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401-3913-00L 5 Credits
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Mathematical Foundations of Finance

Lecturers & Examiners: Prof. em. Dr. Freddy Delbaen, Dr. Sebastian Maass
VVZ CR n/a

Last Updated: 2026-02-05 14:59:44

Abstract

The course aims at providing an introduction to mathematical finance. Starting from a review of probability theory and a summary of martingale theory, an introduction to stochastic integration including its application to mathematical finance will be given. Topics addressed include Ito's formula, Girsanov's Theorem, put-call parity and option pricing in the Black-Scholes Model.

Content

Contents Review of probability theory: probability spaces; random variables, product spaces, conditional expectations (defining properties). The case of finite sigma-algebras. Bayes' rule for conditional expectations. Convergence of random variables. The weak convergence of laws. Basic theorems in probability theory (law of large numbers, CLT) Filtrations, adapted processed and predictable processes. Their relation with financial modelling. Relation between finite filtrations, partitions and "trees" (if you like gardening). Stopping times and the associated sigma-algebras. Definition of adapted and predictable stochastic processes. Martingales, submartingales, supermartingales (discrete time). Doob decomposition, relation with sub-, supermartingales. Stopping time theorem. Snell envelopes and the American option. Continuous time processes. Summary of martingale theory, c\`adl\`ag processes. The stopping time theorem. Predictable and optional processes. Notions from the "general theory". Their relation with financial modelling. Brownian motion and its properties. Starting again after stopping. Basic martingales. Hitting times. More dimensional BM. Lévy's characterization of Brownian motion. Brownian motion with drift. Introduction to stochastic integration. Itô's lemma, Girsanov-Maruyama theorem. How to get rid of the drift. Option pricing in the Samuelson-Black-Scholes world. Martingales and PDE. Stochastic differential equations. Feynman-Kac formula.

General Information

Language
English
Frequency
Yearly recurring

Examination

Type
session examination
Mode
oral 30 minutes

Course Components

Type Title Time & Place Hours
lecture Mathematical Foundations of Finance
  • Fri 09:15-11:00 (HG D 3.2)
2 h weekly
exercise Mathematical Foundations of Finance
  • Fri 11:15-12:00 (HG D 3.2)
1 h weekly

Offered In