This lecture discusses the basic mathematical tools for continuous-time finance and applies these to problems in derivative pricing. In particular we study option pricing in the Black Scholes model.

We will cover the following topics

• Wiener process and compound Poisson processes
• Quadratic variation
• Pathwise Ito calculus, elementary Ito integral and the Ito formula
• Generators and Feynman Kac for one-dimensional diffusions
• Derivative pricing via replication in the Black Scholes model
• Black Scholes formula and application
• Extensions beyond the classical Black Scholes model
• Basic numeric approaches for option pricing

Continuous time finance is concerned with: Brownian motion, stochastic calculus, risk-neutral pricing, stochastic and partial differential equations, exotic options.

After completing this class the student will have the ability to:

• describe the basic concepts and methods of continuous time finance;

• apply and do computational work with the basic concepts and definitions of continuous time finance.

After completing this class the student will also have the ability to:

• confidently apply ideas of continuous time finance in doing analytical work for financial markets.

• solve applied problems where skills are required from continuous time finance.

Standard rules for PIs (80 % of lectures and tutorium)

This class is taught as a lecture complemented with exercises and a tutorium

Midterm Exam (20%)

Exercise Series (25%)

Final exam (55%)

A minimum score of 40% in the final exam and an overall score of 50 % is necessary for passing.

Exercises will be discussed during the tutorium (also additional exercises).

• Advanced Business Mathematics (class Mathematics I of the QFin program);

• Advanced Business Probability theory (class Probability of the QFin program);

.

Probability and Mathematics II from QFin or equivalent