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Hein, Torsten : On Tikhonov regularization for the inverse problem of option pricing in the price-dependent case

Hein, Torsten : On Tikhonov regularization for the inverse problem of option pricing in the price-dependent case

Author(s):
Hein, Torsten
Title:
On Tikhonov regularization for the inverse problem of option pricing in the price-dependent case
Electronic source:
application/postscript
Preprint series:
Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 8, 2004
Mathematics Subject Classification:
35R30 [ Inverse problems for PDE ]
65J20 [ Improperly posed problems; regularization ]
91B24 [ Price theory and market structure ]
Abstract:
This paper deals with analytic studies for solving the inverse problem of identifying purely price-dependent volatilities from given option price data. Using the classical theory of parabolic differential equations we formulate and analyze the forward operator as a mapping between the Hilbert spaces H^1(R)and L^2(R). We investigate continuity and Fréchet differentiability of this operator and prove the discontinuity of the inverse operator. We use Tikhonov regularization and present assertions to the stable solvability of this problem.
Keywords:
inverse problem of option pricing, identification of local volatilities, Black-Scholes model, parabolic equations, fundamental solutions, ill-posed problem, regularizations, convergence rates
Language:
English
Publication time:
4 / 2004