Normal Variance Mixture Distributions as Approximations of Poisson Mixture Sums

By the central limit theorem and its generalisations, normal and - more generally - stable distributions turn up as weak limits of suitable scaled sums of i.i.d. random variables. When the number of summands is itself random... [ view full abstract ]

By the central limit theorem and its generalisations, normal and - more generally - stable distributions turn up as weak limits of suitable scaled sums of i.i.d. random variables. When the number of summands is itself random having a Poisson mixture distribution, then normal variance mixture distributions appear as weak limits. This justifies their use to model asset price returns. We give upper bounds for the quality of the approximation with respect to the Kolmogorov and the Wasserstein distance. We also relate these distances to the risk measures value-at-risk and expected shortfall, respectively.