Coherent chaos interest rate models and the wick calculus in finance

Abstract

This thesis develops new tools in stochastic analysis with applications to finance. The first part presents novel developments in the Wiener chaos approach to the modelling, calibration, and pricing of interest rate derivatives. To price financial instruments it suffices to specify the pricing kernel, which in Brownian models can be represented as the conditional variance of a square-integrable random variable which serves as the “generator" of the pricing kernel. The coefficients of the chaos expansion of the generator act as the parameters of a generic interest-rate model. A special class of generators, arising from “coherent" chaos expansions, is considered, and the resulting interest rate models are investigated. Coherent representations are important since a kernel generator can be expressed as a linear superposition of coherent generators. This property is exploited to derive general expressions for the pricing kernel, along with the associated discount bond and short rate processes. Pricing formulae for bond options and swaptions are obtained in closed form. The pricing kernel of a generic incoherent model is then obtained by use of the underlying coherent elements. Finite-dimensional representations of coherent chaos models are investigated, and used to construct a class of tractable models having the feature that discount bond prices are piecewise-flat processes. In the second part of the thesis, a general theory of the Wick calculus is developed. Novel results concerning the Wick orders of random variables are derived. In the case where the underlying process is a Brownian motion the Wick calculus reduces to the Ito calculus, but the former is not restricted to the Gaussian class, and is applicable to other cases, such as Lévy processes. With financial applications in mind, the Wick calculus is extended to a wider class of stochastic processes. The thesis concludes with a change of measure analysis for Wick exponentials of Lévy processes, indicating that the Wick calculus can be used as a tool for modelling the dynamics of asset prices.