This semester, I'm teaching a graduate course on stochastic programming targeted towards financial engineering students. The syllabus is fairly standard (I'm using the book of Birge and Louveaux, Introduction to Stochastic Programming). As computational aspects are very important, I intend to define a course project in which students implement a classical solution method for a stochastic or robust financial problem. This practice problem should be easy to understand and require a modest theoretical effort as the focus in on the implementation aspects (we will use GAMS).

Right now, I've benders decomposition in mind for the solution method (as it's is the basis of many methods in stochastic programming), but I'm looking for a good practice problem. My own search has led me to the portfolio optimization problem with cardinality constraints (the maximum number of holdings is limited). This could be solved by the generalized benders decomposition.

I would really appreciate it if you could point out other possible candidates for the practice problem. While I prefer benders decomposition as the selected solution method, I'm open to consider other methods. Please note that the practice problem could be rooted in either stochastic programming or robust optimization techniques as I'm covering both.

Thanks in advance for your sharing your expertise and advice.

asked 02 Feb '16, 00:33

Ehsan's gravatar image

Ehsan ♦
accept rate: 16%

edited 02 Feb '16, 16:49

I am assuming that you are looking for theoretically simple and practical examples.

1) Portfolio Problem : it is a old and classical problem. When there are N number of stocks in the market, and you have a fixed amount of money. Stock prices are random, and the transition probability can be estimated by brownian motion model. Let's assume there is no transaction cost. And every morning you observe the prices of all the stocks, and you can sell and buy so that you can change the portfolio everyday. Theoretically very simple. Since we can assume all the money will be invested into the stock market, decision variable will be the proportion of the budget for each stock.

2) Bermudan Option Pricing: Bermuda is located between Europe and America. This option is in-between European Option and American Option. The owner can exercise the option only at multiple but specific time points. Discrete time version of American option. But, If you are not familiar with Option pricing, it might be a burden to learn option pricing.

Since Stochastic Programming is not good for multiple stage (large number of stage) problems, you may want to limit the number of stages.

I believe these two problems (Portfolio and Option pricing) are the most famous and widely studied questions in financial engineering. Of cause, What I mentioned above are very simple versions.


answered 12 Feb '16, 20:06

ksphil's gravatar image

accept rate: 14%

I think it is a nice exercise in graduate school to verify results from important papers in your field. Looking back to grad school, I was interested in robust optimization and inventory control, so early on I looked at the following papers and verified their results using GAMS. I found them to be pretty straightforward to code, so they may be good choices. The second is a Benders' approach, and that paper really helped me understand the issue of overconservatism for some RO models.

[1] Ben-Tal, A., Golany, B., Nemirovski, A., & Vial, J. P. (2005). Retailer-supplier flexible commitments contracts: a robust optimization approach. Manufacturing & Service Operations Management, 7(3), 248-271.

[2] Bienstock, D., & ÖZbay, N. (2008). Computing robust basestock levels. Discrete Optimization, 5(2), 389-414.


answered 02 Feb '16, 14:04

Andreas's gravatar image

accept rate: 12%

Thanks for your suggestions. The first paper is a good one, which I've used in my own research. The second one was new for me, but it seems interesting. Although, I'm more interested in financial optimization problems for the sake of the course.

Your suggestion about using classic papers is right on point. However, as I'm new to the field, this might take some time. Also, I haven't had a good luck there. For example, the most important paper on using BD to solve the asset-liability management problem, to my surprise, does not have any detail on deriving master problem and subproblems.

(02 Feb '16, 15:07) Ehsan ♦

Yes, sorry for listing several papers out of context. Maybe something in chapters 16, 17, 18, or 20 of the following would be useful.

Cornuejols, G., & Tütüncü, R. (2006). Optimization methods in finance (Vol. 5). Cambridge University Press.

(02 Feb '16, 15:25) Andreas

No problem. Also, thanks for the wonderful reference. I just checked it at the library. Although, there is not enough theoretical details in the book, however there are plenty of applied problems with their original references.

(03 Feb '16, 03:10) Ehsan ♦
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Asked: 02 Feb '16, 00:33

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Last updated: 24 Apr '16, 09:14

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