Low-factor market models of interest rates

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Abstract

In this thesis we study three different, but interconnected low-factor market models: LIBOR market model, Markov-functional model, and two-currency Markov-functional model.

The LIBOR market model (LMM) is one of the most popular term structure models. However, it suffers from a major drawback, it is high-dimensional. The problem of highdimensionality can be in part solved imposing a separability condition. We will be interested how the separability condition interacts with time-homogeneity, a desirable property of an LMM. We address this question by parametrising two- and three-factor separable and time-homogeneous LMMs and show that they are of practical interest.

Markov-functional models (MFMs) are a computationally efficient alternative to the LMMs. We consider two aspects of the MFMs, implementation and specification. First we provide two new algorithms that can be used to implement the one-dimensional MFM under the terminal and the spot measure driven by a general diffusion process. Since the existing literature has been focused exclusively on the Gaussian driving processes our algorithms open the scope for new parameterisations. We then prove that the dynamics of the onedimensional MFM are only affected by the time dependence of the driving process, described by a copula, and not by its marginal distributions. We then shift our focus and show that the one-dimensional MFM under the terminal measure is closely related to the one-factor separable local-volatility LMM.

Finally, we move our attention to the models of a two-currency economy. We propose a new three-factor model that we calibrate to the domestic and foreign caplet prices and the foreign exchange call options. To maintain the no-arbitrage condition while calibrating to foreign exchange market we propose a predictor-corrector type step. It is our conjecture that the predictor-corrector step converges, thus the model is well defined.

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