Reliable calculations of financial risk require that the fat-tailed nature of prices changes is included in risk measures. To this end, a non-Gaussian approach to financial risk management is presented, modeling the power-law tails of the returns distribution in terms of a Student-$t$ (or Tsallis) distribution. Non-Gaussian closed-form solutions for Value-at-Risk and Expected Shortfall are obtained and standard formulae known in the literature under the normality assumption are recovered as a special case. The implications of the approach for risk management are demonstrated through an empirical analysis of financial time series from the Italian stock market. Detailed comparison with the results of the widely used procedures of quantitative finance, such as parametric normal approach, RiskMetrics methodology and historical simulation, as well as with previous findings in the literature, are shown and commented. Particular attention is paid to quantify the size of the errors affecting the risk measures obtained according to different methodologies, by employing a bootstrap technique.
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