rate approach by relying on a pure jump process with predetermined jumps times. Let us also mention that stochastic discontinuities also play an important role in credit markets (see [START_REF] Gehmlich | Dynamic defaultable term structure modelling beyond the intensity paradigm[END_REF], [START_REF] Fontana | General dynamic term structures under default risk[END_REF]), while a general view on modelling multiple yield curve markets with stochastic discontinuities is given in [START_REF] Fontana | Term structure modelling for multiple curves with stochastic discontinuities[END_REF].

Structure of the paper. We start in Section 2 with a general view on interest rate markets based on overnight rates. In Section 3, we introduce a general framework for the joint modelling of RFRs and forward term rates, also characterizing the compatibility between forward term rates and setting-inarrears rates by means of BSDEs. In Section 4, we introduce a model based on affine semimartingales. Finally, in Section 5 we study locally risk-minimizing hedging strategies in the presence of stochastic discontinuities. Some of the more technical proofs are postponed to the appendix.

Term structure modelling with overnight rates

Modelling interest rate markets in the presence of overnight rates starts from the numéraire obtained by investing in the overnight rate. Investing in an overnight rate implies that the numéraire is generated by a roll-over strategy and as such is piecewise constant and jumps at predetermined, in this case even deterministic, dates. More generally, we do not assume that the numéraire is constant between successive jump times, thus covering classical interest rate models as a special case. We let pΩ, F , F, Qq be a stochastic basis supporting the processes introduced below.

We fix a time-grid 0 ă t 1 ă ¨¨¨ă t N of finitely many deterministic times at which jumps in the numéraire are expected and we define as follows the measure η on R `:

ηpAq " ż A dt `N ÿ i"1 δ ti pAq, for all A P BpR `q, (2.1)
with BpR `q denoting the Borel sigma-algebra of R `. The numéraire is assumed to be a strictly positive semimartingale S 0 " pS 0 t q tě0 of the form

S 0 " exp ˆż 0 ρ t ηpdtq ˙, (2.2)
where ρ " pρ t q tě0 is an adapted process such that ş T 0 |ρ t |dt ă 8 a.s. for all T ą 0. We will refer to ρ as risk-free rate (RFR).

Remark 2.1 (Classical interest rate models). Clearly, the above framework includes classical interest rate models without stochastic discontinuities, where the numéraire S 0 is obtained as the continuoustime limit of a roll-over strategy. This case can be recovered by setting N " 0 in (2.1), thus yielding S 0 " expp ş 0 ρ t dtq in (2.2), with ρ representing the risk-free short rate.

2.1. Notions of interest rates. In the following, we recall several notions of interest rates which are important in post-LIBOR markets, relying mostly on [START_REF] Lyashenko | Looking forward to backward-looking rates: a modeling framework for term rates replacing LIBOR[END_REF]. We consider a finite family D " tδ 1 , . . . , δ m u of tenors, with δ 1 ă . . . ă δ m , for some m P N. In market practice, the elements of D are typically represented by t1D, 1W, 1M, 2M, 3M, 6M, 1Yu.

Backward-looking and forward-looking rates. For each T ě 0 and δ P D, the compounded setting-in-arrears rate RpT, T `δq for the time interval rT, T `δs is defined as

RpT, T `δq :" 1 δ ´S0 T `δ S 0 T ´1¯.
(2.3)

In our setup, as a consequence of (2.2), we have that RpT, T `δq " 1 δ ´eş pT ,T `δs ρtηpdtq ´1¯.

The setting-in-arrears rate is backward-looking, since its value is only known at the end of the accrual period rT, T `δs (in other words, RpT, T `δq is F T `δ -measurable and not F T -measurable).

The forward-looking rate F pT, T `δq is defined as the fixed rate K such that the swap which delivers the payoff δpRpT, T `δq ´Kq at maturity T `δ has zero value at time T . Note that, differently from the backward-looking rate, the forward-looking rate F pT, T `δq is determined at the beginning of the accrual period (i.e., it is F T -measurable).

Backward-looking and forward-looking forward rates. For T ě 0, δ P D and t P r0, T `δs, the backward-looking forward rate Rpt, T, δq is defined as the fixed rate K such that the swap which delivers the payoff δpRpT, T `δq ´Kq at maturity T `δ has zero value at time t. Notice that, by definition, it holds that RpT, T, δq " F pT, T `δq.

By analogy, we define the forward-looking forward rate F pt, T, δq as the fixed rate K such that the swap which delivers the payoff δpF pT, T `δq´Kq at maturity T `δ has zero value at time t, for T ě 0, δ P D and t P r0, T s. Clearly, the forward-looking forward rate satisfies F pT, T, δq " F pT, T `δq.

By comparing the notions of backward-looking and forward-looking forward rate, we notice that Rpt, T, δq " F pt, T, δq, for all t P r0, T s.

(2.4) However, while the forward-looking forward rate F p¨, T, δq stops evolving at time T , the backwardlooking forward rate Rp¨, T, δq continues to evolve until T `δ, when it reaches the terminal condition

RpT `δ, T, δq " RpT, T `δq.

Remark 2.2. As pointed out in [START_REF] Lyashenko | Looking forward to backward-looking rates: a modeling framework for term rates replacing LIBOR[END_REF], identity (2.4) implies that backwardlooking forward rates and forward-looking forward rates can be consolidated into a single process Rp¨, T, δq. We adopt this approach and generically call Rp¨, T, δq a forward term rate. Depending on whether a forward term rate is considered on r0, T s or on r0, T `δs, it takes the meaning of a forward-looking forward rate or of a backward-looking forward rate, respectively. 2 2.2. Key ingredients of a term structure model. We first observe that the swaps considered in Section 2.1 are actually forward rate agreements (FRAs), since they involve a single cashflow at maturity. For T ě 0, δ P D, t P r0, T `δs and K P R, let us denote by Πpt, T, T `δ, Kq the price at time t of a FRA delivering the cashflow δpRpT, T `δq ´Kq at time T `δ. In the following, we will assume that FRA prices are determined by a linear pricing functional (more precisely, affine in K). This assumption is standard in interest rate modelling. In the following remark, we show that zero-coupon bond (ZCB) prices can be inferred from FRA prices.

Remark 2.3. We first remark that ZCBs can be replicated statically from FRAs. Indeed, a static strategy consisting of a long position in a FRA with arbitrary fixed rate K and maturity T `δ and a short position in a FRA with fixed rate K `1{δ and maturity T `δ delivers the constant payoff 1 at time T `δ. Therefore, the arbitrage-free price P pt, T `δq of a ZCB with maturity T `δ must satisfy P pt, T `δq " Πpt, T, T `δ, Kq ´Πpt, T, T `δ, K `1{δq, for all 0 ď t ď T `δ and δ P D. Since the map K Þ Ñ Πpt, T, T `δ, Kq is affine, it holds that Πpt, T, T `δ, Kq " apt, T, T `δq ´δKP pt, T `δq,

2 Abstracting from a specific definition of the forward term rate has the additional advantage of making our modelling framework applicable to any Libor fallback rate, including the proposals currently under discussion in the market.

for some apt, T, T `δq. Recalling the definition of backward-looking forward rate, we obtain that 0 " apt, T, T `δq ´δRpt, T, δqP pt, T `δq and, hence, Πpt, T, T `δ, Kq " δP pt, T `δq `Rpt, T, δq ´K˘.

(2.5) Equation (2.5) shows the key ingredients of a term structure model in the presence of overnight rates: zero-coupon bond prices tP p¨, T q; T P R `u and forward term rates tRp¨, T, δq; pT, δq P R `ˆDu.

Throughout the paper, we say that the reference probability measure Q is a risk-neutral measure if the processes P p¨, T q{S 0 and P p¨, T `δqRp¨, T, δq{S 0 are local martingales under Q. This suffices to ensure that the large financial market composed of ZCBs for all maturities and FRAs for all maturities and tenors is arbitrage-free in the sense of no asymptotic free lunch with vanishing risk (see (Fontana et al., 2020, Section 6) for a thorough analysis of this condition in a related market setting).

Term structure dynamics of bonds and forward term rates

In this section, we develop a general term structure model in the presence of overnight rates as well as forward term rates associated to a family of tenors D " tδ 1 , . . . , δ m u, with δ 1 ă . . . ă δ m , for some m P N. We introduce the following two families of fixed discontinuity dates: T " tt 1 , . . . , t N u, with t 1 ă . . . ă t N , representing the discontinuities in the numéraire process S 0 and corresponding to the atoms of the measure η introduced in (2.1), and S " ts 1 , . . . , s M u, with s 1 ă . . . ă s M , representing a set of deterministic times at which the RFR and forward term rates are expected to exhibit jumps.

Note that we do not exclude the case S X T ‰ H, meaning that stochastic discontinuities in the term structure of RFRs and term rates can occur simultaneously to the jump times of the numéraire.

Remark 3.1 (Extension to predictable times). The results presented in this section are also valid in the more general setting where S is a countable family of predictable times, see [START_REF] Fontana | General dynamic term structures under default risk[END_REF]. For simplicity of presentation and in order to treat S with the same techniques used for T , we suppose that S is a finite family of fixed dates.

3.1. An extended Heath-Jarrow-Morton framework. This section is devoted to the development of a general framework for term structure modelling when stochastic discontinuities are present. In Theorem 3.8 we state generalized drift conditions characterizing the risk-neutral property of Q.

We work on the stochastic basis pΩ, F , Qq endowed with a filtration F " pF t q tě0 satisfying the usual conditions and supporting a d-dimensional Brownian motion W " pW t q tě0 and an integervalued random measure µpdt, dxq on R `ˆE, with compensator νpdt, dxq " λ t pdxqdt, where λ t pdxq is a kernel from pΩˆR `, Pq into pE, BpEqq, with P denoting the predictable sigma-field on ΩˆR `and pE, BpEqq a Polish space with its Borel sigma-field. The compensated random measure is denoted by μpdt, dxq :" µpdt, dxq ´νpdt, dxq. We refer to [START_REF] Jacod | Limit Theorems for Stochastic Processes[END_REF] for all unexplained notions related to stochastic calculus.

As mentioned in Section 2.2, the two ingredients of a term structure model are zero coupon bond (ZCB) prices and forward term rates. The specification of forward term rates is postponed to equation (3.14). For the ZCBs, we work in a Heath-Jarrow-Morton (HJM) approach with stochastic discontinuities by assuming that

P pt, T q " exp ´´ż pt,T s f pt, uqηpduq ¯,
for all 0 ď t ď T.

(3.1)

We always consider finite maturities (i.e., T ă 8) without further explicit mentioning and adopt the convention that ş pT,T s f pT, uqηpduq " 0. The process f p¨, T q is the instantaneous forward rate and we assume that

f pt, T q " f p0, T q `ż t 0 αps, T qds `ż t 0 ϕps, T qdW s `ż t 0 ż E ψps, x, T qμpds, dxq `V pt, T q, (3.2)
for 0 ď t ď T , where V p¨, T q is a pure jump adapted process such that t∆V p¨, T q ‰ 0u Ď Ω ˆS.

Assumption 3.2. The following conditions hold a.s.: (i) the initial forward curve T Ñ f p0, T q is pF 0 b BpR `qq-measurable, real-valued and satisfies ş T 0 |f p0, uq|du ă 8, for all T ą 0, (ii) the drift process α : Ω ˆR2 Ñ R is progressively measurable 3 , satisfies αpt, T q " 0 for 0 ď T ă t, and ż T 0 ż u 0 |αps, uq|dsηpduq ă 8, for all T ą 0, (iii) the diffusive volatility process ϕ : ΩˆR 2 `Ñ R d is progressively measurable and satisfies ϕpt, T q " 0 for 0 ď T ă t, and

d ÿ i"1 ż T 0 ˆż u 0 |ϕ i ps, uq| 2 ds

˙1{2

ηpduq ă 8, for all T ą 0, (iv) the jump function ψ : ΩˆR 2 `ˆE Ñ R is a pP bBpEqbBpR `qq-measurable function satisfying ψpt, x, T q " 0 for 0 ď T ă t and x P E, and

ż T 0 ż E ż E 0 |ψps, x, uq| 2 ηpduqλ s pdxqds ă 8.
(v) the stochastic discontinuity process V p¨, T q " pV pt, T qq tPr0,T s satisfies ş T 0 |∆V ps, uq|du ă 8 for all s P S and ∆V pt, T q " 0 for all 0 ď T ă t ă 8.

As a consequence of Assumption 3.2, all integrals appearing in the forward rate equation (3.2) are well-defined for η-a.e. T ě 0. Moreover, the integrability requirements in parts (ii)-(iii)-(iv) of Assumption 3.2 ensure that ordinary and stochastic Fubini theorems can be applied, in the versions of (Veraar, 2012, Theorem 2.2) for Brownian integrals and (Björk et al., 1997, Proposition A.2) for stochastic integrals with respect to the compensated random measure μ.

For all 0 ď t ď T and x P E, we define ᾱpt, T q :" ż rt,T s αpt, uqηpduq, φpt, T q :" ż rt,T s ϕpt, uqηpduq, ψpt, x, T q :" ż rt,T s ψpt, x, uqηpduq,

V pt, T q :" ż rt,T s

∆V pt, uqηpduq.

In the proof of the main result of this section, we shall make use of a stochastic exponential representation of S 0 -discounted ZCB prices. This is the content of the following lemma. Lemma 3.3. Suppose that Assumption 3.2 holds. Then, for every T ě 0, P p¨, T q S 0 " P p0, T qE ˆ´ż 0 ᾱps, T qds `1 2 ż 0 } φps, T q} 2 ds ´ż 0 φps, T qdW s ´ż 0 ż E ψps, x, T qμpds, dxq `ż 0 ż E `e´ψps,x,T q ´1 `ψps, x, T q ˘µpds, dxq `ż 0 `f pu, uq ´ρu ˘du `ÿ τ PSYT `e´V pτ,T qδ S pτ q`pf pτ,τ q´ρτ qδT pτ q ´1˘1 rrτ,8rr

˙.

(3.3)

Proof. Similarly as in the proof of (Fontana et al., 2020, Lemma 3.5), Assumption 3.2 ensures that

F pt, T q :" ż pt,T s f pt, uqηpduq " ż T 0 f p0, uqηpduq `ż t 0 ᾱps, T qds `ż t 0 φps, T qdW s `M ÿ i"1 V ps i , T q1 tsiďtu `ż t 0 ż E ψps, x, T qμpds, dxq ´ż t 0 f pu, uqηpduq ": Gpt, T q, (3.4)
for all 0 ď t ă T . Since F pT, T q " 0, to prove that (3.4) holds also for t " T , it suffices to show that ∆GpT, T q :" GpT, T q ´GpT ´, T q " ´F pT ´, T q. Moreover, since µptT u, Eq " 0 a.s. for all T P R `, it is enough to show that ∆GpT, T q " ´F pT ´, T q for all T P S Y T . This holds since ∆GpT, T q " V pT, T qδ S pT q ´f pT, T qδ T pT q " `V pT, T qδ S pT q ´f pT, T q ˘δT pT q " ´f pT ´, T qδ T pT q " ´F pT ´, T q, as a consequence of the definition of F pt, T q. Therefore,

P pt, T q S 0 t " exp ˆ´Gpt, T q ´ż t 0 ρ u ηpduq ˙.
Representation (3.3) then follows from (Jacod and Shiryaev, 2003, Theorem II.8.10).

We aim at characterizing when Q is a risk-neutral measure. To this end, we first characterize the local martingale property of discounted ZCB prices. The proof of the following proposition follows the lines of the proof of (Fontana et al., 2020, Theorem 3.7). Given that the modelling quantities in [START_REF] Fontana | Term structure modelling for multiple curves with stochastic discontinuities[END_REF] are different from those employed in the present setup and the stochastic discontinuities in the numéraire are different from the stochastic discontinuities in the forward rates f pt, T q, which is not the case in [START_REF] Fontana | Term structure modelling for multiple curves with stochastic discontinuities[END_REF], we give a self-contained proof.

Proposition 3.4. Suppose that Assumption 3.2 holds. Then, for every T ě 0, P p¨, T q{S 0 is a local martingale if and only if ż T 0 ż E ˇˇe ´ψps,x,T q ´1 `ψps, x, T qq ˇˇλ s pdxqds ă 8 a.s. for all T ě 0, (3.5)

and the random variable e ´şpτ,T s ∆V pτ,uqηpduqδ S pτ q´∆ρτ δ T pτ q (3.6) is sigma-integrable with respect to F τ ´, for every τ P S Y T , and (i) f pt, tq " ρ t pQ b dtq-a.e., (ii) on r0, T s, it holds pQ b dtq-a.e. that ᾱpt, T q " 1 2 } φpt, T q} 2 `żE `e´ψpt,x,T q ´1 `ψpt, x, T q ˘λt pdxq, (iii) for every j " 1, . . . , N , it holds a.s. that f pt j ´, t j q " ρ tj ´´log `Ere ´∆ρt j |F tj ´s˘, (iv) for every i " 1, . . . , M , it holds a.s. that E " e ´∆ρs i δ T psiq `e´ş ps i ,T s ∆V psi,uqηpduq ´1˘ˇˇˇF si´ı " 0.

Proof. In view of (3.3), P p¨, T q{S 0 is a local martingale if and only if the finite variation process Kp¨, T q :" ż 0 kps, T qds `Kp1q p¨, T q `Kp2q p¨, T q (3.7) is a local martingale, where kpt, T q :" f pt, tq ´ρt ´ᾱpt, T q `1 2 } φpt, T q} 2 , K p1q pt, T q :"

ż t 0 ż E `e´ψps,x,T q ´1 `ψps, x, T q ˘µpds, dxq, K p2q pt, T q :" ÿ τ PSYT
`e´V pτ,T qδ S pτ q`pf pτ,τ q´ρτ qδT pτ q ´1˘1 tτ ďtu , for all t P r0, T s. If Kp¨, T q is a local martingale, it is also of locally integrable variation by (Jacod and Shiryaev, 2003, Lemma I.3.11). Since t∆K p1q p¨, T q ‰ 0u X t∆K p2q p¨, T q ‰ 0u " H, it holds that |∆K piq p¨, T q| ď |∆Kp¨, T q|, for i " 1, 2, and, therefore, both processes K piq p¨, T q, i " 1, 2, are of locally integrable variation. This implies that condition (3.5) holds and the random variable e ´V pτ,T qδ S pτ q`pf pτ,τ q´ρτ qδT pτ q is sigma-integrable with respect to F τ ´, for all τ P S Y T (see (He et al., 1992, Theorem 5.28)).

Taking into account the definition of V pt, T q, equation (3.2) and the fact that pf pτ ´, τ q ´ρτ´q δ T pτ q is F τ ´-measurable, the latter property is equivalent to the sigma-integrability of the random variable (3.6). Denoting by p K piq p¨, T q the compensator of K piq p¨, T q, for i " 1, 2, and making use of (He et al., 1992, Theorem 5.29), the local martingale property of Kp¨, T q is then equivalent to the validity of 0 " ż 0 kps, T qds `p K p1q p¨, T q " ż 0 ˆkps, T q `żE `e´ψps,x,T q ´1 `ψps, x, T q ˘λs pdxq ˙ds, (3.8)

0 " p K p2q p¨, T q " ÿ τ PSYT
´E"`e ´V pτ,T qδ S pτ q`pf pτ,τ q´ρτ qδT pτ q ˘ˇF τ ´‰ ´1¯1 rrτ,8rr , (3.9) up to an evanescent set. Equation (3.8) holds if and only if

f pt, tq ´ρt ´ᾱpt, T q `1 2 } φpt, T q} 2 `żE `e´ψpt,x,T q ´1 `ψpt, x, T q ˘λt pdxq " 0 (3.10)
outside of a set of pQ b dtq-measure zero. Taking T " t in (3.10) yields condition (i), while condition (ii) follows by inserting condition (i) into (3.10). In view of (3.2) and of the definition of V pt, T q, equation (3.9) holds if and only if, for all τ P S Y T , it holds that e pf pτ ´,τ q´ρτ´qδ T pτ q E " e ´şpτ,T s ∆V pτ,uqηpduqδ S pτ q´∆ρτ δ T pτ q ˇˇFτ´ı " 1 a.s.

(3.11)

Taking T " τ in (3.11) directly yields condition (iii), while condition (iv) is then obtained by inserting condition (iv) into (3.11).

Conversely, if condition (3.5) is satisfied and the random variable in (3.6) is sigma-integrable, for every j " 1, . . . , M , then the compensators p K piq p¨, T q are well-defined, for i " 1, 2. It is then straightforward to verify that if conditions (i)-(iv) are satisfied, then equations (3.8)-(3.9) hold true. This implies that the process Kp¨, T q given in (3.7) is a local martingale, thus proving the local martingale property of P p¨, T q{S 0 , for every T P R `.

In the absence of stochastic discontinuities, it is well-known that P p¨, T q{S 0 is a local martingale if and only if the integrability requirement (3.5) and conditions (i)-(ii) of Proposition 3.4 hold, see (Björk et al., 1997, Proposition 5.3). The presence of stochastic discontinuities is reflected in conditions (iii)-(iv) (together with the requirement of sigma-integrability of (3.6)). More specifically, condition (iii) relates the stochastic discontinuities T in the numéraire to the short end of the forward rate, while condition (iv) concerns the stochastic discontinuities S in the forward term rate. Taken together, conditions (iii)-(iv) exclude the possibility of predicting the size (or the direction) of the jump occurring at any discontinuity date, as this would be incompatible with absence of arbitrage (see [START_REF] Fontana | On the existence of sure profits via flash strategies[END_REF] for an analysis of arbitrages possibilities arising with predictable jumps).

The conditions of Proposition 3.4 admit a simplification under the following additional assumption.

Assumption 3.5. The set tpω, tq P Ω ˆT : ∆ρ t pωq ‰ 0u is evanescent and S X T " H.

This assumption corresponds to requiring that the RFR ρ and the forward term rates do not jump in correspondence of the time grids of the numéraire S 0 . In this case, conditions (i) and (iii) can be rewritten in the following compact way

f pt, tq " ρ t pQ b ηq-a.e.
Moreover, under Assumption 3.5 the term ∆ρ si δ T ps i q in condition (iv) vanishes.

Remark 3.6. For later use, we observe that under the conditions of Proposition 3.4 the stochastic exponential representation (3.3) of discounted ZCB prices takes the following form:

P p¨, T q S 0 " P p0, T qE ˆ´ż 0 φps, T qdW s `ż 0 ż E `e´ψps,x,T q ´1˘μ pds, dxq
`ÿ τ PSYT `e´V pτ,T qδ S pτ q`pf pτ,τ q´ρτ qδT pτ q ´1˘1

rrτ,8rr

˙.

(3.12) Under Assumption 3.5, formula (3.12) can be further simplified as follows:

P p¨, T q S 0 " P p0, T qE ˆ´ż 0 φps, T qdW s `ż 0 ż E `e´ψps,x,T q ´1˘μ pds, dxq `M ÿ i"1 `e´V psi,T q ´1˘1 rrsi,8rr
˙.

(3.13)

We complete the description of the model by postulating the following dynamics for forward term rates:

Rpt, T, δq " Rp0, T, δq `ż t 0 α R ps, T, δqds `ż t 0 ϕ R ps, T, δqdW s `ż t 0 ż E ψ R ps, x, T, δqμpds, dxq `V R pt, T, δq, (3.14)
for all 0 ď t ď T ă 8 and δ P D, where V R p¨, T, δq is a pure jump adapted process such that t∆V R p¨, T, δq ‰ 0u Ď Ω ˆpS X r0, T sq.

In the following assumption, we impose minimal technical requirements ensuring the well-posedness of all integrals in (3.14). We recall that L 2 loc pr0, T sq denotes the space of all R d -valued progressively measurable processes h such that ş T 0 }h t } 2 dt ă 8 a.s. and G loc pr0, T sq the space of all pP b BpEqqmeasurable functions g : Ω ˆr0, T s ˆE Ñ R such that ş T 0 ş E ppgps, xq 2 ^|gps, xq|qλ s pdxqds ă 8. Assumption 3.7. For all T ě 0 and δ P D, the following holds:

(i) α R p¨, T, δq is a real-valued adapted process such that ş T 0 |α R ps, T, δq|ds ă 8 a.s.; (ii) ϕ R p¨, T, δq P L 2 loc pr0, T sq; (iii) ψ R p¨, ¨, T, δq P G loc pr0, T sq.
The next theorem provides necessary and sufficient conditions for the reference probability Q to be a risk-neutral measure. It can be regarded as a counterpart to (Fontana et al., 2020, Theorem 4.1), which addresses the case of a multi-curve market model where the modelling quantities are forward Ibor rates.

Theorem 3.8. Suppose that Assumptions 3.2 and 3.7 hold. Then Q is a risk-neutral measure if and only if all conditions of Proposition 3.4 are satisfied and for all T ě 0 and δ P D:

ż T 0 ż E ˇˇψ R pt, x, T, δq `e´ψpt,x,T `δq ´1˘ˇˇλ t pdxqdt ă 8 a.s., (3.15)
and the random variable ∆V R ps i , T, δqe ´şps i ,T `δs ∆V psi,uqηpduq´∆ρs i δ T psiq (3.16)

is sigma-integrable with respect to F si´, for every i " 1, . . . , M , together with the following conditions: (i) on r0, T s it holds pQ b dtq-a.e. that α R pt, T, δq " φpt, T `δq J ϕ R pt, T, δq `żE ψ R pt, x, T, δq `1 ´e´ψpt,x,T `δq ˘λt pdxq, (ii) for every i " 1, . . . , M , it holds a.s. that E " ∆V R ps i , T, δqe ´şps i ,T `δs ∆V psi,uqηpduq´∆ρs i δ T psiq ˇˇFs i´ı " 0.

Proof. Recall that Q is a risk-neutral measure if and only if P p¨, T q{S 0 and Rp¨, T, δqP p¨, T `δq{S 0 are Q-local martingales, for all T ě 0 and δ P D. In view of Proposition 3.4, to prove the theorem it suffices to characterize the local martingale property of Rp¨, T, δqP p¨, T `δq{S 0 , assuming that P p¨, T `δq{S 0 is a local martingale, for all T ě 0 and δ P D. Making use of (3.12) and (3.14), the product rule yields

d ˆRpt, T, δq P pt, T `δq S 0 t ˙" Rpt´, T, δqd P pt, T `δq S 0 t `P pt´, T `δq S 0 t´d Rpt, T, δq `d " Rp¨, T, δq, P p¨, T `δq S 0  t « P pt´, T `δq S 0 t´`α R pt, T, δqdt ´φpt, T `δq J ϕ R pt, T, δq ˘dt `P pt´, T `δq S 0 t ż E ψ R pt, x, T, δq `e´ψpt,s,T `δq ´1˘µ pds, dxq `P pt´, T `δq S 0 t´∆ V R pt, T, δqe ´V pt,T `δq`pf pt,tq´ρtqδ T ptq δ S pdtq ": dK R pt, T, δq,
where « denotes equality up to local martingale terms. This shows that Rp¨, T, δqP p¨, T `δq{S 0 is a local martingale if and only if the finite variation process K R p¨, T, δq is a local martingale. Arguing similarly as in the proof of Proposition 3.4, this holds if and only if K R p¨, T, δq is of locally integrable variation and its compensator p K R p¨, T, δq is null (up to an evanescent set). The fact that K R p¨, T, δq is of locally integrable variation is equivalent to the validity of (3.15) together with the σ-integrability of the random variable in (3.16), for all j " 1, . . . , M . In this case, the compensator p K R p¨, T, δq is well-defined and, by arguments analogous to those used in the proof of Proposition 3.4, it can be verified that p K R p¨, T, δq " 0 holds if and only if conditions (i)-(ii) are satisfied.

3.2. Compatibility between backward-looking forward rates and setting-in-arrears rates.

In Section 3.1, we have provided necessary and sufficient conditions for Q to be a risk-neutral measure in a market described by ZCBs and forward-looking forward rates (see Section 2). Since no relation has been assumed in Section 3.1 between the forward term rates and the RFR, this corresponds to considering the dynamics (3.14) of Rp¨, T, δq on the interval r0, T s (see Remark 2.2). The analysis of backward-looking forward rates requires to consider two additional aspects:

(i) a backward-looking forward rate continues to evolve until the end of the accrual period;

(ii) at the end of the accrual period, the backward-looking forward rate must coincide with the setting-in-arrears rate.

While the first aspect can be addressed by simply extending the dynamics (3.14) up to time T `δ, the second aspect imposes an exogenous terminal condition on the forward term rate process. More specifically, the backward-looking forward rate Rp¨, T, δq " pRpt, T, δqq tPr0,T `δs , introduced in Section 2.1, must satisfy the following terminal condition:

RpT `δ, T, δq " RpT, T `δq.

(3.17)

In this section, we show how the language of BSDEs provides the appropriate framework to characterize the risk-neutral property of Q in a market described by ZCBs and backward-looking forward rates. We assume the validity of the following martingale representation property. Assumption 3.9. There exists a family pξ 1 , . . . , ξ M q of random variables on pΩ, F , Qq taking values in pB, BpBqq such that ξ i is F si -measurable, for each i " 1, . . . , M , and, for all T P R `, every local martingale N " pN t q tPr0,T s on pΩ, F, Qq admits a representation of the following form:

N " N 0 `ż 0 θ t dW t `ż 0 ż E γpt, xqμpdt, dxq `M ÿ i"1 f i pξ i q1 rrsi,T ss , (3.18)
where θ P L 2 loc pr0, T sq, γ P G loc pr0, T sq, and, for all i " 1, . . . , M , f i p¨q : ΩˆB Ñ R is a pF si´b BpBqqmeasurable function such that Erf i pξ i q|F si´s " 0 a.s.

We denote by H the space of all functions f " pf 1 , . . . , f M q satisfying these properties.

Notice that Assumption 3.9 is satisfied if the random measure µ is a multivariate point process and the filtration F is generated by W , the random measure µ and the step process

ř M i"1 ξ i 1 rrsi,8rr .
Definition 3.10. For T ě 0 and δ P D, a special semimartingale Rp¨, T, δq " pRpt, T, δqq tPr0,T `δs is said to be compatible if Rp¨, T, δqP p¨, T `δq{S 0 is a local martingale and condition (3.17) is satisfied.

The next theorem characterizes the notion of compatibility in terms of the solution to a linear BSDE with jumps and stochastic discontinuities. For each T P R `, we denote by S p pr0, T sq the space of special semimartingales X " pX t q tPr0,T s on pΩ, F, Qq. Theorem 3.11. Suppose that the hypotheses of Proposition 3.4 and Assumptions 3.5 and 3.9 hold and consider T ě 0 and δ P D. A forward term rate process Rp¨,

w i pξ i qδ si pdtq, Y T `δ " RpT, T `δq. (3.19)
Proof. Suppose first that Rp¨, T, δq is compatible. Since Rp¨, T, δq P S p pr0, T `δsq, it admits a unique decomposition Rp¨, T, δq " Rp0, T, δq `A `N , where A is a predictable process of finite variation and N a local martingale, with A 0 " N 0 " 0. By Assumption 3.9, it holds that

N " ż 0 z t dW t `ż 0 ż E vpt, xqμpdt, dxq `M ÿ i"1 w i pξ i q1 rrsi,T `δss ,
for some pz, v, wq P L 2 loc pr0, T `δsq ˆGloc pr0, T `δsq ˆH. Since Assumption 3.5 holds, S 0 -discounted ZCB prices are given by (3.13). An application of the product rule then yields and that the random variable pe ´V psi,T `δq ´1qw i pξ i q is sigma-integrable with respect to F si´, for all i " 1, . . . , M . In particular, this implies that the driver of the BSDE (3.19) is well-defined. The process K R p¨, T, δq being of locally integrable variation and also a local martingale, its compensator must be null. In view of the definition of K R p¨, T, δq above, this means that, up to an evanescent set,

d ˆRpt, T,
dA t " ˆφpt, T `δq J z t `żE `1 ´e´ψpt,T `δ,xq ˘vpt, xqλ t pdxq ˙dt´M ÿ i"1 E " e ´V psi,T `δq w i pξ i q ˇˇF si´‰ δ si pdtq,
where we have used the fact that Erw i pξ i q|F si´s " 0 a.s., for all i " 1, . . . , M , since w P H. By compatibility, Rp¨, T, δq satisfies the terminal condition (3.17). We have therefore shown that pRp¨, T, δq, z, v, wq is a solution to the BSDE (3.19).

Conversely, suppose that pY, z, v, wq is a solution to (3.19). In this case, it holds that E " ∆Y e ´V psi,T `δq ˇˇF si´‰ " ´E" e ´V psi,T `δq ˇˇF si´‰ E " e ´V psi,T `δq w i pξ i q ˇˇF si´‰ `E" w i pξqe ´V psi,T `δq ˇˇF si´‰ " 0, since Ere ´V psi,T `δq |F si´s " 1 as a consequence of Proposition 3.4 together with Assumption 3.5. Therefore, considering the structure of the driver of the BSDE (3.19), we see that conditions (i)-(ii) of Theorem 3.8 are satisfied on the time interval r0, T `δs. This implies that Y P p¨, T `δq{S 0 is a local martingale. Since Y T `δ " RpT, T `δq by (3.19), it follows that the process Y is compatible.

Remark 3.12. We mention that the hypotheses of Theorem 3.11 can be weakened by removing Assumption 3.5 and allowing for the presence of the stochastic discontinuities T " tt 1 , . . . , t N u in the martingale representation (3.18). In this case, Theorem 3.11 takes an analogous form, but requires a heavier notation due to the possible intersection between the two sets S and T of discontinuity dates.

In general, there does not exist a unique specification of compatible forward term rates. Equivalently, the BSDE (3.19) does not have a unique solution in the space S p pr0, T `δsq ˆL2 loc pr0, T δsq ˆGloc pr0, T `δsq ˆH considered in Theorem 3.11. This is illustrated by the following remark.

Remark 3.13. (i) Similarly as in [START_REF] Lyashenko | Looking forward to backward-looking rates: a modeling framework for term rates replacing LIBOR[END_REF], ZCB prices can be extended beyond their maturity by setting P pt, T q " S 0 t {S 0 T , for 0 ď T ă t. Letting Rpt, T, δq :"

1 δ ˆP pt, T q P pt, T `δq ´1˙, 0 ď t ď T `δ, (3.20)
always generates a compatible process Rp¨, T, δq as long as it is a special semimartingale. Definition (3.20) corresponds to the usual construction of simply compounded forward rates (see, e.g., (Musiela and Rutkowski, 1997, Chapter 9)).

(ii) Suppose that 1{S 0 T P L 1 pQq, for all T ą 0, and define the process M by

M t :" E " 1 S 0 T ´1 S 0 T `δ ˇˇˇF t  , 0 ď t ď T `δ.
By Assumption 3.9, there exists pz, v, wq P L 2 loc pr0, T `δsq ˆGloc pr0, T `δsq ˆH such that M can be represented as in (3.18). Let us then define the process R 1 p¨, T, δq by

R 1 pt, T, δq :" 1 δ S 0 t P pt, T `δq M t , 0 ď t ď T `δ. (3.21)
Clearly, this process satisfies the terminal condition (3.17). Moreover, if R 1 p¨, T, δq is a special semimartingale, then it can be easily seen that it satisfies compatibility.

(iii) The two specifications described in Remark 3.13 coincide whenever S 0 -discounted ZCB prices are true martingales under Q (see Section 3.3 below for an analysis of this case).

In general, the forward term rate Rp¨, T, δq can differ from the "martingale" solution (3.21), since uniqueness does not hold for the BSDE (3.19). Conditions ensuring uniqueness of the solution of (3.19) are given in the next proposition. We recall that, if P p¨, T q{S 0 is a uniformly integrable Q-martingale, we can define the T -forward measure Q T with density process dQ T | Ft :" P pt, T q{pP 0 pT qS 0 T qdQ| Ft , for all t P r0, T s. Let us also define: ' M 2 fwd pr0, T sq is the space of all square-integrable martingales X " pX t q tPr0,T s under Q T ; ' L 2 fwd pr0, T sq is the space of all R d -valued progressively measurable processes z " pz t q tPr0,T s such that

E Q T r ş T 0 }z t } 2 dts ă 8; ' G 2 fwd pr0, T sq is the space of all pP b BpEqq-measurable functions v : Ω ˆr0, T s ˆE Ñ R such that E Q T r ş T 0 ş E v 2 pt, xqν T pdt, dxqs ă 8, denoting by ν T pdt, dxq the compensator of µ under Q T ; ' H 2,T
fwd is the space of all functions w " pw 1 , . . . , w M q such that w i : Ω ˆB Ñ R is pF si´b Bqmeasurable, w i pξ i q P L 2 pQ T q and E Q T rw i pξ i q|F si´s " 0 a.s., for all i " 1, . . . , M .

Proposition 3.14. Consider T ě 0 and δ P D and assume that (i) P p¨, T `δq{S 0 is a uniformly integrable Q-martingale, and (ii) S 0 T `δ {pS 0 T q 2 P L 1 pQq. Then there exists a unique solution pY, z, v, wq P M 2 fwd pr0, T `δsq ˆL2 fwd pr0, T `δsq ˆG2 fwd pr0, T δsq ˆH2,T `δ fwd to the BSDE (3.19).

Proof. Condition (i) ensures that the definition of the measure Q T `δ is well-posed. Moreover, condition (ii) can be easily seen to be equivalent to requiring that RpT, T `δq P L 2 pQ T `δ q. By Girsanov's theorem (see (Jacod and Shiryaev, 2003, Theorem III.3.24)), the process W T `δ :" W `ş¨0 φps, T `δqds is a Brownian motion under Q T `δ and the compensator ν T `δ is given by ν T `δ pdt, dxq " e ´ψpt,T `δ,xq λ t pdxqdt.

Moreover, notice that

E T `δ rw i pξ i q|F si´s " S 0 siṔ ps i ´, T `δq E " P ps i , T `δq S 0 si w i pξ i q ˇˇˇF si´ " Ere ´V psi,
w i pξ i qδ si pdtq, Y T `δ " RpT, T `δq.
Therefore, it holds that

Y t " E Q T `δ " RpT, T `δq ˇˇF t ‰ ,
for all t P r0, T `δs.

Moreover, the fact that Y P M 2 fwd pr0, T `δsq together with an application of the Burkholder-Davis-Gundy inequality can be shown to imply the uniqueness of the processes z, v and w with respect to the corresponding norms on the spaces L 2 fwd pr0, T `δsq, G 2 fwd pr0, T `δsq and H 2,T `δ fwd , respectively. Remark 3.15. In this section, we have characterized forward term rate processes that satisfy the terminal condition (3.17). Under the current Libor fallback protocol4 , a spread is added to RpT, T `δq to account for risk components that were present in Libor rates and are not reflected in setting-inarrears rates based on RFRs. The BSDE formulation developed in this section can be readily extended to this case, by simply adding a spread to the terminal condition of the BSDE (3.19).

3.3. Risk-neutral pricing and the connection with the classical theory. In most applications, the risk-neutral price of an asset is typically meant to be the (conditional) Q-expectation of the asset's discounted payoff at maturity. In our context, this corresponds to assuming that S 0 -discounted ZCB and FRA prices, as described in Section 2, are true martingales under Q. As mentioned in part (iii) of Remark 3.13, if S 0 -discounted bond prices are true martingales under Q, the two specifications given in Remark 3.13 coincide. In this case, it is easily seen that the first component Y of the solution to the BSDE (3.19) is unique in the class of all adapted processes X " pX t q tPr0,T `δs such that P p¨, T `δqX{S 0 is a true martingale. This observation leads to the following statement, which provides an extension of the classical risk-neutral representation of ZCB prices and forward rates in a setting with stochastic discontinuities. Proposition 3.16. Suppose that S 0 -discounted ZCB and FRA prices are true martingales under Q. Then, for all T ě 0 and δ P D, it holds that

P pt, T q " E " e ´şpt,T s ρsηpdsq ˇˇF t ‰ , for 0 ď t ď T, (3.22) Rpt, T, δq " $ & % 1 δ ´P pt,T q P pt,T `δq ´1¯, for 0 ď t ď T, 1 δ ´eş pT ,ts ρs ηpdsq P pt,T `δq ´1¯, for T ă t ď T `δ.
(3.23)

Proof. Formula (3.22) is a direct consequence of (2.2) together with the assumption that P p¨, T q{S 0 is a martingale, for every T P R `. Under this assumption, equation (2.5) implies that S 0 -discounted FRA prices are martingales if and only if Rp¨, T, δqP p¨, T `δq{S 0 is a martingale. Being a martingale, this process is fully determined by the terminal condition (3.17) and, therefore, making use of representations (2.2) and (3.22), it is necessarily given by (3.23).

Considering the standard case of stochastically continuous processes, in Proposition 3.16 we recover the general formulas for ZCB prices and forward rates stated in [START_REF] Lyashenko | Looking forward to backward-looking rates: a modeling framework for term rates replacing LIBOR[END_REF].

The affine framework

In this section, we present a general modelling framework based on affine semimartingales. Affine semimartingales, as introduced in Keller-Ressel et al. ( 2019), generalize affine processes by allowing for discontinuities at fixed points in time with possibly state-dependent jump sizes.

Let X " pX t q tě0 be an affine semimartingale with X 0 " x taking values in D :" R m `ˆR n . We assume that convpsupppX t qq " D, for all t ą 0, and that X is quasi-regular and infinitely divisible, in the sense that the regular conditional distribution QpX t P dx|X s q is an infinitely divisible probability measure on D a.s. for all 0 ď s ď t ă 8, see [START_REF] Keller-Ressel | Affine processes beyond stochastic continuity[END_REF]. By (Keller-Ressel et al., 2019, Lemma 4.3 and Theorem 3.2), there is no loss of generality in assuming that X is a Markov process and the semimartingale characteristics pB X , C X , ν X q of X with respect to a fixed truncation function h : R d Ñ R d , with d " m `n, have the following structure, where B X,c and ν X,c denote the continuous parts of B X and ν X , respectively:

B X,c t " ż t 0 ´βX 0 psq `d ÿ i"1 X i s´β X i psq ¯ds, C X t " ż t 0 ´αX 0 psq `d ÿ i"1 X i s´α X i psq ¯ds, ν X,c pdt, dxq " ´µX 0 pt, dxq `d ÿ i"1 X i t´µ X i pt, dxq ¯dt, ż D pe xu,xy ´1qν X pttu, dxq " exp ´γX 0 pt, uq `d ÿ i"1 X i t´γ X i pt, uq ¯´1, (4.1)
where

β X i : R `Ñ R d , α X i : R `Ñ S d , γ X i : R `ˆU Ñ C, for i " 0, 1, . . . , d
, and pµ X i pt, ¨qq tě0 is a family of Lévy measures on Dzt0u, with S d denoting the cone of symmetric positive semidefinite d ˆd matrices and U " C m ´ˆiR n . In addition, it holds that γ X i pt, uq " 0, for all i and for all pt, uq P pR `zJ X q ˆU, where J X :" tt P R `: ν X pttu, Dq ‰ 0u represents the set of stochastic discontinuity dates of the affine semimartingale X.

The defining property of an affine semimartingale is that, for all 0 ď t ď T ă 8 and u P U, where the functions F X and R X are of Lévy-Khintchine form, for all i " 1, . . . , d:

E " e xu,
F X pt, uq " xβ X 0 ptq, uy `1 2 xu, α X 0 ptquy `żDzt0u `exx,uy ´1 ´xhpxq, uy ˘µX 0 pt, dxq, R X i pt, uq " xβ X i ptq, uy `1 2 xu, α X i ptquy `żDzt0u
`exx,uy ´1 ´xhpxq, uy ˘µX i pt, dxq.

Second, for all pT, uq P R `ˆU , the discontinuous parts of φ t pT, uq and ψ t pT, uq are determined by ∆φ t pT, uq " ´γX 0 pt, ψ t pT, uqq, ∆ψ t pT, uq " ´γ X pt, ψ t pT, uqq, where γX pt, uq " pγ X 1 pt, uq, . . . , γ X d pt, uqq P R d . Finally, the terminal conditions are given by φ T pT, uq " 0 and ψ T pT, uq " u.

Remark 4.1. In order to be consistent with the framework of Section 3, we restrict our attention to affine semimartingales whose characteristics do not contain singular continuous parts. However, we point out that all results of this section (in particular, Propositions 4.3 and 4.4) can be generalized in a straightforward way to affine semimartingales with singular characteristics.

4.1. An affine term structure model. In this subsection, we specialize the general setup of Section 3.1 to the case where the driving process is an affine semimartingale X. In this regard, we assume that X is a special semimartingale with canonical decomposition

X " X 0 `BX `Xc `x ˚pµ X ´νX q,
where µ X is the jump measure of X and ν X its compensator. We denote by μX " µ X ´νX the compensated measure. We denote with LpXq (resp. LpX i q) the set of all R d -valued (resp. R-valued) predictable processes that are integrable in the semimartingale sense with respect to X (resp. X i ).

Definition 4.2. An term structure model is called affine if

f pt, T q " f p0, T q `ż t 0 ζps, T qdX s , Rpt, T, δq " Rp0, T, δq `ż t 0 ζ R ps, T, δqdX s ,
for all 0 ď t ď T ă 8 and δ P D, where ζ : Ω ˆR2 `Ñ R d and ζ R : Ω ˆR2 `ˆD Ñ R d are predictable processes such that ζp¨, T q, ζ R p¨, T, δq P LpXq and the stochastic integrals ζp¨, T q ¨X and ζ R p¨, T, δq ¨X are special semimartingales, for all T ě 0 and δ P D.

We assume in addition that ζpt, T q " 0, for all 0 ď T ă t, and, for every i " 1, . . . , d,

˜ż T 0 |ζ i p¨, uq| 2 ηpduq ¸1{2 P LpX i q and ż T 0 ż T 0 |ζ i pt, uq|ηpduqdt ă 8 a.s.
The assumptions above imply in particular that Assumptions 3.2 and 3.7 are satisfied and, hence, an affine term structure model is a special case of the framework presented in Section 3.1.

For ease of presentation, we assume that J X " ts 1 , . . . , s M u ": S, with s 1 ă . . . , s M , in line with the notation introduced at the beginning of Section 3. This assumption can be relaxed to allow for countably many stochastic discontinuities (see Remark 3.1) and also for other stochastic discontinuities in J X that are not necessarily included in S. Recall that T denotes the stochastic discontinuities introduced in the model via the atoms of the measure η, see (2.1).

For the numéraire S 0 given in (2.2), we assume that

ρ t " r t `ÿ siPJ X ζ J si ∆X si 1 tsiďtu , t ě 0, (4.3)
with an adapted stochastically continuous process r satisfying ş t 0 |r s |ds ă 8 a.s., for all t ě 0, and d-dimensional F si´-measurable random vectors ζ si , for all s i P S. For all 0 ď t ď T , we denote ζpt, T q :" ż rt,T s ζpt, uqηpduq.

As a consequence of Theorem 3.8, we obtain the following sufficient conditions for Q being a risk-neutral measure which are completely explicit in terms of the characteristics of the affine semimartingale X.

Proposition 4.3. Consider an affine term structure model with differentiable functions u Þ Ñ γ X i p¨, uq, u P U, i " 0, . . . , d, and assume that, for all T ě 0, ż T 0 ż Dzt0u ´ˇe ´ζps,T qx ´1 `ζps, T qx ˇˇ`ˇˇζ R ps, T, δqx `e´ζps,T `δqx ´1˘ˇˇ¯ν X,c pds, dxq ă 8 (4.4)

and that for every τ P J X , k " 1, . . . , d and T ě 0,

ż rxsą1 p1 `|x k |qe ´´ş pτ,T s ζpτ,uqηpduq´ζτ δ T pτ q ¯Jx ν X ptτ u, dxq ă 8. (4.5)
Then Q is a risk-neutral measure if (i) f pt, tq " r t pQ b dtq-a.e., (ii) for every i " 0, . . . , d, it holds that ζpt, T qβ X i ptq "

1 2 } ζpt, T q} 2 α X i ptq `żDzt0u
`e´ψpt,T qx ´1 `ψpt, T qx ˘µX i pt, dxq on r0, T s pQ b dtq-a.e., for all T ě 0, (iii) for every i " 0, . . . , d it holds that

ζ R pt, T, δqβ X i ptq " ζpt, T `δq J ζ R pt, T, δqα X i ptq `żDzt0u ζ R pt, T, δqx `1 ´e´ζpt,T `δqx ˘µX i pt, dxq
on r0, T s pQ b dtq-a.e., for all T ě 0 and δ P D, (iv) for every i " 1, . . . , M , it holds a.s. that

f ps i ´, s i qδ T ps i q " ˆρsi´´´γ X 0 ps i , ´ζsi q `d ÿ l"1 X l si´γ X l ps i , ´ζsi q ¯˙δ T ps i q, (4.6) γ X 0 `si , ´ζsi δ T ps i q ˘´γ X 0 ˆsi , ´ζsi δ T ps i q ´żpsi,T s ζps i , uqηpduq " d ÿ l"1 X l si´« γ X l `si , ´ζsi δ T ps i q ˘`γ X l ˆsi , ´ζsi δ T ps i q ´żpsi,T s ζps i , uqηpduq ˙ff , (4.7) 0 " d ÿ k"1 ζ R,k ps i , T, δq ˜exp ´γX 0 ps i , uq `d ÿ l"1 X l si´γ X l ps i , uq B Bu k γ X 0 ps i , uq `d ÿ l"1 X l si´B Bu k γ X l ps i , uq ¯¸ˇˇˇˇu "´ş ps i ,T s ζpsi,uqηpduq´ζs i δ T psiq . (4.8)
Proof. The present affine framework is a special case of Theorem 3.8 and, hence, the result follows by identifying the corresponding terms in the affine setup and applying conditions from Proposition 3.4 and Theorem 3.8. Thanks to the affine characteristics (4.1) of X, we have ᾱpt, T q " ζpt, T q ´βX 0 ptq `d ÿ

i"1

X i t´β X i ptq ¯,
ψpt, x, T q " ζpt, T qx, and analogously for α R pt, T, δq and ψ R pt, x, T, δq. The expression for the measure λ t pdxq is given in the third equation of (4.1) and the quadratic variation and covariation terms are given by

} φpt, T q} 2 " } ζpt, T q} 2 ˜αX 0 ptq `d ÿ i"1 X i t´α X i ptq ¸, φpt, T `δq J ϕ R pt, T, δq " ζpt, T `δq J ζ R pt, T, δq ˜αX 0 ptq `d ÿ i"1 X i t´α X i ptq ¸.
The stochastic discontinuities in the affine setup are given explicitly by ∆ρ τ " ζ J τ ∆X τ , ∆V pτ, T q " ζpτ, T q J ∆X τ and ∆V R pτ, T, δq " ζ R pτ, T, δq J ∆X τ , (4.9) for all τ P S. Note that condition (4.5) replaces integrability conditions (3.6) and (3.16) in the affine case, taking into account also that ∆ρ τ " 0 for τ P T zS. Conditions (4.6) and (4.7) are obtained using the fourth equation of (4.1). Finally, condition (ii) from Theorem 3.8 is given in the affine setup as follows:

0 " d ÿ k"1 E " ζ R,k ps i , T, δq∆X k si e
´´ş ps i ,T s ζpsi,uqηpduq´ζs i δ T psiq ¯J∆Xs i ˇˇFs i´ı .

By B Bu k pe u J x q " x k e u J x , we obtain the result by differentiating the right-hand side above with respect to the k-th coordinate of X together with the affine characteristics (4.1) of X and predictability of ζ R p¨, T, δq. We get 0 "

d ÿ k"1 ζ R,k ps i , T, δq B Bu k ˜exp ´γX 0 ps i , uq `d ÿ l"1 X l si´γ X l ps i , uq ¯¸ˇˇˇˇu "´ş ps i ,T s ζpsi,uqηpduq´ζs i δ T psiq
, which is exactly condition (4.8) in the statement of the proposition.

4.2. Affine models for overnight rates. In interest rate theory, a widely used approach consists in modelling directly the short rate, computing ZCB and derivative prices by risk-neutral valuation (compare with Section 3.3). If the short rate model is sufficiently tractable (e.g., in the case of affine models as in (Duffie et al., 2003, Section 13)), then explicit valuation formulae for ZCBs and interest rate derivatives can be easily obtained. In this section, we show how this short rate approach can be extended to the case of overnight rates driven by affine semimartingales with stochastic discontinuities.

In analogy to classical affine short-rate models, we model the process ρ " pρ t q tě0 by ρ t " ptq `xΛ, X t y, for all t ě 0, (4.10)

where Λ P R d and : R `Ñ R is a deterministic and càdlàg function such that ş T 0 | puq|ηpduq ă 8, for all T P R `, where the measure η is given in (2.1). The function serves to fit the initially observed term structure of ZCB prices, similarly as in [START_REF] Brigo | A deterministicâĂŞshift extension of analyticallyâĂŞtractable and timeâĂŞhomogeneous shortâĂŞrate models[END_REF].

In view of (4.2)-(4.10), the conditional characteristic function of the RFR is directly available. However, this does not suffice for the valuation of ZCBs and interest rate derivatives, since the discount factor and most types of payoffs depend on the integrated process R :" ş 0 ρ t ηpdtq. The following proposition shows that the joint process pX, Rq is an affine semimartingale on the extended state space D ˆR. This result is a generalization to affine semimartingales of the enlargement of the state space approach of (Duffie et al., 2003, Section 11.2). The proof is postponed to the appendix. In particular, the joint process pX, Rq is an affine infinitely divisible semimartingale.

The explicit characterization of the conditional Fourier transform of the joint process pX, Rq obtained in Proposition 4.4 allows for an efficient valuation of a variety of RFR-based derivatives. In particular, RFR-based zero-coupon bonds can be priced by evaluating (4.11) at pu, wq " p0, ´1q, whenever the expectation is finite. In turn, this leads to explicit pricing formulae for all RFR-based linear interest derivatives such as swaps. Non-linear derivatives can be priced by relying on Fourier methods, analogously to the classical case of affine processes, see (Filipović, 2009, Section 10.3).

4.3.

A Hull-White model for the overnight rate. In this section, we present a tractable specification of an affine model for an overnight rate, as introduced in Section 4.2. More specifically, we generalize the popular Hull-White model to the case of stochastic discontinuities. We recall that T " tt 1 , . . . , t N u represents the time-grid at which the numéraire S 0 jumps, while S " ts 1 , . . . , s M u denotes the set of stochastic discontinuities of the RFR process.

We assume that ρ is the unique strong solution of the following stochastic differential equation: (4.16) where α : R Ñ R is a continuous function, β P R, σ ě 0 and W " pW t q tě0 is a Brownian motion. The process J " pJ t q tě0 in (4.16) is a pure jump process specified as follows: As illustrated in the following example, processes of the form (4.18), despite their simple structure, allow for different types of stochastic discontinuities that are in line with the empirical features of overnight rates, as discussed in the introduction (see Figure 1).

dρ t " `αptq `βρ t ˘dt `σdW t `dJ t ,
J " M ÿ i"1 ξ i 1 rrsi,8rr
Example 4.6 (A two-factor affine specification). Overnight rates are characterized by the presence of spikes and jumps, both potentially occurring at predetermined dates. While affine processes can exhibit a spiky behavior (e.g., shot-noise processes, see Schmidt ( 2008)), they do not accommodate spikes and jumps at predetermined dates. These features can be reproduced by affine semimartingales of the form (4.18). For the purpose of illustration, let ρ 1 and ρ 2 two independent process satisfying (4.18), with parameters ρ 1 0 " 0.01875, β 1 " 0.20, α 1 ptq " α 1 " 0.01, σ 1 " 0.012, ρ 2 0 " 0, β 2 " 80, α 2 ptq " α 2 " 0, σ 2 " 0, for all t ě 0. The process ρ 1 is assumed to have a single discontinuity at time s 1 1 " 150, while ρ 2 has discontinuity dates s 2 1 " 50 and s 2 2 " 100. We assume that all jump sizes are i.i.d. and distributed as N p0.1, 0.4 2 q. Figure 2 shows a simulated path of the process ρ :" ρ 1 `ρ2 . We can observe that ρ exhibits spikes at t " 50 and t " 100 and a jump to a new level at t " 150. The spiky behavior is generated by ρ 2 , which has no diffusive component and very high mean-reversion speed (increasing β 2 further would produce even more pronounced spikes). The jump to a new level at t " 150 is generated by the component ρ 1 , which has a much slower mean-reversion.

In the next results, we shall make use of the following notation: apt, T q :" ż T t e βpT ´sq αpsqds.

(4.19)

For t " 0, we simply write apT q :" ap0, T q. In the following proposition, we compute the conditional characteristic function of the solution to SDE (4.18). While this result can be deduced from the general theory of affine semimartingales, we provide a direct simple proof which exploits the structure of (4.18).

Proposition 4.7. Let ρ " pρ t q tě0 be the solution of the SDE (4.18) and assume that the random variables tξ i ; i " 1, . . . , M u are independent. Then, for every u P iR and 0 ď t ď T , it holds that where ψpu, T ´tq " ue βpT ´tq , φpu, t, T q " u apt, T q `puσq 2 4β `e2βpT ´tq ´1˘`ÿ (4.20) This immediately yields that ψpu, T ´tq " ue βpT ´tq . The result of the proposition then follows by relying on the independence of the random variables tξ i ; i " 1, . . . , M u,

E " e uρ T ˇˇρ t ‰ " exp `φpu, t, T q `ψpu, T ´tqρ t ˘,
Explicit expressions for ZCB prices can be obtained either by applying the general result from Proposition 4.4 or by a direct computation of the conditional characteristic function of the timeintegral R T " ş T 0 ρ t ηpdtq, as illustrated in the next subsection under a specific assumption on the distributional properties of the jump sizes.

4.4.

A Gaussian Hull-White model for the overnight rate. Until the end of this section we assume that the random variables tξ i ; i " 1, . . . , M u are independent and normally distributed. In this case, we immediately see from (4.18) that ρ is a Gaussian process (more precisely, a Markov process with Gaussian increments). Moreover, the random variable R T :" ş T 0 ρ t ηpdtq is also normally distributed. This immediately yields the following proposition.

Proposition 4.8. For all 0 ď t ď T , it holds that

P pt, T q " exp ´´ErR T |ρ t , R t s `Rt `1 2 VarpR T |ρ t , R t q ¯.
We proceed to compute explicitly the conditional mean and variance of R T appearing in Proposition 4.8. The proofs of the following two lemmata are based on rather lengthy computations and, therefore, are deferred to the appendix. As a first step, we compute the conditional mean and covariance of ρ. We denote m i :" Erξ i s and γ 2 i :" Varpξ i q, for each i " 1, . . . , M . Lemma 4.9. For all 0 ď t ď T and 0 ď t ď T 1 , T 2 , let us denote mpt, T q :" Erρ T |ρ t s and cpt, T 1 , T 2 q :" Covpρ T1 , ρ T2 |ρ t q.

Then, it holds that mpt, T q " ρ t e βpT ´tq `apt, T q `N ÿ

i"1

1 tsiPpt,T su m i e βpT ´siq , cpt, T 1 , T 2 q " σ 2 e βpT1`T2q 2β ´e´2βt ´e´2βpT1^T2q ¯`M ÿ i"1
1 tsiPpt,T1^T2su γ 2 i e βpT1`T2´2siq .

(4.21)

By integrating equation (4.18) and applying Fubini theorems, we obtain

R T " R t `ρt B 1 pt, T q`A 1 pt, T q`σ ż T 0 B 1 ps, tqdW s `M ÿ i"1 1 tsiPpt,T su ˆB1 ps i , T q`N ÿ j"1 1 tsi"tj u ˙ξi , (4.22)
for all 0 ď t ď T , where we make use of the notation

B 1 pt, T q :" ż pt,T s e βpu´tq ηpduq " e βpT ´tq ´1 β `N ÿ j"1 1 ttj Ppt,T su e βptj ´tq ": BpT ´tq `N ÿ j"1 1 ttj Ppt,T su e βptj ´tq
and

A 1 pt, T q :" ż pt,T s apt, uqηpduq " ż T t apt, uqdu `N ÿ j"1 1 ttj Ppt,T su apt, t j q ": Apt, T q `N ÿ j"1
1 ttj Ppt,T su apt, t j q.

By relying on (4.22) and on the properties of the random variables tξ i ; i " 1, . . . , M u, the next lemma gives explicit expressions for the conditional expectation and variance of R T . We denote Bpt, T q :" e 2βpT ´tq ´1 2β `N ÿ j"1 1 ttj Ppt,T su e 2βptj ´tq , for all 0 ď t ď T. (4.23)

Lemma 4.10. For all 0 ď t ď T , it holds that

ErR T |ρ t , R t s " R t `ρt ˜BpT ´tq `N ÿ j"1 1 ttj Ppt,T su e βptj ´tq Apt, T q `N ÿ j"1 1 ttj Ppt,T su apt, t j q `M ÿ i"1 1 tsiPpt,T su m i ˜BpT ´si q `N ÿ j"1 1 ttj Prsi,T su e βptj ´siq ¸, (4.24) VarpR T |ρ t , R t q " σ 2 2β B 1 pt, T q 2 ´σ2 β ˜B1 pt, T q ´pT ´tq β `N ÿ j"1 1 ttj Ppt,T su ´B1 pt j , T q `1 2 ´1 β ¯2 M ÿ i"1 1 tsiPpt,T su γ 2 i ˜Bps i , T q ´B1 ps i , T q β `N ÿ j"1
1 ttj Prsi,T su e 2βptj ´siq ´B1 pt j , T q `1 2 ¯¸.

We close this section by presenting an explicit formula for the price of a caplet in the context of the present Gaussian Hull-White model. As discussed in [START_REF] Lyashenko | Looking forward to backward-looking rates: a modeling framework for term rates replacing LIBOR[END_REF], in the post-Libor universe one may consider two distinct types of caplets, depending on whether the payoff is determined by the backward-looking rate RpS, T q or by the forward-looking rate F pS, T q. For illustration, we consider here a forward-looking caplet, whose payoff at date T is given by H " pT ´Sq `F pS, T q ´K˘`, for some K ą 0. We recall that under risk-neutral valuation (see Section 3.3) it holds that F pS, T q " p1{P pS, T q ´1q{pT ´Sq. In view of Proposition 4.8 and Lemma 4.10, we have that:

P pt, Sq " e ´ρtB 1 pt,Sq´Ξpt,Sq , (

for all 0 ď t ď S, where, for brevity of notation, Ξpt, Sq collects all terms appearing in Lemma 4.10 that do not multiply ρ t . For the determination of the caplet price, we need to compute the F tconditional distribution of ρ S under the S-forward measure Q S defined by dQ S {dQ " 1{pS 0 S P p0, Sqq. Lemma 4.11. Under the S-forward measure Q S , the following hold:

(i) the process W S " pW S t q tPr0,Ss defined by W S t :" W t `σ ş t 0 B 1 pu, Sqdu, for all t P r0, Ss, is a Brownian motion; (ii) for each i " 1, . . . , M , it holds that

ξ i " N ˜mi ´1tsiďSu γ 2 i ´BpS ´si q `N ÿ j"1 1 ttj Prsi,Ssu e βptj ´siq ¯, γ 2 i ¸.
Moreover, the random variables tξ i ; i " 1, . . . , M u are independent and independent of the Brownian motion W S .

Proof. Part (i) follows by a standard application of Girsanov's theorem. To prove part (ii), it suffices to consider an arbitrary λ P R M and compute the joint characteristic function of the random variables tξ i ; i " 1, . . . , M u under the measure Q S conditionally on the sigma-field F W :" σpW S u ; u P r0, T sq. Observe that F W " σpW u ; u P r0, T sq. Denoting by E S the expectation under Q S and making use of equation ( 4.22), it holds that

E S " e i ř M i"1 λiξi |F W ‰ " E " e ´RS `i ř M i"1 λiξi |F W ‰ Ere ´RS |F W s " M ź i"1 E " e piλi´1 ts i ďSu pBpS´siq`ř N j"1 1 tt j Prs i ,Ssu e βpt j ´si q qqξi ‰ E " e ´1ts i ďSu pBpS´siq`ř N j"1 1 tt j Prs i ,Ssu e βpt j ´si q qξi ‰ " M ź i"1 e iλipmi´1 ts i ďSu γ 2 i pBpS´siq`ř N j"1 1 tt j Prs i ,Ssu e βpt j ´si q qq´λ 2 i γ 2 i 2 ,
where we have used the independence of the random variables tξ i ; i " 1, . . . , M u together with the fact that ξ " N pm i , γ 2 i q, for each i " 1, . . . , M . As a consequence of Lemma 4.11 and Proposition 4.5, it holds that ρ S " N `ρt e βpS´tq `Γ1 pt, Sq, Γ 2 pt, Sq ˘under Q S , (4.26) conditionally on F t , for t P r0, Ss, where the quantities Γ 1 pt, Sq and Γ 2 pt, Sq are defined as Γ 1 pt, Sq :"

ż S t `αpsq ´σB 1 ps, Sq ˘eβpS´sq ds `M ÿ i"1 1 tsiPpt,Ssu ´mi ´γ2 i ´BpS ´si q `N ÿ j"1 1 ttj Prsi,Ssu e βptj ´siq ¯¯e βpS´siq , Γ 2 pt, Sq :" σ 2 ż S t e 2βpS´sq ds `M ÿ i"1 1 tsiPpt,Ssu γ 2 i e 2βpS´siq " σ 2 e 2βpS´tq ´1 2β `M ÿ i"1 1 tsiPpt,Ssu γ 2 i e 2βpS´siq .
The following proposition gives the arbitrage-free price of a forward-looking RFR caplet. We denote by Φ the cumulative distribution function of a N p0, 1q random variable. Proposition 4.12. Consider a forward-looking RFR caplet delivering the payoff H " pT ´SqpF pS, T qḰ q `at date T , for S P r0, T s and K ą 0. Its risk-neutral price H t at time t P r0, Ss is given by

H t " Gpρ t ,

t, S, T, Kq

where the function Gp¨, t, S, T, Kq is explicitly given for all x P R by Gpx, t, S, T, Kq " e ´xB 1 pt,Sq´Ξpt,Sq ˆΦpd 1 pxqq ´K1 e ´ΞpS,T q´B 1 pS,T qpxe βpS´tq `Γ1pt,Sq´B 1 pS,T qΓ 2 pt,Sq

2 q Φpd 2 pxqq ˙,
with K 1 :" 1 `pT ´SqK and

d 1 pxq " ´logpK 1 q `ΞpS, T q B 1 pS, T q a Γ 2 pt, Sq `xe βpS´tq `Γ1 pt, Sq a Γ 2 pt, Sq and d 2 pxq " d 1 pxq ´B1 pS, T q a Γ 2 pt, Sq.
Proof. Using the definition of the S-forward measure Q S and equation (4.25), we can compute

H t " pT ´SqE " S 0 t S 0 T `F pS, T q ´K˘`ˇˇˇF t  " E « S 0 t S 0 T ˆ1 P pS, T q ´K1 ˙`ˇˇˇˇF t ff " K 1 E « S 0 t S 0 S ˆ1 K 1 ´P pS, T q ˙`ˇˇˇˇF t ff " K 1 P pt, SqE S « ˆ1 K 1 ´P pS, T q ˙`ˇˇˇˇF t ff " K 1 P pt, SqE S « ˆ1 K 1 ´e´ρ S B 1 pS,T q´ΞpS,T q ˙`ˇˇˇˇF t ff .
Under the S-forward measure Q S , the F t -conditional distribution of ρ S is given by (4.26). The result then follows by an application of the Black-Scholes formula.

Hedging in the presence of stochastic discontinuities

The presence of stochastic discontinuities may induce market incompleteness, in the sense that perfect replication of payoffs by means of self-financing strategies is not always possible. This is for instance the case of the affine model of Section 4.3, which is affected by the jump risk generated by the process J. In this section, we aim at determining optimal hedging strategies in the sense of local riskminimization. This corresponds to attaining a perfect replication of a given payoff while relaxing the self-financing requirement and minimizing the cost of the strategy according to a quadratic criterion (see [START_REF] Pham | On quadratic hedging in continuous time[END_REF] and [START_REF] Schweizer | A guided tour through quadratic hedging approaches[END_REF] for an overview of the theory). In Section 5.1 we provide a genereal description of local risk-minimization with stochastic discontinuities, while in Section 5.2 we study an explicit example in the context of the Gaussian model of Section 4.4.

5.1.

Local risk-minimization with stochastic discontinuities. In order to reduce the technicalities in the presentation and to focus on the impact of stochastic discontinuities, we assume the validity of the following assumption. We consider a finite time horizon T .

Assumption 5.1. There exists a family pξ 1 , . . . , ξ M q of random variables on pΩ, F , Qq taking values in pB, BpBqq such that ξ i is F si -measurable, for each i " 1, . . . , M , and every local martingale N " pN t q tPr0,T s on pΩ, F, Qq admits a representation of the following form:

N " N 0 `ż 0 θ t dW t `M ÿ i"1 f i pξ i q1 rrsi,T ss , (5.1)
where θ P L 2 loc pr0, T sq and f i p¨q : Ω ˆB Ñ R is a pF si´b BpBqq-measurable function such that Erf i pξ i q|F si´s " 0 a.s., for each i " 1, . . . , M .

In particular, Assumption 5.1 holds for the model of Section 4.3 if the filtration F is generated by the pair pW, Jq. Note also that the assumption that the discontinuity dates T do not appear in the martingale representation (5.1) is only made for simplicity of presentation (see Remark 3.12).

We suppose that the market contains a traded security with S 0 -discounted price process X " pX t q tPr0,T s , assumed to be a special semimartingale with canonical decomposition X " X 0 `A `M,

(5.2)

where A " pA t q tPr0,T s is a predictable process of finite variation and M " pM t q tPr0,T s a squareintegrable martingale, with A 0 " M 0 " 0. The process X can represent for instance the price process of an RFR future contract, at present the most liquid derivative referencing RFR (see Section 5.2). Note also that in this section we do not necessarily assume that Q is a risk-neutral measure.

As a consequence of Assumption 5.1, the martingale M admits a representation of the form (5.3) where η " pη t q tPr0,T s is a predictable process such that Er ş T 0 η 2 u dus ă 8 and ∆M si " w i pξ i q, where the function w i is as in Assumption 5.1, for each i " 1, . . . , M . We furthermore assume that X has non-vanishing volatility, meaning that η t ą 0 a.s. for all t P r0, T s.

M " ż 0 η u dW u `ÿ siď¨∆ M si ,
By absence of arbitrage, there exists a predictable process λ " pλ t q tPr0,T s such that A " ş 0 λ u dxM y u . In particular, this implies that ∆A si " λ si Erp∆M si q 2 |F si´s , for all i " 1, . . . , N . We furthermore assume that the expected mean-variance tradeoff is finite, i.e., Er ş T 0 λ 2 u dxM y u s ă 8. This corresponds to assuming that X satisfies the structure condition (see [START_REF] Schweizer | A guided tour through quadratic hedging approaches[END_REF]).

Let H be a square-integrable F T -measurable random variable, representing a discounted payoff. By market incompleteness, H may not be attainable by self-financing trading. We then consider non-selffinancing strategies attaining the payoff H, as formalized in the following definition, where we denote by Θ the set of all predictable processes ζ " pζ t q tPr0,T s such that Er ş T 0 ζ 2 u dxM y u `pş T 0 ζ u dA u q 2 s ă 8. Definition 5.2. We call H-admissible strategy a pair ϕ " pζ, V q, where ζ " pζ t q tPr0,T s P Θ and V " pV t q tPr0,T s is an adapted process such that V T " H a.s. We say that an H-admissible strategy ϕ " pζ, V q is locally risk-minimizing if the associated cost process

C t pϕq :" V t ´ż t 0 ζ u dX u ,
for all t P r0, T s, is a square-integrable martingale strongly orthogonal to M .

In Definition 5.2, ζ t and V t represent respectively the positions held in the traded security and the portfolio value at time t, for all t P r0, T s. In the present context, it can be shown that the definition of locally risk-minimizing strategy adopted in Definition 5.2 is equivalent to the original definition of [START_REF] Schweizer | Option hedging for semimartingales[END_REF]. This follows along the lines of the proof of (Schweizer, 2001, Theorem 3.3).

In view of (Schweizer, 2001, Proposition 3.4), finding a locally risk-minimizing strategy ϕ " pζ, V q corresponds to obtaining a decomposition of the payoff H of the form (5.4) where ζ H " pζ H t q tPr0,T s P Θ and L H " pL H t q tPr0,T s is a square-integrable martingale strongly orthogonal to M with L H 0 " 0. Decomposition (5.4) is known as the Föllmer-Schweizer decomposition of H and a locally risk-minimizing strategy is then given by pζ H , V H q, where V H :" H 0 `ş¨0 ζ H u dX u `LH . Under Assumption 5.1, we can explicitly derive decomposition (5.4) for a generic discounted payoff H. To this effect, let us define p Z :" E p´ş 0 λ u dM u q and assume that p Z is a strictly positive squareintegrable martingale under Q. (5.5) where θ " pθ t q tPr0,T s is a predictable process such that ş T 0 θ 2 u du ă 8 a.s. and ∆N si " v i pξ i q, where v i is a function satisfying the requirements of Assumption 5.1, for all i " 1, . . . , M . We are now in a position to state the following proposition.

H " H 0 `ż T 0 ζ H u dX u `LH T ,
N " N 0 `ż 0 θ u dW u `ÿ siď¨∆ N si ,
p Z ´1 " E ˜ż 0 λ u η u dW u `ż 0 λ 2 u η 2 u du `ÿ siď¨λ si ∆M si 1 ´λsi ∆M si ¸.
Therefore, in view of equation (5.5), we can compute

p H t " p H 0 `ż t 0 p Z ´1 u´θu dW u `ż t 0 N u´p Z ´1 u´λu η u dW u `ż t 0 N u´p Z ´1 u´λ 2 u η 2 u du `ż t 0 p Z ´1 u´θu λ u η u du `ÿ siďt ˆp Z ´1 si´∆ N si `Nsi´p Z ´1 si´λ si ∆M si 1 ´λsi ∆M si `p Z ´1 si´λ si ∆M si ∆N si 1 ´λsi ∆M si " ż t 0 p Z ´1 u´`θu `Nu´λu η u ˘`dW u `λu η u du ˘`ÿ siďt ∆ p H si .
Since A c " ş 0 λ t dxM y c t " ş 0 λ t η 2 t dt and t∆X ‰ 0u Ď Ω ˆS, we have that

H " p H T " p H 0 `ż T 0 ζ H u dX u `ÿ siďT `∆ p H si ´ζH si ∆X si ˘, (5.7)
where ζ H " pζ H t q tPr0,T s is defined as in (5.6). We proceed to show that (5.7) provides the Föllmer-Schweizer decomposition (5.4) of H, where L H :" ř siď¨p ∆ p H si ´ζH si ∆X si q is a square-integrable martingale strongly orthogonal to M under Q. To prove that L H is a martingale, it suffices to verify that Er∆L H si |F si´s " 0 a.s. for all i " 1, . . . , M . To this effect, using (5. Proposition 5.3 provides an explicit description of the locally risk-minimizing strategy for a generic payoff H. In particular, formula (5.6) reveals that the locally risk-minimizing strategy consists in a perfect replication at all times t P r0, T szS, when the only active source of randomness is the Brownian motion W . Indeed, the first term on the right-hand side of (5.6) corresponds to the Delta-hedging continuous strategy. On the other hand, in correspondence of the discontinuity dates S " ts 1 , . for all i " 1, . . . , M , as follows from (5.6) using the predictability of the process A. We also remark that the associated cost process Cpϕ H q is generated by the residuals of the regressions (5.8).

An example.

In this section, we illustrate the hedging approach described in Section 5.1 in the case of a forward-looking caplet using an RFR future as hedging instrument. This choice is motivated by the fact that, at the time of writing, SOFR futures represent the most liquidly traded products written on SOFR, while caps/floors are not yet sufficiently developed in the market. We consider the model of Section 4.3, where Q plays now the role of the physical probability measure:

dρ t " pαptq `βρ t qdt `σdW t `dJ t , (5.9)

where J is defined as in (4.17), where the random variables tξ i ; i " 1, . . . , M u are independent and independent of W , with distribution N pm i , γ 2 i q under Q, for each i " 1, . . . , M . For simplicity of presentation, in this subsection we assume that ηpdtq " dt.

As traded security, we consider a futures contract with reference period rS, T s, for some S ă T . We denote by f pt, S, T q the corresponding futures rate at date t, for t P r0, Ss, and define Bpt, S, T q :" BpT ´tq ´BpS ´tq T ´S " e βpT ´tq ´eβpS´tq pT ´Sqβ , for all t P r0, Ss.

We assume that the futures rate f p¨, S, T q satisfies the following dynamics under Q: df pt, S, T q " hptqdt `Bpt, S, T qσdW t `Bpt, S, T qd r J t , (5.10) where r J denotes the compensated jump process defined as r J t :" J t ´řm i"1 1 tsiďtu m i , for all t P r0, T s, and h : r0, T s Ñ R is a bounded deterministic function. The local martingale part M of the discounted futures price process can be written as in (5.3), with η t " pS 0 t q ´1Bpt, S, T qσ and ∆M si " pS 0 si q ´1Bps i , S, T qpξ i ´mi q, (5.11) for all t P r0, Ss and i " 1, . . . , M .

In the present setting, p Z :" E p´ş 0 hpuq{pσBpu, S, T qqdW u q is a square-integrable strictly positive martingale and, therefore, the minimal martingale measure p Q is given by d p Q " p Z T dQ. By Girsanov's theorem, the process x W " p x W t q tPr0,T s defined by x W t " W t `şt 0 hpuq{pσBpu, S, T qqdu, for all t P r0, T s, is a Brownian motion under p Q. Note that the change of measure from Q to p Q leaves invariant all the properties of the random variables tξ i ; i " 1, . . . , M u.

Remark 5.4. In the context of the model of Section 4.3, the futures rate f pt, S, T q can be explicitly computed. Suppose that, in line with the market convention for 1-month RFR futures contracts, the futures contract settles at date T at a rate quoted as pR T ´RS q{pT ´Sq. By risk-neutral valuation under the minimal martingale measure p Q, it holds that f pt, S, T q " p ErR T ´RS |F t s T ´S , for all t P r0, Ss.

Similarly as in Section 4.3, under the minimal martingale measure p Q it holds that

R T ´Rt " ρ t BpT ´tq `p Apt, T q `σ ż T t BpT ´sqd x W s `M ÿ i"1 1 tsiPpt,T su BpT ´si qξ i ,
where p Apt, T q :" ş T t αpsqBpT ´sqds, for all t P r0, T s, with αptq :" αptq ´hptq{Bpt, S, T q denoting the deterministic drift term in the dynamics of ρ under p Q. The futures rate f pt, S, T q admits then the following explicit representation:

f pt, S, T q " ρ t Bpt, S, T q`p Apt, T q ´p Apt, Sq T ´S `M ÿ i"1 1 tsiPpt,Ssu Bps i , S, T qm i `M ÿ i"1 1 tsiPpS,T su BpT ´si q T ´S m i .
We suppose that the payoff H to be hedged corresponds to an RFR caplet with discounted payoff H :" pT ´Sq `F pS, T q ´K˘`{ S 0 T , for some K ą 0, as considered in Section 4.4. To determine the locally risk-minimizing strategy, we first need to compute the price process p H " p p H t q tPr0,T s of the payoff H under the measure p Q. This can be achieved by a direct application of Proposition 4.12, leading to p H t " Gpρ t , t, S, T, Kq{S 0 t , where the function Gpρ t , t, S, T, Kq is explicitly given in Proposition 4.12, replacing Apt, Sq by p Apt, Sq in the definition of the quantity ΞpS, T q and αpsq by αpsq in the definition of Γ 1 pt, Sq.

In view of Proposition 5.3, the component ζ H of the locally risk-minimizing strategy ϕ H is determined by two terms: a first term representing the continuous Delta-hedging strategy and an additional term that takes into account the discontinuities dates S " ts 1 , . . . , s M u.

Proposition 5.5. Suppose that Assumption 5.1 holds. Consider a caplet delivering at date T the payoff pT ´SqpF pS, T q ´Kq `, for S P r0, T s and K ą 0. The locally risk-minimizing strategy ϕ H " pζ H , v H q is determined by the process ζ H " pζ H t q tPr0,T s defined by In view of equations (5.9) and (5.11), it follows that the first component ζ H,c t of the locally riskminimizing strategy is given by (5.12). To compute the second term on the right-hand side of equation ( 5 ErGpρ si´`ξi , s i , S, T, Kqpξ i ´mi qs pS 0 si q ´1Bps i , S, T qγ 2 i , for all i " 1, . . . , M, from which (5.13) follows due to the independence of the random variables tξ i , i " 1, . . . , M u from the Brownian motion W . Finally, it remains to verify that sup tPr0,Ss p H t P L 2 pQq and ζ L P Θ. The first property can be shown to hold since p H t ď P pt, Sq for all t P r0, Ss and by means of standard estimates together with an application of Doob's maximal inequality. The fact that ζ L P Θ follows by noting that the integral Er ş S 0 pζ H u q 2 dxM y u s can be reduced to the integration of continuous functions on the compact domain r0, Ss and that the function h in (5.10) is assumed to be bounded.

ζ H t " ζ H,
Remark 5.6. We mention that, in the context of a Vasiček-type model, [START_REF] Rutkowski | Pricing and hedging of SOFR derivatives under differential funding costs and collateralization[END_REF] derive an explicit replication strategy for a SOFR caplet based on SOFR futures. This is possible since their model is driven by a single source of randomness represented by a standard Brownian motion. In contrast, in our setting the presence of unpredictable jumps at fixed times does not allow for perfect replication, thereby justifying the use of local risk-minimization. In particular, note that γ Y i pt, u, vq " 0 for all pt, u, vq P pR `zpT Y J X qq ˆU ˆiR and i " 0, 1, . . . , d `1. It follows that the parameter set pA Y , β Y , α Y , µ Y , γ Y q is good in the sense of (Keller-Ressel et al., 2019, Definition 3.1). Moreover, since the affine semimartingale X is assumed to be infinitely divisible, (Keller-Ressel et al., 2019, Lemma 4.4) implies that, for all t P J X and i " 0, 1, . . (Keller-Ressel et al., 2019, Definition 5.1). In turn, this implies that the enhanced parameter set of Y , determined by p βY , αY , μY q as defined above, is also admissible. Therefore, by (Keller-Ressel et al., 2019, Theorem 5.7), on the canonical stochastic basis pΩ 1 , F 1 , pF 1 t q tě0 , P 1 q there exists an infinitely divisible Markov process Y 1 " pY 1 t q tě0 with Y 1 0 " px, 0q that is an affine semimartingale with characteristics pB Y , C Y , ν Y q as computed above. Since the two Markov processes Y " pX, Rq and Y 1 have the same characteristics and the process Y 1 is unique in law, it follows that Y " pX, Rq and Y 1 have the same law (compare with (Duffie et al., 2003, Lemmata 10.1 and 10.2)).

Denoting by E 1 the expectation under the measure P 1 , this implies that Ere xw,Y T y s " E 1 re xw,Y 1 T y s " e Φ0pT,wq`xΨ0pT,wq,xy , (A.2) for all w " pu, vq P U ˆiR and 0 ď t ď T ă 8, where the functions Φ 0 pT, wq and Ψ 0 pT, wq are solution to (4.12)-(4.15), as follows from (Keller-Ressel et al., 2019, Theorem 3.1) together with the specific structure of the characteristics pB Y , C Y , ν Y q computed in the first part of the proof. The conditional version of the Fourier transform (4.11) follows from (A.2) by relying on the Markov property of Y on the stochastic basis pΩ, F , pF t q tě0 , P q.

Proof of Lemma 4.9. Firstly, taking expectation of equation (4.18) immediately yields mpt, T q and thus (4.21). Regarding the covariance, for the continuous part we note that 1 ttj Ppt,T su mpt, t j q.

In view of (4.21), the first integral on the right-hand side can be computed as follows:

Figure 1 .

 1 Figure 1. SOFR time series from 01/12/2015 until 30/09/2019. Spikes and jumps occurring at regular points in time are clearly visible. Source: Anbil et al. (2020).

Proposition 4. 4 .

 4 Let the D ˆR-valued process Y :" pX, Rq be defined as above. Then it holds that E " e xu,X T y`vR

Figure 2 .

 2 Figure 2. A simulation of the two-factor affine model considered in Example 4.6.The simulated path exhibits spikes at the discontinuity dates t " 50 and t " 100 and a jump to a new level at t " 150.

Appendix A .

 . Technical proofsProof of Proposition 4.4. We start by computing the semimartingale characteristics pB Y , C Y , ν Y q of the joint process Y " pX, Rq. First, denoting by B Y,c the continuous part of the first characteristic B Y , it holds that

  Ere ´V psi,T `δq w i pξ i q|F si´s δ si pdtq

	to the BSDE
	$ ' ' '	dY t " ˆφpt, T `δq J z t	`żE	`1 ´e´ψpt,T `δ,xq ˘vpt, xqλ t pdxq ˙dt
	'
	'
	'
	'
	'
	'
	' ' ' &	´M ÿ
		i"1
	'
	'
	' ' ' ' '	`zt dW t	`żE	vpt, xqμpdt, dxq	`M ÿ
	'
	' '					i"1
	'
	'
	%

T, δq is compatible if and only if Rp¨, T, δq " Y , where pY, z, v, wq P S p pr0, T `δsq ˆL2 loc pr0, T `δsq ˆGloc pr0, T `δsq ˆH is a solution

  T `δq w i pξ i q|F si´s

	for all i " 1, . . . , M . Consequently, a tuple pY, z, v, wq P M 2 fwd pr0, T `δsq ˆL2 fwd pr0, T `δsq	Ĝ2
	fwd pr0, T `δsq ˆH2,T `δ fwd	solves the BSDE (3.19) if and only if
	$
	' &	dY t " z t dW T `δ t	`żE	vpt, xqpµ ´νT `δ qpdt, dxq	`M ÿ
						i"1
	'
	%

  ,(4.17) where the random variables tξ i ; i " 1, . . . , M u are assumed to be independent of W . The following lemma gives the explicit solution to (4.16) and is an immediate consequence of Itô's formula.

Lemma 4.5. The solution ρ " pρ t q tě0 of the SDE (4.16) satisfies, for all 0 ď t ď T ,

ρ T " ρ t e βpT

´tq `ż T t e βpT ´sq αpsqds `σe βT ż T t e ´βs dW s `M ÿ i"1 1 tsiPpt,T su ξ i e βpT ´siq . (4.18)

  This enables us to define the minimal martingale measure p Er p Z T H|F t s, for all t P r0, T s. As a consequence of Assumption 5.1, it holds that

		Q by
	d p Q " p Z T dQ. We can then define the p Q-martingale p H " p p H t q tPr0,T s by
	p H t :" p ErH|F t s,	for all t P r0, T s,
	where we denote by p E the expectation with respect to p Q. By Bayes' formula, it holds that p H " N { p Z,
	with N t :"

  If ζ H P Θ, then an H-admissible locally risk-minimizing strategy is given by ϕ H " pζ H , V H q, where V H t " p H t , for all t P r0, T s.

	Proof. By the product rule, it holds that
	p H t " p Z ´1 t N t " p H 0	`ż t	p Z ´1 u´d N u	`ż t	N u´d p Z ´1 u	`r p Z ´1, N s t ,
		0			0
	for all t P r0, T s. An application of Itô's formula yields
	Proposition 5.3. Suppose that Assumption 5.1 holds. Let H be an F T -measurable random variable
	and suppose that sup tPr0,T s p H t P L 2 pQq. Define the predictable process ζ H " pζ H t q tPr0,T s by
	ζ H t :" `p Z ´1 t´η ´1 t θ t `p H t´λt ˘δS c ptq	`Er∆ p H t ∆M t |F t´s Erp∆M t q 2 |F t´s	δ S ptq.	(5.6)

  2) we can compute Er∆ p H si |F si´s ´ζH si Er∆X si |F si´s " Er∆ p H si |F si´s ´ζH si ∆A si " Er∆ p H si |F si´s ´ζH si λ si Erp∆M si q 2 |F si´s " Erp1 ´λsi ∆M si q∆ p H si |F si´s Square-integrability of L H under Q follows from the assumptions. Finally, L H and M are strongly orthogonal under Q if and only if rL H , M s is a Q-martingale. In turn, the latter property is equivalent to Er∆L H si ∆M si |F si´s " 0 a.s., for all i " 1, . . . , M . This can be shown to hold since Er∆L H si ∆M si |F si´s " Er∆ p H si ∆M si |F si´s ´ζH si Er∆X si ∆M si |F si´s " Er∆ p H si ∆M si |F si´s ´ζH si Erp∆M si q 2 |F si´s " 0.

	" p Er∆ p H si |F si´s " 0,
	where in the last step we used the fact that 1 ´λsi ∆M si " p Z si { p Z si´a nd the p Q-martingale property
	of p H.

  . . , s M u, the strategy ζ H si is determined by a linear regression of ∆ p H si onto ∆X si , conditionally on F si´. Indeed, we have that

	ζ H si "	Covp∆ p H si , ∆X si |F si´q Varp∆X si |F si´q	,	(5.8)

  Proof. For brevity of notation, let us denote Gpx, tq :" Gpx, t, S, T, Kq, for all px, tq P R ˆr0, Ss. As follows from the proof of Proposition 5.3, the first term on the right-hand side of (5.6), corresponding to ζ H,c , is determined by the diffusive part of the process p H. To this effect, we compute BGpx, tq Bx " ´BpS ´tqGpx, tq `BpT ´Sqe βpS´tq `e´xBpS´tq´Ξpt,Sq Φpd 1 pxqq ´Gpx, tq " ´Gpx, tqBpT ´tq `BpT ´Sqe βpS´tq e ´xBpS´tq´Ξpt,Sq Φpd 1 pxqq.

			t	c	δ S c ptq `ζH,d t	δ S ptq,	for all t P r0, T s
	where, for all i " 1, . . . , M ,
	ζ H,c t	"	´Gpρ t , t, S, T, KqBpT ´tq `BpT ´Sqe βpS´tq P pt, SqΦpd 1 pρ t qq Bpt, S, T q	,	(5.12)
	ζ H,d si	"	ErGpy `ξi , s i , S, T, Kqpξ i ´mi qs| y"ρs i B i ps i , S, T qγ 2 .	(5.13)

  si ∆M si |F si´s Erp∆M si q 2 |F si´s " Er p H si ∆M si |F si´s Erp∆M si q 2 |F si´s " ErGpρ si , s i , S, T, Kqpξ i ´mi q|F si´s Bps

		.6), corresponding to ζ H,d t	, observe that, in view of equation (5.11),
	ζ H,d si	"	Er∆ p H

i , S, T q Varpξ i |F si´q , for all i " 1, . . . , M . Due to the independence of the random variables tξ i ; i " 1, . . . , M u, it holds that ζ H,d si "

  where β Y 0 psq :" pβ X 0 psq, psqq, β Y i psq :" pβ X i psq, Λ i q, for all i " 1, . . . , d, and β Y d`1 psq :" 0. For the second characteristic C Y , we have that for all i " 0, 1, . . . , d, and α Y d`1 psq :" 0.The compensator ν Y pdt, dx, drq of the jump measure of the joint process Y " pX, Rq satisfies ν Y,c pdt, dx, drq " ν X,c pdt, dxqδ p0q pdrq " ˆµY 0 pt, dx, drq `d`1 ÿ dx, drq " µ X i pt, dxqδ p0q pdrq, for all i " 0, 1, . . . , d, and µ Y d`1 pt, dx, drq " 0. Moreover, for all pt, u, vq P R `ˆU ˆiR,(Jacod and Shiryaev, 2003, Proposition II.1.17) together with (4.10) and the fact that X is an affine semimartingale implies that δ T ptq ż DˆR `exu,xy`vr ´1˘ν Y pttu, dx, drq " δ T ptqE " e xu,∆Xty`vρt ´1|F t´‰ " δ T ptq ´evp ptq`xΛ,Xt´yq Ere xu`vΛ,∆Xty ´1|F t´s `evp ptq`xΛ,Xt´yq ´1" δ T ptq ˆevp ptq`xΛ,Xt´yq ż pt, u, vq :" δ T c ptqγ X i pt, uq `δT ptq `vΛ i `γX i pt, u `Λvq ˘, for all i " 1, . . . , d, γ Y d`1 pt, u, vq :" 0.

	C Y t "	ˆCX t 0	0 0 ˙" ż t 0	ˆαY 0 psq	i"1 `d`1 ÿ	Y i s´α	Y i psq ˙ds,
	where
	α Y i psq :"	ˆαX i psq 0 0 0	˙, i"1	Y i t´µ Y i pt, dx, drq ˙dt
	where
	µ Y i pt, In turn, this leads to
	ż
	0 pt,uq`ř d i"1 X i t´γ i pt,uq ´1¯`δ X	T ptq ´ev ptq`γ X 0 pt,u`vΛq`ř d i"1 X i t´p vΛi`γ X i pt,u`vΛqq ´1"
	e γ Y 0 pt,u,vq`ř d`1 i"1 Y i t´γ i pt,u,vq ´1, Y
	where
	γ Y 0 pt, u, vq :" δ T c ptqγ X 0 pt, uq `δT ptq `v	ptq `γX 0 pt, u `Λvq	˘,
	γ Y i
	B Y,c t	"	˜BX,c t ş t 0 ρ s ds ¸" ż t 0	ˆβY 0 psq	i"1 `d`1 ÿ	Y i s´β	Y i psq ˙ds,

D

pe xu`vΛ,xy ´1qν X pttu, dxq `evp ptq`xΛ,Xt´yq ´1"

δ T ptq ´ev ptq`γ X 0 pt,u`vΛq`ř d i"1 X i t´p vΛi`γ X i pt,

u`vΛqq ´1¯. DˆR `exu,xy`vr ´1˘ν Y pttu, dx, drq " δ T c ptq ż D `exu,xy ´1˘ν X pttu, dxq `δT ptq ż DˆR `exu,xy`vr ´1˘ν Y pttu, dx, drq " δ T c ptq ´eγ X (A.1)

  Making use of the notation w " pu, vq P D ˆR and y " px, rq and in view of (A.1), this implies that γ Y for all t P T Y J X and i " 0, 1, . . . , d `1, where we set βY 0 ptq :" ˆβ X 0 ptq p ptq `x βX 0 ptq, Λy `şD p hd`1 pxΛ, xyq ´xΛ, hpxqyqμ X 0 pt, dxqqδ T ptq Λy `şD p hd`1 pxΛ, xyq ´xΛ, hpxqyqμ X i pt, dxqqδ T ptq ˙, for all i " 1, . . . , d, ptqδ T ptq Λ J αX i ptqΛδ T ptq ˙P S d`1 , for all i " 0, 1, . . . , d, and μY i pt, dyq " μY i pt, dx, drq :" μX i pt, dxq `δxΛ,xy pdrqδ T ptq `δ0 pdrqδ T c ptq ˘, with h : R d`1 Ñ R d`1 being a truncation function satisfying hi pyq " h i pxq, for all i " 1, . . . , d. Moreover, we set βY d`1 ptq :" 0, αY d`1 ptq :" 0 and μY d`1 pt, dyq :" 0 for all t P R `. For all i " 0, 1, . . . , d, the measure μY i pt, dyq is a Lévy measure on pDzt0uq ˆR. This follows by observing that, as a consequence of Cauchy-Schwarz inequality, Dzt0u p1 ^}x} 2 qμ X i pt, dxq ă 8, for all t P T and i " 0, 1, . . . , d. Since the affine semimartingale X satisfies by assumption the conditions of (Keller-Ressel et al., 2019, Proposition 5.2), its associated enhanced parameter set is admissible, in the sense of

						˙,
	βY i ptq :"	ˆβ X i ptq pΛ i `x βX
	αY i ptq :"	ˆα X i ptq Λ J αX	αX i ptqΛδ T ptq
		ż				ż
			p1 ^}y} 2 qμ Y i pt, dyq ď p1 `}Λ} 2 q
		pDzt0uqˆR
							. , d,
	γ X i pt, uq " x βX i ptq, uy	`1 2	xu, αX i ptquy	`żDzt0u	`exx,uy ´1 ´xhpxq, uy ˘μ X i pt, dxq,	for all u P U,
	for suitable βX

i ptq P R d , αX i ptq P S d and Borel measures μX i pt, ¨q on Dzt0u. i pt, wq " x βY i ptq, wy`1 2 xw, αY i ptqwy`ż pDzt0uqˆR pe xy,wy ´1´x hpyq, wyqμ Y i pt, dyq, for all w P UˆiR, i ptq, i

  Proof of Lemma 4.10. By Fubini's theorem, we have that ErR

	E	"	ż T1 t	e βpT1´uq dW u	¨ż T2 t	e βpT2´vq dW v		"	ż T1^T2 t	e βpT1`T2´2uq du "	e βpT1`T2q 2β	´e´2βt ´e´2βpT1^T2q	¯.
	Next, we compute the conditional covariance of the jumps:
	Cov	´M ÿ	1 tsiPpt,T1su e βpT1´siq ξ i ,	M ÿ	1 tsiPpt,T2su e βpT2´siq ξ i ˇˇρt	¯" M ÿ	1 tsiPpt,T1^T2su e βpT1`T2´2siq Varpξ i q.
				i"1				i"1					i"1
	Putting the two parts together we obtain (4.21).	l

T |ρ t , R t s " R t `żpt,T s mpt, uqηpduq " R t `żpt,T s mpt, uqdu `N ÿ j"1

See https://www.isda.org/protocol/isda-2020-ibor-fallbacks-protocol.

Financial support from the University of Padova (research programme BIRD190200/19, "is gratefully acknowledged. Data sharing not applicable to this article as no

datasets were generated or analysed during the current study.

1 See the FCA announcement on future cessation and loss of representativeness of the Libor benchmarks (https: //www.fca.org.uk/publication/documents/

1 tsiPpt,usu m i e βpu´siq ¯du " ρ t BpT ´tq `Apt, T q `M ÿ

i"1

1 tsiPpt,T su m i BpT ´si q, while the second term is given by

1 ttj Ppt,T su mpt, t j q " N ÿ j"1 1 ttj Ppt,T su ´ρt e βptj ´tq `apt, t j q `M ÿ

i"1

1 tsiPpt,tj su m i e βptj ´siq ¯.

Regarding the conditional variance, we observe that by Fubini's theorem it holds that

We first compute

Then, observe that

The last integral in (A.5) reduces to

Applying a reasoning analogous to (A.5), we can rewrite as follows the second term in equation (A.3), omitting to write the term γ 2 i for simplicity of presentation:

1 tsiPpt,usu e 2βpu´siq ηptuuqηpduq.

The first integral appearing in the last line can be computed as follows

1 tsiPpt,vsu e βpu`v´2siq ηpdvqηpduq

1 tsiPpt,T su 1 ttj Prsi,T su e 2βptj ´siq Ipt j , T q, while the second integral in this line reduces to