The aim of these lectures at MITACS-PIMS-UBC Summer School in Risk Man- agement and Risk Sharing is to discuss risk controlled approaches for the pricing and hedging of financial risks. We will start with the classical dual approach for financial markets, which al- lows to rewrite super-hedging problems in terms of optimal control problems in standard form. Based on this, we shall then consider hedging and pricing prob- lems under utility or risk minimization criteria. This approach will turn out to be powerful whenever linear (or essentially linear) problems are considered, but not adapted to more general settings with non-linear dynamics (e.g. large investor models, high frequency trading with market impact features, mixed finance/insurance issues). In the second part of this lecture, we will develop on a new approach for risk control problems based on a stochastic target formulation. We will see how flexible this approach is and how it allows to characterize very easily super- hedging prices in term of suitable Hamilton-Jacobi-Bellman type partial differ- ential equations (PDEs). We will then see how quantile hedging and expected loss pricing problems can be embeded into this framework, for a very large class of financial models. We shall finally consider a simple example of optimal book liquidation in which the control is a continuous non-decreasing process, as an illustration of possible practical developments in optimal trading under risk constraint.These lectures are organized in small chapters, each of them being focused on a particular aspect.
