Content:
Financial derivatives have become an indispensable tool in the financial world for controlling and hedging risks in recent years. The challenging problem is the "fair" valuation of financial instruments based on modern mathematical methods. The basis for the valuation of simple models is the Black-Scholes equation, which has a closed solution formula. For more complex models, however, closed formulas no longer exist and the model equations must be solved numerically. Both problems, the mathematical modelling and the numerical simulation of financial derivatives, are dealt with in detail in this lecture. We will cover a wide range of topics from modelling and analysis to the simulation of realistic financial products. We restrict ourselves predominantly to continuous-time (i.e. not discrete-time) models, which can be described by stochastic or partial differential equations. For the valuation of financial derivatives, three classes of methods can be roughly distinguished: Binomial methods, Monte Carlo simulations and methods for solving partial differential equations and free boundary value problems. We explain these techniques in detail and work out their algorithmic implementation using Matlab programmes in an accompanying practical course.

Target Audience:
The course is aimed at students of the master programme Mathematics, teacher candidates SII, Bachelor/Master students IT (specialisation Computing) and especially Master students of Business Mathematics. The course can be taken as a compulsory elective subject/module in mathematics or in the field of numerical mathematics by students of the above degree programmes.

Remarks:
Exercises and computer practicals are integrated into the lecture. The lecture covers the modules SKap.NAaA and SKap.WM of the Master's programme in Mathematics (specialisation Numerical Analysis and Algorithms as well as Business Mathematics). Building on this course, master's theses can be assigned, gladly also in cooperation with banks.

Prior Knowledge:
Analysis I-II, Lineare Algebra I-II, Einführung in die numerische Mathematik, Finanzmathematik

Literature:

• Y. Achdou und O. Pironneau, Computational Methods in Option Pricing, SIAM, 2005.
• M. Günther und A. Jüngel, Finanzderivate mit MATLAB, Vieweg, 2003.
• D. Higham, An Introduction to Financial Option Valuation: Mathematics, Stochastics and Computation, Cambridge University Press, 2004. (with many MATLAB files)
• J.C. Hull, Options, Futures and other Derivatives, 3rd edition, Prentice Hall, 1997.
• Y.K. Kwok, Mathematical Models of Financial Derivatives, Springer, Singapur, 1998.
• L.C.G. Rogers und D. Thalay, Numerical Methods in Finance, 1997.
• R. Seydel, Computational Finance, 4th edition, Springer, 2009.
• P. Wilmott, S. Howison und J. Dewyenne, The Mathematics of Financial Derivatives, Cambridge University Press, Cambridge 1996.
• U. Wystup, FX Options and Structured Products, Wiley Finance, 2006.
• P. Zhang, Exotic Options, World Scientific, Singapure 1997.

Links

• Computational Finance domain www.compfin.de (R. Seydel)
• Modellierung und Numerische Methoden von Finanzderivaten (Wikiversity-Kurs)
• Financial Numerical Recipes in C++ (Bernt Arne Ødegaard)
• DAAD Project Fin-Diff-Fin: Finite differences for Financial derivative models
• Master's program in Financial Mathematics (Numerical Methods), Halmstad University, Schweden.
• PREMIA - pricing financial derivatives Mathematics (MATHFI-Team at INRIA)
• Black-Scholes in Multiple Languages (Espen Haug)