Numerical Methods in Financial Engineering
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
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. Density approximation techniques
- Polynomial models and calculation of moments
- Option pricing via density approximation
6. Rough Volatility
- Motivation for rough volatility models and examples
- The rough Heston model
7. 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)
· 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.