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Article

An Interest Rate Tree Driven by a Lévy Process

Donatien Hainaut and Renaud MacGilchrist
The Journal of Derivatives Winter 2010, 18 (2) 33-45; DOI: https://doi.org/10.3905/jod.2010.18.2.033
Donatien Hainaut
is an assistant professor at ENSAE-CREST in Paris, France.
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  • For correspondence: donatien.hainaut@ensae.fr
Renaud MacGilchrist
is an associate professor at ESC Rennes in Rennes, France.
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  • For correspondence: renaud.macgilchrist@esc-rennes.fr
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Abstract

The lognormal diffusion process is mathematically tractable and incorporates the kind of continuous random evolution of the price by small increments that seems to characterize most security prices. But market microstructure studies have shown that a lognormal diffusion does not describe very well price formation at the shortest intervals. This is especially true of short-term bond returns. Bond price changes are mostly small, but the tails of the distribution are fatter than the lognormal allows and occasional non-diffusive jumps do seem to occur. Also, the intervals between price changes vary considerably in length. Alternative distributions have been proposed, but they do not have the convenient mathematical properties of the lognormal, so implementation can be challenging. Hainaut and MacGilchrist propose using the normal inverse Gaussian (NIG) distribution that arises from a particular Lévy process and develop a lattice implementation for pricing. A pentanomial tree incorporates the NIG by matching its first four moments. In a simulation exercise, the NIG consistently outperforms the lognormal, largely due to its ability to capture skewness in returns.

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The Journal of Derivatives: 18 (2)
The Journal of Derivatives
Vol. 18, Issue 2
Winter 2010
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An Interest Rate Tree Driven by a Lévy Process
Donatien Hainaut, Renaud MacGilchrist
The Journal of Derivatives Nov 2010, 18 (2) 33-45; DOI: 10.3905/jod.2010.18.2.033

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An Interest Rate Tree Driven by a Lévy Process
Donatien Hainaut, Renaud MacGilchrist
The Journal of Derivatives Nov 2010, 18 (2) 33-45; DOI: 10.3905/jod.2010.18.2.033
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  • Article
    • Abstract
    • THE NIG PROCESS
    • SOME EMPIRICAL EVIDENCE
    • THE INTEREST RATE MODEL
    • THE CHARACTERISTIC FUNCTION OF Yt AND ITS MOMENTS
    • THE PENTANOMIAL TREE
    • NUMERICAL TESTS
    • CONCLUSIONS
    • APPENDIX
    • ENDNOTES
    • REFERENCES
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