- Lecturer
- Christoph Schwab
- Coordinators
- Lukas Herrmann, Diyora Salimova

A closed-book, computer-based exam will take place

**date:**June 1, 2018**time:**13:00 until 15:00 (must arrive before 12:50)**place:**HG E 19, HG E26.3, HG E 27**duration:**120 minutes

There will be weekly homework assignments, which are due in the break between the two hours of lecture on Wednesday, i.e. at 14:15.

Solutions to the theoretical questions can be handed in in the lecture room in paper or scanned and submitted via e-mail before the deadline. Code must be handed in online using the submission interface. Only in case the submission does not work: send your codes via E-Mail to your assistant and contact Lukas Herrmann to update the configuration of the submission interface.

Submissions of problem sheets in a group are not allowed.

Each problem will be marked according to the following scheme:

- 0 - no submission
- 0.5 - incomplete or insufficient submission
- 1 - sufficient submission
- 1.5 - excellent submission.

MUST take the written exam at the end of the semester. Students who acquire at least 70% of the points attainable by doing the weekly problem sheets, i.e., in average 0.7 points per exercise, are given an additive bonus of 0.25 on their final grade (e.g. grade 4.5 (without bonus) will be grade 4.75 (with bonus)).

(i.e. only require a "pass" grade, which includes D-MATH PhD students at ETH) must achieve at least 70% of the maximal number of points attainable by sufficient submission of the weekly homework problem sheets, i.e. in average 0.7 points must be achieved per problem. Students who did not achieve the required percentage of points in the weekly homework problem sheets can still achieve a "pass" by taking the final written exam.

exercise sheet | due by | code templates | solutions |
---|---|---|---|

Exercise sheet 1 | March 07 | templates01 | distributed on March 9 |

Exercise sheet 2 | March 21 | templates02 | distributed on March 23 |

Exercise sheet 3 | March 28 | templates03 | distributed on April 13 |

Exercise sheet 4 | April 11 | templates04 | distributed on April 13 |

Exercise sheet 5 | April 18 | templates05 | distributed on April 20 |

Exercise sheet 6 | April 25 | templates06 | distributed on April 27 |

Exercise sheet 7 | May 2 | templates07 | distributed on May 4 |

Exercise sheet 8 | May 9 | templates08 | distributed on May 11 |

Exercise sheet 9 | May 16 | templates09 | distributed on May 18 |

Exercise sheet 10 | May 23 | templates10 | distributed on May 25 |

time | room | assistant | distribution by surnames |
---|---|---|---|

Fr 14-15 | HG D1.2 | Diyora Salimova (diyora.salimova@sam.math.ethz.ch) and Lukas Herrmann (lukas.herrmann@sam.math.ethz.ch) | A - K |

Fr 14-15 | HG G5 | Lukas Gonon (lukas.gonon@math.ethz.ch) | L - Z |

The main methods of option pricing for efficient numerical valuation of derivative contracts in a Black-Scholes as well as in incomplete markets due to Levy processes or due to stochastic volatility models with emphasis on PDE-based methods are introduced. Further, implementation of pricing methods in MATLAB is developed.

- Foundations and Implementation of efficient valuation of European and exotic contracts on jump-diffusions.
- Enable participants to develop and use MATLAB implementations of these methods for the solution of pricing problems.
- Reformulation of the pricing problem as deterministic partial (integro) differential equation, for general Levy price processes and numerous types of contracts.
- Contracts covered range from algorithms for classical Black-Scholes pricing of European Vanillas to American puts to most recent, advanced methods for pricing in incomplete markets, with prices governed by jump-diffusion processes, and to pricing in the presence of stochastic volatility.
- Modelling, analysis and implementation of the algorithms will be emphasized throughout.

- Continuous time financial modeling: Black-Scholes models, basic types of contracts: European, American call/put (equivalent to the course "Mathematical Foundations of Finance").
- Basic stochastic calculus (Brownian Motion, Ito's Lemma ... ), some knowledge about jump-diffusion processes as e.g., in the book "Financial Modeling with Jump Diffusions".
- Basic Numerical Mathematics (A.Quarteroni, R. Sacco and F. Saleri: Numerical Mathematics, Springer, 2000).
- Basic knowledge of MATLAB.

- Review of option pricing. Wiener and Lévy price process models. Deterministic, local and stochastic volatility models.
- Finite Difference methods for option pricing.
- Finite Difference methods for Asian, American and Barrier type contracts.
- Finite Element methods for European and American style contracts.
- Pricing under local and stochastic volatility in Black-Scholes markets.
- Finite Element methods for option pricing under Lévy processes. Treatment of integro-differential operators.
- Stochastic volatility models for Lévy processes.
- Techniques for high-dimensional problems. Baskets in a Black-Scholes setting and stochastic volatility models in Black-Scholes and Lévy markets.
- Introduction to sparse grid techniques.

Students of ETH can download Matlab via Stud-IDES for free (product name 'Matlab free')

The course will mainly be based on the following book:

- Y. Achdou, O. Pironneau: Computational Methods for Option Pricing, SIAM, 2005.
- R. Cont, P. Tankov: Financial Modelling with Jump Processes, Chapman and Hall, 2004.
- D. Lamberton, B. Lapeyre: Introduction to Stochastic Calculus Applied to Finance, Chapman and Hall, 1997, Chapters 4 and 5.
- T. von Petersdorff, C. Schwab: Wavelet Discretizations of Parabolic Integrodifferential Equations, SIAM J. Numer. Anal. Vol.41, No.1, 2003 (reference for slides 3).
- W. Schoutens: Levy Processes in Finance, Wiley, 2003.
- R. Seydel: Tools for Computational Finance, Springer, 2002.
- P. Wilmott, J. Dewynne, S. Howison: Option Pricing: Mathematical Models and Computation, Oxford Financial Press, 1993.