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On the Dynamics of the Forward Interest Rate Curve and the Evaluation of Interest Rate Derivatives and their Sensitivities

  • Christian Croitoru
  • Christian Fries
  • Willi Jäger
  • Jörg Kampen
  • Dirk-Jens Nonnenmacher

Abstract

We present an overview of in project developed new techniques in computing key quantities of financial markets. Our approach is generic in the sense that the techniques apply essentially in the general frame work of markets which are described by systems of stochastic differential equations. We exemplify our methods in the LIBOR market model which is a standard interest rate market model widely used in practice and has its name from the daily quoted London interbank offered rates. The LIBOR market model has been developed in recent years beyond the classical framework in the direction of incomplete market models (with stochastic volatility and with jumps). Particular challenges are the high dimensionality (up to 20–40 factors), the calibration, and related problems of derivative prices evaluation and computation of sensitivities. We show how advanced Monte-Carlo techniques can be combined with analytic results about transition densities in order to obtain highly efficient and accurate numerical schemes for computing some of the key quantities in financial markets, especially hedging parameters.

Keywords

Market Model Transition Density Stochastic Volatility High Order Approximation Zero Coupon Bond 
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References

  1. 1.
    Croitoru, C., Kampen, J.: Accurate numerical schemes and computation of a class of linear parabolic initial value problems. (in preparation)Google Scholar
  2. 2.
    Duffie, D.: Dynamic Asset Pricing Theory. Princeton, Princeton University Press (2001)zbMATHGoogle Scholar
  3. 3.
    Fries, C., Kampen, J.: Proxy Simulation Schemes for generic robust Monte-Carlo sensitivities, process oriented importance sampling and high accuracy drift approximation (with applications to the LIBOR market model). Journal of Computational Finance, Vol. 10, Nr. 2Google Scholar
  4. 4.
    Fries, C., Kampen, J.: Proxy Simulation Schemes for generic robust Monte-Carlo sensitivities based on dimension reduced higher order analytic expansions of transition densities. (in preparation)Google Scholar
  5. 5.
    Fries, C.: Mathematical Finance. Theory, Modeling, Implementation. Wiley, Hoboken (2007), http://www.christian-fries.de/finmath/ bookzbMATHGoogle Scholar
  6. 6.
    Kampen, J.: The WKB-Expansion of the fundamental solution of linear parabolic equations and its applications. Book, submitted to Memoirs of the American Mathematical Society (electronically published at SSRN 2006)Google Scholar
  7. 7.
    Kampen, J.: How to compute the length of a geodesic on a Riemannian manifold with small error in arbitrarily regular norms. WIAS preprint (to appear)Google Scholar
  8. 8.
    Kampen, J.: Regular polynomial interpolation and global approximation of global solutions of linear partial differential equations. WIAS preprint (2007)Google Scholar
  9. 9.
    Kampen, J., Kolodko, A., Schoenmakers, J.: Monte Carlo Greeks for callable products via approximative Greenian Kernels. WIAS preprint, revised version to appear in SIAM Journal of computation (2007)Google Scholar
  10. 10.
    Krylov, N.V.: Lectures on Elliptic and Parabolic Equations in Hölder Spaces. Graduate Studies in Mathematics, Vol. 12, American Mathematical Society (1996)Google Scholar
  11. 11.
    Schoenmakers, J.: Robust Libor Modelling and Pricing of Derivative Products. Financial Mathematics. Chapman & Hall/CRC (2005)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Christian Croitoru
    • 1
  • Christian Fries
    • 2
    • 3
  • Willi Jäger
    • 4
  • Jörg Kampen
    • 1
    • 5
  • Dirk-Jens Nonnenmacher
    • 2
    • 6
  1. 1.INF 368Interdisciplinary Center for Scientific Computing (IWR) HeidelbergHeidelbergGermany
  2. 2.Dresdner Bank AGFrankfurt am MainGermany
  3. 3.now at DZBank AGFrankfurt am MainGermany
  4. 4.IAMUniversität HeidelbergHeidelbergGermany
  5. 5.now at Weierstrass Institute for Applied Analysis and StochasticsBerlinGermany
  6. 6.now at HSH Nordbank AGHamburgGermany

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