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.
||Peirce 302 (Tuesdays 4:30–6:00 pm OR by appointment)
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
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
9. Can formulate two-stage risk-averse portfolio optimization problem.
MA 230 Multivariate analysis and Optimization; MA 222 Probability and statistics
Main reference Lecture Notes distributed in class.
Supplementary references (not required):
• D.G. Luenberger, Investment Science, Oxford University Press, New York
• G. Cornuejols, R. Tutuncu, Optimization Methods in Finance, Cambridge University Press, 2007.
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
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.
||Review of linear programming optimality conditions and duality;
Non-arbitrage conditions and state probabilities.
| Sep 5
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;
| Sep 26
||Non-linear optimization. Review of optimality conditions and duality;
Utility models. The concept of risk.
| Oct 3
Two-fund and one-fund theorems.
| Oct 10
||Value at risk;
Conditional value at risk.
| Oct 17
| 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.