Abstract

The aim of this course is to study several aspects of risk measures particularly in the context of financial applications. The primary framework that we use is that of coherent risk measures. We study the problem of pricing financial positions with coherent risk measures and accordingly we introduce a Good Deal as a natural alternative to Arbitrage in this framework. We also compare coherent risk measures with Value at Risk. This is to show the practical advantages and disadvantages of using coherent risk measures against Value at Risk. Assessing financial risks is an ever present concern in economics and mathematical finance. The mathematical framework that incorporates a quantifiable financial risk was originally defined in terms of the language of von Neumann-Morgenstern expected utility theory, i.e., at the individual level, risk has always been characterized in terms of preference relations. Yet, from a risk management perspective, profits or losses are what define and quantify risk. The groundbreaking work of Artzner, Delbaen, Eber and Heath (1999) is the cornerstone of a sound mathematical theory of risk measures that is compatible with risk management applications. They introduce the notion of risk measure as a real-valued function that assigns a meaningful numerical value to any given financial model. Their construction is axiomatic and it allows for a rich mathematical theory with room for practical applications. In fact, many examples of axiomatic risk measures are readily applied in practice and appear naturally in mathematical finance. Nowadays, risk measures have found their place as a relevant field in financial mathematics. The theory of risk measures is built with tools from well-developed fields of mathematics like probability and convex analysis. One element behind this success is that, although the axiomatic construction of these objects is dictated by the mathematical tools behind the theory, these also respond to financial intuition and needs. These axiomatic risk measures have mathematical representations that, far from being mere artifacts, have economical meaning. This brings new insight into the discussion.