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2021/2022  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
  • Paul Whelan - Department of Finance (FI)
Main academic disciplines
  • Finance
  • Mathematics
  • Statistics and quantitative methods
Teaching methods
  • Blended learning
Last updated on 21/12/2021

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.
  • You could also write something about "use the correct terminology and communicate the contents of the relevant models
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.
The course includes a voluntary Introductory Python Workshop with pre-recorded video lectures and exercises.
Mathematical Finance 2: Continuous Time Finance:
Exam ECTS 7.5
Examination form Home assignment - written product
Individual or group exam Individual exam
Size of written product Please see text below
The home assignment has a duration of 3 hours and no maximum of pages.
Assignment type Written assignment
Duration Written product to be submitted on specified date and time.
Grading scale 7-point grading scale
Examiner(s) One internal examiner
Exam period Autumn
Make-up exam/re-exam
Same examination form as the ordinary exam
If the number of registered candidates for the make-up examination/re-take examination warrants that it may most appropriately be held as an oral examination, the programme office will inform the students that the make-up examination/re-take examination will be held as an oral examination instead.
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 intro- duction 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 the following sections:


Discrete Time Finance Revision

- Binomial model

- Derivative pricing and hedging

- Risk-neutral measure

- Self-financing strategies

- Absence of arbitrage


Introduction to Stochastic Processes and Stochastic Calculus

- Continuous time limits

- Brownian motion

- Ito’s lemma

- Change of probability 

- Simulation

- Feynman-Kac Theorem



Pricing and Hedging in Continuous Time

- Arbitrage pricing and hedging theory

- Black-Scholes-Merton (BSM) equation

- Relatives of BSM

- Hedging in the BSM model

- Affine term structure models 



- The implied volatility surface

- Stochastic volatility models

- The rise of volatility products and the VIX

- Option trading strategies 


Description of the teaching methods
The pedagogical approach of the course is applied in nature. That means will 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:

- Real life trading resource: At the beginning of the course you will all receive an invitation for a real-time trading account within the student lab at Interactive Brokers:


Your accounts will be charged with USD 1,000,000 in paper trading equity. Account equity will fluctuate as if trades were executed in the real market and you can trade stocks, options, futures, bonds and currencies, and credit default swaps. We will use this as a resource for real world pricing, calibration exercises, risk management and speculative (trading) exercises

- Coding: The financial industry is increasingly adopting Python and recent recruitment trends suggest that Python will soon become the de-facto programming language for quantitative analysis in finance. The benefits of Python with respect to its competitors is are that it is (a) easy to learn, read and write; (b) is a high-level programming language; (c) fast and easily scalable; (d) has vast library support; (e) is free and open source. This course includes an introductory set of lectures in Python for financial analysis. We will begin with the basics such as installation, an overview of the programming environment and tips on how to code ‘Pythonically’. You will learn the fundamentals of Python variables, loops, functions, data structures and learn powerful methods to efficiently practice data science and visualise data. The course will be pre-recorded online allowing students to progress at their own pace and will be accompanied by a series of exercises designed to implement techniques learned in the lectures.
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

Drop-in office hours will be held weekly

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:


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

``Analytical Finance: Volume I: The Mathematics of Equity Derivatives, Markets, Risk Valuation'' Jan Röman (2017)

(The books are available online through CBS library)



Hand outs from selected chapters of:


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

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

``Arbitrage Theory in Continuous Time'', Björk (2004)

``Stochastic Calculus for Finance: Continuous-time Models'' , Shreve


Last updated on 21/12/2021