INSTITUTE OF APPLIED MATHEMATICS
Last Updated:
28/08/2017 - 21:10

IAM614 - Methods of Computational Finance

Credit: 3(3-0); ECTS: 8.0
Instructor(s): Ömür Uğur
Prerequisites: Consent of Instructor(s)

Course Catalogue Description

Numerical Methods for Discrete Time Models: binomial method for options; discrete time optimal control problems. Reminders on Continuous Models: Ito process and its applications in stock market, Black-Scholes equation and its solution; Hedging, Volatility smile. Monte Carlo Method for Options: generating random numbers, transformation of random variables and generating normal variates; Monte Carlo integration; pricing by Monte Carlo integration; variance reduction techniques, quasi-random numbers and quasi-Monte Carlo method. Finite Difference Methods for Options: explicit and implicit finite difference schemes, Crank-Nicolson method; Free-Boundary Problems for American options. Finite Difference Methods for Control Problems: Markov Chain approximation method, elliptic Hamiltion-Jacobi-Bellman equations, computational methods.

Course Objectives

At the end of this course, the student will learn:
  • the basics of fixed income securities and portfolio optimisation under discrete time models
  • European and American type option pricing via Binomial (Lattice or Tree) method
  • how to derive and solve the famous Black-Scholes differential equation for options
  • Monte Carlo methods and variance reduction techniques in option pricing
  • to generate pseudo-random numbers from a given distribution, in particular, normal distribution
  • the basics of numerical solutions of stochastic differential equations, Euler-Maruyama scheme
  • finite-difference methods to solve partial differential equations (PDEs) and apply the techniques in valuation of options
  • the basic principles of pricing American options using PDEs and hence free boundary problems
  • basic principles of control problems

Course Learning Outcomes

Student, who passed the course satisfactorily will be able to:
  • apply basic optimisation algorithms to portfolio management and optimisation problems
  • approximately price simple as well as complex (exotic) options by Binomial method
  • use the famous Black-Scholes pricing formulae for vanilla options that are European type
  • simulate stochastic differential equations using Euler-Maruyama scheme
  • price options by Monte Carlo approach with variance reduction techniques
  • price European and American options using finite difference approximation for the underlying PDE
  • understand basic principles of control problems

Tentative (Weekly) Outline

  1. Fixed Income Securities
  2. Portfolio Optimisation
  3. Option Pricing by Binomial Method
  4. Stochastic Differential Equations
  5. Black-Scholes PDE
  6. Black-Scholes Formulae
  7. Generating Random Samples (Numbers)
  8. Monte Carlo Methods for Options
  9. Variance Reduction Techniques
  10. Finite-Difference Methods for Diffusion Equations
  11. Option Pricing by Partial Differential Equations
  12. Finite Difference Methods for American Options
  13. Finite Difference Methods for Control Problems
  14. Hamilton-Jacobi-Bellman Equations

Course Textbook(s)

  1. Uğur, Ö., An Introduction to Computational Finance, Imperial College Press, 2009
  2. Seydel, R., Tools for Computational Finance, 5th edition, Springer-Verlag, 2012

Supplementary Materials and Resources

  • Books:
    • Paolo Brandimarte, Numerical Methods in Finance and Economics (2nd ed.), 2006
  • Readings:
    • http://www.mathworks.com/support/learn-with-matlab-tutorials.html
  • Resources:
    • MATLAB Student Version is available to download on MathWorks website, http://www.mathworks.com, or METU FTP Servers (Licenced)

More Info on METU Catalogue

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