# GRA 6550 Stochastic Calculus for Finance

## GRA 6550 Stochastic Calculus for Finance

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.

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.

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 MATLAB

- Model the stochastic processes underlying the dynamics of financial securities and apply the main stochastic calculus tools for the valuation of derivative instruments.

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

- 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

The main course element is the lectures which cover theoretical concepts and the derivations of fundamental results in stochastic calculus, 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 solution of these exercises are discussed in class and the students are encouraged to present their solution. Finally, the course includes a graded mid-term exam that provides the students with interim assessment of their standing.

Please note that while attendance is not compulsory in all courses, it is the student’s own responsibility to obtain any information provided in class.

This is a course with continuous assessment (several exam components) and one final exam code. Each exam component is graded by using points on a scale from 0-100. The components will be weighted together according to the information in the course description in order to calculate the final letter grade for the examination code (course). Students who fail to participate in one/some/all exam elements will get a lower grade or may fail the course. You will find detailed information about the point system and the cut off points with reference to the letter grades when the course starts.

At resit, all exam components must, as a main rule, be retaken during next scheduled course.

__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.

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 spesific prerequisites and will assume that students have followed normal study progression. For double degree and exchange students, please note that equivalent courses are accepted.

**Covid-19 **

Due to the Covid-19 pandemic, there may be deviations in teaching and learning activities as well as exams, compared with what is described in this course description.

**Teaching**

Information about what is taught on campus and other digital forms will be presented with the lecture plan before the start of the course each semester.

Exam category | Weight | Invigilation | Duration | Support materials | Grouping | Comment exam |
---|---|---|---|---|---|---|

Exam category:Submission Form of assessment:Written submission Exam code:GRA65501 Grading scale:Point scale Grading rules:Internal examiner Resit:All components must, as a main rule, be retaken during next scheduled course | 50 | Yes | 2 Hour(s) | - BI-approved exam calculator
- Simple calculator
- Bilingual dictionary
| Individual | Written mid-term examination under supervision |

Exam category:Submission Form of assessment:Written submission Exam code:GRA65501 Grading scale:Point scale Grading rules:Internal and external examiner Resit:All components must, as a main rule, be retaken during next scheduled course | 50 | Yes | 2 Hour(s) | - BI-approved exam calculator
- Simple calculator
- Bilingual dictionary
| Individual | Written examination with supervision |

A course of 1 ECTS credit corresponds to a workload of 26-30 hours. Therefore a course of 6 ECTS credits corresponds to a workload of at least 160 hours.