We investigate a method based on risk minimization to hedge observable but non-tradable source of risk on financial or energy markets. The optimal portfolio strategy is obtained by minimizing dynamically the Conditional Value-at-Risk (CVaR) using three main tools: a stochastic approximation algorithm, optimal quantization and variance reduction techniques (importance sampling (IS) and linear control variable (LCV)) as the quantities of interest are naturally related to rare events. We illustrate our approach by considering several portfolios in connection with energy markets.

Keywords : VaR, CVaR, Stochastic Approximation, Robbins-Monro algorithm, Quantification[-]

Non-convex random sets of admissible positions naturally arise in the setting of fixed transaction costs or when only a finite range of possible transactions is considered. The talk defines set-valued risk measures in such cases and explores the situations when they return convex result, namely, when Lyapunov's theorem applies. The case of fixed transaction costs is analysed in greater details.
Joint work with Andreas Haier (FINMA, Switzerland).[-]