Elliptic Distributed Risk Factors in Quadratic Portfolio of Securities

Title: Elliptic Distributed Risk Factors in Quadratic Portfolio of Securities

Abstract: This research focuses on estimating the Value-at-Risk (VaR) of a quadratic portfolio of securities without Delta and Gamma, using elliptic distributed risk factors. The study employs a numerical method by Alan Genz to reduce the estimation of the quadratic VaR to a resolution of one-dimensional integral equations. The research illustrates the method using mixture of normal distribution and mixture of t-student distribution examples. The results provide an explicit equation with a solution for VaR, applicable to multivariate elliptic distributions. The implications of this study are significant for the financial literature, as it extends the concept of quadratic Delta-Gamma Portfolio VaR and offers a more generalized approach to risk management.

Main Research Question: How can we estimate the Value-at-Risk of a quadratic portfolio of securities using elliptic distributed risk factors?

Methodology: The study uses numerical estimation to calculate the VaR of a quadratic portfolio of securities. The method involves reducing the estimation of the quadratic VaR to a resolution of one-dimensional integral equations using Alan Genz's numerical method. The research applies this method to mixture of normal distribution and mixture of t-student distribution examples to demonstrate its effectiveness.

Results: The research provides an explicit equation with a solution for VaR, applicable to multivariate elliptic distributions. The equation is derived using the given examples and can be used to estimate the VaR of a quadratic portfolio of securities with elliptic distributed risk factors.

Implications: The study's findings are significant for the financial literature as it extends the concept of quadratic Delta-Gamma Portfolio VaR and offers a more generalized approach to risk management. The research provides a practical method for estimating the VaR of a quadratic portfolio of securities, which can be applied to various financial scenarios. Additionally, the use of elliptic distributed risk factors broadens the scope of the study, making it applicable to a wider range of risk factors in the financial market.

Link to Article: https://arxiv.org/abs/0310043v1 Authors: arXiv ID: 0310043v1