Modelling interest rates by correlated multi-factor CIR-like processes

arXiv:0807.3898v1 [q-fin.GN] 24 Jul 2008 Modelling interest rates by correlated multi-factor CIR-like processes Lorenzo Bertini · Luca Passalacqua Abstract We investigate the joint description of the interest-rate term stuctures of Italy and an AAA-rated European country by mean of a –here proposed– correlated CIR-like bivariate model where one of the state variables is interpreted as a benchmark risk-free rate and the other as a credit spread. The model is constructed by requiring the strict positivity of interest rates and the asymptotic decoupling of the joint distribution of the two state variables on a long time horizon. The second condition is met by imposing the reversibility of the process with respect to a product measure, the first is then implemented by using the tools of potential theory. It turns out that these conditions select a class of non-affine models, out of which we choose one that is quadratic in the two state variables both in the drift and diffusion matrix. We perform a numerical analysis of the model by investigating a cross section of the term structures comparing the results with those obtained with an uncoupled bivariate CIR model. Keywords Interest rates · Multidimensional CIR processes · Potential theory JEL Classification E43 Mathematics Subject Classification (2000) 62P05 · 60J45 1 Introduction The difficulty to model the evolution of the term structure of interest rates is witnessed by the existence of a large number of models present in the academic literature and in the financial practice, see e.g. [3,22] for a review. Broadly speaking, these models can Lorenzo Bertini Dipartimento di Matematica, Universit`a di Roma “La Sapienza” Piazzale A. Moro 2, 00185 Roma (Italy) E-mail: bertini@mat.uniroma1.it Luca Passalacqua (fi) Dipartimento di Scienze Attuariali e Finanziarie, Universit`a di Roma “La Sapienza”, Via Nomentana 41, 00161 Roma (Italy) Tel.: +39-06-49919559 Fax: +39-06-44250289 E-mail: luca.passalacqua@uniRoma1.it 2 be grouped in financially oriented arbitrage models, whose main objective is pricing interest rate sensitive contracts and measuring risk associated with the time evolution of the term structure, and economically oriented models that are embedded in more complex market equilibrium models. Among equilibrium models that of Cox, Ingersoll and Ross (hereafter CIR) is certainly one of the most attractive. This model, introduced in [5,6], is characterized by two main properties: mean-reversion to an asymptotic state and absence of negative interest rates. Moreover, as Gaussian-like models (i.e. models founded on Ornstein-Uhlenbeck processes) generally develop numerically relevant tails in region of negative interest rates with growing time horizons, the CIR formulation is particularly popular in financial applications having as underlying portfolios composed of government bonds and long time horizons, such as the strategic asset allocation of life insurance segregated funds. However, well known limits of the CIR model are that the term structure can assume (see, e.g. [16]) only the following three shapes: monotonically increasing, monotonically decreasing and humped (i.e. increasing to a maximum and then decreasing), the need to allow the model parameters to vary with time in order to capture the observed evolution (see, e.g. [3]), and the difficulty to describe simultaneously all types of interest rate sensitive contracts, such as interest rate swaps, caps and swaptions (see, e.g. [15]). Moreover a single factor model is unable to describe simultaneously the evolution of the term structure of real and nominal interest rates. All the above difficulties lead quite naturally to multi-factor extensions of the basic univariate CIR model. For example, already in the original model proposed by Cox, Ingersoll and Ross in [6], the instantaneous nominal interest rate is a linear combination of two independent state variables, the real interest rate and the expected instantaneous inflation rate, each evolving in time according to univariate diffusion processes, thus realizing the stochastic version of the well-known Fisher equation. Another example is the two-factor extension proposed by Longstaffand Schwartz [18], where the two factors are used to express the short rate and its volatility. A different interpretation proposed for the two factor model is that the factors are linked to the short and long (w.r.t. the maturity of the contract) rates, as in the Brennan and Schwartz model [2]. A three-factor extension has also been considered and empirically investigated, among others, by Chen and Scott [4] on U.S. market data. The three factor setting is often motivated by the findings of Litterman and Scheinkman [17] according to whom the empirical description of the intertemporal variation of the term structure needs the use of three factors: the general level of interest rates, the slope of the yield curve and its curvature, that is associated with the volatility. For the euro market, a recen

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