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
• 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.

 no formal requirements but attendance is strongly recommended (due to distance learning)

The course consists of several parts: overview presentations of the key elements via ZOOMduring lecture hours (usually Tuesday, 10-12.15), a tutorium where exercises are discussed (usually Tuesday 9-10), solution of exercises and self-study of the course material provided (slides and lecture notes).

Midterm Test 25% (remote take home)

Exercise Series (25%) (remote take home)

Final exam is written (online supervision) and an optional oral exam  (50%)

A minimum score of 45% in the final exam and an overall score of 50 % is necessary for passing. The final exam consists of two parts: a mandatory written test and an oral test. The oral test is optional and can be taken only by students who obtained at least 80% of the points in  the rest of the course and who want to obtain the grade "excellent"  (but note that a bad performance in the final can also lead to a deterioration of your grade) .

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