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Education, Master
Class, Master Class 2000/2001, Detailed Course Content In this course an introduction will be given to the fundamental economic and financial principles which form the corner stone of modern financial theory. In the first half of the course the modelling, pricing and hedging of derivative contracts will be treated using partial differential equation methods. Special emphasis will be given to the analysis of American-style options. In the second part of the course, martingale methods will be used to give a concise yet thorough introduction to the economical theory of pricing in discrete time models, and especially to the pricing theory for contingent claims. Lecturers: Prof.dr. Arun Bagchi and Dr. Michel Vellekoop (Twente) Course Material:
An important part of the applications of mathematics in finance deals with the modelling, analysis and implementation issues of interest rate models. The goal of this course is twofold:
Key concepts include: Coupon Bonds, Caps and Floors, Swaps, Short Rate Models (Vasicek, Ho-Lee, CIR, Hull-White), Forward Rate models and the Heath-Jarrow-Morton equation. Lecturers: Prof.dr. Arun Bagchi and Dr. Michel Vellekoop (Twente) Course Material:
In this course we
will first lay the measure-theoretic foundations of probability theory.
Concepts such as events, probability, random variable, expectation, conditional
probability, will be put on a firm axiomatic ground. We will then study
martingales, a type of stochastic processes originally invented asmodels
for fair games and presently of great importance in mathematical finance.
The main cornerstone of discrete-parameter martingale theory, such as
optional sampling, martingale transforms, stopping times and convergence
result, will be presented, along with a number of examples and applications. Course material:
Financial data such
as prices of risky assets (share prices, stock indices, exchange rates)
are time series. The analysis of financial time series is grounded on
the same theoretical bases as general time series analysis. Therefore
a large part of the course will be devoted to the analysis of general
time series. We first consider standard models (stationary processes,
ARMA (autoregressive moving average) processes) and study their dependence
structure in terms of the autocorrelations and autocovariances as well
as their periodic behavior in terms of their spectral distribution. Then
special attention is given to standard financial time series models, including
the ARCH (autoregressive conditionally heteroscedastic) processes and
the stochastic volatility models. The latter two classes of processes
are most popular in applications. The statistical analysis of such processes
(fitting of models, estimation of parameters and order) and their prediction
are addressed. In addition to these topics of classical time series an
introduction to the extremal behavior of time series is given. The theory
is illustrated by examplesd of simulated and real-life data. Course material: Stochastic Calculus Stochastic calculus
is based on the 'stochastic integral' of It\^{o}, in which integration
is defined w.r.t. a stochastic process (e.g. Brownian motion). This leads,
in a somewhat surprising way, to a new type of differential calculus.
With the help of this calculus, stochastic differential equations can
be solved. Many stochastic processes satisfy such equations. Examples
are: diffusion processes (in physics and chemistry), electronic signals
with noise, transport and queueing systems, stock market and options,
insurance policies. The course develops the basic theory and describes
some applications. References: |