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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 Java or matlab are probably the best choices
but if you have some C/C++ knowledge then you may well be able to use
this also.
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:
[6].
3.1 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 [8] and, if you are taking
it, the module MA3976.
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.
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).
Dealing with time varying interest rates and volatility.
Quantitative assessment of the effect of boundary
truncation.
For very high marks you will cover the extension of
the Black-Scholes theory to American options and
implement an approximate solution algorithm in matlab.
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], [8], the paper
[4] and, if you are taking
it, the module MA3976.
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'.
The price of an option can be viewed as an expected value of a
random variable that describes the price of the underlying asset.
By using a computer to simulate this random variable the expected
value can be estimated and hence the price of the option obtained (at
least approximately). This is called a Monte Carlo method because
we are using a game of chance in order to solve a problem. In this
project you will will use matlab to generate the random process
and implement various enhancements that will allow your code
to price various types of financial options. The main starting
point is the book [5].
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
its lognormal statistics.
Matlab simulations of the asset price process. Illustration of
the effect of volatility and drift. Cogent description of the
option price as a discounted expected value and the construction
of the Monte Carlo pricing technique.
Working implementation of the pricing algorithm for a variety
of options. Comparison of estimated price with exact value.
Estimation of the `Greeks'.
For very high marks you will derive confidence intervals for
your computations and implement advanced techniques of variance
reduction based on antithetic variables or control variates.
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. 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
[5] 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.
4.1 Theory and animation of wave propagation in elastic rods
When a long thin rod is struck at one end, when a hammer strikes 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'. 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,
[3] (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.
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
[2,7]
(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.
Desmond J. Higham.
An introduction to financial option valuation; mathematics,
stochastics and computation.
Cambridge University Press, New York, 2004.
