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.

For further information see the academic catalog: IAM614

### 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

### Instructional Methods

The following instructional methods will be used to achieve the course objectives: lecture, questioning, discussion, group work, simulation.

### Tentative Weekly Outline

1.  Fixed Income Securities
2. Portfolio Optimisation
3. Option Pricing by Binomial Method
4. Stochastic Differential Equations
5. Black-Scholes PDE and Formulae
6. Generating Random Samples (Numbers)
7. Black-Scholes PDE and Formulae
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)

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

Seydel, R., Tools for Computational Finance, 5th edition, Springer-Verlag, 2012

#### Material(s)

Lecture Notes will be available on ODTUClass (Moddle)

Deriving the solutions for European vanilla options from the Black-Scholes PDE: Chapter 4, The Black-Scholes Equation (Uğur, Ö., Introdution to Computational Finance, Imperial College Press, 2009)

Paolo Brandimarte, Numerical Methods in Finance and Economics (2nd ed.), 2006

#### Resources

If not all, most of the programming will be by using MATLAB language. See the website, http://www.mathworks.com, for more about the software.

A Short introduction to MATLAB is also available on the author's website, http://www.metu.edu.tr/~ougur, of the textbook, An Introduction to Computational Finance

#### Other

MATLAB programs (m-files) of the textbook, An Introduction to Computational Finance, can be obtained from the author's website: http://www.metu.edu.tr/~ougur

There are other software, in place of MATLAB, that are available for free:

• Scilab: Scilab is a scientific software package for numerical computations providing a powerful open computing environment for engineering and scientific applications
• Octave: GNU Octave is a high-level language, primarily intended for numerical computations