Markov Chains and the Potential Approach to Modelling Interest Rates and Exchange Rates

Within the mathematical finance literature, there have been several distinct classes of interest-rate model. The first historically was the family of spotrate models, where one proposes a model for the evolution of the spot rate of interest under the pricing measure, and then attempts to find expressions for the prices of derivatives; the models of Vasicek [16], Cox, Ingersoll & Ross [7], Black, Derman & Toy [3] and Black & Karasinski [4] are well-known examples of this type. Next came the whole-yield models, starting with Ho & Lee [10] in a discrete setting, and then in the continuous setting Babbs [1] and Heath, Jarrow & Morton [9]. Lately, there has been much interest in so-called market models, whose chief characteristic is the choice of some suitable numéraire process, relative to which the prices of various derivatives have some particularly tractable form; see Miltersen, Sandmann & Sondermann [12] and Brace, Gatarek & Musiela [5] for examples of such models. These three classes of models have been developed extensively; a thorough survey would be outside the aims of this paper, but we refer the reader to the excellent recent monograph of Musiela & Rutkowski [11] for more details and references.