Contents
Stochastic programming applied to finance and economics problems. Mathematics content: linear and nonlinear stochastic programming, dynamic programming. Discrete and continous time models. Context and applications: economics analysis, portfolio theory and financial economics.
This course is divided in the following 4 segments.
1) Long Introduction. Dynamic optimization in finance and economics, deterministic and stochastic. Review of deterministic programming in dynamic contexts. Introduction to linear stochastic dynamic programming theory. Feasibility of constraints, nonanticipativity, recourse, two-stage and multistage programs. Probabilistic constraints. Scenarios.
2) Optimal Portfolio Theory. Markowiz mean-variance theory. Two-stage mean-risk models. Portfolio stochastic multistage linear models. Discussion of reference examples (to be solved during the next segment): asset allocation and index funds (deterministic integer programming), CVaR (two-stage stochastic programming and probabilistic constraints), asset/liability management (multistage stochastic programming).
3) Dynamic Economics and Business Cycles. Reference example: optimal consumption/investment choice. Dynamic programming, deterministic and stochastic. Main solution approaches: Lagrange/Euler and Bellman. Analytic and numerical solution of the reference example. Relationship of dynamic programming with control theory.
4) Formal Theory. Constrained deterministic linear and nonlinear, integer and mixed programming (including Lagrangian techniques) in dynamic contexts, solution algorithms and decomposition (Benders, Dantzig-Wolfe) techniques. Stochastic programming and recourse, two-stage and multistage, linear and nonlinear. Recourse function. Probabilistic constraints. Duality. Relaxation. Stochastic dual dynamic programming. Importance of information indicators: EVPI and VSS. Numerical solution techniques.
Objectives
D1 – KNOWLEDGE AND UNDERSTANDING
After the end of the course, the student should be able to
1 Define and identify the structure of a stochastic program.
2 Discuss the role of time in a dynamic stochastic program.
3 Recognize and describe the role of uncertainty and time in the constraints of a dynamic stochastic program.
4 Recognize the specific features of dynamic programming in a stochastic programming context.
5 Review and discuss the main solution techniques in linear and nonlinear stochastic programming.
6 Review and discuss the main solution techniques in dynamic programming.
7 Outline and give examples of portfolio theory and asset/liability management models.
8 Outline and discuss dynamic programming in the context of dynamic optimal consumption theory.
D2 – APPLYING KNOWLEDGE AND UNDERSTANDING
After the end of the course, the student should be able to
1 Formulate an optimization program for a choice of risky portfolios, and propose and discuss an algorithm for its solution.
2 Derive the Bellman equation from first principles, and relate it to the general stochastic programming theory.
3 Generalize a deterministic program to a stochastic setting.
4 Propose and evaluate different solution schemes for a given stochastic program.
5 Propose a real word situation, not studied in the course, and associate to it a suitable stochastic program and its solution.
D3 – MAKING JUDGMENTS
After the end of the course, the student should be able to
1 Evaluate different stochastic program types for one real world situation proposed by the instructor.
D4 – COMMUNICATION SKILLS
D5 – LEARNING SKILLS
After the end of the course, the student should be able to
1 Locate, read and understand research literature on stochastic programming and dynamic programming.
Prerequisites
Required: Applied Mathematics 3.
Suggested courses: Probability and Stochastic Processes, Mathematical Analysis 4.
Teaching Methods
Lectures. Use of handout slides written by the instructor. Class exercises. Use of Matlab, Lingo, MPL, and/or other software platforms/programs to implement solution algorithms.
Verification of learning
M0 – Participatory class discussions, stimulated by questions posed at the end of the preceding class. M1 - Class exercises. M2 - Individual oral exam, also in written form, in front of the instructor.
Specifically, the learning outcomes D1, D2 and D3 will be assessed by M1 and M2. D5 will be assessed by M0.
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Exam sessions will be held in the no-teaching window between the first and the second semester, at the end of the second semester, and after Summer vacations in September and at the beginning of October. No exams during the teaching periods.
Texts
J.R.Birge, F. Louveaux, Introduction to Stochastic Programming, Springer. G Cornuejols, R. Tütüncü, Optimization Methods in Finance, Cambridge. F.J. Fabozzi, P.N. Kolm, D.A. Pachamanova, S. M. Focardi, Robust Portfolio Optimization and Management, J. Wiley & Sons. B. Heer, A. Maußner, Dynamic General Equilibrium Modelling, Springer. Slide handouts.
