Hein, Torsten : Neue Aspekte zur Tikhonov-Regularisierung des inversen Problems der Optionspreisbildung im preisabhängigen Fall

Author(s):

Hein, Torsten

Title:

Neue Aspekte zur Tikhonov-Regularisierung des inversen Problems der Optionspreisbildung im preisabhängigen Fall

Preprint series:

Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 4, 2003

Mathematics Subject Classification:

35R30			[ Inverse problems for PDE ]
65J20			[ Improperly posed problems; regularization ]
49K20			[ Problems involving partial differential equations ]
35K10			[ General theory of second-order, parabolic equations ]
91B24			[ Price theory and market structure ]

Abstract:

This paper deals with analytic studies for solving the inverse problem of identifying purely pricedependent volatilities from given optionprice data. Using the classical theory of parabolic differential equations we formulate the forward operator as a mapping in the Hilbert Space L^2(R). We investigate the Fréchet differtentiability of these operator and prove the discontinuity of the inverse operator. We use Tikhonov Regularization and develope a gradient method for solving the corresponding minimization problems. A change of the solution space from L^2(R) to H^1(R) enables us to present assertions on the stable solvability of this problem.

Keywords:

inverse problem of optionpricing, identification of local volatilities, Black-Scholes model, parabolic equations, fundamental solution, ill-posed problem, regularization, optimal control

Language:

English

Publication time:

6 / 2003