Achievements of Research Area H

Insurance and financial market risk.

An important line of research focused on valuation, hedging, and risk management of exotic contracts. Sandmann and Wittke derive a unified approach to chooser options in [H:SW10]. An Chen, Pelser and Vellekoop present a utility based analysis in [H:CPV]. An Chen and Suchanecki study Parisian exchange option in [H:CS]. Following a joint workshop in Königswinter of Research Area H and Research Area J 2009, An Chen, Suchanecki and Evangelia Petrou (IAM) started a collaboration on rainbow barrier options.
Another topic of high current importance is the interplay of risk management and market regulation both in the financial and insurance sectors. Together with several collaborators, An Chen studied dynamic modeling of financial market regulation and bankruptcy. New approaches to the dynamic modeling of Solvency II and to optimal regulation are given among others in [H:BC09]. The relation between insurance and financial market risk and the pricing and hedging of hybrid derivatives, e.g. equity linked life insurance contracts were analysed by Albeverio, Steblovskaya and Wallbaum [H:ASW09]. Ankirchner (Bonn Junior Fellow) and Heyne derive optimal hedging strategies in incomplete markets, allowing for stochastic correlation between different risk factors [H:AH]. In a more abstract context, Ankirchner, Imkeller, and Reis [H:AIR10] analise pricing and hedging based on non-tradable underlyings.

Decision theory.

In recent years, a successful interplay between mathematical finance and decision making theory has emerged in the context of optimal investment planning. Using dynamic programming methods, Ankirchner and Dermoune solve mean variance optimization problems with an application to a discrete portfolio investment model in [H:AD]. Applying methods of game theory, Szimayer, Desmettre and Gould clarify the relation between own-company shareholding and work effort preferences in [H:DGS10].

Econometric analysis of financial data.

In addition to theoretical research, the econometric analysis of financial data has become a major topic of research. This was emphasized by the workshop Financial Mathematics meets Econometrics jointly organized by Ankirchner and Pigorsch in 2009. Pigorsch, together with several collaborators, analyses discrete-time models for daily stock returns and elaborates a new econometric approach incorporating stochastic volatility in order to model financial markets (see [H:BTPP09] and [H:PS09]). Benko, Härdle and Kneip use methods of functional data analysis to analyse implied volatility surfaces of European options in [H:BHK09]. In [H:PKH] Paluch, Kneip and Hildenbrand analyse budget elasticities of demand. It is shown that there are significant differences between properly defined aggregate elasticities and mean individual elasticities on the micro level.

Heterogeneity.

A central point of methodological research has been econometric factor models for analysing panel data. From an economic point of view, factor models are able to reflect heterogeneity of individual behavior, which is the core of the aggregation problem explained in the original grant proposal. It is assumed that there exists ‘unobserved heterogeneity’ characterizing individual trajectories over time, which can be described by individually different, linear combinations of unknown factors. These factors and the model dimension then have to be estimated from the data. Investigators from this RA have clarified various problems in this context. Substantial methodological contributions are provided by Breitung and Das, Breitung and Eickmeier as well as in Breitung and Tenhofen [H:BD08], [H:BE11] and [H:BT].
When dealing with heterogeneous populations, a major task is to model the time evolutions of the distributions of important characteristics, or of their corresponding density and regression functions. In the last decades, functional data analysis has been a very active field of international statistical research. The aim is to develop methods for analysing data representing samples or time series of related functions. In [H:KSS], Kneip, Sickles and Song combine econometric factor models and methods of functional data analysis in order to analyse heterogeneity in time trends. Functional regression problems are studied by Crambes, Kneip and Sarda in [H:CKS09], while Kneip and Ramsay present a new approach to the so-called registration problem in [H:KR08].