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Erschienen in:

2013 | OriginalPaper | Buchkapitel

12. Calibrating Models – Inverse Problems

verfasst von : Hansjoerg Albrecher, Andreas Binder, Volkmar Lautscham, Philipp Mayer

Erschienen in: Introduction to Quantitative Methods for Financial Markets

Verlag: Springer Basel

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Abstract

In the previous chapters we studied several model choices to describe stock price and interest rate dynamics. When using models to valuate derivatives or to obtain a hedging strategy, the used parameters will greatly impact the results. While there is broad agreement of how to model many problems in physics (such as the thermal conductivity of copper at room temperature), financial markets are fundamentally different. Many market participants have different views on the distributions of market variables, and market prices of liquid assets only represent an economic equilibrium resulting from those different views.

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Vorheriges Kapitel Simulation Methods

Nächstes Kapitel Case Studies: Exotic Derivatives

Integral equations have been studied extensively, see e.g. Engl [27].

This is a general property of integral operators with bounded integration region and which satisfy weak conditions (e.g. quadratic integrability) for the integration kernel (which is an exponential function in our case). Under such conditions, the integral operator is found to be a compact operator whose singular values tend to 0.

We have used exactly this property for the construction of the above example when determining the forward short rates.

bp = 1 basis point = 0.01 %.

This penalty term can contain a priori information on a guess for the true function f ^∗, for example, by measuring the distance to f ^∗.

The proof of this statements requires some profound techniques of Functional Analysis (see Engl, Hanke & Neubauer [28]). In this case the optimization problem (12.2) (in the infinite-dimensional setting) will be equivalent to the solution of \(({K}^{{\ast}}K +\alpha I)f_{\delta }^{\alpha } = {K}^{{\ast}}g_{\delta },\) where K ^∗ is the operator adjoint to K and I the identity operator.

The Fokker-Planck equation, which describes the time evolution of the probability density function of the transition distribution of the stock price under the risk-neutral measure, offers one possible way of deriving this dual equation. Hereby one takes a Dirac-delta distribution as starting distribution. Under the risk-neutral measure the local volatility function is uniquely determined by the call prices (for all strike/maturity pairs).

Note that market data will always be noisy due to bid-ask spreads.

12.

P. Brémaud. Markov Chains. Gibbs Fields, Monte Carlo Simulation and Queues. Springer, New York, 2008.

18.

R. Cont and P. Tankov. Financial Modelling with Jump Processes. Chapman & Hall/CRC, Boca Raton, FL, 2004.MATH

25.

H. Egger and H. W. Engl. Tikhonov regularization applied to the inverse problem of option pricing: convergence analysis and rates. Inverse Problems, 21(3):1027–1045, 2005.MathSciNetMATHCrossRef

27.

H. W. Engl. Integralgleichungen. Springer-Verlag, Vienna, 1997.MATHCrossRef

28.

H. W. Engl, M. Hanke, and A. Neubauer. Regularization of Inverse Problems. Kluwer, Dordrecht, 1996.MATHCrossRef

29.

H. W. Engl. Calibration problems—an inverse problems view. WILMOTT Magazine, pages 16–20, 2007.

44.

B. Kaltenbacher, A. Neubauer, and O. Scherzer. Iterative Regularization Methods for Nonlinear Ill-Posed Problems. Radon Series on Computational and Applied Mathematics, de Gruyter, Berlin, 2008.

Titel: Calibrating Models – Inverse Problems
verfasst von: Hansjoerg Albrecher
Andreas Binder
Volkmar Lautscham
Philipp Mayer
Verlag: Springer Basel
Buch: Introduction to Quantitative Methods for Financial Markets
Print ISBN: 978-3-0348-0518-6

Electronic ISBN: 978-3-0348-0519-3

Copyright-Jahr: 2013
DOI: https://doi.org/10.1007/978-3-0348-0519-3_12