Abstract
We describe four methods to approximate the delta-gamma distribution, commonly used in Value-at-Risk calculations, and evaluate the methods for accuracy and speed. The best techniques are Partial Monte Carlo and Fourier inversion of the moment generating function. The Fourier inversion is the best unless the number of risk factors is very large (1000 . 5000 depending on the confidence level of VaR).
Introduction
Non-linear positions, such as options, produce portfolio returns which are frequently fat-tailed and skewed. Consequently, knowledge of the mean and variance is not enough to characterize the distribution of returns and measure Value-at-Risk (VaR). In order to calculate VaR for a non-linear portfolio, we need to obtain a percentile of the distribution of changes in portfolio value, but in general, it is impossible to obtain a closed form for the return distribution, and therefore the VaR, of a non-linear portfolio.
There are two main approaches used to calculate VaR in the non-linear case: The first involves Monte Carlo simulation to obtain a numerical estimate of VaR. This method is very accurate but can be computationally expensive for large portfolios. The second approach consists of analytical approximations of the true distribution of changes in the portfolio value. This approach can provide an approximate but fast parametric solution to the problem (e.g. [8, 1]). Hybrid approaches rely on delta-gamma methods to dramatically reduce the time to calculate VaR by judiciously selecting which random trials to evaluate explicitly (see [2]). In this article, we evaluate four different methods to obtain an analytical delta-gamma approximation of the distribution of portfolio returns using Johnson transformations (Section 3), Cornish-Fisher expansions (Section 4), Fourier methods (Section 5), and partial Monte-Carlo. Results are presented in Section 6. We conclude in Section 7.
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