Shortfall Risk Minimization in Discrete Time Financial Market Models

Abstract

In this paper, 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 on $L^{p}(\mathbb{P})$ in terms of shortfall risk. Under mild assumptions, namely, the absence of arbitrage opportunity and the nondegeneracy of the price process, we prove the existence of an optimal strategy by performing a dynamic programming argument in a non-Markovian framework. In a Markovian framework, the shortfall risk and optimal dynamic strategies are estimated using three main tools: Newton--Raphson Monte Carlo--based procedure, stochastic approximation algorithm, and Markovian quantization scheme. Finally, we illustrate our approach by considering several shortfall risk measures and portfolios inspired by energy and financial markets.