A chaotic approach to interest rate modelling
AbstractThis 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
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Bibliographic InfoArticle provided by Springer in its journal Finance and Stochastics.
Volume (Year): 9 (2005)
Issue (Month): 1 (January)
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Web page: http://www.springerlink.com/content/101164/
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- Jir\^o Akahori & Takahiro Tsuchiya, 2006.
"What is the natural scale for a L\'evy process in modelling term structure of interest rates?,"
- Jirô Akahori & Takahiro Tsuchiya, 2006. "What is the Natural Scale for a Lévy Process in Modelling Term Structure of Interest Rates?," Asia-Pacific Financial Markets, Springer, vol. 13(4), pages 299-313, December.
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