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 stochastic processes. 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: Friday from 13:15-15 in HFG610 (formally the room was BBG 308)

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: Friday November 13 from 13-16 in HFG 611. The Midterm consists of presentation of problems by each student. A collection of problems will be given ahead of time, and on November 13 a lottery will be performed to decide which problem a student will present.

Final: Friday 29/01/2016 from 13:30-16:30 in BBG 201. The final is NOT open book, but you are allowed to bring an A4 with whatever information you would like on it.

Material Covered:

-Friday September 11: Chapter 1. Exercises p.41: 1.4, 1.5, 1.10, 1.12, 1.13

-Friday September 18: Chapter 2. Exercises p.77: 2.4, 2.5, 2.6, 2.7, 2.10

-Friday September 25: Chapter 3 sections 3.1-3.2. Exercises p.117: 3.1, 3.2, 3.3, 3.4.

-Friday October 2: Chapter 3 sections 3.3, 3.4 Exercises p. 118: 3.5, 3.6,

-Friday October 9: Chapter 3 sections 3.5, 3.6, 3.7 Exercises p. 118: 3.7, 3.8.

-Friday October 16: Rest of Chapter 3 and Chapter 4 sections 4.1, 4.2. Exercises p. 189: 4.1, 4.2, 4.3.

-Friday October 23: Chapter 4 sections 4.3. Exercises p.190: 4.5, 4.6.

-Friday October 27: Chapter 4 section 4.4. Exercises p. 190: 4.8, 4.9.

-Friday November 6: No classes.

-Friday November 13: Rest of Chapter 4. Exercises p. 190: 4.11, 4.14

-Friday November 20: Presentations.

-Friday November 27: Chapter 5 sections 5.1, 5.2: Exercises p. 251: 5.1, 5.2, 5.3.

-Friday December 4: Chapter 5 subsections 5.2.2, 5.2.3, 5.2.4, 5.2.5, 5.3.1, 5.3.2: Exercises p. 251: 5.5, 5.8

-Friday December 11: Chapter 5 subsections 5.4.1, 5.4.2. Exercises p. 254: 5.7, 5.9, 5.10.

-Friday December 18: Chapter 5 subsections 5.4.3, 5.4.4, 5.5. Exercises p. 256: 5.11, 5.12, 5.13.