Efficient Simulation and Approximation of Value at Risk under GARCH Model

Abstract

Value-at-risk (VaR) is a the most commonly known measure for the risk of a portfolio. Many methods are used to evaluate the VaR, such as Delta-Normal, historical simulation and Monte Carlo simulation methods. We use importance sampling to reduce the variance of Monte Carlo simulation method. The probabilities considered in VaR typically are of moderate deviations. However, the variance reduction techniques developed in the literature for VaR computation are based on large deviations consideration. Modeling risk factors by GARCH(1,1) model and develop a new moderate-deviations method for VaR computation. We use some empirical data, such as DAX30, FTSE250 index and a portfolio to evaluate the VaR by those methods. We find the log return of empirical data do not follow normal distribution. The back testing shows that the proposed method is consistently more efficient than others.