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Non-parametric calibration of the local volatility surface for European options using a second-order Tikhonov regularization. (English) Zbl 1294.91187

Summary: We calibrate the local volatility surface for European options across all strikes and maturities of the same underlying. There is no interpolation or extrapolation of either the option prices or the volatility surface. We do not make any assumption regarding the shape of the volatility surface except to assume that it is smooth. Due to the smoothness assumption, we apply a second-order Tikhonov regularization. We choose the Tikhonov regularization parameter as one of the singular values of the Jacobian matrix of the Dupire model. Finally we perform extensive numerical tests to assess and verify the aforementioned techniques for both volatility models with known analytical solutions of European option prices and real market option data.

MSC:

91G60 Numerical methods (including Monte Carlo methods)
91G20 Derivative securities (option pricing, hedging, etc.)
65F22 Ill-posedness and regularization problems in numerical linear algebra
65M30 Numerical methods for ill-posed problems for initial value and initial-boundary value problems involving PDEs
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