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Due to font typesetting problems with mathematical symbols,
printing this web page is NOT recommended.
Where possible a PostScript and/or PDF
file is available for hardcopy download.
Currently available projects are listed in the following sections.
Each one indicates whether or not it is purely theory, or a mix of
theory and computing. In some cases the mix between computer programming
and theory can be varied to suit the student's preference and/or course
but the computing aspect of the project should not be regarded as
optional. For these projects matlab is probably the best choice
but if you have some Java, C/C++ or other knowledge then you may well be
able to use those also. This will need to be discussed.
Note that a significant number of marks are allocated or closely connected
to the quality of the written document. For a guide on how to
`write mathematics' at a professional level see Higham's book:
[7].
3.1 Four Projects on the Numerical Solution of the Black-Scholes Partial
Differential Equation
In the `Black-Scholes world' the value of an option can be found by solving
the Black-Scholes partial differential equation (BSPDE). Although
exact solutions are known for the simplest types of option
(European calls and puts for example), the BSPDE cannot be solved analytically
for many types of exotic option. In such cases
it needs to be solved numerically using, say, finite
difference methods and matlab. The main reference for this project
is the book [9] and, if you are taking
it, the module MA3667 (the MA3976 replacement).
Broadly speaking, your final marks
will reflect how far down this list of achievements you
successfully go:
Discovery of relevant literature and theory.
Graphical illustration of prices for vanilla
call and put options as given, for example, at
en.wikipedia.org/wiki/Black-scholes.
Finite difference approximation of the BSPDE.
Matlab implementation and verification of accuracy
using options for which the exact solutions are
known.
High quality graphical output from the BSPDE solver
and its extension to various option strategies (e.g.
straddles, strangles and butterfly spreads).
Successful description and implementation of the numerical solution
for one of the following.
For very high marks you will cover advanced material such as, for example,
detailed analyses of the numerical methods, dealing with time varying interest
rates and volatility, quantitative assessment of the effect of boundary
truncation and so on. This type of advanced study will be agreed on a
case-by-case basis between the student and the supervisor.
Binomial trees provide an approximate means of pricing
financial options that does not involve solving the
Black-Scholes partial differential equation. The basic
idea is to build a recombining binary tree of possible
asset prices at discrete times that extend from now until
the expiry date of the option. At expiry the value of the
option (the known payoff) is calculated at the
tree's final node layer, and then
discounted risk-neutral expectation is used
to recursively value the option backward in time until
the current time is reached. The main references for
this project are the books
[1], [9], the paper
[5] and, if you are taking
it, the module MA3667 (the MA3976 replacement).
You can also look at
en.wikipedia.org/wiki/Binomial_options_pricing_model
to get a flavour of what is involved.
Broadly speaking, your final marks
will reflect how far down this list of achievements you
successfully go:
Discovery of relevant literature and theory.
Graphical illustration of prices for vanilla
call and put options.
A recap of the Black-Scholes theory, the partial
differential equation and its solution as a
risk-neutral discounted expectation.
A description of the binomial world and the
population of the tree with log-normal asset
prices. Implementation (e.g. matlab) is required.
An implementation of backward recursion to determine
the current value of a European option given the
payoff function. Your code should generate high
quality graphical output.
For very high marks you will decribe and implement
the extension of this theory to American options
and also use your computer progam to estimate
all of the so-called `greeks'.
In the Black Scholes theory of option pricing the price of the
underlying asset is assumed to follow a lognormal random walk
with the `randomness' controlled by a parameter, s, called
the volatility. The value of this parameter is important for the
accuracy of the option price but is not observable in the market.
In this project you will study two common ways to obtain this
parameter for a given asset. The first is the use of Newton's
method where, given the option price and all other observable
data, the volatility is given by finding the root of a nonlinear
equation (see e.g. en.wikipedia.org/wiki/Implied_volatility).
The second is the use of historical asset price data
where one hopes to extract this volatility as a standard deviation
of past prices. The main starting point is the book
[6] and we note that historical data adequate
for the purposes of this project can be obtained from,
for example, uk.finance.yahoo.com/q/hp?s=VOD
Broadly speaking, your final marks
will reflect how far down this list of achievements you
successfully go:
Discovery of relevant literature and theory. Comprehension of
the asset price process used in the Black-Scholes theory and
of its lognormal statistics.
Cogent discussion of Newton's method for implied volatility
and a demonstrably correct matlab implementation.
Acquisition of a selection of historical data and a matlab
implementations of historical volatility calculations.
For very high marks you will give the theoretical details behind
the robustness and convergence of Newton's mathod for this problem
as well as provide confidence intervals and further theoretical
developments for the historical voilatility calculation.
The orbit of the earth around the sun is described by
the branch of mechanics known as Central Force Theory
and is encapsulated in Kepler's laws. Analytical solutions are
known for the case of one large body (the sun) and one small
body (the earth). When the other planets are brought into
the picture the differential
equations of Newton's theory of gravity can, in general, no
longer be solved analytically and numerical methods (using,
for example, matlab) must be used. The simulated solutions
can then be used to produce animations. See the links
at the bottom of
www.ams.org/featurecolumn/archive/orbits1.html
for some examples of what you can achieve.
Broadly speaking, your final marks
will reflect how far down this list of achievements you
successfully go:
Discovery of relevant literature and theory.
Coverage of classical central force theory and Kepler's
laws (e.g. sun and earth), together with computer animations.
Extension of the theory to `many (N) body problems' in
two-dimensions, each body having a different mass and position.
Numerical simulation of the N-body problem with computer
animations.
For very high marks you will extend the N-body theory,
algorithm and graphics to 3D, use an implicit time stepper and compare
its performance to an explicit one,
and you will illustrate the effect of discretisation stability and error
on the physics.
Involves: theory and computing.
4.2 Theory and Animation of Wave Propagation in Elastic Rods
When a long thin rod is struck at one end, a hammer striking a
nail for example, the impact is not felt instantaneously along the
rod but rather travels through it at a fixed speed as an impact
stress wave. Once it reaches the other end it will typically bounce
off and travel back the way it came. The mathematics of this problem
consists, essentially, of Hooke's law of elasticity and Newton's
second law of motion. Once combined they produce the partial
differential equation known as the `wave equation' (see e.g.
en.wikipedia.org/wiki/Wave_equation). In this project you
will study this background theory and derive the exact solution to
some representative examples. These solutions will be coded in matlab
and graphical animations will be produced in order to illustrate the
travelling waves. The main starting reference is Graff's book,
[4] (Brunel library: QC176.8.W3G73).
Broadly speaking, your final marks
will reflect how far down this list of achievements you
successfully go:
Discovery of relevant literature and theory. Coverage of elasticity
theory and derivation of the basic equation.
Review and development of appropriate Laplace transform theory.
Derivation of exact solutions to some model problems
Illustration of the derived solution in matlab. Graphical animation
of the solution as well as derived quantities such as stress and
strain. Discussion of properties of the exact solution.
For very high marks you will have extended this work to more
challenging wave propagation problems (e.g. involving dispersion)
and give a deep consideration to the derivation and illustration
of the solution.
Involves: theory and computing.
4.3 Theory and Computation of Impact Stresses in Elastic Tubes
What happens if all of the people in a lift jump up and land together?
The sudden force felt by the lift cable is much larger than just
the weight of the people. The `impact' causes a magnification of stress.
This is the reason why a hammer can drive a nail into wood, why
a sudden twist can undo a stubborn bottle top and why even a low
speed collision can kill and maim. The aim of this project is for the
student to discover the relevant mathematics
and mechanics of impact through a
literature search, explain a few illustrative models and - if time
allows - use software approximations to the underlying equations
so as to obtain approximate solutions to realistic problems. A
particular example is when a steel tube, clamped somewhere along its
length, is struck suddenly at one end. What happens at the other end?
What effect does the clamp size and postion have? And does how
much difference does the tube's wall thickness make? To answer
this means considering the problem as a two-space dimensional
system of partial differential equations in polar coordinates
and then employing a computer algorithm (using the finite element
technique) to approximate these physics. Matlab can be used
to write this code, although other choices of programming languages
can be considered after proper discussion.
To get some idea of the basic ideas behind impact you can take
a look at the old book by Case and Chilver (if you can find
it anywhere):
[2], or at Hibbeler's book Mechanics of Materials
(Brunel library: TA405.H47 2008).
These online notes www.freestudy.co.uk/dynamics/impulse%20and%20momentum.pdf
also contain some useful starting material.
Broadly speaking, your final marks
will reflect how far down this list of achievements you
successfully go:
Discovery of relevant literature and theory. Coverage of elasticity
theory and derivation of the basic equations of impact.
Cogent explanations and derivations of a set of interesting and
realistic examples with, possibly, some supporting code and graphics.
Demonstrated appreciation of how the partial differential equation
known as the wave equation governs impact waves in a 1D structure.
For very high marks you will have extended this work to more
challenging wave propagation problems in a tube by using a
finite element code. This will be supplied. You will use and
configure it for examples, but are not required to understand
how it is derived and from what.
Pursuit problems arise in mechanics when one
particle (A), travelling on a given trajectory, is pursued
by another (B) in such a way that B's velocity is always
toward A. Examples include a missile homing in on a plane
in flight, a robot `hand' trying to pick up a moving object,
or even a dog chasing a rabbit. The mathematical
model of a pursuit problem is a system of differential
equations that may, or may not, be amenable to analytic
solution. This project will convey the theory of pursuit
problems and use matlab to arrive at numerical solutions to
some example problems as well as to produce animations. For an
example of what is possible see the java applet at
curvebank.calstatela.edu/pursuit2/pursuit2.htm
(and imagine how much more interesting it would be if the green
particle travelled on a curved path between wall bounces).
Two starting references are
[3,8]
(Brunel library: Chorlton, QA846.C45; Smith & Smith, QA801.S63 1990)
but more focussed supervisory guidance will be given once the
project commences.
Broadly speaking, your final marks
will reflect how far down this list of achievements you
successfully go:
Discovery of relevant literature and theory.
Coverage of pursuit theory in 2D and worked solutions to some
example problems. Graphical illustrations.
Numerical solutions and animations of the pursuit curves
for 2D problems where A travels a straight line.
Extension of the above to the case where A travels a
non-straight, perhaps even random, trajectory.
For very high marks you will have extended the pursuit theory,
algorithm and graphics to 3D, and you will illustrate the
effect of discretisation error on the physics.
John Case and A. H. Chilver.
Strength of materials and structures: an introduction to the
mechanics of solids and structures.
Edward Arnold (Publishers) Ltd., second edition, 1981.
Desmond J. Higham.
An introduction to financial option valuation; mathematics,
stochastics and computation.
Cambridge University Press, New York, 2004.
