Coherent Chaos Interest Rate Models

Abstract

The Wiener chaos approach to interest rate modelling arises from the observation that the pricing kernel admits a representation in terms of the conditional variance of a square-integrable random variable, which in turn admits a chaos expansion. When the expansion coefficients factorise into multiple copies of a single function, then the resulting interest rate model is called coherent, whereas a generic interest rate model will necessarily be incoherent. Coherent representations are nevertheless of fundamental importance because incoherent ones can always be expressed as a linear superposition of coherent elements. This property is exploited to derive general expressions for the pricing kernel and the associated bond price and short rate processes in the case of an n-th order chaos model for each $n$. The pricing formulae for bond options and swaptions are obtained in closed forms for a number of examples. An explicit representation for the pricing kernel of a generic---incoherent---model is then obtained by use of the underlying coherent elements. Finally, finite-dimensional realisations of the coherent chaos models are investigated in detail. In particular, it is shown that a class of highly tractable models can be constructed having the characteristic feature that the discount bond price is given by a piecewise flat (simple) process.