Adaptive Wavelet Methods for Structured Financial Products

近年来，金融市场要求引入越来越复杂的金融结构。到目前为止，这些结构的风险管理和估值还没有得到充分的理解，正如当前的信贷危机所显示的那样。我们项目的主题是开发高效、可靠和快速的评估方法，用于所谓的债务抵押债券（cdo）和基于cdo的奇异衍生品。我们的策略将基于偏微分方程（PDEs），使用自适应小波方法进行估值。要考虑的方程将是偏积分微分方程（PIDE）。外来选择导致障碍问题，我们计划通过自适应投影方案来处理。此外，我们研究了参数依赖性和灵敏度分析（如果可能的话）和自动微分。最后，我们将此方法与（拟）蒙特卡罗模型进行比较，以研究基于pde的模型的性能。

In recent years financial market required the introduction of increasingly complex financial structures. Up to now the risk management and valuation of these structures is not sufficiently understood as the current credit crisis shows. Subject of our project is the development of efficient, reliable and fast valuation methods for so-called Collateralized Debt Obligations (CDOs) and exotic derivatives on tranches of CDOs. Our strategy will be based on Partial Differential Equations (PDEs) using an adaptive wavelet method for valuation. The equations to be considered will be Partial Integro-Differential Equations (PIDE). Exotic options lead to obstacle problems which we plan to treat by means of an adaptive projection scheme. Moreover, we investigate parameter dependencies and sensitivities both analytical (if possible) and by Automatic Differentiation. Finally, we compare this approach with (quasi-)Monte Carlo models in order to investigate the performance of PDE-based models.