On a Class of Semi-Elliptic Diffusion Models - Part I: A Constructive Analytical Approach for Global Solutions, Densities and Numerical Schemes with Applications to the LIBOR Market Model

28 Pages Posted: 5 Apr 2010

See all articles by Christian P. Fries

Christian P. Fries

Ludwig Maximilian University of Munich (LMU) - Faculty of Mathematics; DZ Bank AG

Joerg Kampen

Weierstrass Institute for Applied Analysis and Stochastics

Date Written: March 17, 2010

Abstract

Computational parsimony makes reduced factor LIBOR market models popular among practitioners. However, value functions and sensitivities of such models are described by degenerate parabolic (i.e. semi-elliptic) equations where the existence of regular global solutions is not trivial. In this paper, we show that for a considerable class of degenerate equations including equations corresponding to reduced LIBOR market models of practical interest) regular global solutions can be constructed. The result is also of interest for the theory of degenerate parabolic equations. In addition, the constructive proof of the global existence result allows to derive explicit approximations for the transition probabilities. These transition probabilities then lead to sophisticated Monte-Carlo schemes for semi-elliptic diffusion models (subsuming projective Markovian models). Moreover, recent results on bounded variance estimators for Greeks of valuations under such schemes are generalized to reduced factor models. The emphasis in the present part of our treatment of reduced factor models is on conceptual and (constructive) analytical issues. A more detailed analysis of numerical and computational issues, as well as quantitative experiments will be content of the second part.

Keywords: Degenerate parabolic equations, financial derivatives, sensitivities, Monte-Carlo methods

Suggested Citation

Fries, Christian P. and Kampen, Joerg, On a Class of Semi-Elliptic Diffusion Models - Part I: A Constructive Analytical Approach for Global Solutions, Densities and Numerical Schemes with Applications to the LIBOR Market Model (March 17, 2010). Available at SSRN: https://ssrn.com/abstract=1582414 or http://dx.doi.org/10.2139/ssrn.1582414

Christian P. Fries (Contact Author)

Ludwig Maximilian University of Munich (LMU) - Faculty of Mathematics ( email )

Theresienstrasse 39
Munich
Germany

DZ Bank AG ( email )

60265 Frankfurt am Main
Germany

Joerg Kampen

Weierstrass Institute for Applied Analysis and Stochastics ( email )

Mohrenstr. 39
Berlin, Berlin 10117
Germany

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