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 E Azmoodeh
Mathematical Sciences
Ehsan.Azmoodeh@liverpool.ac.uk
Year CATS Level Semester CATS Value
Session 2021-22 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 familiarise students with sample generating methods, including acceptance-rejection and variance reduction, and its application in finance


Learning Outcomes

(LO2) Ability to analyse a simple numerical method for convergence and stability

(LO3) Ability to formulate approximations to derivative pricing problems numerically.

(LO4) Ability to generate a sample for a given probability distribution and its use in finance

(LO5) Awareness of the major issues when solving mathematical problems numerically.

(S1) Problem solving skills

(S2) Numeracy


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

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 equa tions - 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.

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

 

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(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
final assessment open book and remote There is a resit opportunity. Standard UoL penalty applies for late submission. This is an anonymous assessment. Assessment Schedule (When) :Examinatio  1 hour time on task    50       
CONTINUOUS Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
class test open book and remote Standard UoL penalty applies for late submission. This is not an anonymous assessment. Assessment Schedule (When) :bi-weekly  around 60-90 minutes    50