Portfolio Theory

Course manual 2024/2025

Course content

In this course, we delve into the stochastic foundations of mathematical finance, with a focus on the general problem of pricing and hedging European contingent claims and portfolio optimization. We rigorously define the concept of a financial market and establish two fundamental relationships: one between arbitrage-free markets and the existence of equivalent martingale measures, and the other concerning the completeness of a market and the uniqueness of the equivalent martingale measure. We explore various methodologies for pricing and hedging contingent claims in both complete and incomplete markets, including risk-neutral, variance-optimal, and utility and risk-indifferent approaches. We also examine how market participants' preferences can be quantified using utility funcitons and risk measures, and how these concepts are applied in portfolio optimization. Furthermore, we discuss the formulation of consumption-investment problems as stochastic optimal control problems and introduce dynamic programming as a general solution method. This course provides students with a thorough understanding of the core principles of mathematical finance and their application in financial decision-making and risk management under uncertainty

Study materials

Literature

• Lecture notes   (download)

Syllabus

• All relevant information on the webpage

Objectives

• The general objective of this course is to equip students with a deep understanding of the stochastic foundations of mathematical finance in a finite discrete-time setting, with a particular focus on portfolio selection and optimization techniques. Specifically, students will: Develop the ability to rigorously prove a carefully chosen set of theorems that are central to mathematical finance. Demonstrate mastery of the theory through practical assignments, where they will apply theoretical concepts to solve problems related to the pricing and hedging of European contingent claims or portfolio optimization.
• Students will gain a solid understanding of the core principles of financial mathematics. In particular, students know the two fundamental theorems of asset pricing; know how to price European contingent claims using risk-neutral pricing and are familiar with the differences between complete and incomplete markets.
• Students will acquire a comprehensive understanding of the pricing and hedging problem in an incomplete market setting and know strategies to address and solve pricing and hedging problems in this context.
• Students will develop a thorough understanding of expected utility theory and risk measures, enabling them to apply these concepts effectively for optimizing portfolios
• Students know how to construct variance optimal hedging strategies.
• Students know how to apply dynamic programming to investment-consumption problems.

Teaching methods

• Lecture
• Excercises

In this course, lectures serve as the foundation, introducing students to the core principles, methods, and proofs in stochastic finance within a discrete time framework. Lectures also provide examples for immediate application of theoretical concepts. Practical training through biweekly exercise sets reinforces theoretical understanding while building problem-solving skills essential for applications in stochastic finance.

Learning activities

Attendance

This programme does not have requirements concerning attendance (TER-B).

Assessment

The grade for the exercises is the average of the biweekly grades, this grade counts for 30% of the final grade. The other 70% is the grade for the oral exam. The weekly assignments have to be made in pairs, this is mandatory!

During the oral exam we review the content of the course globally. The student is asked to study his/her choice of four theorems with their proofs in detail. If one fails the first attempt then we reschedule for a resit. The exercises also contribute to the final grade after the resit.

Assignments

Biweekly assignments consisting of self-study modules and exercises to be solved and handed in (in groups of two). The exercise assignments are graded and a short feedback is given individually on each assignments.

Fraud and plagiarism

The 'Regulations governing fraud and plagiarism for UvA students' applies to this course. This will be monitored carefully. Upon suspicion of fraud or plagiarism the Examinations Board of the programme will be informed. For the 'Regulations governing fraud and plagiarism for UvA students' see: www.student.uva.nl

Course structure

Additional information

Recommended prior knowledge: Measure Theoretic Probability.

Contact information

Coordinator

• dr. Sven Karbach

Staff

• Konstantinos Chatziandreou