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Affine processes Time-inhomogeneity
Motivated by financial applications, we study convex analysis for modules over the ordered ring L0 of random variables. We establish a module analogue of locally convex vector spaces, namely locally L0-convex modules. In this context, we prove hyperplane separation theorems. We investigate continuity, subdifferentiability and dual representations of Fenchel–Moreau type for L0-convex functions from L0-modules into L0. Several examples and applications are given.
Research monograph providing appropriate consistency conditions for and examples of blended models for the term structure of interest rates within the Health-Jarrow-Morton framework, combining curve-fitting methods and factor models. Softcover.
Affine processes are distinguished by their rich structural properties, which makes them favorite when it comes to computations in financial applications of all kind. This fact has been explored and illustrated for the time-homogeneous case in a recent paper by Duffie, Filipovic and Schachermayer. However, there are many situations which require time-dependent parameters, such as when it comes to model calibration. This paper provides a rigorous treatment and complete characterization of time-inhomogeneous affine processes
Changing interest rates constitute one of the major risk sources for banks, insurance companies, and other financial institutions. Modeling the term-structure movements of interest rates is a challenging task. This volume gives an introduction to the mathematics of term-structure models in continuous time. It includes practical aspects for fixed-income markets such as day-count conventions, duration of coupon-paying bonds and yield curve construction; arbitrage theory; short-rate models; the Heath-Jarrow-Morton methodology; consistent term-structure parametrizations; affine diffusion processes and option pricing with Fourier transform; LIBOR market models; and credit risk. The focus is on a mathematically straightforward but rigorous development of the theory. Students, researchers and practitioners will find this volume very useful. ...
In this paper we provide the characterization of all finite-dimensional Heath-Jarrow-Morton models that admit arbitrary initial yield curves. It is well known that affine term structure models with time-dependent coef- ficients (such as the Hull-White extension of the Vasicek short rate model) perfectly fit any initial term structure. We find that such affine models are in fact the only finite-factor term structure models with this property. We also show that there is usually an invariant singular set of initial yield curves where the affine term structure model becomes time-homogeneous. We also argue that other than functional dependent volatility structures - such as local state dependent volatility structures - cannot lead to finite-dimensional realizations. Finally, our geometric point of view is illustrated by several examples.
Viability and invariance problems related to a stochastic equation in a Hilbert space H are studied. Finite dimensional invariant C2 submanifolds of H are characterized. We derive Nagumo type conditions and prove a regularity result: Any weak solution, which is viable in a finite dimensional C2 submanifold, is a strong solution. These results are related to finding finite dimensional realizations for stochastic equations. There has recently been increased interest in connection with a model for the stochastic evolution of forward rate curves.
We study a problem posed in Bjork and Christensen (1999): does there exist any nontrivial interest rate model which is consistent with the Nelson-Siegel family? They show that within the HJM framework with deterministic volatility structure the answer is no.In this paper we give a generalized version of this result including stochastic volatility structure. For that purpose we introduce the class of consistent state space processes, which have the property to provide an arbitrage-free interest rate model when representing the parameters of the Nelson-Siegel family. We characterize the consistent state space Ito processes in terms of their drift and diffusion coefficients. By solving an inverse problem we find their explicit form. It turns out that there exists no nontrivial interest rate model driven by a consistent state space Ito pro...
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