An efficient Fourier expansion method for the calculation of value-at-risk: Contributions of extra-ordinary risks

This paper proposes a new method to compute Value-at-Risk (VaR) of a given portfolio, in which the returns of different assets are assumed to come from the log-stable Paretian family with and without identical characteristic exponents (extra-ordinary risk). The advantage of stable Paretian is its “stability” under sum, i.e., assets with the same characteristic exponent will generate portfolios with the same characteristic exponent, the self-similar property. Empirically, assets may have different characteristic exponent estimates that may be caused by estimation errors or in fact, different assets have different return generating properties. Theories to justify the latter has yet to be developed. McCulloch (1985, 2006), Shephard (1994)’s local scale models, etc. however are consistent with this “unstable” behavior. For the former, we believe that it can be captured by larger standard errors of the estimates. Our Fourier-cosine based method is first applied to compute VaR of log-stable Paretian distributions with different characteristic exponents. In this study, we provide an error analysis for the method. We also prove the properties for some of our models. A series of numerical results based on empirical data and simulations are showed to support the efficiency of this new method.