Over the past few decades, asymptotic study of financial systems has become an integral part of practical decision making and analytics. Realistic financial systems are complex, and undesirable events in them are often rare. In order to gain more insights into their behaviour, it is important to develop structural simplifications and efficient computational algorithms for rare event analysis. In this talk, we undertake a detailed study of these aspects, and develop structural insights on a number of financial systems of practical interest.
In the first part, we consider two problems related to rare event analysis:
1) The estimation of default probabilities of financial firms from data is an important problem which has received significant attention over the past two decades. We discuss the development of a closed form, interpretable parameter estimation technique for predicting defaults of financial firms. Typically, one uses Maximum Likelihood Estimation (MLE) for predicting the firm default probabilities. We prove that our estimator is almost as accurate as the MLE for a realistic sample of financial data. Further since our estimator is closed form, it is significantly faster than MLE. Finally, we demonstrate that unlike the MLE our estimator also gives interesting structural insights - specifically, we show that from the standpoint of default prediction, collecting covariate data just before occurrence of default is sufficient to estimate probabilities.
2) Building upon the well established notions of multivariate regular variation and large deviations theory, we derive a unified framework, in which tail analysis of a large class of stochastic loss functions can be performed. Within this framework, assuming the underlying stochasticity is heavy tailed, we develop a data-driven estimator for tail exceedences of a large class of financial losses, and applying it to a tail risk-constrained portfolio optimisation problem, showcase superior performance over the state of the art. Additionally, assuming oracle access to the densities of loss causing covariates, we develop a self-replicating, provably accurate importance sampling algorithm to estimate rare event probabilities over a variety of covariate/loss structures.
In the latter part of the talk, we develop a limiting representation for an interconnected banking network in presence of partial information. Practical banking networks are large and complicated, and one searches for simple limiting representations (as the network size goes to infinity). We characterise the wealths of banks in a large network in terms of a simple, one dimensional distributional fixed point, which we show is amenable to simulation. While such fixed point representations have been well studied when the network is of a finite size, to the best of our knowledge, our work is the first to provide a simplified limiting representation.
