Introduction to convex optimization in financial markets

Convexity arises quite naturally in financial risk management. In risk preferences concerning random cash-flows, convexity corresponds to the fundamental diversification principle. Convexity is a basic property also of budget constraints both in classical linear models as well as in more realistic models with transaction costs and constraints. Moreover, modern securities markets are based on trading protocols that result in convex trading costs. The first part of this paper gives an introduction to certain basic concepts and principles of financial risk management in simple optimization terms. The second part reviews some convex optimization techniques used in mathematical and numerical analysis of financial optimization problems.