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This is an introductory course on Brownian motion, Ito calculus and its application to nancial mathematics. The course is designed for students with a background in calculus based probability and an exposure to bachelor level measure theory and stochastic processes. We start with a quick review of all the necessary background, we then move to the concept of Brownian motion and discuss some of its important properties like quadratic variation, Markov property and the reflection principle. We then discuss stochastic calculus, in particular the Ito-Doeblin formula and the derivation of the Black-Scholes-Merton formula for pricing European options. After making this link with financial mathematics, we introduce risk-neutral pricing and discuss Girsanov's Theorem and Fundamental Theorems of Asset Pricing. We end the course, with a quick discussion of stochastic partial differential equations.
This course is one of the four modelling courses in the bachelor programme with major 'mathematics' or 'mathematics and applications'. Please find more information on the students website.
Learning goals:
At the end of this course, the student
- has an in-depth knowledge of Brownian motion, their essential properties
- has in-depth knowledge in Ito-Calculus, in particular the Ito-Doeblin Formula
- has an understanding of the derivation and the application of Black and Scholes formula in Financial Mathematics
- .has the knowledge and usage of the risk-neutral measure, the Girsanov's Theorem, and the Fundamental Theorem of asset pricing as well as a decent exposure to stochastic dierential equations.
Teaching methods:
Lectures (2x per week 2 hours) and exercise classes (2x per week 2 hours)
Assessment:
Hand-in exercises (4 sets) 20%, a mid-term 30% and a final exam 50%.
Retake:
Students with a final grade lower than 4 are eligible to do the retake exam only if they submitted all the hand-in assignments and were present at the final exam.
Language of instructions:
Dutch or English, depending on the participants.
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