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A chaotic approach to interest rate modelling

Author

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  • Lane Hughston

    ()

  • Avraam Rafailidis

    ()

Abstract

This paper presents a new approach to interest rate dynamics. We consider the general family of arbitrage-free positive interest rate models, valid on all time horizons, in the case of a discount bond system driven by a Brownian motion of one or more dimensions. We show that the space of such models admits a canonical mapping to the space of square-integrable Wiener functionals. This is achieved by means of a conditional variance representation for the state price density. The Wiener chaos expansion technique is then used to formulate a systematic analysis of the structure and classification of interest rate models. We show that the specification of a first-chaos model is equivalent to the specification of an admissible initial yield curve. A comprehensive development of the second-chaos interest rate theory is presented in the case of a single Brownian factor, and we show that there is a natural methodology for calibrating the model to at-the-money-forward caplet prices. The factorisable second-chaos models are particularly tractable, and lead to closed-form expressions for options on bonds and for swaptions. In conclusion we outline a general “international” model for interest rates and foreign exchange, for which each currency admits an associated family of discount bonds, and show that the entire system can be generated by a vector of Wiener functionals. Copyright Springer-Verlag Berlin/Heidelberg 2005

Suggested Citation

  • Lane Hughston & Avraam Rafailidis, 2005. "A chaotic approach to interest rate modelling," Finance and Stochastics, Springer, vol. 9(1), pages 43-65, January.
  • Handle: RePEc:spr:finsto:v:9:y:2005:i:1:p:43-65
    DOI: 10.1007/s00780-004-0135-6
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    Cited by:

    1. Stephane Crepey & Andrea Macrina & Tuyet Mai Nguyen & David Skovmand, 2015. "Rational Multi-Curve Models with Counterparty-Risk Valuation Adjustments," Papers 1502.07397, arXiv.org.
    2. repec:taf:quantf:v:17:y:2017:i:2:p:275-288 is not listed on IDEAS
    3. repec:gam:jrisks:v:6:y:2018:i:1:p:18-:d:134969 is not listed on IDEAS

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