Term Structure Modelling With Overnight Rates Beyond Stochastic Continuity

31 Pages Posted: 15 Feb 2022

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 2, 2022

Abstract

In the current reform of interest rate benchmarks, a central role is played by risk-free rates (RFRs), such as SOFR (secured overnight financing rate) in the US. A key feature of RFRs is the presence of jumps and spikes at periodic time intervals as a result of regulatory and liquidity constraints. This corresponds to stochastic discontinuities (i.e., jumps occurring at predetermined dates) in the dynamics of RFRs. In this work, we propose a general modeling framework where RFRs and term rates can have stochastic discontinuities and characterize absence of arbitrage in an extended HJM setup. When the term rate is generated by the RFR itself, we show that it solves a BSDE, whose driver is determined by the HJM drift restrictions. In general, this BSDE may admit multiple solutions and we provide sufficient conditions ensuring uniqueness. We develop a tractable specification driven by affine semimartingales, also extending the classical short rate approach to the case of stochastic discontinuities. In this context, we show that a simple specification allows 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 study hedging in the sense of local risk-minimization when the underlying term structures have 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 2, 2022). 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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