Numerical Analysis of Stochastic ODEs (Comp. Meth. Quant. Fin. I: Monte Carlo Methods)

Start of Lectures: Wednesday, September 22, 2010.

Start of Exercises: Friday, October 1, 2010.

Grading Policy: 

A NUMERICAL GRADE for the course is based only on the written End-of-Semester final examination.
Participation in the written End-of-Semester final examination requires ''TESTAT''.
A ''TESTAT'' constitutes ''successful participation in course''.
It is NOT a numerical grade. ''TESTAT'' is given if correct solution of at least
70 per cent of COURSE HOMEWORK ASSIGNMENTS has been achieved.

VVZ-Link

Examination Details:

End-of-Semester examination will be *closed book*, 2hr in class,
and will involve theoretical as well as MATLAB programming problems.
Examination will take place on ETH-workstations running MATLAB
under LINUX. Own computer will NOT be required for the examination.

Aims of the course

• Theory and Computer Implementation
  of Random Number Generators, Mathematical
  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 (Levy) market models.
• Applications of stochastic ODEs in the sciences:
  finance (option pricing), chemistry and biology
  (master equation), material science (polymeric flow models).

Prerequisites

• Mandatory:
  Elementary Probability, Probability Theory I, Found. Math. Finance.
• Recommended: course on
  Introduction to Parallel Computing, Stochastic Processes.

Contents

• 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 of processes.
  Simulation and MC for these processes.
  Application to pricing of basic financial contracts
  (call, put, european, american, asian),
  on single underlying and baskets,
  Error analysis and computer implementation.
• Application to Computational Finance:
  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 SODEs I:
  MC for Ito-SDEs:
  Existence, Uniqueness of solutions of Ito-SODEs,
  Numerical solution: Euler-Maruyama, Milstein and
  higher order schemes,
  weak, strong and pathwise convergence.
  Applications:
  MC based Option Pricing in Black-Scholes Setting.
  Stochastic Volatility Models.
  Master Equation.
• Numerical Solution of SODEs II:
  Jump Diffusions and Levy Driven SDEs,
  Theory of Levy SDEs: Existence, Path regularity,
  Numerical solution: fast increment generation,
  Euler-Maruyama, extrapolation.
  Application:
  MC based Option Pricing in Incomplete Markets.
• Convergence Acceleration techniques for MC:
  Variance Reduction, Extrapolation Techniques
  Quasi MC, Adaptive Sampling Methods,

Matlab Links

Matlab Primer

Practical Introduction to Matlab

Literature

• Rama Cont & Peter Tankov:
  Financial Modelling With Jump Processes.
  Chapman & Hall/CRC Financial Mathematics Series, Boca Raton 2004,
  ISBN 1-5848-8413-4
• I. I. Gihman & A. V. Skorohod
  Stochastic differential equations, Springer Publ. 1972.
• P. Glassermann:
  Monte Carlo Methods in Financial Engineering, Springer Publ. 2004.
• Jun S. Liu
  Monte Carlo Strategies in Scientific Computing, Springer Publ. 2001.
• Philip E. Protter:
  Stochastic Integration and Differential Equations, 2nd Ed., Springer Publ. 2004 (additional).
• G. S. Fishman
  Monte Carlo: Concepts, Algorithms, and Applications, Springer Publ. 1996.
• Soren Asmussen & Peter W. Glynn
  Stochastic Simulation: Algorithms and Analysis, Springer Publ. 2007.
• X. Mao & C. Yuan
  Stochastic Differential Equations With Markovian Switching, World Scientific Pub. 2006.
• P. E. Kloeden & E. Platen
  Numerical Solution of Stochastic Differential Equations, Springer Publ. 1992.