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Syllabus:

# Financial Mathematics, 7.5 Credits

Swedish name: Finansiell matematik

This syllabus is valid: 2023-01-02 and until further notice

Course code: 5MA208

Credit points: 7.5

Education level: Second cycle

Main Field of Study and progress level: Mathematics: Second cycle, has only first-cycle course/s as entry requirements
Computational Science and Engineering: Second cycle, has only first-cycle course/s as entry requirements
Mathematical Statistics: Second cycle, has only first-cycle course/s as entry requirements

Established by: Faculty Board of Science and Technology, 2022-09-12

## Contents

The course covers the basic mathematical theory for modeling and pricing of financial instruments in discrete and continuous time. The focus in the course is on modeling stocks and pricing of stock options leading up to the Black Scholes model, built on geometric Brownian motion. The course also covers the theory of interest rates and pricing of different interest rate instruments.

## Expected learning outcomes

For a passing grade, the student must be able to

Knowledge and understanding

• explain the notions self-financing portfolio and arbitrage
• describe the connection between price setting of derivatives contracts and hedging
• explain the connection between arbitrage and the existence of so called martingale measures - First Fundamental Theorem of Financial Mathematics
• define stochastic differential equations and explain how they can be use to model financial assets
• explain the connection between market completeness and uniqueness of martingale measures - Second Fundamental Theorem of Financial Mathematics
• account for commonly occuring instruments for interest rate derivatives and models for short term interests

Skills and abilities

• independently apply the theory for price setting of financial derivatives, based on the no-arbitrage principle, in discrete and continuous time
• compute option prices as discounted expectations under a martingale measure
• use the Ito formula to transform stochastic dynamics
• critically apply the Black-Scholes equation for price setting of European buy- and sell options
• apply the theory for price setting of some exotic options
• solve simpler partial and stochastic differential equations
• derive and apply the put-call parity
• calculate and interpret the sensitivities (Greeks) for european buy- and sell options

## Required Knowledge

The course requires 90 ECTS including a course in Multivariable Calculus and Differential Equations and a basic course in Mathematical Statistics, minimum 6 ECTS. Proficiency in English equivalent to the level required for basic eligibility for higher studies. Where the language of instruction is Swedish, applicants must prove proficiency in Swedish to the level required for basic eligibility for higher studies.

## Form of instruction

The teaching takes the form of lectures.

## Examination modes

The course is assessed through written examinations in the form of two tests. From the first test students can award bonus points that can be added to the result on the mandatory final test. The bonus points are only valid on the ordinary final exam and on the first re-exam connected to the course occasion on which the bonus points were awarded. For the whole course, one of the following grades is assigned: Fail (U), Pass (3), Pass with merit (4) or Pass with distinction (5). The grade is only set once all compulsory elements have been assessed.

A student who has been awarded a passing grade for the course cannot be reassessed for a higher grade. Students who do not pass a test or examination on the original date are given another date to retake the examination. A student who has sat two examinations for a course or a part of a course, without passing either examination, has the right to have another examiner appointed, provided there are no specific reasons for not doing so (Chapter 6, Section 22, HEO). The request for a new examiner is made to the Head of the Department of Mathematics and Mathematical Statistics. Examinations based on this course syllabus are guaranteed to be offered for two years after the date of the student's first registration for the course.

Examiners may decide to deviate from the modes of assessment in the course syllabus. Individual adaption of modes of assessment must give due consideration to the student's needs. The adaption of modes of assessment must remain within the framework of the intended learning outcomes in the course syllabus. Students who require an adapted examination must submit a request to the department holding the course no later than 10 days before the examination. The examiner decides on the adaption of the examination, after which the student will be notified.

Credit transfer
All students have the right to have their previous education or equivalent, and  their working life experience evaluated for possible consideration in the corresponding education at Umeå university. Application forms should be adressed to Student services/Degree evaluation office. More information regarding credit transfer can be found on the student web pages of Umeå university, http://www.student.umu.se, and in the Higher Education Ordinance (chapter 6). If denied, the application can be appealed (as per the Higher Education Ordinance, chapter 12) to Överklagandenämnden för högskolan. This includes partially denied applications

## Other regulations

In a degree, this course may not be included together with another course with a similar content. If unsure, students should ask the Director of Studies in Mathematics and Mathematical Statistics. The course can also be included in the subject area of computational science and engineering.

In the event that the syllabus ceases to apply or undergoes major changes, students are guaranteed at least three examinations (including the regular examination opportunity) according to the regulations in the syllabus that the student was originally registered on for a period of a maximum of two years from the time that the previous syllabus ceased to apply or that the course ended.

## Literature

### Valid from: 2023 week 1

Pliska Stanley R.
Introduction to mathematical finance : discrete time models
Oxford : Blackwell : 1999 : 262 s. :
ISBN: 1557869456
Search the University Library catalogue

Björk Tomas
Arbitrage theory in continuous time
Fourth edition. : Oxford : Oxford University Press : 2019 : xxi, 561 sider :
ISBN: 9780198851615
Search the University Library catalogue