### Contents

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.

### Aim and 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.

### Specific objectives to be met at the end of the course:

- 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.

### Prerequisites

"Portfolio Theory" is generally structured as a second-year course in the Master's programmes of Mathematics and Stochastics and Financial Mathematics at the University of Amsterdam. Students should be familiar with fundamental concepts of probability and measure theory, as covered in the "Measure Theoretic Probability" course.

### Literature

A set of lecture notes will be the main source for this course.

Recommended book: Stochastic Finance - An Introduction in Discrete Time by H. FĂ¶llmer and A. Schied.

### Examination Details

**Assessment Components:**Mandatory take-home exercises and an oral examination.**Homework:**- All submissions must be made within two weeks of being assigned.
- Collaborative work in pairs is compulsory.
- Submit homework via the course's Canvas page.

**Oral Examination:**- Assesses comprehension of key definitions, results (like lemmas and theorems), and four chosen theorems with their proofs.
- Scheduling will be available by early December 2024.

**Final Grades:**- 70% - Oral examination
- 30% - Assignments