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

Start of Lectures: Wednesday, September 21, 2011.

Start of Exercises: Friday, September 30, 2011.

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 a ''TESTAT''.
A ''TESTAT'' constitutes of ''successful participation in course''.
It is NOT a numerical grade. The ''TESTAT'' is given if correct solution of at least
70 per cent of COURSE HOMEWORK ASSIGNMENTS has been achieved.

The exam will take place on Friday, December 16, 2011 9.00-11.00 am.

VVZ-Link

Examination Details:

The 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. An own computer is NOT required for the examination.

Aims of the course

• Theory and 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.
  As well as 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 and
  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 underlyings and baskets,
  error analysis and implementation.
• Application to computational finance:
  Option pricing with
  Black Scholes (BS) market models,
  no arbitrage principle, changes of measure.
  Basic types of derivative contracts:
  plain vanilla, barrier, European, Asian.
  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 the 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

• R. Cont & P. Tankov
  Financial Modelling With Jump Processes.
  Chapman & Hall/CRC Financial Mathematics Series, Boca Raton 2004.
  
• I. I. Gihman & A. V. Skorohod
  Stochastic differential equations, Springer Publ. 1972.
• P. Glassermann:
  Monte Carlo Methods in Financial Engineering, Springer Publ. 2004.
• J. S. Liu
  Monte Carlo Strategies in Scientific Computing, Springer Publ. 2001.
• P. 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.
• S. Asmussen & P. 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.
• S. M. Ross
  Simulation, Academic Press, 2006.