Module Details

The information contained in this module specification was correct at the time of publication but may be subject to change, either during the session because of unforeseen circumstances, or following review of the module at the end of the session. Queries about the module should be directed to the member of staff with responsibility for the module.
Title NUMERICAL ANALYSIS FOR FINANCIAL MATHEMATICS
Code MATH371
Coordinator Dr H Assa
Mathematical Sciences
H.Assa@liverpool.ac.uk
Year CATS Level Semester CATS Value
Session 2016-17 Level 6 FHEQ Second Semester 15

Aims

1.

To provide basic background in solving mathematical problems numerically, including understanding of stability and convergence of approximations to exact solution.

2.

To acquaint students with two standard methods of derivative pricing: recombining trees and Monte Carlo algorithms.

3.

To familiarize students with implementation of n umerical methods in a high level programming language.


Learning Outcomes

1.

Awareness of the major issues when solving mathematical problems numerically.

2.

Ability to analyse a simple numerical method for convergence and stability

3.

Ability to formulate approximations to derivative pricing problems numerically.

4.

Ability to read and understand algorithms in a high level programing language.


Syllabus

1

Basics
- Root finding & minimisation for functions of one and several variables
- Least squares method
- Iterative methods for solving linear systems (Gaussian elimination & LU decomposition assumed known)
- Polynomial interpolation, theorem on error estimates, Runge phenomena, linear spline & error estimate
- Numerical integration: Newton-Cotes formulae and composite methods
- Computer implementations of the above and applications to finance e.g. bond yields and curve fitting

2

Binomial and trinomial tree methods in mathematical option pricing
- Problem specification and model formulation
- Application to pricing European, Asian and American call and put options on one risky asset
- Relation to continuous time models. Convergence.  

3

Basics of Monte Carlo methods in mathematical option pricing
- Convergence in distribution, law of large numbers, central limit theorem
- Numerical integration using Monte Carlo methods.
- Pricing European put and call options using Monte Carlo methods.

4

Numerical methods for ordinary and stochastic differential equations
- Revision of ordinary differential equations (ODEs), existence, uniqueness
- Implicit and explicit Euler methods for ODEs: stability, accuracy, convergence
- Introduction to stochastic differential equations (SDEs)
- Introduction to approximations of SDEs, issues with approximating Ito integral, simulation


Recommended Texts

Reading lists are managed at readinglists.liverpool.ac.uk. Click here to access the reading lists for this module.
Explanation of Reading List:

Pre-requisites before taking this module (other modules and/or general educational/academic requirements):

MATH264; MATH101; MATH102; MATH103; MATH111; MATH162; MATH201  

Co-requisite modules:

 

Modules for which this module is a pre-requisite:

 

Programme(s) (including Year of Study) to which this module is available on a required basis:

Programme:G1N3 Year:3

Programme(s) (including Year of Study) to which this module is available on an optional basis:

 

Assessment

EXAM Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
Written Exam  2.5 hours  100  Standard University Policy    Assessment 1 Notes (applying to all assessments) Written Exam  
CONTINUOUS Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes