# Numerical Methods in Financial Mathematics

## Prof. Dr. Christian Fries, Dr. Alessandro Gnoatto, Juan Miguel Montes

## Schedule and Venue

Programming LecturesDr. Alessandro Gnoatto |
From 7.10.2013 to 11.10.2013 Every morning at 9:00 - 13: 00 |
quantLabRoom B 121 |

LecturesProf. Dr. Christian Fries |
Thu 14:15 - 15:45 and Fri 08:15 - 09:45 | |

ExercisesJuan Miguel Montes |
Wed 12:00 - 14:00 | |

quantLab Tools and Technology TutoriumJuan Miguel Montes |
Wed 14:00 - 16:00 and Thu 10:00 - 12:00 | |

Mid-term Project Review |
to be announced | |

Final Written Exam |
to be announced |

*Note: It is obligatory for students to attend the programming lectures, during which an introduction to Object-Oriented Programming in Java will be given.*

## Course Description

The lecture gives an introduction to some of the most important numerical methods in financial mathematics. A central topic of this lecture is the Monte Carlo method and its applications to stochastic differential equations, as used for example in the valuation of derivatives. In this context pseudo-random number generation, Monte Carlo simulation of stochastic processes and variance reduction methods are discussed. For low dimensional models, existing alternatives to Derivatives pricing by numerical solutions of partial differential equations (PDEs) will be discussed, albeit with less emphasis. In addition, numerical methods for financial mathematics are addressed as they are used in the processing of market data, model calibration and calculation of risk parameters. With time permitting, the object-oriented implementation of some numerical methods in the context of a (mathematical) application will be discussed (to follow this course it is obligatory to attend the programming lectures on Introduction to Object-Oriented Programming in Java).

## References

Asmussen, Søren; Glynn, Peter W.: Stochastic Simulation: Algorithms and Analysis. Springer, 2007. ISBN 978-0387306797.

Fries, Christian P.: Mathematical Finance. Theory, Modeling, Implementation. John Wiley & Sons, 2007. ISBN 0-470-04722-4.

http://www.christian-fries.de/finmath/book

## Exercises

Active participation in the Exercises is necessary to pass the Exam. Moreover, correcting your answers and thinking through the exercises is the best preparation for the exam. The solutions need not be submitted, but if you wish them to be corrected, please submit your exercise solutions.

**Exercise Handouts:** The problem sheets will be uploaded here during the course.

Exercise Sheet 1 (deadline: 30.Oct.2013) | Solution Sheet 1 summation False Position Method |

Exercise Sheet 2 (deadline: 6.Nov.2013) | Solution Sheet 2 histogram.pdf histogram.ods piErrorsPseudo.pdf piErrorsQuasi.pdf piErrors.ods |

Exercise Sheet 3 (deadline: 13.Nov.2013) | Solution Sheet 3 |

Exercise Sheet 4 (deadline: 20.Nov.2013) | Solution Sheet 4 add-on to Solution Sheet 4: Exercise 12 Solution |

Exercise Sheet 5 (deadline: 27.Nov.2013) | Solution Sheet 5 Example Spreadsheet for Solution Sheet 5 |

Exercise Sheet 6 (deadline: 11.Dec.2013) | Solution Sheet 6 |

Exercise Sheet 7 (deadline: 18.Dec.2013) | Solution Sheet 7 |

Exercise Sheet 8 supplement (deadline: 8.Jan.2014) | Solution Sheet 8 supplement |

Exercise Sheet 8 (deadline: 8.Jan.2014 extended: 15.Jan.2014) |
Solution Sheet 8 |

Exercise Sheet 9 (deadline: 22.Jan.2014) | Solution Sheet 9 |

Exercise Sheet 10 (deadline: 7.Feb.2014) | Solution Sheet 10 to be uploaded |

## Exam

The exam of this lecture will consist of two parts both of which have to be passed: a successful review of a mid term project and a written exam at the end of the lecture. The final grade shall be computed from 70% of the written exam grade and 30% from the mid term project grade.

**Mid term project: **Euler-Scheme and Hedge Simulation Project.

**Written exam:** The written exam is open-book, that is, all notes, books, solutions of exercises etc. may be used. Electronic devices of any kind are not allowed. To participate, please bring to the exam your ID card or passport and your student card. Please be on time.