In this lecture we give an introduction to important tools in optimization and convex analysis that are needed in various parts of quantitative  finance.

Part one of the lecture is devoted to unconstrained optimization problems. Part two is devoted to the solution of constrained optimization problems via Lagrange multiplier theory and methods based on calculus. The third part of the lecture deals with convex analysis and duality theory for convex optimization problems.

In order to illustrate the methods we will study several applications in economics and finance including Markowitz portfolio optimization, optimal production plans, portfolio optimization via expected utility maximization and cost-minimal superreplication.

After completing this course the student will have the ability to:

• understand and interpret classic models in financial economics that build on an optimization argument;
• construct economic models that imply an optimizing decision maker and perform analytical and /or numerical analysis.
• communicate and discuss possible approaches to a certain problem in class;
• work in groups and contribute to the implementation of economic optimization models. Defend the chosen approach in class.
• apply methods of static and dynamic optimization to questions arising in financial economics;

Full attendance is compulsory. This means that students should attend at least 80% of all lectures, at most one lecture can be missed.

• written final exam (weight: 60%)
• 2 minitests  (weight of each test 15%)
• project (weight 10%)

• Basic knowledge of Analysis and linear Algebra as in Mathematics I
  
• An understanding of derivative pricing in one-period financial models will be helpful
  

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