Modeling the Term Structure of Interest Rates provides a comprehensive review of the continuous-time modeling techniques of the term structure applicable to value and hedge default-free bonds and other interest rate derivatives. The authors offer a unifying framework in which most continuous-time term structure models can be viewed and compared in terms of their similarities, their idiosyncratic features, and their main contributions and limitations. Modeling the Term Structure of Interest Rates is organized as follows: Section 1 presents the main objectives and basic definitions and notation; Section 2 proposes an interest rate models' taxonomy; and Section 3 introduces the mathematical framework used throughout the survey. Section 4 presents the main economic theories of the term structure of interest rates. Section 5 provides a unifying framework to present the family of short-term rate-based term structure models, which encompasses some of the most popular one and multi-factor short-term rate models developed at earlier stages of the term structure modeling literature. Section 6 reviews the second family of forward rate-based models grouped under the name the Heath, Jarrow and Morton (1992) family of term structure models. Section 7 presents the recently developed Libor or market models while preserving the same unifying mathematical framework. Section 8 surveys the empirical evidence on interest rate models' estimation and calibration issues and list selected empirical references which are useful for practitioners interested in the validity and performance of these models. Section 9 introduces a novel approach to characterize and quantify the degree of model misspecification associated with interest rate models. Section 10 discusses some of the challenges posed by running simulations of continuous-time term structure models. Section 11 concludes the survey. Finally, some useful mathematical results can be found in the Appendices.

Modeling the Term Structure of Interest Rates: A Review of the Literature (Foundations and Trends(r) in Finance)

Business & Money

Talay, Denis

Lhabitant, François-Serge

Gibson, Rajna

Economics

Now Publishers

In various scientific and industrial fields, stochastic simulations are taking on a new importance. This is due to the increasing power of computers and practitioners’ aim to simulate more and more complex systems, and thus use random parameters as well as random noises to model the parametric uncertainties and the lack of knowledge on the physics of these systems. The error analysis of these computations is a highly complex mathematical undertaking. Approaching these issues, the authors present stochastic numerical methods and prove accurate convergence rate estimates in terms of their numerical parameters (number of simulations, time discretization steps). As a result, the book is a self-contained and rigorous study of the numerical methods within a theoretical framework. After briefly reviewing the basics, the authors first introduce fundamental notions in stochastic calculus and continuous-time martingale theory, then develop the analysis of pure-jump Markov processes, Poisson processes, and stochastic differential equations. In particular, they review the essential properties of Itô integrals and prove fundamental results on the probabilistic analysis of parabolic partial differential equations. These results in turn provide the basis for developing stochastic numerical methods, both from an algorithmic and theoretical point of view. The book combines advanced mathematical tools, theoretical analysis of stochastic numerical methods, and practical issues at a high level, so as to provide optimal results on the accuracy of Monte Carlo simulations of stochastic processes. It is intended for master and Ph.D. students in the field of stochastic processes and their numerical applications, as well as for physicists, biologists, economists and other professionals working with stochastic simulations, who will benefit from the ability to reliably estimate and control the accuracy of their simulations.

Stochastic Simulation and Monte Carlo Methods: Mathematical Foundations of Stochastic Simulation (Stochastic Modelling and Applied Probability, 68)

Science & Math

Graham, Carl

Mathematics

Springer

In various scientific and industrial fields, stochastic simulations are taking on a new importance. This is due to the increasing power of computers and practitioners’ aim to simulate more and more complex systems, and thus use random parameters as well as random noises to model the parametric uncertainties and the lack of knowledge on the physics of these systems. The error analysis of these computations is a highly complex mathematical undertaking. Approaching these issues, the authors present stochastic numerical methods and prove accurate convergence rate estimates in terms of their numerical parameters (number of simulations, time discretization steps). As a result, the book is a self-contained and rigorous study of the numerical methods within a theoretical framework. After briefly reviewing the basics, the authors first introduce fundamental notions in stochastic calculus and continuous-time martingale theory, then develop the analysis of pure-jump Markov processes, Poisson processes, and stochastic differential equations. In particular, they review the essential properties of Itô integrals and prove fundamental results on the probabilistic analysis of parabolic partial differential equations. These results in turn provide the basis for developing stochastic numerical methods, both from an algorithmic and theoretical point of view. The book combines advanced mathematical tools, theoretical analysis of stochastic numerical methods, and practical issues at a high level, so as to provide optimal results on the accuracy of Monte Carlo simulations of stochastic processes. It is intended for master and Ph.D. students in the field of stochastic processes and their numerical applications, as well as for physicists, biologists, economists and other professionals working with stochastic simulations, who will benefit from the ability to reliably estimate and control the accuracy of their simulations.

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