Stochastic Processes for Interest Rates and Equilibrium Bond Prices

Marsh, Terry A.; Rosenfeld, Eric

doi:10.2307/2328002

Cited by 83 publications

(53 citation statements)

References 10 publications

(12 reference statements)

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“…These processes are referred to in statistical finance as the Ornstein-Uhlenbeck processes when the distribution is Gaussian [20], while there are non-Gaussian generalizations as well [21]. These processes are widely adopted models for interest rates and currency exchange rates [20]. Among other applications, one can name derivative securities [22], electricity pricing [23], and pairs trading [24].…”

Section: A Scopementioning

confidence: 99%

“…For the generalized increment processes and , we have that i) and are stationary processes; ii) if , then the random variables and are independent; iii) if , then and are independent; iv) if represents the probability density function of , then we have that (20) where is the Lévy exponent of the innovations. Proof: i) First, we express the generalized increments in terms of the innovation process.…”

Section: Preliminariesmentioning

confidence: 99%

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On the Linearity of Bayesian Interpolators for Non-Gaussian Continuous-Time AR(1) Processes

Amini

Thévenaz

Ward

et al. 2013

IEEE Trans. Inform. Theory

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Abstract-Bayesian estimation problems involving Gaussian distributions often result in linear estimation techniques. Nevertheless, there are no general statements as to whether the linearity of the Bayesian estimator is restricted to the Gaussian case. The two common strategies for non-Gaussian models are either finding the best linear estimator or numerically evaluating the Bayesian estimator by Monte Carlo methods. In this paper, we focus on Bayesian interpolation of non-Gaussian first-order autoregressive (AR) processes where the driving innovation can admit any symmetric infinitely divisible distribution characterized by the Lévy-Khintchine representation theorem. We redefine the Bayesian estimation problem in the Fourier domain with the help of characteristic forms. By providing analytic expressions, we show that the optimal interpolator is linear for all symmetric -stable distributions. The Bayesian interpolator can be expressed in a convolutive form where the kernel is described in terms of exponential splines. We also show that the limiting case of Lévy-type AR(1) processes, the system of which has a pole at the origin, always corresponds to a linear Bayesian interpolator made of a piecewise linear spline, irrespective of the innovation distribution. Finally, we show the two mentioned cases to be the only ones within the family for which the Bayesian interpolator is linear.

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Section: A Scopementioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

On the Linearity of Bayesian Interpolators for Non-Gaussian Continuous-Time AR(1) Processes

Amini

Thévenaz

Ward

et al. 2013

IEEE Trans. Inform. Theory

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“…Model 5 is the constant elasticity of variance (CEV) process introduced by Cox (1975) and by Cox and Ross (1976). The application of this process to interest rates is discussed in Marsh and Rosenfeld (1983). Model 6 is relatively recent compared to Model 1-5.…”

Section: Transformation Functions and Sdesmentioning

confidence: 99%

Modeling Multivariate Interest Rates Using Time-Varying Copulas and Reducible Nonlinear Stochastic Differential Equations

Giet

Hadri

et al. 2010

Journal of Financial Econometrics

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We propose a new approach for modeling non-linear multivariate interest rate processes based on time-varying copulas and reducible stochastic differential equations (SDEs). In the modeling of the marginal processes, we consider a class of non-linear SDEs that are reducible to Ornstein-Uhlenbeck (OU) process or Cox, Ingersoll, and Ross (1985) (CIR) process. The reducibility is achieved via a non-linear transformation function. The main advantage of this approach is that these SDEs can account for non-linear features, observed in short-term interest rate series, while at the same time leading to exact discretisation and closed form likelihood functions. Although a rich set of specifications may be entertained, our exposition focuses on a couple of nonlinear constant elasticity volatility (CEV) processes, denoted OU-CEV and CIR-CEV, respectively. These two processes encompass a number of existing models that have closed form likelihood functions. The statistical properties of the two processes are investigated. In order to obtain more flexible functional form over time, we allow the transformation function to be time-varying. Results from our study of US and UK short term interest rates suggest that the new models outperform existing parametric models with closed form likelihood functions. We also find the time-varying effects in the transformation functions statistically significant. We study the conditional dependence structure of the two rates using Patton (2006a) time-varying Symmetrised JoeClayton copula. We find evidence of asymmetric dependence between the two rates, and that the level of dependence is positively related to the level of the two rates. (1985) (CIR). La reduction est réalisée via une fonction de transformation non-linéaire. L'avantage principal de cette approche consiste en ce que ces SDES peuvent représenter des caractéristiques non-linéaires, observées dans les séries de taux d'intérêt a court terme, tout en conduisant en même tempsà une discretisation exacte età fonctions de vraisemblance analytique. Bien qu'une riche palette de spécifications puisseêtre considérée, nous nous concentrons sur deux fonctions de volatilitéàélasticité constante nonlinéaire (CEV), dénotées respectivement OU-CEV et CIR-CEV. Ces deux processus enveloppent un nombre important de modèles existants qui possèdent des fonctions de vraisemblance analytiques. Nous examinons les propriétés statistiques de ces deux processus. Afin d'obtenir des formes fonctionnelles flexibles, nous autorisons la fonction de transformationà varier dans le temps. Nos résultats empiriques portant sur les taux courts américains et britaniques suggèrent que nos nouveaux modèles dominent les modèles existants ayant des fonctions de vraisemblance analytiques. Nous trouvons aussi que les effets variables dans le temps de la fonction de transformation sont statistiquement significatifs. Nousétudions la structure de dépendance conditionnelle des deux taux au moyen de la copule de Joe-Clayon symétriséeà paramètres variables proposée par Patt...

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“…However, even higher sensitivity has been recommended in an empirical study by Chan et al [15], who find that such models do a better job predicting future rates-precisely our goal. This particular approach, with a logarithmic transformation, is a straightforward way to insure our discrete-time simulations remain positive and is equivalent to the geometric Brownian motion model used by Black and Scholes [9] and Marsh and Rosenfeld [33]. As suggested by Lubrano [31], there continues to be disagreement over the best empirical model of interest rate behavior.…”

Section: Estimation Of Interest Rate Behaviormentioning

confidence: 99%

Discounting the distant future: how much do uncertain rates increase valuations?

Newell

Pizer

2003

Journal of Environmental Economics and Management

340

121

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We demonstrate that when the future path of the discount rate is uncertain and highly correlated, the distant future should be discounted at significantly lower rates than suggested by the current rate. We then use two centuries of US interest rate data to quantify this effect. Using both random walk and meanreverting models, we compute the ''certainty-equivalent rate'' that summarizes the effect of uncertainty and measures the appropriate forward rate of discount in the future. Under the random walk model we find that the certainty-equivalent rate falls continuously from 4% to 2% after 100 years, 1% after 200 years, and 0.5% after 300 years. At horizons of 400 years, the discounted value increases by a factor of over 40,000 relative to conventional discounting. Applied to climate change mitigation, we find that incorporating discount rate uncertainty almost doubles the expected present value of mitigation benefits. r

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Stochastic Processes for Interest Rates and Equilibrium Bond Prices

Cited by 83 publications

References 10 publications

On the Linearity of Bayesian Interpolators for Non-Gaussian Continuous-Time AR(1) Processes

On the Linearity of Bayesian Interpolators for Non-Gaussian Continuous-Time AR(1) Processes

Modeling Multivariate Interest Rates Using Time-Varying Copulas and Reducible Nonlinear Stochastic Differential Equations

Discounting the distant future: how much do uncertain rates increase valuations?

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