# 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 simpler random walk and geometric Brownian motion all the way to models with jumps and models for bubbles.

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

- Definition of a Brownian motion
- Arithmetic Brownian motion, geometric Brownian motion, Ornstein-Uhlenbeck process
- Ito's lemma and Multivariate Ito's lemma
- Girsanov's theorem
- Evaluation of stochastic differential equations
- No arbitrage models
- Black - Sholes model
- Models with jumps
- Models with bubbles

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

- be comfortable in applying Ito's lemma in its various forms to derive the dynamic process of a random variable
- be able to solve the more common stochastic differential equations
- be able to switch between equivalent probability measures
- apply the main stochastic calculus tools for the valuation of securities
- code up the main continuos-time models including Geometric Brownian motion, mean-reverting processes and processes with Poisson discontinuities.

The students by the end of the course are expected to be able to reflect on the workings and limitations of the theories taught during the course.

- Review of probability theory and discrete time models
- Continuous time finance
- Brownian motion
- Continuous-time models for asset prices
- Introduction to stochastic calculus
- Ito’s lemma
- Girsanov’s theorem
- SDEs
- Security valuation and advanced topics in stochastic calculus
- No arbitrage
- Black-Sholes model
- PDEs
- Multivariate Ito
- Jumps

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 that is not included on the course homepage/itslearning or text book.

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. The expected behaviour and honour code is outlined here.

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 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.

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.