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2024/2025  KAN-CMECV1250U  Mathematical Finance 2: Continuous Time Finance

English Title
Mathematical Finance 2: Continuous Time Finance

Course information

Language English
Course ECTS 7.5 ECTS
Type Elective
Level Full Degree Master
Duration One Quarter
Start time of the course Second Quarter
Timetable Course schedule will be posted at calendar.cbs.dk
Max. participants 80
Study board
Study Board for HA/cand.merc. i erhvervsøkonomi og matematik, MSc
Course coordinator
  • Lars Christian Larsen - Department of Finance (FI)
Main academic disciplines
  • Finance
  • Mathematics
  • Statistics and quantitative methods
Teaching methods
  • Blended learning
Last updated on 13-03-2024

Relevant links

Learning objectives
By the end of the course, students will be confident with the probabilistic techniques required to understand the most widely used models in finance, from the Black-Scholes model to stochastic volatility models and affine term structure models.

To achieve the grade 12, students should meet the following learning objectives with no or only minor mistakes or errors:
  • The student should be able to account for selected asset pricing theories (or models).
  • The student should be able to discuss the strength and weakness in those theories (or models).
  • The student should be able to independently apply the analytical methods and models taught in the course to analyze relevant financial issues and challenges.
  • The student should be able to reflect on the implications of alternative models on a given issue.
  • The student should be able to use stochastic calculus and probability theory to derive and describe financial models.
  • The student should be able to use arbitrage pricing, martingale pricing and risk-neutral valuation to price financial contracts.
Course prerequisites
Prerequisites equivalent to the level of the HA(mat.) program. Particularly, students must have taken a basic derivatives course. It is an advantage to have followed Matematisk Finansiering 1 but it is not a prerequisite. Undergraduate knowledge of probability theory, algebra and calculus is required.
Mathematical Finance 2: Continuous Time Finance:
Exam ECTS 7,5
Examination form Written sit-in exam on CBS' computers
Individual or group exam Individual exam
Assignment type Written assignment
Duration 4 hours
Grading scale 7-point grading scale
Examiner(s) One internal examiner
Exam period Autumn
Aids Limited aids, see the list below:
The student is allowed to bring
  • USB key for uploading of notes, books and compendiums in a non-executable format (no applications, application fragments, IT tools etc.)
  • Any calculator
  • In Paper format: Books (including translation dictionaries), compendiums and notes
The student will have access to
  • Access to Canvas
  • Access to the personal drive (S-drive) on CBS´ network
  • Advanced IT application package
Make-up exam/re-exam Oral Exam
Duration: 30 min. per student, including examiners' discussion of grade, and informing plus explaining the grade
Preparation time: No preparation
Examiner(s): If it is an internal examination, there will be a second internal examiner at the re-exam. If it is an external examination, there will be an external examiner.
Description of the exam procedure

At the oral reexam, a topic is drawn from a pre-determined list. The students then have approximately 10 minutes to present the topic. The students can use prepared slides for this presentation. Questions may be asked about the presentation during and after it. Following the presentation, the remaining course material is tested without the aid of slides. Here, the students can be asked to go to the whiteboard and solve questions or exercises similar to those seen in the course.

Course content, structure and pedagogical approach

Over the last few decades, financial analysts have started using more and more sophisticated mathematical concepts and models to describe the behaviour of markets and to derive computing methods. This course provides an introduction to stochastic calculus and shows how it is applied to the pricing and hedging of financial contracts such as equity options and interest rate derivatives. The goal is to make students confident with the probabilistic techniques required to understand the most widely used financial models. The course is mainly suitable for students who would like to become quantitative analysts, asset managers, traders, risk managers, structurers, or who are simply generally interested in financial markets and want to gain the technical skills needed to understand and model their behaviour.


The course is split into 4 sections:


1. Discrete Time Finance: the Binomial Model

  • Introduction to measure theoretic probability
  • Derivative pricing and hedging in the generalized market model
  • Risk-neutral measure
  • Self-financing strategies
  • Absence of arbitrage    


2. Introduction to Stochastic Processes and Stochastic Calculus

  • Continuous time limits
  • Brownian motion
  • Stochastic integrals
  • Ito’s lemma
  • Change of probability and the risk neutral measure
  • Simulation


3. Pricing and Hedging in Continuous Time

  • Arbitrage pricing and hedging theory
  • Martingale approach to arbitrage theory
  • Black-Scholes-Merton equation
  • Feynman-Kac Theorem
  • Hedging in the BSM model
  • The implied volatility surface


4. Advanced material (time permitting)

  • Stochastic volatility models
  • Change of numeraire
  • Currency derivatives
  • Term structure models


Description of the teaching methods
The lectures will include exercise sessions where the students have (limited) time to work on problem sets that will then be discussed afterwards.

The pedagogical approach of the course is applied in nature. That means that we will abstract from much of the (deep) technical rigour underlying continuous time theory and instead concentrate on applications and problem solving. In other words, we will be ‘learning-by-doing’, which will require investing time in problem sets and exercises. We will also discuss few programming exercises to illustrate the how the theoretical models taught in the course are implemented in practice.
Feedback during the teaching period
The students are encouraged to actively participate in the lectures and problem-solving sessions.

Exercise sets will be regularly distributed and selected problems solved in class. Solutions will be provided. There will be feedback during the problem-solving sessions and the students can compare their own exercise solutions with the lecturer's solutions and students can receive hints and comments on their proposed solutions.

The course will also incorporate blended learning, where a portion of the feedback will take place online.

Student workload
Lectures 36 hours
Preparation for lectures and solving exercises 134 hours
Exam and exam preparation 36 hours
Further Information

The course can be taken by any interested student



Expected literature

The main textbook is:

``Arbitrage Theory in Continuous Time'', Björk (2020)


Selected chapters from the following textbooks will also be covered:

``Stochastic Calculus for Finance: Discrete-time Models'', (2004) Shreve

``Stochastic Calculus for Finance II: Continuous-time Models'' , (2004) Shreve


(all books are available online through CBS library)


Last updated on 13-03-2024