Financial Models of Interaction Based on Marked Point Processes and Gaussian Fields

Abstract

This thesis deals with interaction phenomena in marked point processes, with particular attention being paid to the extreme value theory framework and with application to high-frequency financial data. While classical low-frequency financial returns are measured at equally spaced temporal locations, transaction data occur at random and irregularly spaced points in time. Due to possible dependencies with the traded prices, the pattern of trading times might already contain information about the price process. We propose different characteristics, referred to as conditional mean marks, to measure interaction effects in marked point processes, based on second-order moment measures. While for general models, these characteristics are analytically intractable, they can be calculated for suitable Poisson process based models. Applying standard estimators to real data, the models' ability to capture interaction effects can be verified. We conduct this comparison for a commonly-used GARCH model applied to transaction data from the German stock index DAX. Another part of this thesis focuses on non-ergodic or non-stationary processes, by which structural breaks in financial data can be captured. When different days of trading, for instance, exhibit different stochastic behavior, multiple possibilities arise of aggregating information. We discuss alternative definitions of our characteristics tailored to the non-ergodic case and give examples in which different definitions of 'average' have a sensible meaning at the same time. Via the mean excess representation of the tail index of an extreme-value distribution, the concept of conditional mean marks is transferred to the marks' tail behavior. For estimation of these tail characteristics, clearly only the extremal observations should be taken into account which may further strengthen the importance of non-ergodic modeling. Some asymptotics of the respective estimators are discussed. In order to be eventually able to define second- and higher-order MPP characteristics measuring extremal dependence, we provide some contribution to a new multivariate peaks-over-threshold approach of inference for max-stable processes.