具有金融衍生品的金融网络的强逼近与非理性
Strong Approximations and Irrationality in Financial Networks with Financial Derivatives
摘要：
金融网络模拟了一系列通过义务相互联系的金融机构（公司）。最近的研究在这个模型中引入了一类债务，称为信用违约互换，一种金融衍生品。这类系统的主要计算挑战被称为清算问题，即确定哪些公司违约，并计算它们对系统风险的敞口，技术上称为它们的回收率。众所周知，回收率形成一个简单函数的不动点集合，并且这些不动点可以是无理数。此外，Schuldenzucker等人（2016）已经证明，找到弱（或“几乎”）近似（有理）不动点是PPAD完全的。我们进一步从非理性和逼近强度的角度研究了出清问题。首先，我们观察到弱近似解可能会歪曲机构的实际财务状况。在此基础上，我们研究了求强（或“近”）近似解的复杂性，并证明了FIXP-完备性。然后，我们研究了非理性所需的结构特性，我们给出了非理性解决方案出现的必要条件：金融网络中某些类型的周期的存在迫使恢复率采取非线性多项式的根的形式。在没有一个大的子类，这样的周期，我们研究的复杂性，找到一个确切的不动点，这是一个问题，虽然外面的，PPAD。
英文摘要：
Financial networks model a set of financial institutions (firms) interconnected by obligations. Recent work has introduced to this model a class of obligations called credit default swaps, a certain kind of financial derivatives. The main computational challenge for such systems is known as the clearing problem, which is to determine which firms are in default and to compute their exposure to systemic risk, technically known as their recovery rates. It is known that the recovery rates form the set of fixed points of a simple function, and that these fixed points can be irrational. Furthermore, Schuldenzucker et al. (2016) have shown that finding a weakly (or "almost") approximate (rational) fixed point is PPAD-complete. We further study the clearing problem from the point of view of irrationality and approximation strength. Firstly, we observe that weakly approximate solutions may misrepresent the actual financial state of an institution. On this basis, we study the complexity of finding a strongly (or "near") approximate solution, and show FIXP-completeness. We then study the structural properties required for irrationality, and we give necessary conditions for irrational solutions to emerge: The presence of certain types of cycles in a financial network forces the recovery rates to take the form of roots of non-linear polynomials. In the absence of a large subclass of such cycles, we study the complexity of finding an exact fixed point, which we show to be a problem close to, albeit outside of, PPAD.
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