Lecturer: Karma Dajani

Course Description: This is an introductory course on Brownian motion, Ito calculus and its application to financial mathematics. The course is designed for students with a background in calculus based probability and a bachelor level financial mathematics. We begin with 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 and some applications to Exotic options.

Time: Monday from 11:00-12:45.

Place: BBG 075

Credits: 7.5 ECTS

Prerequisites: Probability theory and discrete financial mathematics. An exposure to stochastic processes and measure theory is recommended but not required.

Literature: Shreve, S.E., Stochastic Calculus for Finance II: Continuous-time models, Springer. ISBN 0-387-40101-6

Assessment: A mid-term (40%) and a final exam (60%).

Exam Schedule:

Mid-term: take-home exam due date is November 28. (The midterm will be available online till Wednesday November 9, if you missed the deadline, write me then an e-mail)

Final: Presentations of 30 minutes per student. This will take place on January 9, 16 and 23. Each student will be assigned a section from the book that they have to present. For long sections two students are assigned, so they have to coordinate together who is going to present what in 60 minutes. The sections come from chapters 6 and 7. Here is the division:

January 9

(1) Alexander Borchert- section 6.4

(2) Kilian Bakker- section 6.5

(3) Winfried van den Dool- section 7.2

January 16

(1) Jori Hoenkamp- subsections 7.3.1+7.3.2

(2) Folkert Kuipers- subsection 7.3.3

(3) Mathijs de Lepper- subsections 7.4.1+7.4.2+7.4.3

January 23

(1) Juriaan Parie + Dustin van Weersel: subsection 7.4.4 (60 minutes)

(2) Stefan Franssen + Luminita Maxim: section 7.5 (60 minutes)

Announcements: There is no class on Monday October 17 and November 21. On Monday January 30, two alumni will tell us of their master thesis project that was done as an internship. The time is 11-13 in BBG 005.

Material Covered:

-Monday September 12: Chapter 1. Exercises p.41: 1.4, 1.5, 1.10, 1.12, 1.13

-Monday September 19: Chapter 2. Exercises p.77: 2.4, 2.5, 2.6, 2.7, 2.10

-Monday September 26: rest of Chapter 2 and Chapter 3 sections 3.1-3.2. Exercises p.117: 3.1, 3.2, 3.3, 3.4.

-Monday October 3: Chapter 3 sections 3.3, 3.4 Exercises p. 118: 3.5, 3.6,

-Monday October 10: Chapter 3 sections 3.5, 3.6, 3.7 Exercises p. 118: 3.7, 3.8.

-Monday October 17: No class

-Monday October 24: Chapter 4 sections 4.1-4.3: Exercises p. 189:4.1, 4.2, 4.3, 4.4.

-Monday October 31: Chapter 4 sections 4.4: Exercises p. 190: 4.8, 4.9.

--Monday November 7: Chapter 4 sections 4.5: Exercises p. 190: 4.11, 4.14

--Monday November 14: Chapter 4 sections 4.6: No exercises work on the midterm.

--Monday November 21: No class, work on the midterm.

--Monday November 28: Chapter 5 sections 5.1, 5.2: Exercises p. 251: 5.1, 5.2, 5.3. (Midterm is due today!)

--Monday December 5: Chapter 5 sections 5.2.5, 5.3, 5.4.1: Exercises p. 251: 5.5, 5.7, 5.8, 5.9.

--Monday December 12: Chapter 5 sections 5.4.2, 5.4.3, 5.4.4: Exercises p. 256: 5.11, 5.12, 5.13.

--Monday December 19: Chapter 5 section 5.6. No new exercises, finish all the exercises from the past weeks.