MA575 Optimization Models in Quantitative Finance

Course Catalog Description


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.


Professor Email Office
Darinka Dentcheva Peirce 302 (Tuesdays 4:30–6:00 pm OR by appointment)

More Information

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.

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.

Lecture Outline

Date Topic Reading
Aug 29 Review of linear programming optimality conditions and duality; Non-arbitrage conditions and state probabilities.
Sep 5 Matrix games; Cash matching problems; dedication.
Sep 12 Bond portfolio duration and immunization; Logical bounds and knapsack constraints, combinatorial optimization problems.
Sep 19 Index funds and the use of combinatorial techniques; Combinatorial auction.
Sep 26 Non-linear optimization. Review of optimality conditions and duality; Utility models. The concept of risk.
Oct 3 Mean-Variance models; Two-fund and one-fund theorems.
Oct 10 Value at risk; Conditional value at risk.
Oct 17 Midterm Test
Oct 24 Coherent measures of risk. Dual representation; Optimization with coherent measures of risk.
Oct 31 Stochastic order relation; Relations to utility theories and measures of risk.
Nov 7 Sequential decision making for Markov models; Belman principle and dynamic programming equation.
Nov 14 Dynamic programming algorithms; Risk-neutral option pricing.
Nov 21 Dynamic optimization for non-Markov models; Multi-stage optimization and asset-liability management.
Dec 5 Dynamic measures of risk. Time consistency; Risk-averse dynamic portfolio optimization.