Perturbation Methods in Credit Derivatives

CHAPTER 6Extension to Multi‐Factor Modelling

6.1 MULTI‐FACTOR PRICING EQUATION

Having demonstrated successfully how the methods we introduced in Chapter 3 can be used to derive pricing kernels for one‐factor short‐rate models, and further how these pricing kernels can be used to calculate derivative prices, we turn our attention to multi‐factor modelling, where our real interest lies in the context of the present work. There are four different ways which we can consider additional stochastic factors as influencing the calculations we performed in the previous two chapters.

Multi‐factor yield curve. It may be that the rate used for discounting, instead of being a function of a single stochastic variable, is a function of several stochastic variables (which, without loss of generality, can be considered as independent).

Additional factors not impacting rates. The pricing equation we consider, and therefore the pricing kernel produced, may specify the co‐evolution of other stochastic variables which evolve without impacting on the evolution of the rate itself. The co‐evolution may be trivial, but usually there is at least the potential for correlated diffusion. An example here would be a diffusion for which the governing diffusion was correlated with the risk‐free interest rate, and the drift of which depended on the same. Equally, we could consider LIBOR of a given tenor being given by the risk‐free rate plus an additional stochastic spread. This spread impacts the LIBOR ...