During the financial and economic crisis of 2008, it was noticed that the amount of capital required for banks' trading portfolio was significantly less than the real losses. To understand the causes of this low capital requirement, it seemed important to estimate the reliability of the market risk models and to propose stress testing methodologies for the management of extreme risks. The objective is to measure the capital requirement on a trading portfolio, composed of shares and commodities by the measure of the Value at Risk (VaR) and Expected Shortfall. To achieve this goal, we use the Generalized Pareto Distribution (GPD) and two internal models commonly used by banks: historical simulation method and model of the normal law. A first evaluation of the reliability made on the three risk models under the hypothesis of constant volatility, shows that the internal banks' models and the GPD model do not measure correctly the risk of the portfolio during the crisis periods. However, GPD model is reliable in periods of low volatility but with a strong overestimation of the real risk; it can lead banks to block more capital requirement than necessary. A second evaluation of the reliability of the risk models was made under the hypothesis of the change of the volatility and by considering the asymmetric effect of the financial returns. GPD model is the most reliable of all, irrespective of market conditions. The performance of the internal banks' risk models improves when considering the change of the volatility. The integration of the historic and hypothetical scenarios in the risk models, improves the estimation of the extreme risk, while decreasing the subjectivity blamed to the stress testing techniques. The stress testing realized with the internal models of banks does not allow a correct measure of the extreme risk. GPD model is better adapted for the stress testing techniques. We developed an algorithm of stress testing which allow banks to estimate the extreme risk of their portfolios and to identify the risk factors causing this risk. The calculation of the capital requirement based on the sum of the VaR and the stress VaR is not logical and leads to doubling the capital requirement of banks. Consequently, it conducts to a credit crunch in the economy. We observe that the multiplier coefficient and the principle of square root of time of the Basel's agreement lead banks to make arbitration in favor of risk models that are not reliable.
