# Portfolio Theory

### University of Amsterdam, 2023-2024

Syllabus
• Week 1: Modelling Financial Markets
• Content:
• Course introduction, history and overview of research trends in mathematical finance (not relevant for the exam)
• Section 1.1: Basic Stochastic Concepts of Mathematical Finance
(up to Assumption 1.5 of course lecture notes)
• Assignments:
• Week 2: Arbitrage
• Content:
• Section 1.2: Portfolios and Trading Strategies
• Section 1.3: Arbitrage and Equivalent Martingale Measures
• Assignments:
• Exercise Set 2
• Read Theorem 8.9 (the so called Doob system theorem) and its proof in Peter's lecture notes (note the different notation)!
• Week 3: Pricing and Hedging
• Content:
• First Fundamental Theorem of Asset Pricing (Thm 1.18) with proof
• Section 2.1 up to Theorem 2.8
• Assignments:
• Week 4: Complete Markets
• Content:
• Section 2.1: Proof of Theorem 2.8 and Theorem 2.9
• Section 2.2 up to Proposition 2.13
• Assignments:
• Week 5: The complete CRR Model and the incomplete market setting
• Content:
• Section 2.3: Pricing and Hedging in the CRR Model
• Section 3: Introduction to the incomplete market setting
• Assignments:
• Week 6: Preferences and Lotteries
• Content:
• Week 7: Expected Utility Theory
• Content:
• Week 8: Utility Optimal Portfolios
• Content:
• Section 5: Static Portfolio Optimization
• Section 6.1.2: Utility Optimal Strategies and Utility Indifferent Pricing and Hedging with CRRA
• Assignments:
• Week 9: CARA Utility Indifferent Pricing and Hedging
• Content:
• Section 6.2.1: Utility Indifferent Pricing and Hedging with CARA
• Motivating Section 6.3: Variance-optimal Hedging
• Assignments:
• Read the proof of Theorem 6.11
• Week 10: Variance-optimal Hedging
• Content:
• Section 6.3: Variance-optimal Hedging
• Section 4.1: Monotonicity, Cash Invariance, Convexity and Coherence
• Assignments:
• Week 11: Risk Measures
• Content:
• Section 4.2: The Acceptance Sets of Risk Measures
• Section 4.3: Examples for risk measures
• Section 4.4: Utility-based shortfall risk
• Section 6.4: Efficient Hedging with Risk Measures
• Assignments:
• Week 12: Portfolio Optimal Control
• Content:
• Section 7.1: Optimal Consumption-Investment Problems
• Section 7.2: Optimal Control Problems
• Assignments:
• Week 13: Optimal Consumption-Investment and Martingale Method
• Content:
• Section 7.3.1-7.3.3: Optimal Consumption-Investment Problems with DP
• Section 7.4: The Martingale Method
• Assignments:
• Week 14: Q&A Session
• Infos:
Course Information

### Contents

In this course we treat the foundations of mathematical finance, its fundamental concepts like arbitrage, equivalent martingale measures, and fundamental economic notions as preference relations and utility functions and show how these are applied in portfolio optimization. This will be done first for static markets and extended later on to a dynamic setting, where time is discrete. Finally we will show how stochastic control theory and dynamic programming can be applied to problems of portfolio optimization.

### Aim and objectives

The general aim is to make students familiar with the mathematical foundations of mathematical finance in discrete time and the fundamentals of portfolio selection. In particular, students will able to prove a number of well selected theorems and demonstrate in assignments that they master the theory.

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

• Students are familiar with fundamental concepts of financial mathematics and know how to apply them.
• Students know the theory behind portfolio optimization and know how to solve such optimisation problems.
• Students are able to optimize under order restrictions.
• Students know results about dynamic arbitrage theory and completeness in multi-period (discrete time) models.
• 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. Prospective students should be conversant with fundamental concepts of probability and measure theory, as covered in the "Measure Theoretic Probability" course.

### Literature

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

The course is mainly based on the 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 a week of being assigned.
• Collaborative work in pairs is compulsory, barring exceptional circumstances.
• 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 2023.