Accounting for risk of non linear portfolios: a novel Fourier approach

The presence of non linear instruments is responsible for the emergence of non Gaussian features
in the price changes distribution of realistic portfolios, even for Normally distributed risk factors. This is
especially true for the benchmark Delta Gamma Normal model, which in general exhibits exponentially
damped power law tails. We show how the knowledge of the model characteristic function leads to Fourier
representations for two standard risk measures, the Value at Risk and the Expected Shortfall, and for
their sensitivities with respect to the model parameters. We detail the numerical implementation of our
formulae and we emphasize the reliability and efficiency of our results in comparison with Monte Carlo
simulation.