CLASSIFICATION OF TWO- AND THREE-FACTOR TIME-HOMOGENEOUS SEPARABLE LMMs

Abstract

The flexibility of parametrizations of the LIBOR market model (LMM) comes at a cost, namely the LMM is high-dimensional, which makes it cumbersome to use when pricing derivatives with early exercise features. One way to overcome this issue for short- and medium-term time horizons is by imposing the separability condition on the volatility functions and approximating the model using a single time-step approximation. In this paper, we examine the flexibility of separable LMMs under the relaxed assumption that the driving Brownian motions can be correlated. In particular, we are interested in how the separability condition interacts with time-homogeneity, a desirable property of a LMM. We show that the two concepts can be related using a Levi-Civitá equation and provide a characterization of two- and three-factor separable and time-homogeneous LMMs and show that they are of practical interest. The results presented in this paper are also applicable to local-volatility LMMs. These separable volatility structures can be used for the driver of a two- or three-dimensional Markov-functional model — in which case no (single time-step) approximation is needed and the resultant model is both time-homogeneous and arbitrage-free.