A multiobjective optimization approach for threshold determination in extreme value analysis for financial time series

Abstract

The literature on the extreme value theory threshold optimization problem for multiple time series analysis does not consider determining a single optimal tail probability for all marginal distributions. With multiple tail probabilities, their discrepancy results in a differing number of exceedances, which may favour a particular marginal series. In this study, we propose a single optimal tail probability by integrating trade-offs among multiple time series within an MOO framework. Mathematically, our approach links the peaks-over-threshold technique and goal programming technique by developing a set of regression functions, which represent continuous paths of possible tail areas for multiple time series, and we formulate them at the desired levels within a multiobjective optimization framework. The optimal solution is found as the minimum Chebyshev variant weighted value. Our approach advances the development of the peaks-over-threshold method by considering the characteristics of a group of time series collectively instead of independently. The proposed optimal tail probability can be considered an optimal reference point for practical risk investment portfolio analysis that employs an identical tail size across multiple time series data. The daily log returns of four U.S. stock market indices, namely, S&P 500, NASDAQ Composite, NYSE Composite, and Russell 2000, from 1 July 1992 to 30 June 2022 are studied empirically.