It is the main objective of this paper to compare different approaches to analytically calculate value-at-risk (VaR) for portfolios that include options. We focus on approaches that are based on a second order Taylor-series approximation of the nonlinear option pricing relationship. The main difficulty common to all these methods is the estimation of the required quantile of the profit and loss distribution, since there exists no analytical representation of this distribution. In our analysis we examine different moment matching approaches and methods to directly approximate the required quantile. For this purpose, we perform a backtesting procedure based on randomly generated risk factor returns which are multivariate normal. The VaR-numbers calculated by a specific methodology are then compared to the simulated actual losses. We conclude that the accuracy of methodologies that rely only on the first four moments of the profit and loss distribution is rather poor. The inclusion of higher moments, e.g. through a Cornish-Fisher expansion seems to be appropriate. 1
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