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.

In line with WU regulations for lectures in PI format  full attendance is required (at most one lecture can be missed)

The course consists of several parts the on-site lecture  (usually Tuesday, 10-13.00), an on-site tutorium where exercises are discussed (usually Tuesday 9-10), solution of exercises in groups and self-study of the course material provided (slides and lecture notes).

Midterm Test 30% (written, on site)

Exercise Series (25%) (remote take home)

Final exam (45%)  (written, on site)

An an overall score of 50 % and a minimum score of 40% in the final 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);
• Discrete time stochastic processes and financial mathematics (class Mathematics II of the QFin program)

It is possible to attend the course even without successfully passing the above classes, but familiarity with the content of these classes is expected.

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Probability and Mathematics II from QFin or equivalent