Dynamic Uncertainty Modeling in Finance

Since the beginning of the financial crisis in 2007, stability of financial markets has become a major topic attracting a lot of attention from experts in finance, economy and politics. In the field of mathematical finance, this led for instance to the emergence of a branch called robust finance, which aims at making financial modeling more solid in times of crises. The goal of this project is to establish two important aspects in this area: introducing dynamic modeling ideas and jointly capturing model risk and information risk. Mathematically, we incorporate model risk via so-called mixture-models and non-linear Markov processes. In both approaches parameter uncertainty and its dynamic nature due to incoming information is explicitly taken into account. In other words we accommodate the view that model risk is among other things a consequence of insufficient or even wrong information. This information risk is modeled via two filtrations. The smaller filtration contains the information actually available to market participants, while the larger filtration also includes (idealized) information on unobservable quantities. Prices are supposed to be adapted to the larger filtration, whereas actual observations can only be done in the smaller filtration, because of unreliable data sources and discrete and noisy signals. This allows us to go beyond the usual assumptions taken in mathematical finance for example, price processes do not need to be semimartingales any longer. In this general two-filtration setup in continuous time we analyze all foundational questions, like fundamental theorems, superhedging, stochastic integration and model calibration. Our main field of application are fixed income markets with multiple yield curves, which became due to the financial crisis highly important. These markets are a prototypical example for model uncertainty being caused by unobservable but important factors, namely liquidity and credit risk in this case. Beyond that they show the necessity of a new formulation of the mathematical modeling setup within which we aim to lay the theoretical foundations to answer questions of model calibration, pricing and hedging.

In view of the current economic situation with high inflation, rising interest rates and the fear of a recession, robust and realistic modeling of financial markets is more relevant than ever. Indeed, the stability of the financial system has become a major topic attracting a lot of attention from experts in finance, economy and politics. In the field of Mathematical Finance, this led already right after the financial crisis of 2007 to the emergence of a branch called "robust finance", which aims at making financial modeling more solid in times of crises. The goal of this project was to establish two important aspects in this area: first, measuring risk by incorporating the financial regulators' beliefs on the appropriateness of different models, thus enabling a reasonable and adequate risk assessment. Second, introducing classes of dynamic and universal models, based on modern data-driven machine learning methods, that open the door to robust and reliable model selection mechanisms, while first principles from finance like "no arbitrage" can still be guaranteed. Mathematically, we incorporate model risk via so-called mixture-models where parameter uncertainty and its dynamic nature due to incoming information is explicitly taken into account. In other words we accommodate the view that model risk is among other things a consequence of insufficient or even wrong information. By adopting dynamic model classes that satisfy so-called universal approximation properties meaning that essentially all classical models can be approximated, we account in principal for all possible sources of model risk which in turn can be narrowed by new incoming information. Our main fields of application are modeling of large indices, like S&P 500 and the related VIX volatility index, as well as fixed income markets. The latter are a prototypical example where model uncertainty can be caused by unobservable but important factors, such as liquidity or credit risk. For such markets we have found new financial mathematical formulations and developed within our framework the theoretical foundations for robust model calibration, pricing and hedging.