Term structure modeling with overnight rates beyond stochastic continuity

Overnight rates, such as the Secured Overnight Financing Rate (SOFR) in the United States, are central to the current reform of interest rate benchmarks. A striking feature of overnight rates is the presence of jumps and spikes occurring at predetermined dates due to monetary policy interventions and liquidity constraints. This corresponds to stochastic discontinuities (i.e., discontinuities occurring at ex ante known points in time) in their dynamics. In this work, we propose a term structure modeling framework based on overnight rates and characterize absence of arbitrage in a generalized Heath–Jarrow–Morton (HJM) setting. We extend the classical short‐rate approach to accommodate stochastic discontinuities, developing a tractable setup driven by affine semimartingales. In this context, we show that simple specifications allow to capture stylized facts of the jump behavior of overnight rates. In a Gaussian setting, we provide explicit valuation formulas for bonds and caplets. Furthermore, we investigate hedging in the sense of local risk‐minimization when the underlying term structures feature stochastic discontinuities.


INTRODUCTION
The discontinuation of the publication of Libor rates for the majority of currencies and tenors on January 1, 2022, and the cessation of the US dollar Libor panel on June 30, 2023, mark a major transition for interest rate markets. 1 In the reform of interest rate benchmarks, overnight rates play a central role, such as Secured Overnight Financing Rate (SOFR) in the United States, Sterling overnight index average (SONIA) in the United Kingdom and Euro short-term rate (€STR) in the Euro zone, sometimes generically referred to as risk-free rates (RFRs).
A distinctive feature of overnight rates is the presence of stochastic discontinuities in their dynamics: jumps and spikes occurring at predetermined dates or at regular intervals, as a result of monetary policy interventions as well as regulatory and liquidity constraints.In particular, overnight rates tend to jump in correspondence with meetings of the monetary policy authority, and these meetings usually follow a set schedule.This is confirmed by the analysis in Backwell and Hayes (2022) documenting that most of the variation in the SONIA rate over the years 2016-2020 occurs in correspondence with the meeting dates of the Monetary Policy Committee of the Bank of England.The recent analysis in Schlögl et al. (2023) provides evidence of a similar phenomenon for SOFR, highlighting the importance of modeling scheduled jumps that coincide with the Federal Open Market Committee (FOMC) meeting dates.
In this work, the predetermined dates at which the overnight rate (and, potentially, forward rates) is expected to exhibit discontinuities will be called expected jump dates and denoted by  = { 1 , … ,   }.In the case of SOFR, the presence of expected jump dates is well illustrated by Figure 1.As a particular example, consider the spike observed on September 17, 2019.According to Anbil et al. (2020), "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 Anbil et al. (2020) suggests that the date of this spike was known well in advance (a corporate tax payment date coinciding with a Treasury coupon settlement), while the magnitude of the jump was obviously not predictable.
Starting from these observations, we develop a general framework for interest rate markets in the presence of overnight rates.A first key point is that the natural choice for the numéraire asset in this context corresponds to a rolled-over investment in the overnight rate, according to a schedule  = { 1 ,  2 , …} of roll-over dates.The numéraire can, therefore, exhibit jumps in correspondence with such roll-over dates, which represent an additional source of stochastic discontinuities, besides the expected jump dates mentioned above.Typically, the roll-over dates will be quite frequent, while expected jump dates will be less frequent, and it is not excluded that some expected jump dates coincide with roll-over dates.In this context, building on the results of Fontana et al. (2020), we extend the Heath-Jarrow-Morton (HJM) approach by allowing for two different types of stochastic discontinuities.We characterize absence of arbitrage by means of generalized drift conditions, with specific no-arbitrage restrictions related to the stochastic discontinuities.
As a second main contribution, we develop a tractable class of models based on affine semimartingales, that is, affine processes, which allow for stochastic discontinuities (see Keller-Ressel et al. (2019)).We show that affine semimartingale models for an overnight rate provide a natural extension of classical affine short-rate models 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.Towards practical applications, 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 induces 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 expected jump dates.We exemplify this result by considering the problem of hedging a SOFR-caplet by trading in a SOFR futures contract, the most liquidly traded contract written on SOFR at the time of writing.

Related literature
The reform of interest rate benchmarks is receiving considerable attention and, therefore, we limit our review of the literature to some contributions that are specifically related to our work, referring to Henrard (2019), Piterbarg (2020), Klingler and Syrstad (2021), and Huggins and Schäller (2022) for a general analysis of the challenges of the Libor reform.One of the earliest and most influential contributions is Lyashenko and Mercurio (2019), based on an extension of the Libor market model.Classical short-rate models have been revisited in the post-Libor universe by several authors, starting from Mercurio (2018).Several recent works employ the classical Hull-White model (see, e.g., Hofman (2020), Turfus (2020), Hasegawa (2021)).Always in a short-rate setup, Skov and Skovmand (2021) propose a multifactor Gaussian model in order to analyze SOFR futures, while Fontana (2023) develops a model driven by general affine processes in view of pricing applications.Rutkowski and Bickersteth (2021) adopt a Vasiček 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 Macrina and Skovmand (2020), who adopt a linear rational model for the savings account and derive several pricing formulas.We also mention Willems (2020), where an extended SABR model is applied to the pricing of caplets in the post-Libor setup.
The papers mentioned in the previous paragraph do not take into account stochastic discontinuities in the dynamics of overnight rates.To the best of our knowledge, the earliest works acknowledging this fact are Piazzesi (2001Piazzesi ( , 2005)).In Kim and Wright (2014), a term structure model with jumps occurring in correspondence of macroeconomic announcements dates is presented.In recent works, stochastic discontinuities are playing an increasingly important role.In particular, Andersen and Bang (2020) develop a model that can generate spikes in the SOFR dynamics, both at totally inaccessible times and at anticipated times.Gellert and Schlögl (2021) and Brace et al. (2022) 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 Backwell and Hayes (2022), the SONIA rate is modeled via a short-rate approach by relying on a pure jump process with both unexpected and predetermined jump dates.Schlögl et al. (2023) make use of a short-rate affine jump-diffusion framework to provide a model, which is able to jointly fit the overnight US policy rate, SOFR and SOFR futures rates.Harju (2023) models US overnight rates, such as the effective Fed funds rate and the USD overnight Libor, in the setup of short rate models, also including jumps at fixed times.Finally, we mention that stochastic discontinuities also play an important role in credit markets (see Gehmlich &Schmidt, 2018 andFontana andSchmidt, 2018), while a general framework for multicurve markets with stochastic discontinuities is developed in Fontana et al. (2020).

Structure of the paper
In Section 2, we present a general view on post-Libor interest rate markets based on overnight rates.In Section 3, we develop a modeling framework based on the HJM approach extended to the case of stochastic discontinuities.In Section 4, we introduce a model based on affine semimartingales for overnight rates and provide explicit valuation formulas in an extended Hull-White model.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.

INTEREST RATE MARKETS WITH OVERNIGHT RATES
In this section, we give a fundamental description of an interest rate market in the presence of overnight rates.We replace the classical assumption of the existence of a savings account generated by an instantaneous short rate by a more general structure, which in particular allows for stochastic discontinuities.The first key point of our analysis is, therefore, a systematic study of the implications of a numéraire with stochastic discontinuities generated by overnight investment.

The numéraire
The numéraire asset obtained by investing according to an overnight rate is generated by a rollover strategy and as such is piecewise constant, being updated at every roll-over date.Those dates are predetermined (hence, deterministic) and we denote them by  1 <  2 < ⋯, usually corresponding to business days.We call those dates roll-over dates, as mentioned in the introduction, and collect them in the set  ∶= {  ∶  ∈ ℕ}.Moreover, we set  0 ∶= 0.
In this situation, the value at time  ≥ 0 of the overnight numéraire takes the following form: with    representing the overnight rate applicable to the time period [  ,  +1 ].
To allow for greater generality, and in particular to include the classical framework into our setting (see Remark 2.1 below), we assume that the numéraire process  0 is given by where  is a Borel measure on ℝ + with the following structure: Here    denotes the Dirac measure in {  } and  is an adapted process satisfying ∫  0 |  |() < ∞ a.s., for all  > 0. We will refer to  as the RFR process.2), with  representing the risk-free short rate.

Notions of interest rates
The second key point in our analysis is the new features of post-Libor markets.We, therefore, introduce several notions of interest rates, which are important in such markets, relying mostly on Lyashenko and Mercurio (2019).We denote by (, ) the price at date  of a zero-coupon bond with unit payoff at maturity  ≥ .In the following, we will refer to zero-coupon bonds simply as bonds.

2.2.1
The backward-looking rate For each 0 ≤  < , the setting-in-arrears rate (, ) is the rate that is achieved over the period [, ] by a rolled-over investment according to the roll-over dates  .This yields the rate where (, ) ∶= { ∈ ℕ ∪ {0} ∶  ≤   and  +1 ≤ } denotes the set of indices of the roll-over dates   for which the interval The setting-in-arrears rate (, ) is said to be backward-looking, since its value can be determined only at the end of the accrual period [, ] and not at its beginning (as it would be the case for a forward rate).Backward-looking rates play a central role in post-Libor markets, having been adopted as the reference fallback rates for most Libor-based contracts and transactions.
The generality of our setup enables us to work with the exact definition (4) of the setting-inarrears rate regardless of the structure of  0 (in particular, also when  0 is absolutely continuous).
Remark 2.2 (Relation to the numéraire).The overnight rate    mentioned in Equation (1) can be obtained from bond prices via 1 +    ( +1 −   ) = 1∕(  ,  +1 ).Hence, if  0 is the overnight numéraire given in Equation (1), then the setting-in-arrears rate (, ) can be directly written in terms of  0 as In the literature (see, e.g., Lyashenko &Mercurio, 2019 andSkov &Skovmand, 2021), the numéraire  0 is usually assumed to be absolutely continuous (see part (ii) of Remark 2.1) and Equation ( 5) is adopted as an approximation of the setting-in-arrears rate (, ).As mentioned above, this approximation is not necessary in our setting with stochastic discontinuities.

2.2.2
The forward-looking rate The forward-looking rate (, ) is defined as the rate  that makes equal to zero the market value at time  of the payoff ( − )((, ) − ) delivered at maturity , for 0 ≤  < .Such an agreement is called a single-period swap.In contrast to the backward-looking rate, the forwardlooking rate (, ) is determined at the beginning of the accrual period.
Remark 2.3 (On the notion of forward-looking rate).The most widely adopted forward-looking rate is the CME term SOFR rate, which has been approved by the Alternative Reference Rates Committee (ARRC) in 2021 for use in cash products and, with some restrictions, derivatives.Moreover, since June 30, 2023, the CME term SOFR rates (plus the respective ISDA fixed spread adjustment) are used for calculation of the temporary synthetic 1-, 3-and 6-month USD-indexed Libor rates meant to facilitate the transition of the contracts that reference USD-indexed Libor (see the footnote on page 1).In theory, forward-looking rates should be determined as discussed above from market quotes of overnight index swaps (OIS), of which single-period swaps are the basic building blocks.However, the CME term SOFR rate is currently determined with a specific methodology, based on the work of Heitfield and Park (2019), that relies on market quotes of SOFR futures.This is due to the fact that liquidity in SOFR OIS is not deemed sufficient, while SOFR futures are traded in much larger volumes. 2 Using futures prices to compute forward rates is a modeldependent procedure and also incurs into the issue of convexity adjustments.Therefore, in the current market environment of imperfect liquidity, it may happen that the CME term SOFR rate is not perfectly aligned with the forward-looking rate derived from SOFR swap quotes.In this paper, we shall not consider this issue, which will be addressed in a separate work.

2.2.3
The forward term rate Single-period swaps can be considered the basic contracts written on backward-looking and forward-looking rates in post-Libor markets, analogously to forward rate agreements in classical interest rate markets.For 0 ≤  <  and  ∈ [0, ], the backward-looking forward rate (, , ) is defined as the rate  that makes equal to zero the value at time  of a single-period swap delivering the payoff ( − )((, ) − ) at maturity .Note that, differently from the classical concept of a forward rate, the backward-looking forward rate (, , ) is also defined inside the accrual period (i.e., for  ∈ [, ]), due to the backward-looking nature of (, ).
In an analogous way, we define the forward-looking forward rate (, , ) for any  ∈ [0, ] as the rate  that makes equal to zero the value at time  of a single-period swap delivering the payoff ( − )((, ) − ) at maturity .The forward-looking forward rate satisfies (, , ) = (, ), where (, ) is the forward-looking rate introduced above.
Comparing the notions of backward-looking and forward-looking forward rate, we notice that However, while the forward-looking forward rate (⋅, , ) stops evolving at time , the backwardlooking forward rate (⋅, , ) continues to evolve until time , when it reaches the terminal condition (, , ) = (, ).Identity (6), which has been first pointed out in Lyashenko and Mercurio (2019), therefore implies that backward-looking forward rates and forward-looking forward rates can be consolidated into a single process (⋅, , ).In this work, we adopt this viewpoint and generically call the process (⋅, , ), considered in the whole time interval [0, ], the forward term rate.
Remark 2.4 (On the validity of Equation ( 6)).The discussion in Remark 2.3 implies that, if (, ) is assumed to coincide with the CME term SOFR rate, then the identity (, ) = (, , ) may fail to hold in the current market environment, due to the CME term SOFR rate calculation methodology.In turn, this implies that violations to Equation ( 6) cannot be excluded a priori.Allowing for violations to Equation ( 6) would require to model (⋅, , ) separately from (⋅, , ), with the introduction of an intrinsic multicurve dimension in the model.We will develop this aspect in a forthcoming work, where we will also provide a mathematical description of violations to Equation ( 6) in terms of the strict local martingale property of solutions to a suitable BSDE.Here, however, we do not account for possible violations to Equation ( 6), assuming implicitly that liquidity in SOFR swaps is sufficient for a robust determination of forward term rates, coherently with the viewpoint of Lyashenko and Mercurio (2019).
It is important to note that forward term rates can be expressed in terms of bond prices.Indeed, in view of formula (4), the backward-looking rate (, ) can be replicated by holding a static position in a bond with maturity  and investing the payoff (, ) = 1 received at time  into the overnight numéraire until time .Since (, ) represents the floating leg of a single-period swap referencing the backward-looking rate, this implies that the forward term rate can be written as The analysis developed in this section shows that, together with the numéraire  0 , the family of bond prices {(⋅, );  > 0} constitutes the fundamental basis of a term structure model for a post-Libor market as considered here.This is the approach that we are going to adopt and develop in the next section, highlighting the role of stochastic discontinuities.

AN EXTENDED HEATH-JARROW-MORTON FRAMEWORK
In this section, we develop a general term structure model based on overnight rates in the presence of stochastic discontinuities.The main result of this section is Theorem 3.5, which provides a set of necessary and sufficient conditions for the risk-neutral property of a given probability measure.We recall that, as mentioned in the introduction, in the considered interest rate market, two different types of stochastic discontinuities arise naturally: (i) roll-over dates  = { 1 ,  2 , … }, corresponding to the discontinuities in the numéraire process  0 , encoded in the atoms of the measure  introduced in Equation (3); (ii) expected jump dates  = { 1 , … ,   }, representing a set of deterministic times at which the RFR process  and forward rates are expected to exhibit jumps.
We do allow for an overlap of these sets (i.e.,  ∩  ≠ ∅), meaning that stochastic discontinuities in the dynamics of the term structure of RFRs can occur simultaneously to some of the roll-over dates.In comparison to the roll-over dates in  , the expected jump dates in  are much less frequent, so that we consider only a finite number of them.
Remark 3.1 (Extension to predictable times).The results of this section are also valid in the more general setting where  is a countable family of predictable times, see Fontana and Schmidt (2018).
For simplicity of presentation and in order to treat  with the same techniques used for  , we suppose that  is a finite family of fixed dates.
The analysis of Section 2.2 shows that the key ingredient of a term structure model in the post-Libor framework as considered here is the family of bond prices {(⋅, );  > 0} together with the numéraire  0 defined in Equation (2).To introduce a general modeling framework, we consider an extension of the HJM approach allowing for discontinuous term structures.To this end, we assume that with the convention ∫ (,] (, )() = 0, for all  ≥ 0.Moreover, we assume that the instantaneous forward rates (⋅, ), for  ≥ 0, are given by (, ) = (0, ) + ∫ Remark 3.2.Integration with respect to the measure  in Equation ( 7) is justified by the fact that, since the numéraire  0 jumps in correspondence of the atoms of , bond prices are expected to be discontinuous (in maturity) at those points.More precisely, absence of arbitrage implies that bond prices have necessarily to be of the form ( 7) with respect to the same measure  appearing in Equation ( 2).This fact has been first pointed out in Gehmlich and Schmidt (2018), albeit in the context of default modeling.
To proceed further, we introduce the following technical requirements on Equation (8).
For all 0 ≤  ≤  and  ∈ , we define The probability measure  is a risk-neutral measure if (⋅, )∕ 0 is a local martingale under , for every  > 0. The existence of a risk-neutral measure suffices to ensure absence of arbitrage, in the sense of no asymptotic free lunch with vanishing risk (NAFLVR, see Cuchiero et al. (2014)), for the large financial market where bonds of all maturities are traded.In turn, this ensures the validity of NAFLVR in the post-Libor market described in Section 2.2.We refer to Klein et al. (2016) and (Fontana et al., 2020, Section 6) for a detailed analysis of absence of arbitrage in interest rate markets.
Theorem 3.5.Suppose that Assumption 3.3 holds.Then,  is a risk-neutral measure if and only if and the random variable is sigma-integrable with respect to ℱ − , for every  ∈  ∪  , and the following four conditions hold: (iii) for every  = 1, … , , it holds a.s. that (iv) for every  = 1, … , , it holds a.s. that Proof.In view of Equation ( 9), (⋅, )∕ 0 is a local martingale if and only if the finite variation process is a local martingale, where ) is a local martingale, it is also of locally integrable variation by (Jacod and Shiryaev, 2003, Lemma I.3.11).Since , and, therefore, both processes  () (⋅, ),  = 1, 2, are of locally integrable variation.This implies that condition (11) holds and the random variable is sigma-integrable with respect to ℱ − , for all  ∈  ∪  (see (He et al., 1992, Theorem 5.28)).
Taking into account the definition of V(, ), Equation ( 8) and the fact that ((−, ) −  − )  () is ℱ − -measurable, the latter property is equivalent to the sigma-integrability of the random variable (12).Denoting by K() (⋅, ) the compensator of  () (⋅, ), for  = 1, 2, and making use of (He et al., 1992, Theorem 5.29), the local martingale property of (⋅, ) is then equivalent to the validity of ) , (14) up to an evanescent set.Equation ( 14) holds if and only if outside of a set of ( ⊗ )-measure zero.Taking  =  in Equation ( 16) yields condition (i), while condition (ii) follows by inserting condition (i) into Equation ( 16).In view of Equation ( 8) and the definition of V(, ), Equation ( 15) holds if and only if for all  ∈  ∪  .Taking  =  in Equation ( 17) yields condition (iii), while condition (iv) is then obtained by inserting condition (iii) into Equation ( 17).Conversely, if condition ( 11) is satisfied and the random variable in Equation ( 12) is sigmaintegrable, for every  = 1, … , , then the compensator K() (⋅, ) is well-defined, for  = 1, 2. It is then straightforward to verify that if conditions (i)-(iv) are satisfied, then Equations ( 14) and (15) hold true.This implies that the process (⋅, ) given in Equation ( 13) is a local martingale, thus proving the local martingale property of (⋅, )∕ 0 , for every  ≥ 0. □ In the absence of stochastic discontinuities, it is well-known that (⋅, )∕ 0 is a local martingale if and only if the integrability requirement (11) and conditions (i) and (ii) of Theorem 3.5 hold, see (Björk et al., 1997, Proposition 5.3).The presence of stochastic discontinuities is reflected in conditions (iii) and (iv) (together with the requirement of sigma-integrability of Equation 12).More specifically, condition (iii) relates the stochastic discontinuities  in the numéraire to the short end of the forward rate, while condition (iv) concerns the stochastic discontinuities  in the forward term rate.Taken together, conditions (iii) and (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 Fontana et al. (2019) for an analysis of the arbitrage possibilities arising with predictable jumps).
The conditions of Theorem 3.5 admit a simplification under the following additional assumption.
This assumption corresponds to requiring that the RFR  and the forward rates do not jump in correspondence of the roll-over dates of the numéraire  0 .In this case, conditions (i) and (iii) can be rewritten in the following compact way: Moreover, under Assumption 3.6, the term Δ     (  ) in condition (iv) vanishes.
Remark 3.7 Relation to Fontana et al. (2020).The presence of the two distinct sets  and  of discontinuity dates distinguishes the present setup from the framework used for modeling multicurve term structures in Fontana et al. (2020).For this reason, we have given full self-contained proofs of Lemma 3.4 and Theorem 3.5, which cannot be deduced in a straightforward way from Fontana et al. (2020).At the same time, we have simplified some of the original techniques of Fontana et al. (2020).
The next corollary provides the dynamics of the short end (, ) of the instantaneous forward rate.We omit the proof, which follows the same arguments of (Filipović, 2009, Proposition 6.1).
Corollary 3.8.Suppose that Assumption 3.3 holds.Assume furthermore that (0, ), (, ), (, ), (, , ) are differentiable in  with ∫  0 |  (0, )| < ∞, for all  ≥ 0, and such that conditions (ii), ( iii Remark 3.9.In the classical HJM setup, the short end of the instantaneous forward rate is equal to the short rate, that is, we have   = (, ).In the presence of stochastic discontinuities, this must not necessarily be the case and absence of arbitrage implies only the equality (, ) =   ( ⊗ )-a.e., as shown in Theorem 3.5 (i).It is interesting to see what happens if one assumes that the RFR  is defined as the short end of the instantaneous forward rate by imposing   ∶= (, ), for all  ≥ 0.
Condition (i) of Theorem 3.5 is of course automatically satisfied.Conditions (iii) and (iv) are equivalent to the validity of condition (15) in the proof of Theorem 3.5.Under Equation ( 18), this condition becomes This in particular entails that, if all stochastic discontinuities in the model correspond only to rollover dates  (meaning that  = ∅), then setting  equal to the short end of the forward rate as in Equation ( 18) implies that conditions (i), (iii), and (iv) of Theorem 3.5 hold and  is a risk-neutral measure if and only if condition (ii) is satisfied, as in the classical HJM setup.
To illustrate the flexibility of the extended HJM approach, we show how a generalized version of the popular Cheyette model with stochastic discontinuities fits into our framework.In Section 4, we will develop an approach based on the philosophy of short-rate modeling.
In this example, we illustrate an extension of the Cheyette model to a setup with stochastic discontinuities.We consider only the presence of the expected jump dates , assuming  = ∅ for simplicity of presentation, so that  0 = exp(∫ ⋅ 0   ).We specify as follows the forward rates: where , ,   ,   , for  = 1, … , , are deterministic functions and   ∼  (  ,  2  ), for  = 1, … , , are independent normally distributed random variables which are also independent of . The where the explicit form of the function   (, ) can be deduced from condition (iv) of Theorem 3.5: with Ā(  , ) ∶= ∫    (), for all  = 1, … ,  and  ≥ 0. If in addition (⋅) = (⋅), then combining Equations ( 19) and ( 20) we directly obtain that (, ) = (0, ) + () ()   + (, ), where   ∶=    + ()  and (, ) ∶=   (, ) +   (, ).Moreover, making use of Equation (2.20) in Beyna (2013) and by means of straightforward computations, it can be proved that  is a mean-reverting Gaussian Markov jump-diffusion process with speed of mean reversion   log(()), jumping only in correspondence with the predetermined stochastic discontinuities dates  1 , … ,   .This example illustrates that the Cheyette model class can be easily extended to the case of stochastic discontinuities, retaining a remarkable level of analytical tractability.

THE AFFINE FRAMEWORK FOR OVERNIGHT RATE MODELING
In this section, we present a modeling framework based on affine semimartingales.We focus here on the direct modeling of the overnight rate to provide a setup for developing extensions of the Hull-White and other affine models, which are able to include stylized facts of overnight markets, namely spikes and jumps occurring at predetermined dates.Note that an affine specification of the HJM framework presented in Section 3 is also possible, leading to explicit conditions in Theorem 3.5.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 and are, therefore, tailor-made to interest rate markets in the presence of overnight rates.We shortly introduce some general notions on affine semimartingales and then develop an affine model for overnight rates.

Affine semimartingales
Consider a semimartingale  taking values in the state space where the functions   (, ) and   (, ) take values in ℂ and ℂ  , respectively.We assume that conv(supp(  )) = , for all  > 0, and that  is quasi-regular and infinitely divisible, in the sense that the regular conditional distribution (  ∈ |  ) is an infinitely divisible probability measure on  a.s.for all 0 ≤  ≤  < ∞.
The property that affine processes have affine semimartingale characteristics has a natural extension to the semimartingale case.More precisely, by (Keller-Ressel et al., 2019, Lemma 4.3 and Theorem 3.2), there is no loss of generality in assuming that  is a Markov process and the semimartingale characteristics (  ,   ,   ) of  with respect to a fixed truncation function ℎ ∶ ℝ  → ℝ  , with  =  + , have the following structure, where  , and  , denote the continuous parts of   and   , respectively: where γ (, ) = (  1 (, ), … ,    (, )) ∈ ℝ  .Finally, the terminal conditions are given by   (, ) = 0 and   (, ) = .
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 can be generalized in a straightforward way to that case.

Affine models for overnight rates
A widely used approach in interest rate theory consists in modeling the short rate and computing bond and derivative prices by risk-neutral valuation.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 formulas for bond prices and interest rate derivatives can be obtained.In this section, we show how this short-rate approach can be extended to the case of affine semimartingales with stochastic discontinuities.
We assume that the RFR rate  is given by where Λ ∈ ℝ  and  ∶ ℝ + → ℝ is a deterministic càdlàg function such that ∫  0 |()|() < ∞, for all  > 0, with measure  from Equation (3).The function  serves to fit the initially observed term structure of bond prices, similarly as in Brigo and Mercurio (2001).
In view of Equations ( 21) and ( 22), the conditional characteristic function of the RFR is directly available.However, this does not suffice for the valuation of bonds and interest rate derivatives, since the discount factor and most payoffs depend on the integrated process  ∶= ∫ ⋅ 0   ().
The following proposition shows that the joint process (, ) is an affine semimartingale on the extended state space  × ℝ.This result, which can be considered of independent interest, represents a generalization to affine semimartingales of the enlargement of the state space approach of (Duffie et al., 2003, Section 11.2).The proof is technical and therefore postponed to the appendix.
In particular, the joint process (, ) is an affine infinitely divisible semimartingale.
The explicit characterization of the conditional Fourier transform of the joint process (, ) obtained in Proposition 4.2 allows for an efficient valuation of a variety of interest rate derivatives in the post-Libor environment.In particular, bonds can be priced by evaluating Equation ( 23) at (, ) = (0, −1), whenever the expectation is finite.In turn, this leads to explicit pricing formulas for all linear derivatives such as swaps.Nonlinear derivatives can be priced by relying on Fourier methods, analogously to the case of affine processes, see (Filipović, 2009, Section 10.3) and Fontana (2023).

A Hull-White model for the overnight rate
In this section, we present a tractable specification of an affine model for the overnight rate.More specifically, we generalize the Hull-White model to the case of stochastic discontinuities.We consider a finite set  = { 1 , … ,   } representing the roll-over dates of the numéraire  0 , while  = { 1 , … ,   } denotes the set of expected jump dates of the RFR process.
Let  be the unique strong solution of the following stochastic differential equation: where  ∶ ℝ → ℝ is a continuous function,  ∈ ℝ,  ≥ 0 and  = (  ) ≥0 is a Brownian motion.
The process  = (  ) ≥0 in Equation ( 28) is a pure jump process specified as follows: where the random variables {  ;  = 1, … , } are assumed to be independent of .The following lemma gives the explicit solution to Equation ( 28) and is an immediate consequence of Itô's formula.(30) As illustrated in the following example, processes of the form (30), 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).
We can observe that  exhibits spikes at  = 50 and  = 100 and a jump to a new level at  = 150.The spiky behavior is generated by  2 , which has no diffusive component and very high meanreversion speed (increasing  2 further would produce even more pronounced spikes).The jump to a new level at  = 150 is generated by the component  1 , which has a much slower meanreversion.
In the next results, we shall make use of the following notation: For  = 0, we simply write () ∶= (0, ).In the following proposition, we compute the conditional characteristic function of the solution to SDE (30).While this result can be deduced from the general theory of affine semimartingales, we provide a direct proof exploiting the structure of Equation ( 30).Proof.From representation (30), we directly obtain that .
This immediately yields that (,  − ) =  exp(( − )).The result of the proposition follows by relying on the independence of the random variables {  ;  = 1, … , }.□ Explicit expressions for bond prices can be obtained either by applying the general result from Proposition 4.2 or by a direct computation of the conditional characteristic function of the timeintegral ∫  0   (), as illustrated in the next subsection under a specific assumption on the distribution of the jump sizes.

A Gaussian Hull-White model for the overnight rate
Until the end of this section, we assume that the random variables {  ∶  = 1, … , } are independent and normally distributed.In this case, we immediately see from Equation ( 30) that  is a Gaussian process (more precisely, a Markov process with Gaussian increments).Moreover, the random variable   ∶= ∫  0   () is also normally distributed for all  > 0. This immediately yields the following proposition.) .
We proceed to compute explicitly the conditional mean and variance of   appearing in Proposition 4.6.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 .
By integrating equation ( 30) and applying Fubini theorems, we obtain By relying on Equation ( 32) and on the properties of the random variables {  ∶  = 1, … , }, the next lemma gives explicit expressions for the conditional expectation and variance of   .We denote ) , for all 0 ≤  ≤ . ) , ) ) .
Remark 4.9 (Impact of expected jump dates).While   is the jump in the overnight rate at date   , its impact on bond prices is modulated by ( −   ) + ∑  =1  {  ∈(  ,]}  (  −  ) in Equation ( 34), with ( −   ) defined in Equation (33).In particular, since bond prices determine forward term rates, this allows generating forward term rates that exhibit less pronounced stochastic discontinuities than the overnight rate, in line with the empirical evidence in post-Libor markets based on overnight rates.
Remark 4.10.In the present affine setup, the availability of exponentially affine formulas for bond prices implies that the backward-looking rate (, ) admits a tractable representation, being defined in Equation ( 4) as a product of bond prices.This highlights the fact that our setting enables us to work with the exact definition of the backward-looking rate, while retaining complete analytical tractability.
We close this section by deriving an explicit formula for the price of a caplet in the context of the present Gaussian Hull-White model.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 (, ) or by the forward-looking rate (, ), see for instance, Lyashenko and Mercurio (2019) and Fontana (2023).For illustration, we consider here a forward-looking caplet, whose payoff at date  is given by for some  > 0. We recall from Section 2.2.3 that (, ) = (, , ) = (1∕(, ) − 1)∕( − ).
In view of Proposition 4.6 and Lemma 4.8, we have that: where, for brevity of notation, Ξ(, ) collects all terms appearing in Lemma 4.8 that do not multiply   .For the determination of the caplet price, we need to compute the ℱ  -conditional distribution of   under the -forward measure   defined by   ∕ = 1∕( 0  (0, )).
Moreover, the random variables {  ∶  = 1, … , } are mutually independent and independent of the Brownian motion   .

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 .In this section, we aim at determining optimal hedging strategies in the sense of local risk-minimization.This corresponds to attaining perfect replication of payoffs while relaxing the self-financing requirement and minimizing the cost of the strategy according to a quadratic criterion (see Pham (2000) and Schweizer (2001) for an overview of the theory).In Section 5.1, we provide a general description of local risk-minimization with stochastic discontinuities, while in Section 5.2, we study an explicit example in the context of the Gaussian Hull-White model of Section 4.4.

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 .
Assumption 5.1 is for instance satisfied in the model of Section 4.3 if the filtration  is generated by the pair (, ).Note that the assumption that the discontinuity dates  do not appear in the martingale representation (37) is only made for simplicity of presentation.
We suppose that the financial market contains a traded security with  0 -discounted price process  = (  ) ∈[0,] , assumed to be a special semimartingale with canonical decomposition where  = (  ) ∈[0,] is a predictable process of finite variation and  = (  ) ∈[0,] a squareintegrable martingale, with  0 =  0 = 0.The process  can represent for instance the price process of a SOFR future contract, at present the most liquid product referencing SOFR (see Section 5.2).Note also that in this section, we do not necessarily assume that  is a risk-neutral measure.
As a consequence of Assumption 5.1, the martingale  admits a representation of the form where  = (  ) ∈[0,] is a predictable process such that [∫  0  2  ] < ∞ and Δ   =   (  ), where the function   is as in Assumption 5.1, for each  = 1, … , .We, furthermore, assume that  has nonvanishing volatility, in the sense that   > 0 a.s.for all  ∈ [0, ].
Let  be a square-integrable ℱ  -measurable random variable, representing a discounted payoff.By market incompleteness,  may not be attainable by self-financing trading.We then consider nonself-financing strategies attaining the payoff , as formalized in the following definition, where we denote by Θ the set of all predictable processes  = (  ) ∈[0,] such that is a square-integrable martingale strongly orthogonal to .
In Definition 5.2,   and   represent, respectively, the positions held in the traded security and the portfolio value at time , for all  ∈ [0, ].By (Schweizer, 2001, Theorem 3.3), the definition of locally risk-minimizing strategy adopted in Definition 5.2 is equivalent to the original definition of Schweizer (1991) if the process  in Equation ( 38) is continuous, as in the case of the example considered in Section 5.2.For general , Definition 5.2 corresponds to the so-called pseudo-locally risk-minimizing strategy.
Under Assumption 5.1, we can explicitly derive decomposition (40) for a generic discounted payoff .To this effect, let us define Ẑ ∶= ℰ(− ∫ ⋅ 0     ) and assume that Ẑ is a strictly positive square-integrable martingale under .This enables us to define the minimal martingale measure Q by  Q = Ẑ .We can then define the Q-martingale Ĥ = ( Ĥ ) ∈[0,] by
Proof.By the product rule, it holds that for all  ∈ [0, ].An application of Itô's formula yields Therefore, in view of Equation ( 41), we can compute where   = (   ) ∈[0,] is defined as in Equation ( 42).We proceed to show that Equation ( 43

□
Theorem 5.3 provides an explicit description of the locally risk-minimizing strategy for a generic payoff .In particular, formula (42) shows that the locally risk-minimizing strategy consists in a perfect replication at all times  ∈ [0, ] ⧵ , when the only active source of randomness is the Brownian motion .The first term on the right-hand side of Equation ( 42) corresponds to the Delta-hedging continuous strategy.On the other hand, in correspondence of the expected jump dates  = { 1 , … ,   }, the strategy     is determined by a linear regression of Δ Ĥ  onto Δ   , conditionally on ℱ   − .Indeed, we have that for all  = 1, … , , as follows from Equation (42) using the predictability of the process .We also remark that the associated cost process (  ) is generated by the residuals of the regressions (44).

An example
In this section, we illustrate the hedging approach described in Section 5.1 in the case of a forwardlooking 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 less liquid in the market.We consider the model of Section 4.3, with  playing now the role of the physical probability measure: where  is defined as in Equation ( 29), where the random variables {  ∶  = 1, … , } are independent and independent of , with distribution  (  ,  2  ) under , for each  = 1, … , .For simplicity of presentation, in this subsection, we assume that () =  (i.e., there are no roll-over dates).
Remark 5.4.In the context of the model of Section 4.3, the futures rate (, , ) can be explicitly computed.Suppose that, in line with the market convention for 1-month RFR futures contracts, the futures contract settles at date  at a rate quoted as (  −   )∕( − ).By risk-neutral valuation under the minimal martingale measure Q, it holds that We suppose that the payoff  to be hedged corresponds to an RFR caplet with discounted payoff  ∶= ( − ) ( (, ) −  ) + ∕ 0  , for some  > 0, as considered in Section 4.4.To determine the locally risk-minimizing strategy, we first need to compute the price process Ĥ = ( Ĥ ) ∈[0,] of the payoff  under the measure Q.This can be achieved by a direct application of Proposition 4.12, leading to Ĥ = (  , , , , )∕ 0  , where the function (  , , , , ) is explicitly given in Proposition 4.12, replacing (, ) by Â(, ) in the definition of the quantity Ξ(, ) and () by α() in the definition of Γ 1 (, ).