Tikhonov Regularization

Abstract An important issue in quantitative finance is model calibration . The calibration problem is the inverse of the pricing problem. Instead of computing prices in a model with given values for its parameters, one wishes to compute the values of the model parameters that are consistent with observed prices. Now, it is well known to physicists that such inverse problems are typically ill posed . So, if one perturbs the data (e.g., if the observed prices change by some small amount between today and tomorrow it is quite typical that a numerically determined best fit solution of the calibration problem switches from one “basin of attraction” to the other; thus the numerically determined solution is unstable . To achieve a robust calibration, we need to introduce some regularization . The most widely known and applicable regularization method is Tikhonov(–Phillips) regularization method. In this paper we provide a survey of the Tikhonov regularization and illustrate it by applying the regularization to the problem of calibrating a local volatility model.