### 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 randomly, 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 lecture and the exercise classes will be given in English. 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"),
- probability theory (cf. lecture "Wahrscheinlichkeitstheorie"), and
- programming in MATLAB.

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.

In the problem class the students are supposed to solve small exercises which illustrate the contents of the lecture, and to write short MATLAB programs in order to test and apply the algorithms which will be presented in the lecture. Therefore, half of the problem classes will take place in the computer pool instead of the seminar room. MATLAB skills are required to solve the programming exercises.

**Perspective:** A second part of the course will probably be taught in summer 2015.

### Exams

The final exams (oral, 30 min) will take place on 27.2.2015, 18.3.2015 or 20.3.2015 in office 4C-11. Questions about the lecture **and** the exercises discussed in the problem class will be asked. If you want to take an exam, please

- send an e-mail to Sonja Becker (sonja.becker@kit.edu) with your name and immatriculation number and let us know which date (27.2., 18.3., or 20.3.2015) and language (English or German) you prefer for the exam, and

- register on the official KIT web page (Studienportal/QISPOS).