# GRA 6550 Stochastic Calculus for Finance

## GRA 6550 Stochastic Calculus for Finance

Course code:
GRA 6550
Department:
Finance
Credits:
6
Course coordinator:
Tatyana Marchuk
Course name in Norwegian:
Stochastic Calculus for Finance
Product category:
Master
Portfolio:
MSc in Quantitative Finance
Semester:
2023 Autumn
Active status:
Active
Level of study:
Master
Teaching language:
English
Course type:
One semester
Introduction

The objective of the course is to provide the students with knowledge of the stochastic calculus that underlies the pricing and hedging of derivative instruments, including stochastic integrals and stochastic differential equations. The course is focused on the application of stochastic calculus methods in finance with both discrete-time and continuous-time stochastic models of financial markets, starting from the simple random walk and geometric Brownian motion all the way to models with jumps. The course is designed to deepen the students’ understanding of pricing techniques used in international derivatives markets.

Learning outcomes - Knowledge

By the end of the course, the students are expected to learn about:

• Stochastic processes, their theoretical foundation and properties; stochastic calculus tools to treat stochastic processes
• Leading models of stochastic processes in modern finance, their application in pricing derivatives and the underlying fundamental theorems of asset pricing.
Learning outcomes - Skills

By the end of the course, the students are expected to:

• Solve common stochastic differential equations both in closed form and numerically via simulation in R
• Model the stochastic processes underlying the dynamics of financial securities and apply the main stochastic calculus tools for the valuation of derivative instruments.
General Competence

The students are expected to master the stochastic calculus techniques to manipulate stochastic processes, reflect on the assumptions and limitations of the main stochastic models used in finance and confidently apply the studied methodology in asset pricing.

Course content
• Review of probability theory, deterministic calculus, and discrete-time models
• Martingale processes, stochastic integrals, and stochastic differential equations (SDEs)
• Ito’s lemma
• Major models of SDEs, solving SDEs analytically and using simulations
• Fundamental theorems of asset pricing, the Girsanov theorem
• Equivalent martingale measure
• Black-Scholes model and partial differential equations
• Jumps
Teaching and learning activities

The main course element is the lectures which cover theoretical concepts and the derivations of fundamental results in stochastic calculus and discuss the underlying intuition and relevance of these results in financial engineering. The lecture material follows the structure of the course textbook (Shreve) with a greater focus on economic intuition and a lower degree of technical rigor. During the lectures, the students work with practical applications and examples to strengthen their understanding of the material. The simulation of stochastic processes in R and Python represents a helpful visualization tool to learn about their dynamics.

The students will be able to assess their progress in achieving the learning objectives using weekly sets of practice exercises. The solutions to these exercises are discussed in class and the students are encouraged to present their solutions. Finally, the course includes a graded mid-term exam that provides the students with an interim assessment of their standing.

Software tools
R

The exam for this course has been changed starting academic year 2023/2024. The course now has two exam codes instead of one. It is not possible to retake the old version of the exam. Please note new exam codes in the Exam section of the course description.

It is the student’s own responsibility to obtain any information provided in class.

Honour Code
Academic honesty and trust are important to all of us as individuals and represent values that are encouraged and promoted by the honour code system. This is a most significant university tradition. Students are responsible for familiarizing themselves with the ideals of the honour code system, to which the faculty are also deeply committed. Any violation of the honour code will be dealt with in accordance with BI’s procedures for cheating. These issues are a serious matter to everyone associated with the programs at BI and are at the heart of the honour code and academic integrity. If you have any questions about your responsibilities under the honour code, please ask.

Qualifications

All courses in the Masters programme will assume that students have fulfilled the admission requirements for the programme. In addition, courses in second, third and/or fourth semester can have specific prerequisites and will assume that students have followed normal study progression. For double degree and exchange students, please note that equivalent courses are accepted.

### Disclaimer

Deviations in teaching and exams may occur if external conditions or unforeseen events call for this.

Exam categoryWeightInvigilationDurationSupport materialsGroupingComment exam
Exam category:
Submission
Form of assessment:
Written submission
Exam code:
GRA 65502
ECTS
Internal examiner
Resit:
Examination when next scheduled course
40Yes2 Hour(s)
• BI-approved exam calculator
• Simple calculator
• Bilingual dictionary
Individual Written mid-term examination with supervision
Exam category:
Submission
Form of assessment:
Written submission
Exam code:
GRA 65503
ECTS
Internal examiner
Resit:
Examination when next scheduled course
60Yes3 Hour(s)
• BI-approved exam calculator
• Simple calculator
• Bilingual dictionary
Individual Written final examination with supervision
Exams:
 Exam category: Submission Form of assessment: Written submission Weight: 40 Invigilation: Yes Grouping (size): Individual Support materials: BI-approved exam calculator Simple calculator Bilingual dictionary Duration: 2 Hour(s) Comment: Written mid-term examination with supervision Exam code: GRA 65502 Grading scale: ECTS Resit: Examination when next scheduled course
 Exam category: Submission Form of assessment: Written submission Weight: 60 Invigilation: Yes Grouping (size): Individual Support materials: BI-approved exam calculator Simple calculator Bilingual dictionary Duration: 3 Hour(s) Comment: Written final examination with supervision Exam code: GRA 65503 Grading scale: ECTS Resit: Examination when next scheduled course
Type of Assessment:
Ordinary examination
All exams must be passed to get a grade in this course.
Total weight:
100