Research On Measuring Options Portfolio VaR Under Multivariate Mixture Normality

Since China has opened the finance market and getting more marketability in the financial profession, financial organs like banks, investment funds and negotiable securities companies will face the market risk progressively. Since nontradable shares reform in 2005, China has continued the thorough research on the options market and made substantive progress in the institutional construction, the demand development and the technology system design. Facing the derivative products like options with giant leverage, how to manage the risk is especially important.This paper makes use of nonlinear VaR model to measure the options portfolio risk based on the domestic and foreign study of the options pricing and risk management. Analyzing the development condition of options market, we can get several primary impact factors of the market risk of options. Then the paper makes the improvement on the VaR model, and propose a new VaR model with the returns of underlying described by multivariate mixture normality.This paper introduces mixture normality to describe the heavy tail of the returns of underlying. First introduces the definition, the density function, the digital characteristic and some related nature of mixture normality, but the parameter estimation of mixture normality is a difficulty, therefore this paper uses EM algorithm to estimate the parameters, and the result is demonstrated good. Next we carry on second-order Taylor to the option value and unify the Black-Scholes options pricing model, constructe the Delta, Gamma and Theta, and propose the Delta-Gamma-Theta models based on normality and mixture normality. Separately, this paper calculates the VaR with Cornish Fisher method, Monte Carlo Simulation method and Fourier Inversion method. However Fourier Inversion method needs the moment generating function of the change value of option portfolio under the normal distribution and the mix normal distribution, this is another key and difficulty. Through the model transformation and the correlation theories we can infer the moment generating function and the characteristic function, and carry on the Fourier's integral transformation, get the VaR value using the numerical method. As introducing multivariate mixture normality into the Delta-Gamma-Theta model, it has to propose the moment generating function of the value change of options portfolio, then this paper uses Fourier Inversion method to carry on the Fourier's integral transformation to calculate the VaR with the numerical method.Finally, this paper carries on the empirical analysis to warrants portfolio in China, and compares the nonlinear VaR of options portfolio under multivariate mixture normality to that of multivariate normality in these models. The empirical analysis indicated that the result is better using multivariate mixture normality, the latent loss is bigger, it is a good sign to explain the heavy tail. Cornish the Fisher method is simpler, makes use of several step moment of the sample to match the distribution of the value change of options portfolio, but this method gets partial VaR, has neglected some information. Monte Carlo simulation method is also an easy method to calculate VaR value, but has the model risks, it can be compared with other methods. Fourier Inversion method has certain superiority in the computation accuracy, but its operation is more tedious, is not easy to implement.