Computational Methods for Quantitative Finance: PDE Methods

Exam date:
Location and time: HG E19 and HG E 27, Wednesday, May 30, 2012.

Question Time : Wed. 15-16 in HG G 52.6 and Thu. 14-15 in HG G 50.1

Grading Policy:

There will be 12 homework assignments. A passing grade (Testat) will require at least 9 solved assignments.
To get the ECTS credit points, all students (except PhD students) must participate in the end-of-semester exam and must pass it.
Participation in the exam will require the Testat.

Solved exercises can be turned in during the exercise class or into the box
located at the entry to room HG G53.

Aims of the course

Introduce 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. Develop implementation of pricing methods in MATLAB.

• 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.

Prerequisites

• Continuous time financial modeling: Black Scholes world, basic types of contracts: European, American call/put
• 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.

Contents

• 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.

Matlab Links

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

• Matlab Online Documentation from MathWorks
• Matlab Primer

Literature

• 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.
• M. Guenther, A. Juengel, Finanzderivate mit MATLAB, Vieweg, 2003
• 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.