Calibrating Financial Mathematical Models Reliably

Models Should Be Versatile: Systematically Investigate Calibration Quality

The starting point for our research is a well-established financial mathematics model for capital markets, used in practice to simulate interest rate and share price trends, which is employed, amongst other things, in the »Classification of Private Retirement Provision«.

We are examining existing standard procedures for calibrating financial market models and supplementing them with methods that allow the quality of the results to be analysed and evaluated more precisely. In doing so, we are considering, amongst other things:

• model-independent criteria by which the quality of a calibration can be measured
• the suitability of the financial market data used, in particular interest rate derivatives
• the optimisation issue arising during calibration
• the systematic distinction between model errors and calibration and estimation errors
• different estimation and calibration methods and a comparison of these

The goal is not only to ask how well a model reproduces specific market data. Equal importance is placed on how reliable the estimated parameters are and how robust the calibrated model remains for different applications.

Diagnostic Methods Make Calibrations More Transparent

A small calibration error does not automatically mean that the model parameters have been determined reliably. Different combinations of parameters can, for example, lead to very similar model prices. Similarly, individual financial instruments used for calibration can have a particularly strong influence on the result.

A key focus of the thesis is therefore on diagnostic methods with which we examine the stability, identifiability and information content of a calibration. To this end, we analyse, amongst other things, how sensitive model prices are to changes in individual parameters. Furthermore, we apply concepts from statistical regression diagnostics to the non-linear calibration of financial market models.

Key metrics such as leverage, measures of influence, Fisher information and effective degrees of freedom provide insight into which market data actually contribute to the determination of the model parameters. At the same time, they help to identify potential identification problems. This allows for a better assessment of how robust the results of a calibration actually are.

Calibration Designed to Match the Application

We examine the methods developed using extensive real-world financial market data from different market phases. The analyses show that the quality of a calibration does not depend solely on how accurately the model replicates the market data used. The stability of the model parameters and the transferability to other application contexts also play an important role.

Which characteristics are particularly relevant depends on the intended use. When valuing financial products, for example, the focus is on reproducing current market prices as accurately as possible. For long-term simulations, on the other hand, historical trends and expectations regarding the future development of the financial markets must also be taken into account. That is why, in this thesis, we examine and combine various calibration approaches.

Reliable Models for Retirement Plans and Life Insurance

This work ties in directly with our department’s long-term research and collaboration projects in the area of »Retirement Provision and Life Insurance«. This includes, for example, our collaboration with the Product Information Centre for Pension Provision (PIA).

The methods developed enable the calibration of financial market models to be analysed more transparently and potential weaknesses to be identified at an early stage. In this way, we create a more sound basis for deciding whether a model is suitable for a particular area of application. This is particularly beneficial for applications where financial market developments need to be modelled and reliably assessed over long periods of time.