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401-4657-00L 6 Credits BSC , MSC , NDS D-MATH
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Computational Methods for Quantitative Finance I: Monte Carlo and Sampling Methods

Lecturers: Prof. Dr. Christoph Schwab, Prof. Dr. Tobias von Petersdorff
Examiners: Prof. Dr. Christoph Schwab
VVZ CR n/a

Last Updated: 2026-02-05 15:23:57

Abstract

Random Number Generation and Monte Carlo Error Estimation.Numerical Solution of SDEs I: Diffusion Driven Ito-SDEs for Black-Scholes Markets -Implementation and Convergence Analysis.Numerical Solution of SDEs II: Jump Diffusions and Levy Driven SDEsImplementation and Convergence Analysis.Variance Reduction, Quasi MCMethods for Barrier Contracts and Exotic Contracts

Objective

Mathematical Theory and Computer Implementation of Random Number Generators, Error Analysis of Monte Carlo Methods, Numerical Solution of Ito-SDEs with Diffusion, Jump-Diffusion and Levy Noise driving processes: fast generation of Levy increments. Implementation of SDE-integrators and convergence analysis. Valuation of basic derivative contracts [European vanilla, barrier, Asian] on possibly large baskets under complete (Black-Scholes) as well as under incomplete market models: basic financial theory and efficient numerical valuation. Advanced computational techniques: Variance Reduction techniques, Quasi Monte Carlo methods. Sparse Tensor Product Sampling Techniques.

Content

Contents (tentative): Basic Monte-Carlo (MC) Techniques: Random Number Generators, MC for a scalar random variable (RV): Implementation and error estimation. MC for stochastic processes: Markov Processes: Wiener, Poisson, Compound Poisson, Levy Processes (single and multivariate), Path regularity. Simulation and MC for these processes. Application to pricing of the basic contracts, single underlying and baskets, Error analysis and computer implementation. Introduction to Option Pricing: Black Scholes (BS) Market Model, No arbitrage principle, Changes of Measure. Basic types of derivative contracts: plain vanilla, barrier, Europeans, Asians. Incomplete markets and equivalent martingale measures. Numerical Solution of SDEs I: MC for Diffusion Driven Ito-SDEs: Theory of Ito-SDEs, Numerical solution: Euler-Maruyama, Milstein. Weak, strong and pathwise convergence rates. Implementation: MC based Option Pricing in Black-Scholes Setting. Numerical Solution of SDEs II: Jump Diffusions and Levy Driven SDEs: Theory of Levy SDEs: Existence, Path properties, Flow and Semigroup Numerical solution: Euler-Maruyama. MC based Option Pricing in Incomplete Markets. Implementation and Convergence Analysis. Application to option pricing in Levy markets. Convergence Acceleration for MC: Variance Reduction, Extrapolation Techniques Quasi MC, Adaptive Sampling Methods,

Resources

Literature

Soren Asmussen and Peter W. Glynn: Stochastic Simulation: Algorithms and Analysis. Springer Publ. 2007. ISBN 038730679X, 9780387306797 Rama Cont & Peter Tankov: Financial Modelling With Jump Processes. Chapman & Hall/CRC Financial Mathematics Series, Boca Raton 2004, ISBN 1-5848-8413-4 P. Glassermann: Monte Carlo Methods in Financial Engineering, Springer Publ. 2004. Philip E. Protter: Stochastic Integration and Differential Equations, 2nd Ed., Springer Publ. 2004.

General Information

Language
English
Levels
BSC , MSC , NDS
Frequency
Yearly recurring

Examination

Type
session examination
Mode
oral 20 minutes

Course Components

Type Title Time & Place Hours
lecture Computational Methods for Quantitative Finance I: Monte Carlo and Sampling Methods
  • Fri 13:15-15:00 (HG F 3)
2 h weekly
exercise Computational Methods for Quantitative Finance I: Monte Carlo and Sampling Methods
  • Wed 15:15-16:00 (HG F 3)
1 h weekly

Offered In