Introduction to Mathematical Finance Spring 2020

Dr. Christoph Czichowsky
Bálint Gersey

The recordings of the lectures are available here .

Lectures Mondays 13-15 HG D 1.2
Thursdays 15-17 HG E 33.3
Exercice classes Wednesdays 14-15 --> Bálint Gersey HG D 3.1
Wednesdays 15-16 --> Zhouyi Tan HG D 3.1

This is an introductory course on the mathematics for investment, hedging, portfolio management, asset pricing and financial derivatives in discrete-time financial markets. We discuss arbitrage, completeness, risk-neutral pricing and utility maximisation. We prove the fundamental theorem of asset pricing and the hedging duality theorems, and also study convex duality in utility maximization.

A related course is 401-3913-01L Mathematical Foundations for Finance (3V+2U, 4 ECTS credits). Although both courses can be taken independently of each other, only one will be recognised for credits in the Bachelor and Master degree. In other words, it is not allowed to earn credit points with one for the Bachelor and with the other for the Master degree.


A knowledge of measure-theoretic probability theory (as taught e.g. in the course "Probability Theory").
This course is the first of a sequence of two courses on mathematical finance. The second course "Mathematical Finance" (MF II), 401-4889-00, focuses on continuous-time models. It is advisable that the present course, MF I, is taken prior to MF II. For an overview of courses offered in the area of mathematical finance, see link.

Office hours

The assistants of Group 3 (Probability Theory, Insurance Mathematics and Stochastic Finance) offer regular office hours for questions on courses and exercises taught by the professors in the group. During the semester, the assistant hours take place Mondays and Thursdays, 12:00–13:00 in HG G 32.6.

Lecture notes

Lecture notes covering the material taught two years ago are available in the Polybox folder. The link to the Polybox folder has been sent to you by mail. You will receive the password from the lecturer. Please note that these are unofficial notes, and in particular, the author has no responsibility for potential typos or mistakes. If you find any typos or mistakes, we would be very grateful if you could let us know by mail.

The exercise sheets will be posted before the lecture.

Exercise Sheet Due Date Solutions Solutions with questions Topics covered Related chapters
Exercise sheet 1 28/02/2020 Solution 1 Questions + Solutions 1 Basic concepts in one-period model: assets, agents, consumption, budget set, preference order and numerical representation, arbitrage of first and second kind Chapter I.1-I.3
Exercise sheet 2 06/03/2020 Solution 2 Questions + Solutions 2 Basic concepts in one-period models (continued): numerical representation of preference orders, binomial and trinomial markets Chapter I.1-I.3
Exercise sheet 3; Facebook; S&P500; 13/03/2020 Solution 3 Questions + Solutions 3 CAPM.r Price systems and attainability, complete markets, hedging duality, CAPM Chapter I.4-I.7
Exercise sheet 4 20/03/2020 Solution 4 Questions + Solutions 4 Seller's/buyer's price (super/sub-replicating prices), arbitrage of first and second kind, put-call parity Chapter I.3 and I.7
Exercise sheet 5 27/03/2020 Solution 5 Questions + Solutions 5 Basic properties of stopping times and corresponding sigma-algebras, multiplicative models, self-financing property + invariance under change of numeraire, properties of the call price surface Chapter II.1-II.4
Exercise sheet 6 stylized_facts.R 03/04/2020 Solution 6 Questions + Solutions 6 Empirical properties of financial data and GARCH models, change of numeriaire, martingales in multiplicative models, admissibility and arbitrage, dividend paying stocks Chapter II.1-II.4
Exercise sheet 7 24/04/2020 Solution 7 Questions + Solutions 7 Doubling strategy, density processes, local martingales bounded from below are supermartingales, change of nemeraire formula, alternative proof of Dalang Morton Willinger Theorem Chapter II.1-II.4
Exercise sheet 8 04/05/2020 Solution 8 Questions + Solutions 8 Trinomial market, EMMs, essential supremum, pasting of measures Chapter II.1, II.2, II.6
Exercise sheet 9 08/05/2020 Solution 9 Questions + Solutions 9 Backwards induction, Snell envelope, Doob decomposition, Asian and European Call options Review of Chapter II + introduction to backwards induction arguments which will be very useful in Chapter III (Dynamic Programming Principle)
Exercise sheet 10 15/05/2020 Solution 10 Questions + Solutions 10 Martingale Optimality Principle, Dynamic Programming Principle Chapter III.2
Exercise sheet 11 22/05/2020 Solution 11 Questions + Solutions 11 Dynamic Programming, Bellman's equation for controlled dynamic systems in discrete time Chapter III.2
Exercise sheet 12 29/05/2020 Solution 12 Questions + Solutions 12 Utility from terminal wealth and duality Chapter IV.1-IV.4


Separation theorems for convex sets in $\mathbb{R}^d$ Separation theorems
Optimisation under constraints, the Kuhn–Tucker theorem Kuhn-Tucker
Martingales and stochastic integrals in discrete time Martingale-results
The Kreps–Yan theorem Kreps–Yan
The essential supremum Ess-sup
The Komlós-type lemma Komlos
Main reference for Chapter IV D. Kramkov and W. Schachermayer, The asymptotic elasticity of utility functions and optimal investment in incomplete markets, Annals of Applied Probability 9 (1999), 904-950