### Exams

The oral exams (25 min) will take place on 20/03/2013 and 10/04/2013 in my office (4C-11). You can choose if you want to answer in German or English. If you want to take an exam, please send an e-mail containing your full name, the date of the exam (20/03 or 10/04), and your program (diploma, master, lectureship/Lehramt, or bachelor) to jahnke@kit.edu. In addition, master students have to register via QISPOS.

**Important:** In the exam, questions about the lecture **and questions about the exercise classes** will be asked.

Diploma students only have to take an exam if they want to obtain an Übungsschein. In this case, the oral exam will focus on the exercises discussed in the exercise classes.

### Evaluation

Results of the evaluation: lecture, exercise classes

### Aims and scope of this lecture

An option is a contract which gives its owner the right to buy or sell an underlying asset at a certain time at a fixed price. The underlying asset is often a stock of a company, and since its value varies in a random way, computing the fair price of the corresponding option is an important and interesting problem which yields a number of mathematical challenges. This lecture provides an introduction to the most important models for option pricing. The main goal, however, is the construction and analysis of numerical methods which approximate the solution of the corresponding differential equations in a stable, accurate and efficient way.

The following topics will be treated:

- Mathematical models for pricing stock options
- Ito integral, Ito formula, stochastic differential equations, Black-Scholes equation
- Binomial methods
- Monte-Carlo methods
- Numerical methods for stochastic differential equations
- Random number generators
- Finite difference methods for parabolic partial differential equations
- Numerical methods for free boundary value problems

Participants should be familiar with

- ordinary differential equations and the corresponding numerical methods (cf. lecture "Numerische Methoden für Differentialgleichungen"), and
- probability theory (cf. lecture "Wahrscheinlichkeitstheorie").

Knowledge about stocks, options, arbitrage and other aspects from mathematical finance are **not** required, because the lecture will provide a short introduction to these topics.

The lecture and the exercise classes will be given in English.

**Perspective:** A second part of the course will probably be taught in summer 2013.

**Diploma and master theses:** Since mathematical finance is a rather popular topic, I get a lot of requests to supervise diploma and master theses. Therefore, attending this lecture does not give you a guarantee that I can supervise your thesis.