Abstract.

This paper includes a marked Hawkes process in the original Heath–Jarrow–Morton (HJM) setup and investigates the impact of this assumption on the pricing of the popular vanilla fixed-income derivatives. Our model exhibits a smile that can fit the implied volatility of swaptions for a given key rate (tenor). We harness the log-normality of the model, conditionally with respect to jumps, and derive formulae to evaluate both caplets/floorlets and swaptions. Our model exhibits negative jumps on the zero-coupon (hence positive on the rates). Therefore, its behavior is compatible with the situation where globally low interest rates can suddenly show a cluster of positive jumps in case of tensions on the market. One of the main difficulties when dealing with the HJM model is to keep a framework that is Markovian. In this paper we show how to preserve the relevant features of the Hull and White version, especially the reconstruction formula that provides the zero-coupon bonds in terms of the underlying model factors.

Keywords

  1. Heath–Jarrow–Morton model
  2. forward rates
  3. Hawkes processes
  4. jumps clustering
  5. swaptions
  6. caplets
  7. floorlets

MSC codes

  1. 60G55
  2. 60J60
  3. 91G05
  4. 91G10
  5. 93E20

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Ackowledgments.

The authors thank an anonymous referee and the associate editor for valuable comments and hints, which provided helpful support in improving the paper.

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Information & Authors

Information

Published In

cover image SIAM Journal on Financial Mathematics
SIAM Journal on Financial Mathematics
Pages: 1062 - 1079
ISSN (online): 1945-497X

History

Submitted: 13 June 2022
Accepted: 14 June 2023
Published online: 17 October 2023

Keywords

  1. Heath–Jarrow–Morton model
  2. forward rates
  3. Hawkes processes
  4. jumps clustering
  5. swaptions
  6. caplets
  7. floorlets

MSC codes

  1. 60G55
  2. 60J60
  3. 91G05
  4. 91G10
  5. 93E20

Authors

Affiliations

Guillaume Bernis
BPCE Assurances, 75013 Paris, France.
Matthieu Garcin
Léonard de Vinci Pôle Universitaire, Research Center, 92 916 Paris La Défense, France.
Simone Scotti
Università di Pisa, 56124 Pisa, Italy.
Carlo Sgarra Contact the author
Politecnico di Milano, 20133 Milano, Italy.

Funding Information

Funding: The third author received financial support from the Institut Louis Bachelier under the grant “The Impact of Information on Financial Markets: Detecting, Analysing and Modelling Abrupt Changes in Prices Resulting from News Arrivals.” The research of the third author was also supported by University of Pisa research project PRA-2022-21.

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