A novel nonlinear value-at-risk method for modeling risk of option portfolio with multivariate mixture of normal distributions

Abstract

This paper proposes a novel nonlinear model for calculating Value-at-Risk (VaR) when the market risk factors of an option portfolio are heavy-tailed. A multivariate mixture of normal distributions is used to depict the heavy-tailed market risk factors and accordingly a closed form expression for the moment generating function that can reflect the change in option portfolio value can be derived. Moreover, in order to make use of the correlation between the characteristic function and the moment generating function, Fourier-Inversion method and adaptive Simpson rule with iterative algorithm of numerical integration into the nonlinear VaR model for option portfolio are applied for calculation of VaR values of option portfolio. VaR values of option portfolio obtained from different methods are compared. Numerical results of Fourier-Inversion method and Monte Carlo simulation method show that high accuracy VaR values can be obtained when risk factors have multivariate mixture of normal distributions than when they have normal distributions. Moreover, VaR values obtained by using the Fourier-Inversion method are not obviously different from VaR values obtained by using Monte Carlo simulation when market risk factors have normal distributions or multivariate mixture of normal distributions. However, the speed of computation is obviously faster when using Fourier-Inversion method, than when using Monte Carlo simulation method. Besides, Cornish Fisher method is faster and simpler than Monte Carlo simulation method or Fourier-Inversion method. However, this method does not offer high accuracy and cannot be used to calculate VaR values of option portfolio when market risk factors have heavy-tailed distributions.