Objective

This course introduces the students to mathematical models and computational methods for static and dynamic optimization problems occurring in finance. We shall discuss linear and non-linear optimization models of finance, dynamic (se- quential) optimization, optimization under uncertainty, mathematical models of risk and their application. Additionally, duality theory and its use in economics and finance will be stresses. The students will be familiarized with concept of risk and risk-aversion. The models involve knowledge of probability, optimality conditions, duality, and basic numerical methods. Special attention will be paid to portfolio optimization and to risk management problems.

Course Outcomes

After successful completion of the class, the students should be able to:
1. Formulate optimization problems associated with various problems in quan- titative finance such as dedication problems, immunized bond portfolio model, portfolio optimization using mean-variance models or coherent measures of risk and/or risk constraints, tracking an index, etc.
2. Interpret the economic meaning of dual variables and use it in sensitivity analysis.
3. Use typical modeling techniques of combinatorial optimization such as log- ical bounds and constraints.
4. Understand the concept of risk and be able to formulate and apply several mathematical models of risk based on utility functions, coherent measures of risk, and risk-constraints.
5. Calculate the efficient frontier determined by a mean-risk model; use the one-fund and two-fund theorems.
6. Use the concept of stochastic orders, be aware of their relation to risk mea- sures and utility functions.
7. Formulate finite-horizon dynamic optimization problems based on Markov and non- Markov discrete time processes.
8. Apply stochastic optimization methods for option pricing and for asset/liability management.
9. Can formulate two-stage risk-averse portfolio optimization problem.

Prerequisites:
MA 230 Multivariate analysis and Optimization; MA 222 Probability and statistics or equivalent.

Main reference Lecture Notes distributed in class.
Supplementary references (not required):
• D.G. Luenberger, Investment Science, Oxford University Press, New York 1998.
• G. Cornuejols, R. Tutuncu, Optimization Methods in Finance, Cambridge University Press, 2007.

Grading Policies

Seven or eight homework assignments will be given. In addition, a mid-term and a final exam will be conducted. The final grade will be based on the following score:
0.35 × Homework + 0.3 × Midterm Exam + 0.35 × Final Exam

The syllabus is subject to change. All announcements about this class will be posted on Canvas.