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

Abstract

Following an approach introduced by Lagnado and Osher [J. Comput. Finance, 1 (1) (1997), pp. 13--25], 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.