# NMAK24001U Mathematical Finance (MathFin)

Volume 2024/2025
Education

MSc Programme in Actuarial Mathematics
MSc Programme in Mathematics-Economics

Content

This course examines mathematics, modeling, concepts, and approaches for pricing derivatives securities in continuous time financial models. The topics include

• Introduction to continuous time stochastic processes
• Definition and properties of Brownian motion
• Stochastic integration: properties, Itô (change of variable) formula, theorems with applications in finance, semimartinales
• Stochastic differential equations
• Absence of arbitrage and martingale measures
• Fundamental theorem of asset pricing
• Pricing and hedging of financial derivatives
• Black-Scholes option pricing model and formula
• Complete and incomplete models
• Continuous time interest rate models
• Pricing intereste rate derivatives

Learning Outcome

Knowledge:

• Stochastic calculus for semimartingales
• Fundamental concepts in mathematical finance
• Pricing financial derivatives in diffusion models
• Short rate models
• Pricing interest rate derivatives

Skills: Ability to

• Apply theorems from stochastic calculus including theorems such as: Ito's formula, Feynman-Kac representations, martingale representations, Girsanov's theorem
• Determine arbitrage free prices of financial derivatives  and to determine hedge portfolio in complete markets
• Apply the first and second fundamental theorems of asset pricing including the determination of martingale measures
• Apply methods from incomplete markets for pricing interest rate derivatives

Compentencies: Ability to

• Discuss and apply central methods from stochastic calculus for pricing financial derivatives
• Evaluate main characteristics of a financial market from a mathematical perspective

See Absalon for a list of course literature.

Sandsynlighedsteori 2(Sand 2) and Finansiering 1 (Fin1).
Academic qualifications equivalent to a BSc degree is recommended.
4 hours of lecture pr week for 15 weeks (Blok 1 and Blok 2)
2 hours of exercises for week 1-8 (Blok 1)
4 hours of exercises for week 9-15 (Blok 2)
The seven first week of this course is equivalent to NMAK24000U Stochastic Processes in Continuous Time
• Category
• Hours
• Lectures
• 60
• Preparation
• 303
• Theory exercises
• 44
• Exam
• 5
• Total
• 412
Continuous feedback during the course of the semester

Individual feedback can be received at the exercise classes upon active participation in exercise classes.

Credit
15 ECTS
Type of assessment
On-site written exam, 4 hours under invigilation
Continuous assessment, 1-hour quiz
Type of assessment details
The exam is composed of a quiz and a written exam. To pass the course, the student must participate in both elements.

The quiz will be in week 7 of Block 1. The duration is 1 hour and it is under invigilation.
The written exam will be in the exam week of Blok 2. The duration is 4 hours and it is under invigilation.

The quiz will count 30% of the final grade
The written exam will count 70% of the final grade
Aid
All aids allowed
Marking scale
Censorship form
External censorship
Re-exam

The re-exam is composed of a quiz and an oral exam. The two parts of the exam do not need to be passed separately, but the student must participate in both exams.

• The quiz, one hour, will be under surveillance.
• The oral exam, 30 minutes without preparation time and no aids.