To include models of financial decisions under uncertainty we need to extend the basic theory incorporating adequate objects. Utility Theory is a first approximation. The basic objects are lotteries that represent situations of choice under risk. A set of axioms for a theory of choice under risk is presented. The results of such a theory can be easily understood by their geometrical meanings. The financial applications require a generalization of that theory and some additional assumptions. They are presented in both intuitive and semiformal way. A simple but powerful model is presented. It encompasses the main features of more advanced ones