COMP0043 Numerical Methods for Finance
This database contains the 201819 versions of syllabuses. These are still being finalised and changes may occur before the start of the session.
Syllabuses from the 201718 session are available here.
Academic session 
201819 
Module 
Numerical Methods for Finance 
Code 
COMP0043 
Module delivery 
1819/A7P/T1/COMP0043 Postgraduate 
Related deliveries 
None 
Prior deliveries  
Level 
Postgraduate 
FHEQ Level 
L7 
FHEQ credits 
15 
Term/s 
Term 1 
Module leader 
Germano, Guido 
Contributors 
Germano, Guido 
Module administrator 
Nolan, Martin 
Aims
An introduction to numerical/computational methods with code examples (Matlab, C++, Python) and an emphasis on applications in finance (derivatives pricing, model calibration).
Learning outcomes
Programming proficiency and demonstrable skills in turning mathematical equations and models into working code; capacity to solve practical problems in financial mathematics applying modern numerical techniques.
Availability and prerequisites
This module delivery is available for selection on the belowlisted programmes. The relevant programme structure will specify whether the module is core, optional, or elective.
In order to be eligible to select this module as optional or elective, where available, students must meet all prerequisite conditions to the satisfaction of the module leader. Places for students taking the module as optional or elective are limited and will be allocated according to the department’s module selection policy.
Programmes on which available: 

Prerequisites: 

Content
 Introduction: Bibliography, programming languages, programming basics: data types, operators, expressions, control structures (iteration i.e. forloop, conditional execution i.e. ifthenelse, etc.), vector/array operations, input/output, plots, etc.
 Fundamental probability distributions: Normal, exponential, lognormal, chi square, etc; plot of the probability distribution function, sampling with pseudorandom numbers, histograms, transformation from uniform to other distributions using the quantile function, i.e. the inverse cumulative distribution function.
 Random numbers: Linear congruential generators, requirements and statistical tests, pathologic cases, more advanced generators; inversion and transformation in one and more dimensions, acceptancerejection method, BoxMuller method for normal deviates, polar method by Marsaglia, Ziggurat algorithm by Marsaglia and Tsang, correlated normal random variates, quasirandom numbers.
 Monte Carlo methods: Introduction, riskneutral valuation of options, EulerMaruyama method for the numerical solution of a stochastic differential equation (SDE), approximation error, strong and weak solution, Milstein method.
 Important stochastic differential equations: Arithmetic and geometric Brownian motion, OrnsteinUhlenbeck process and the Vasicek model, CoxIngersollRoss process, constant elasticity of variance processes, Brownian bridge, Heston model of stochastic volatility. Model calibration.
 Stochastic processes with jumps: Poisson process, normal compound Poisson process, Gamma process, jumpdiffusion processes (Merton, Kou), timechanged Brownian motion (variance Gamma process), Lévy processes.
 BlackScholes option pricing: A simple program that prices European calls and puts with the analytical solution, the analytical solution provided by Matlab's Financial Toolbox, the Fourier transform method, and Monte Carlo.
 Model calibration: Implied volatility, NewtonRaphson method, Jäckel's equivalent form, Jäckel's modification, complex initial guess and fractals.
 Fourier transform methods: Definitions of the Fourier transform, inverse transform, notable transform pairs (normals, double exponential/Lorentz, Dirac delta/1), discrete and fast Fourier transform, Laplace transform, transform of the derivative, solution of the standard diffusion equation by Fourier transform and in FourierLaplace space, fractional derivatives, spacetime fractional diffusion equation and its solution in FourierLaplace space, characteristic function, momentgenerating function, cumulantgenerating function, Lévy processes, correlation/convolution theorem, auto/crosscovariance and correlation, Parseval/Plancherel theorem, shift theorem, use in option pricing.
 Exotic options: Fourier methods for the numerical pricing of discretely and continuously monitored pathdependent options like barrier and lookback.
 Partial differential equations: Classification, secondorder PDEs, notable examples of elliptic, parabolic and hyperbolic PDEs, diffusion equation, BlackScholes equation, FeynmanKac theorem and relationship with SDEs, finite difference schemes.
A reading list is available on http://readinglists.ucl.ac.uk/departments/comps_eng.html.
Delivery
The module is delivered through a combination of lectures (3 hours per week for 10 weeks), homework assignments, and selfdirected learning.
Assessment
This module delivery is assessed as below:
# 
Title 
Weight (%) 
Notes 
1 
Written examination (2hrs 30mins) 
60 

2 
InClass Test 1 (1hr 15mins) 
20 

3 
InClass Test 2 (1hr 15mins) 
20 

In order to pass this module, students must achieve an overall weighted mark of 50%.