- Informations scientifiques
- Informations pratiques
In this presentation, we study theoretical and computational aspects of risk minimization in financial market models operating in discrete time. To define the risk, we consider a class of convex risk measures defined in terms of shortfall risk. Under simple assumptions, namely the absence of arbitrage opportunity and the non-degeneracy of the price process, we prove the existence of an optimal strategy by performing a dynamic programming argument in a non-Markovian framework. Optimal strategies are shown to satisfy a first order condition involving the constructed Bellman functions.
In a Markovian framework, we propose and analyze algorithms to estimate the shortfall risk and optimal dynamic strategies based on several numerical probabilistic tools: Stochastic Approximation algorithm, optimal vector quantization, Newton-Raphson's optimization algorithm. Finally, we illustrate our approach by considering several shortfall risk measures and portfolios inspired by energy and financial markets.