  We introduce the formalism of generalized Fourier transforms in the context
of risk management. We develop a general framework to efficiently compute the
most popular risk measures, Value-at-Risk and Expected Shortfall (also known as
Conditional Value-at-Risk). The only ingredient required by our approach is the
knowledge of the characteristic function describing the financial data in use.
This allows to extend risk analysis to those non-Gaussian models defined in the
Fourier space, such as Levy noise driven processes and stochastic volatility
models. We test our analytical results on data sets coming from various
financial indexes, finding that our predictions outperform those provided by
the standard Log-Normal dynamics and are in remarkable agreement with those of
the benchmark historical approach.
