Extreme risk in finance: analysis and modeling.

Summary This thesis studies risk management and hedging using Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR) as risk measures. The first part proposes a price evolution model that we confront with real data from the Paris stock exchange (Euronext PARIS). Our model takes into account the probabilities of occurrence of extreme losses and the regime changes observed in the data. Our approach consists in detecting the different periods of each regime by constructing a hidden Markov chain and estimating the tail of each regime distribution by power laws. We show empirically that the latter are more suitable than normal and stable distributions. The VaR estimation is validated by several backtests and compared to the results of other classical models on a base of 56 stock assets. In the second part, we assume that stock prices are modeled by exponential Lévy processes. First, we develop a numerical method for computing the cumulative VaR and CVaR. This problem is solved using the formalization of Rockafellar and Uryasev, which we evaluate numerically by Fourier inversion. In a second step, we focus on minimizing the hedging risk of European options, under a budget constraint on the initial capital. By measuring this risk by the CVaR, we establish an equivalence between this problem and a Neyman-Pearson type problem, for which we propose a numerical approximation based on the relaxation of the constraint.