Fixed Income Derivatives: Risk Management and Financial Institutions (F)

Course content

In the world of today, both public and private institutions rely heavily on bond issuance to raise capital, and fixed income markets have come to play a central role in the global economy. This development has led to a rapid increase in the use of ever more sophisticated derivatives playing a dual role on one hand as means to insure against losses and on the other as tools of risky speculation. Interest rate derivatives have often played a central role in times of financial distress highlighting the need for financial actors to have a solid framework for pricing, hedging and risk management of these instruments.


Throughout this course, students will develop a thorough understanding of how fixed income markets can be modeled and how pricing and hedging of the most commonly traded fixed income derivatives can be performed within these models. Much of the course will be set in continuous time, and we will begin by covering the basics of stochastic calculus including Brownian motion, stochastic differential equations, Ito's formula, etc. This portion of the course will be somewhat technical, however emphasis will be on application of the methods and results we introduce. Once we have laid the mathematical foundation for the course, we will proceed to study dynamic models for the short rate and how the term structure of interest rates evolves in these types of models. As part of our discussion, we will learn how to fit term structure models to market data and how forward rate agreements, interest rate swaps and exchange options can be priced in the context of these models. Next, we will study the pricing, and hedging of more complicated interest rate options such as caps, floors, digital options and swaptions as well as how the “greeks” can be used for hedging and risk management of such contracts. Finally, we will cover more exotic financial derivatives including currency contracts such as FX forwards, FX swaps and cross currency swaps and credit derivatives such as asset swaps and credit default swaps.


The course will be somewhat technical and quantitative in nature, but emphasis will be placed on developing results that have applications in practice. The many methods and tools, we will develop, will be implemented using Python and hence, experience with a scripting language such as Python, Matlab, Julia or R will be helpful. By the end of this course, we will have developed a substantial library in Python containing some of the methods and algorithms most commonly used by financial practitioners.


MSc programme in Economics – elective course


The course is part of the Financial line at the MSc programme in Economics,   symbolized by ‘F’.


The course is open to:

  • Exchange and Guest students from abroad
  • Credit students from Danish Universities
  • Open University students
Learning outcome

- Develop an intuition for the mathematical framework underlying continuous time models.
- Know some of the most widely used dynamic models of the term structure of interest rates.
- Understand the properties of a wide range of interest rate derivatives.
- Deduce the risks associated with a wide range of derivatives commonly traded in financial markets.

- Choose an appropriate model to price and/or hedge commonly traded interest rate derivatives.
- Critically asses a financial model including its limitations and applicability in practice.
- Determine methods to price interest rate derivatives within the context of a dynamic model.
- Identify why a given model might not fit market data and suggest how to improve the model.

- Implement and fit a given dynamic term structure model to market data using Python.
- Calculate prices of a wide range of commonly traded interest rate derivatives.
- Dynamically compute a replicating strategy to hedge an interest rate derivative in practice.

The course, will consist of lectures, exercise classes and assignments. Students are not required to hand in the assignments posted throughout the course but are strongly encouraged to work on these to better understand the material and as preparation for the exam.

All teaching is conducted physically on campus.

Arbitrage Theory in Continuous Time (4th edition), Thomas Bjôrk, Oxford University Press, December 5. 2019, Chapters 4-5 and 20-25, Online ISBN: 9780191886218, Print ISBN: 9780198851615,

Fixed Income Derivatives Lecture Notes, Martin Linderstrøm, University of Copenhagen, February 3. 2013 Interpolation Methods for Curve Construction, Patrick S. Hagan and Graeme West, Applied Mathematical Finance, June 2006, Vol 13, No 2., pages 89-129,

Managing Smile Risk, Patrick S. Hagan, Deep Kumar, Andrew S. Lesniewski, Diana E. Woodward, Wilmott Magazine, January 2002, Vol 1, pages 84-108

Pricing Derivatives on Financial Securities Subject to Credit Risk, Robert Jarrow and Stuart M. Turnbull, Journal of Finance, March 1995,

Valuation of Credit Default Swaps, Dominic O'Kane and Stuart Turnbull, Fixed Income Quantitative
Research, Lehman Brothers, April 2003

Lecture notes and slides

Supplementary reading:
Arbitrage Theory in Continuous Time (4th edition), Thomas Bjôrk, Oxford University Press, December 5. 2019, Chapters 1-3 and 6-8, Online ISBN: 9780191886218, Print ISBN: 9780198851615,

Stochastic Calculus for Finance II: Continuous-Time Models, Steven Shreve, Springer Finance, June 28. 2005, Chapters 1-6, ISBN-10: 0387249680, ISBN-13: 978-0387249681


Pricing Derivatives on Financial Securities Subject to Credit Risk, Robert Jarrow and Stuart M. Turnbull,
Journal of Finance, March 1995,


Valuation of Credit Default Swaps, Dominic O'Kane and Stuart Turnbull, Fixed Income Quantitative
Research, Lehman Brothers, April 2003

This course is not an introductory course and students are expected to have a basic knowledge of derivatives pricing including Black-Scholes formula and fixed income markets. It is therefore recommended that students have a followed either the courses 'Financial Decision Making' or 'Corporate Finance and Incentives' offered at the Economics program at University of Copenhagen, or a similar course.

Furthermore, it is important to stress that an integral part of this course will involve programming in Python. Though no prior knowledge of Python is assumed, students are expected to have some basic programming experience.

Some of the models used in this course are set in continuous time, and though students are not assumed to have experience with continuous time finance from a previous courses, it would be a plus.

There will be a total of 4 hours of teaching a week from week 36 to 40 (except week 42).

Timetable and venue:
To see the time and location of lectures please press the link under "Timetable"/​"Se skema" at the right side of this page (E means Autumn).

You can find the similar information partly in English at
-Select Department: “2200-Økonomisk Institut” (and wait for respond)
-Select Module:: “2200-E24; [Name of course]”
-Select Report Type: “List – Weekdays”
-Select Period: “Efterår/Autumn”
Press: “ View Timetable”

Please be aware:
- The schedule of the lectures can change without the participants´ acceptance. If this occurs, you can see the new schedule in your personal timetable at KUnet, in the app myUCPH and through the links in the right side of this course description and the link above.
- It is the students´s own responsibility continuously throughout the study to stay informed about their study, their teaching, their schedule, their exams etc. through the curriculum of the study program, the study pages at KUnet, student messages, the course description, the Digital Exam portal, Absalon, the personal schema at KUnet and myUCPH app etc.

Continuous feedback during the course of the semester


Individual feedback can be received at the exercise classes.

7,5 ECTS
Type of assessment
Written assignment, 12 hours
Type of assessment details
Individual take-home exam. It is not allowed to collaborate on the assignment with anyone.
The exam assignment is given in English and must be answered in English.
Exam registration requirements

There are no requirements during the course that the student has to fulfill to be able to sit the exam.


All aids allowed at the written exams.


Use of AI tools is permitted. You must explain how you have used the tools. When text is solely or mainly generated by an AI tool, the tool used must be quoted as a source.

Marking scale
7-point grading scale
Censorship form
No external censorship
for the written exam.
Exam period

Exam information:

More information is available in Digital Exam from the middle of the semester.

More information about examination, rules, aids etc. at Master (UK) and Master (DK)


Same as the ordinary exam. 


Reexam info:

More information in Digital Exam in February. 

More info at Master(UK), and Master(DK)

Criteria for exam assessment

Students are assessed on the extent to which they master the learning outcome for the course.


In order to obtain the top grade "12", the student must with no or only a few minor weaknesses be able to demonstrate an excellent performance displaying a high level of command of all aspects of the relevant material and can make use of the knowledge, skills and competencies listed in the learning outcomes.


To obtain the grade 12 in this course, students are required to demonstrate a thorough understanding of all aspects surrounding fixed income and credit derivatives – from the basic legal framework to the practical implementation of pricing models using Excel and VBA.


In order to obtain the passing grade  “02”, the student must in a satisfactory way be able to demonstrate a minimal acceptable level of  the knowledge, skills and competencies listed in the learning outcomes.

Single subject courses (day)

  • Category
  • Hours
  • Lectures
  • 42
  • Class Instruction
  • 30
  • Preparation
  • 86
  • Exam
  • 48
  • English
  • 206


Course number
7,5 ECTS
Programme level
Full Degree Master

1 block

- Go to 'Signup' for information about registration and enrollment.

Information about admission and tuition fee:  Master and Exchange Programme, credit students and guest students (Open University)

and venue:
- For teaching: Go to 'Remarks'.
- For exam and re-sits: Go to 'Exam'.
Department of Economics, Study Council
Contracting department
  • Department of Economics
Contracting faculty
  • Faculty of Social Sciences
Course Coordinator
  • Jacob Lundbeck Serup   (20-4f6668746733517a7369676a687033586a777a75456a68747333707a336970)

See "Course Coordinators".

Please read "Remarks" regarding the schedule of the teaching.

Saved on the 17-05-2024

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