## Courses

##### Last Updated:

10/03/2019 - 15:02

## 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

- Fixed Income Securities
- Portfolio Optimisation
- Option Pricing by Binomial Method
- Stochastic Differential Equations
- Black-Scholes PDE
- Black-Scholes Formulae
- Generating Random Samples (Numbers)
- Monte Carlo Methods for Options
- Variance Reduction Techniques
- Finite-Difference Methods for Diffusion Equations
- Option Pricing by Partial Differential Equations
- Finite Difference Methods for American Options
- Finite Difference Methods for Control Problems
- Hamilton-Jacobi-Bellman Equations

#### Course Textbook(s)

- Uğur, Ö., An Introduction to Computational Finance, Imperial College Press, 2009
- 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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