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2023/2024  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 23-08-2023

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 apply the appropriate model on a given issue.
  • The student should be able to reflect on the implications of alternative models on a given issue.
Course prerequisites
Prerequisites at the level of HA(mat.). 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 3 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
  • Any calculator
  • Language dictionaries in paper format
The student will have access to
  • Advanced IT application package
Make-up exam/re-exam Oral Exam
Duration: 20 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

Three hour closed book written exam

Course content, structure and pedagogical approach

Over the last few years, financial analysts have used more and more sophisticated mathematical concepts to describe the behaviour of markets or to derive computing methods. This course provides an introduction to stochastic calculus and shows how it can be 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
  • Risk-neutral measure
  • Self-financing strategies
  • Absence of arbitrage    


2. Introduction to Stochastic Processes and Stochastic Calculus

  • Continuous time limits
  • Brownian motion
  • Ito’s lemma
  • Change of probability and the risk neutral measure
  • Martingale approach to arbitrage theory
  • Simulation
  • Feynman-Kac Theorem


3. Pricing and Hedging in Continuous Time

  • Arbitrage pricing and hedging theory
  • Black-Scholes-Merton equation
  • Hedging in the BSM model
  • The implied volatility surface


4. Advanced material

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


Description of the teaching methods
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 getting our hands dirty with data and coding exercises.
Feedback during the teaching period
Each lecture will consist of a set of slides

Exercise sets will be regularly distributed and selected problems solved in class

Solutions will be provided

There will be a voluntary mid-term assignment that will test students on course material. Feedback will be given.
Student workload
Lectures 32 hours
Preparation 138 hours
Exam and Preparation 36 hours
Further Information

The course can be taken by any interested student



Expected literature

Main course textbooks are:


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

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


(The books are available online through CBS library)


Hand outs from selected chapters of:


``Options, Futures and Other Derivatives'' (8th edition or later), John Hull

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

``A Course in Derivative Securities'', Kerry Back (2005)

 ``Fixed Income Securities: Valuation, Risk, and Risk Management'', Pietro Veronesi (2010)


Last updated on 23-08-2023