Securities with Mixture of Elliptic Distributed Risk Factors

Title: Securities with Mixture of Elliptic Distributed Risk Factors

Abstract: This research explores the estimation of Value-at-Risk (VaR) and Expected Shortfall (ES) for a quadratic portfolio of securities, specifically equities, without using Delta and Gamma Greeks. The study focuses on elliptic distributed risk factors, which are a generalization of normal distributions. The authors propose a method to estimate VaR and ES using Monte Carlo simulations and elliptic distribution assumptions. The study provides explicit equations for calculating VaR when the joint log-returns follow specific mixtures of elliptic distributions, such as multivariate Student t-distributions and mixture of normal distributions. The results suggest that the proposed method can be used to estimate VaR and ES for quadratic portfolios of equities without relying on Delta and Gamma Greeks.

Main Research Question: How can we estimate Value-at-Risk (VaR) and Expected Shortfall (ES) for a quadratic portfolio of securities without using Delta and Gamma Greeks, using Monte Carlo simulations and elliptic distribution assumptions?

Methodology: The study uses Monte Carlo simulations to estimate VaR and ES for a quadratic portfolio of securities. The method relies on the assumption that the joint log-returns follow an elliptic distribution, which is a generalization of normal distributions. The authors introduce a method to estimate VaR and ES using multiple integral equations and symmetric interpolar rules for multiple integrals over hyperspheres. The study provides explicit equations for calculating VaR when the joint log-returns follow specific mixtures of elliptic distributions, such as multivariate Student t-distributions and mixture of normal distributions.

Results: The study provides explicit equations for calculating VaR when the joint log-returns follow specific mixtures of elliptic distributions, such as multivariate Student t-distributions and mixture of normal distributions. The results suggest that the proposed method can be used to estimate VaR and ES for quadratic portfolios of equities without relying on Delta and Gamma Greeks.

Implications: The study's findings have implications for the financial industry, as they provide a method to estimate VaR and ES for quadratic portfolios of securities without using Delta and Gamma Greeks. This can be particularly useful for portfolios of non-normally distributed assets or those containing derivatives instruments. The study also contributes to the literature by generalizing the concept of ∆-Elliptic VaR, which was previously limited to linear portfolios.

Link to Article: https://arxiv.org/abs/0310043v3 Authors: arXiv ID: 0310043v3