Department of Mathematics

Efficient Numerical Methods for Option Pricing

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Lecturer Dr. Oleg Reichmann
Ort HG G 5
Zeit Thursday 15:00-17:00
Beginnt am 19.09.2013
Vorbesprechung 21.08.2013, 15.00-16.30, HG G19.1
Kontakt Dr. Oleg Reichmann
Voraussetzungen Required:
Completed course in Probability Theory or/and in Numerical Solution of elliptic & parabolic PDEs or hyperbolic PDEs.
Some basic knowledge of R or Matlab or willigness to learn.
Courses in Functional Analysis and/or
Computational Methods for Quantitative Finance and/or Numerical Solution of SODEs.
Beschreibung In recent years, many efficient computational methods in the area of option pricing have been developed. The need for efficient methods stems from the increasing complexity of the considered financial products as well as from the high dimensionality of the considered models. In this seminar different computational methods will be analysed. We will discuss the mathematical formulation of the arising PDEs and SDEs and the analysis of the corresponding approximation schemes. Calibration and hedging shall also be discussed. The aim of the seminar is to give an overview over up-to-date methods and carry out implementations in Matlab or R.

Topics to be discussed include:

- Fourier-cosine methods
- Multilevel Monte Carlo techniques
- Reduced Basis methods
- Sparse Grids
Literatur - Yves Achdou, Govindaraj Indragoby, and Olivier Pironneau, Volatility calibration with American options, Methods Appl. Anal. Volume 11, Number 4 (2004), pp. 533-556.
- Yves Achdou and Olivier Pironneau, Numerical Procedure for Calibration of Volatility with American Options, Applied Mathematical Finance, 2005, vol. 12, issue 3, pp. 201-241.
- C. Kaebe, J. H. Maruhn, E. W. Sachs, Adjoint-based Monte Carlo calibration of financial market models, Finance and Stochastics, September 2009, Volume 13, Issue 3, pp 351-379.
- D. Heath, E. Platen and M. Schweizer: A Comparison of Two Quadratic Approaches to Hedging in Incomplete Markets, Mathematical Finance 11, pp. 385-413.
- D. Heath, E. Platen and M. Schweizer: Numerical Comparison of Local Risk-Minimisation and Mean-Variance Hedging in: E. Jouini, J. Cvitanic, M. Musiela (eds.), "Option Pricing, Interest Rates and Risk Management", Cambridge University Press, pp. 509-537.
- S. B. Hamida and R. Cont: Recovering Volatility from Option Prices by Evolutionary Optimization, Journal of Computational Finance, Vol. 8, No. 4, Summer 2005.
- M.B. Giles: Improved multilevel Monte Carlo convergence using the Milstein scheme, pp.343-358, in Monte Carlo and Quasi-Monte Carlo Methods 2006, Springer, 2008.
- M.B. Giles:Multi-level Monte Carlo path simulation, Operations Research, 56(3):607-617, 2008.
- Haasdonk, B., Salomon, J. Wohlmuth, B.: A Reduced Basis Method for Parametrized Variational Inequalities. SIAM Journal on Numerical Analysis, 50(5):2656-2676, 2012.
- Haasdonk, B., Salomon, J., Wohlmuth, B.: A Reduced Basis Method for the Simulation of American Options. ENUMATH Proceedings, 2012.
- N. Hilber, S. Kehtari , Ch. Schwab, C. Winter, Wavelet finite element method for option pricing highdimensional diffusion market models, SAM Report 2010-01.
- K. Oosterlee and F. Fang: Pricing early-exercise and discrete barrier options by Fourier-cosine series expansions, Numerische Mathematik 114: 27-62 (2009).
- K. Oosterlee and F. Fang: A novel pricing method for European options based on Fourier-cosine series expansions, SIAM J. Sci. Comput. 31: 826-848, 2008.

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