A Chaotic Approach to Interest Rate Modeling” Finance and Stochastic

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 mo-tion 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 den-sity. 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. Key words: Interest rate models, arbitrage free term-structure dynamics, Wiener chaos, Heath-Jarrow-Morton theory, Flesaker-Hughston framework. JEL Classification: E43