A multi-curve HJM factor model for pricing and risk management

Abstract

In this paper, we introduce a multi-curve model under the historical probability based upon multiplicative relative spreads, inspired by the HJM and affine factor approaches, which implies positive and ordered spreads. In particular, we focus upon ${\delta }_i$-XIBOR relative (instantaneous) forward rates and appropriate XIBOR HJM drift constraints, and we describe the dynamics of the different forward rates and spreads under different measure changes (including forward measures). We introduce an explicit model satisfying both the XIBOR HJM drift constraints as well as the property of positive and ordered spreads. We demonstrate the flexibility of this model for derivative pricing by focusing upon the price of a caplet and of options with a payoff based upon XIBOR forward prices with different tenors. We perform on one hand a calibration of the model based upon cap prices. On the other hand, we do an estimation of a spread curve in our proposed model under the historical probability by using a Kalman filter approach. Numerical results are included, and they confirm that the model performs very well.

Keywords:

• XIBOR rate
• Multiple yield curves
• Multiplicative spread
• Affine processes

JEL Classifications:

• E43
• G12

Acknowledgments

The authors would like to thank the anonymous referees for their useful comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Following the lines of Konikov and McClelland (2020) and the related SSRN-id6073 version, approximate quasi-explicit expressions for swaption prices can also be derived using some usual ‘freezing’ methods and Fourier inversion techniques.

2 In figure 3, the notation $\ln P_{\text{6m}}^S$ and $s_{\text{6m}}$ has been used since ${\delta }_1$ equals 6 months.

Additional information

Funding

Griselda Deelstra acknowledges support from the EU Framework Program for Research and Innovation Horizon 2020 (H2020-MSCA-ITN-2018, Project 813261, EID ABC-EU-XVA), as well as of the ARC grant IAPAS.