Term structure modelling with overnight rates beyond stochastic continuity

TL;DR

This paper develops a new term structure model for overnight rates that incorporates stochastic discontinuities at known dates, capturing jumps due to monetary policy actions, and provides explicit valuation and hedging methods.

Contribution

It extends the classical HJM framework to include stochastic discontinuities driven by affine semimartingales, offering a tractable approach for modeling overnight rate jumps.

Findings

The model captures key stylized facts of overnight rate jumps.

Explicit formulas for bond and caplet valuation in Gaussian settings.

Analysis of hedging strategies under stochastic discontinuities.

Abstract

Overnight rates, such as the SOFR (Secured Overnight Financing Rate) in the US, 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 modelling framework based on overnight rates and characterize absence of arbitrage in a generalised 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.