Term structure modeling with overnight rates beyond stochastic continuity

Abstract

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.

Discretely Distributed Scheduled Jumps and Interest Rate Derivatives: Pricing in the Context of Central Bank Actions

“…Recognizing this limitation, recent studies have proposed ground-breaking approaches that introduce jumps with scheduled deterministic times into interest rate models. The use of Gaussian distribution for the size of scheduled jumps has been implemented in several studies, such as Heidari and Wu (2009), Kim and Wright (2014), Schloegl et al (2023), and Fontana et al (2024). By entering these deterministic jump times triggered by monetary authority meetings into the framework of interest rate models, the aim is to better encapsulate scheduled adjustments in interest rates triggered by central bank meetings.…”

“…The class of models developed here, named the AJD-Skellam model, can be viewed as a forward step in reference to Piazzesi (2005); Heidari and Wu (2009); Kim and Wright (2014); Backwell and Hayes (2022); Schloegl et al (2023), andFontana et al (2024), as it presents the following novel features.…”

“…The model class introduced herein advances beyond existing models in the literature, such as Heidari and Wu (2009), Kim andWright (2014), andFontana et al (2024), by encompassing not only diffusive dynamics but also integrating instantaneous interest rate jumps according to a discrete distribution at predetermined times. Furthermore, this class enables analytical derivation of the characteristic function, facilitating computationally efficient calculations for both bonds and derivative instruments.…”