Dynamic uncertainty modeling in Finance

自2007年金融危机爆发以来，金融市场的稳定成为金融、经济、政治等领域专家关注的重大话题。例如，在数学金融领域，这导致了一个名为“稳健金融”的分支的出现，该分支旨在使金融模型在危机时期更加可靠。该项目的目标是建立这一领域的两个重要方面：引入动态建模思想和联合捕获模型风险和信息风险；在数学上，我们通过所谓的混合模型和非线性马尔可夫过程来整合模型风险。在这两种方法中，都明确考虑了由于输入信息而引起的参数不确定性及其动态性质。换句话说，我们接受了这样一种观点，即模型风险是信息不足甚至错误的结果。这种信息风险通过两次过滤进行建模。较小的过滤包含市场参与者实际可用的信息，而较大的过滤还包括关于不可观测数量的(理想化的)信息。价格应该适应较大的过滤，而实际观察只能在较小的过滤中进行，因为不可靠的数据源和离散且有噪声的信号。这使我们能够超越数学金融学中的常见假设--例如，价格过程不再需要是半令牌。在这个连续时间的一般双过滤模型中，我们分析了所有的基本问题，如基本定理、超套期保值、随机积分和模型校准。我们的主要应用领域是具有多条收益率曲线的固定收益市场，这在金融危机中变得非常重要。这些市场是模型不确定性的典型例子，这些不确定性是由不可观察但重要的因素造成的，在这种情况下是流动性和信用风险。除此之外，它们表明了建立新的数学模型体系的必要性，我们的目标是为回答模型校准、定价和套期保值等问题奠定理论基础。

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.