Numerical Methods in Financial Engineering

Objective

Participants of this course will master computational techniques frequently used in mathematical finance applications.

Prerequisites: Stochastic processes / stochastic calculus, Numerical Analysis, Derivatives, Command of Python

TDs: In the form of ipython notebooks

Planning

1. Brief introduction to asset price modeling and option pricing

• Basic stochastic models in finance
• Overview of numerical methods for option pricing

2. Monte Carlo methods for pricing

• Basics of MC simulation and price estimation in the Black-Scholes Model
• Variance reduction techniques
• Euler schemes and applications to pricing in local volatility models
• Computing greeks

3. Option pricing via PDEs

• Finite difference approximation of Black-Scholes PDE
• American options and free boundary problems

4. Transformation based methods

• Affine models
• Option pricing via Fourier transforms

5. Probabilistic numerical methods for non-linear pricing

• American options (Tsitsiklis-Van Roy, Longstaff-Schwarz methods)
• Market with imperfections and non-linear pricing (BSDEs)
• Dynamic programming vs. shooting method (Non linear regression)

References

·      Achdou, Yves; Pironneau, Olivier. Computational methods for option pricing. Frontiers in Applied Mathematics, 30. SIAM, Philadelphia, PA, 2005.

·      Björk, Tomas. Arbitrage theory in continuous time. Third edition, OUP Oxford, 2009.

·      Gatheral,  Jim. Volatility Surface: A Practitioner’s Guide. Wiley, 2006.

·      Glasserman, Paul. Monte Carlo methods in financial engineering. Springer, 2003.

·      Hilber, Norbert; Reichmann, Oleg; Schwab, Christoph; Winter, Christoph. Computational methods for quantitative finance. Springer, 2013.

·      Hirsa, Ali. Computational methods in finance. Chapman & Hall/CRC Financial Mathematics Series. CRC Press, Boca Raton, FL, 2013.

·      Lamberton, Damien; Lapeyre, Bernard. Introduction to stochastic calculus applied to finance. Second revised edition. Chapman & Hall/CRC, 2008.

·      Seydel, Rüdiger U. Tools for computational finance. Fourth edition. Universitext. Springer-Verlag, Berlin, 2009.

·      Shreve, Steven E. Stochastic calculus for finance II: Continuous-Time models, Volume 11. Springer Science & Business Media, 2004.

·      Additional lecture material will be provided by the instructors.