Calibration of the Local Volatility in a Generalized Black--Scholes Model Using Tikhonov Regularization

Abstract

Following an approach introduced by Lagnado and Osher (1997), we study Tikhonov regularization applied to an inverse problem important in mathematical finance, that of calibrating, in a generalized Black-Scholes model, a local volatility function from observed vanilla option prices. We first establish W 1,2 p estimates for the Black-Scholes and Dupire equations with measurable ingredients. Applying general results available in the theory of Tikhonov regularization for ill-posed nonlinear inverse problems, we then prove the stability of this approach, its convergence towards a minimum norm solution of the calibration problem (which we assume to exist), and discuss convergence rates issues.

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