Term Structure Modelling With Overnight Rates Beyond Stochastic Continuity

27 Pages Posted: 15 Feb 2022 Last revised: 12 Feb 2023

See all articles by Claudio Fontana

Claudio Fontana

University of Padova, Department of Mathematics

Zorana Grbac

Université Paris VII Denis Diderot

Thorsten Schmidt

University of Freiburg

Date Written: February 11, 2023

Abstract

Overnight rates, such as the SOFR (Secured Overnight Financing Rate) in the US, play a central role in the current reform of interest rate benchmarks. A striking feature of overnight rates is the presence of jumps and spikes occurring at predetermined dates, as a result of monetary policy interventions and liquidity constraints. This corresponds to stochastic discontinuities (i.e., discontinuities occurring at ex-ante known points in time) in the dynamics of overnight rates.

In this work, we propose a term structure modelling framework in the presence of overnight rates and characterize absence of arbitrage in a generalised Heath-Jarrow-Morton (HJM) setting. By extending the classical short-rate approach to the case of stochastic discontinuities, we develop 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. Finally, we investigate hedging in the sense of local risk-minimization when the underlying term structures exhibit stochastic discontinuities.

Keywords: Libor reform, alternative risk-free rate, SOFR, SONIA, €STR, affine processes, semimartingales, stochastic discontinuities, BSDE, local risk-minimization

JEL Classification: C02, C60, E43, G12, G13

Suggested Citation

Fontana, Claudio and Grbac, Zorana and Schmidt, Thorsten, Term Structure Modelling With Overnight Rates Beyond Stochastic Continuity (February 11, 2023). Available at SSRN: https://ssrn.com/abstract=4023920 or http://dx.doi.org/10.2139/ssrn.4023920

Claudio Fontana

University of Padova, Department of Mathematics ( email )

Via Trieste 63
Padova, 35121
Italy

Zorana Grbac

Université Paris VII Denis Diderot ( email )

2, place Jussieu
Paris, 75005
France

Thorsten Schmidt (Contact Author)

University of Freiburg ( email )

Fahnenbergplatz
Freiburg, D-79085
Germany

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