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Abstract

The article aims to measure the market risk beyond the basic risk measures like the Value-at-Risk (VaR) and the Expected Shortfall (ES). The Entropic Value-at-Risk is selected among the available measures based on its advantages -- it is the coherent upper bound of the VaR and ES. This risk measure is applied to the classical Black-Scholes model as well as to some more realistic ones, such as the exponential tempered stable model (the log-returns are presented by a tempered stable L\'evy process), the stochastic volatility model of Heston, its jump extension of Bates, and another stochastic volatility model but with tempered stable jump behavior.  Closed or semi-closed form formulas are obtained for all these models. The derived theoretical results are applied to a historical sample for the S\&P 500 index. There are several advantages of the use of a stochastic model instead of the direct application of the risk measure to the statistical data. The main advantage is that an appropriate model can capture the inherent features of the financial asset that are hidden in the time series due to an insufficient number of observations. This is especially true for the distribution tails -- terms closely related to the risk measures.

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