Optimization with Applications in Portfolio Choice

  • Summer schools
  • Department of Mathematics
  • Application code SS-ME201
  • Starting TBC
  • Short course: Closed
  • Location: Houghton Street, London

Please note: This course will not be running as part of the 2021 programme. However, you may be interested in our confirmed courses. 

The aim of this course is to impart knowledge and develop understanding of optimization theory. Although emphasis is placed on problems of optimal investment and consumption, the techniques developed are applicable to problems arising in economics and several other areas.

What is the best composition of assets in a portfolio? What is the minimal amount of fuel needed to launch a spacecraft to orbit? How much daily exercise is optimal? These seemingly unrelated questions have one thing in common: to answer them one has to optimize. Optimization permeates our daily life, and the ability to act optimally distinguishes success from failure. This is especially true in the world of finance, where bankruptcy may be the result of wrong investment decisions.

The course studies several approaches to solving constrained and unconstrained static as well as dynamic optimization problems. The theory covered is exemplified by applications such as the Markowitz portfolio selection problem and the Merton optimal investment problem. To introduce the optimal investment problem, the multi-period binomial tree model for a financial market is introduced and the concepts of arbitrage and self-financing portfolios are discussed.

The course is relevant to students pursuing a degree in a quantitative subject who are interested in finance, as well as to students majoring in finance or economics who want to hone their quantitative skills. The course is largely self-contained and relevant mathematical background, such as elementary probability theory and properties of functions, is thoroughly reviewed. 


Session: TBC
Dates: TBC
Lecturers:  Dr Albina Danilova and Professor Mihail Zervos


Programme details

Key facts

Level: 200 level. Read more information on levels in our FAQs

Fees:  Please see Fees and payments

Lectures: 36 hours 

Classes: 18 hours

Assessment*: Two written examinations

Typical credit**: 3-4 credits (US) 7.5 ECTS points (EU)

*Assessment is optional

**You will need to check with your home institution

For more information on exams and credit, read Teaching and assessment


Multivariate calculus and linear algebra at lower undergraduate level.

Programme structure

  • Unconstrained optimization
  • Numerical methods for finding a local minimum or maximum
  • Optimization with equality constraints - Lagrange theorem
  • Numerical methods for solving a constrained optimization problem
  • The one-period Markowitz portfolio selection problem
  • Bellman's dynamic programming principle
  • Multi-period binomial tree model for a financial market
  • Arbitrage and self-financing portfolios
  • Agent preferences and the Merton optimal investment problem

Course outcomes

Students learn how to approach a wide range of optimization problems both theoretically and numerically. They also get exposure to financial modelling and basic financial concepts, such as arbitrage and self-financing portfolios. 


The LSE Department of Mathematics is internationally recognised for its teaching and research. Located within a world-class social science institution, the department aims to be a leading centre for Mathematics in the Social Sciences. The Department of Mathematics was submitted jointly to REF 2014 with LSE's Department of Statistics: 84% of the research outputs of the two departments were classed as either world-leading or internationally excellent in terms of originality, significance and rigour.

The Department has more than doubled in size over the past few years, and this growth trajectory reflects the increasing impact that mathematical theory and mathematical techniques are having on subjects such as economics and finance, and on many other areas of the Social Sciences.

On this three week intensive programme, you will engage with and learn from full-time lecturers from the LSE’s mathematics faculty.

Reading materials

-   D. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods, Athena Scientific, 1996. [recommended]

-   J. E. Ingersoll, Theory of Financial Decision Making, Rowman & Littlefield, 1987. [recommended]

-   R. K. Sundaram, A First Course in Optimization Theory, CUP, 1996. [required]

*A more detailed reading list will be supplied prior to the start of the programme

**Course content, faculty and dates may be subject to change without prior notice

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