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 $L^{p}$ 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 several algorithms based on Monte Carlo simulations to estimate the shortfall risk and optimal dynamic strategies. Finally, we illustrate our approach by considering several shortfall risk measures and portfolios inspired by energy and financial markets.