Calibration problems - An inverse problems view

The White Paper titled "Calibration Problems – An Inverse Problems View" by Heinz W. Engl explores the challenges associated with calibrating financial models, particularly focusing on the calibration of parameters in derivative pricing models. The paper frames these challenges as inverse problems, a class of mathematical problems where solutions are not directly obtainable but must be inferred from indirect measurements, often leading to issues of "ill-posedness." This ill-posed nature can result in solutions that are highly sensitive to noise in the data, making the calibration process both difficult and unstable.

Engl discusses various regularization techniques to stabilize the solution of these ill-posed problems. These techniques help in balancing the trade-off between accuracy and stability in the presence of noisy data. The paper highlights the application of these methods to both simple and complex financial models, such as the Hull-White model and local volatility models, showing that appropriate regularization can significantly improve the robustness of the calibration process.

The document also provides examples from both finance and industrial processes, such as the cooling of steel in continuous casting, to illustrate the broader applicability of these mathematical techniques. By drawing on a well-established mathematical theory of regularization, Engl demonstrates that these methods are essential for effectively solving inverse problems in computational finance.