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On filtering in Markovian term structure models: an approximation approach

Part of: Parametric inference Stochastic systems and control Inference from stochastic processes Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Carl Chiarella ,
Sara Pasquali  and
Wolfgang J. Runggaldier
Carl Chiarella*
Affiliation:
University of Technology Sydney
Sara Pasquali*
Affiliation:
Universitá degli Studi di Parma
Wolfgang J. Runggaldier*
Affiliation:
Universitá di Padova
*
∗ Postal address: School of Finance and Economics, University of Technology Sydney, PO Box 123, Broadway, NSW 2007, Australia. Email address: carl.chiarella@uts.edu.au
∗∗ Postal address: Dipartimento di Matematica, Universitá degli Studi di Parma, 85, Via M. D'Azeglio, I-43100 Parma, Italy.
∗∗∗ Postal address: Dipartimento di Matematica Pura ed Applicata, Universitá di Padova, 7 Via Belzoni, I-35131 Padova, Italy.
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Abstract

We consider a parametrization of the Heath-Jarrow-Morton (HJM) family of term structure of interest rate models that allows a finite-dimensional Markovian representation of the stochastic dynamics. This parametrization results from letting the volatility function depend on time to maturity and on two factors: the instantaneous spot rate and one fixed-maturity forward rate. Our main purpose is an estimation methodology for which we have to model the observations under the historical probability measure. This leads us to consider as an additional third factor the market price of interest rate risk, that connects the historical and the HJM martingale measures. Assuming that the information comes from noisy observations of the fixed-maturity forward rate, the purpose is to estimate recursively, on the basis of this information, the three Markovian factors as well as the parameters in the model, in particular those in the volatility function. This leads to a nonlinear filtering problem, for the solution of which we describe an approximation methodology, based on time discretization and quantization. We prove the convergence of the approximate filters for each of the observed trajectories.

Keywords

Filter approximationsHeath-Jarrow-Morton modelmarket price of interest rate riskMarkovian representationsmeasure transformationnonlinear filteringterm structure of interest rates

MSC classification

Primary: 93E11: Filtering 60G35: Signal detection and filtering 62F15: Bayesian inference 62M05: Markov processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 33 , Issue 4 , December 2001 , pp. 794 - 809
Copyright
Copyright © Applied Probability Trust 2001 

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