Math 623 - Computational Finance
Page Updated: 1/3/2008
- Prerequisites: A solid background in probability theory,
differential equations, mathematical finance and computer programming.
- Credit: 3 credit hours
- Recommended Texts: Interest Rate Models: Theory and Practice by Brigo and
Mercurio, Springer 2007.
The Mathematics of Financial Derivatives by
Wilmott, Howison and Dewynne, Cambridge 1998.
Monte Carlo Methods in Finance by Jaeckel, Wiley 2002.
- Background: Sophisticated mathematical methods are
becoming ever more important
in the financial industry. This trend started with the advent of
exchange traded options in the 1970's and the fundamental research
by Black, Merton and Scholes, later awarded the Nobel Prize. Today,
Financial Mathematics is used not only by investment banks
and hedge funds on Wall Street, but also by energy companies and
large corporations in need of managing financial risk.
- Goals:
Using Financial Mathematics (like many branches of applied mathematics)
in practice involves two tasks.
First, one has to develop mathematical models that
accurately describe the ``real'' objects that one wishes to
study. In our case, this typically means finding models, based on
probability theory, for the evolution of stock prices, interest rates
and other underlying financial quantities. It also means to derive
theoretical equations or formulas for prices
of derivative securities (options, caps, etc).
The second task is to get actual numbers out of these equations.
Doing so involves both calibration, ie finding the parameters
in the models (eg the volatility of a stock) from financial data,
and numerically solving the equations that were obtained
theoretically.
While the two tasks cannot, in practice, be separated from each other,
we will in this course emphasize the second of them.
Namely, the students are assumed to be familiar with
the more theoretical parts of mathematical finance
(at the level of Math 542/IOE 552) and we will
focus on the numerical implementing and calibration of
financial models.
- Contents: .
The course has four components. In the first part we study
finite difference methods. This is a technique for solving
partial differential equations (PDEs) numerically. We will apply
this to the Black-Scholes equation as well as PDEs appearing
in fixed income. Both explicit and implicit schemes will
be discussed, as will concepts such as stability and convergence.
After that we will turn to Monte Carlo simulations
a quite general method for computing expected values numerically.
The importance of this method stems from risk-neutral pricing,
an important principle saying that prices of derivative securities
can often be expressed as a the discounted expected payoff under
a risk-neutral measure. In addition to ``vanilla'' Monte Carlo,
which is normally quite easy to understand and implement,
we will study variance reduction techniques, which are often
necessary for obtaining accurate results.
The third part involves lattice and tree methods. The idea behind
this method is to approximate a continuous-time stochastic process
by discrete time process and then compute prices using this
approximate model. Alternatively, lattice models can be used
as financial models in their own right.
Finally, we shall discuss calibration, that is, finding
parameters in the models from available financial data.
As we shall see, calibration to stock prices or options
data is done quite differently.
In the homework, which forms an integral part of the class,
you will implement many of the models yourself in a
computer language of your choosing.
- Grading: The grade for the course will be determined from
performances on 6 homeworks (45%) and a final exam (55%).
Additional References:
- My hand written lecture notes.
- Additional hand written lecture notes.
- Notes on using MATLAB.
postscript
- Numerical Methods for PDE.
Math 572
- Review article on interest rate models.
Review
- Article on the BDT interest rate model.
BDT Model
Winter 2007 Homework and Solutions:
- Homework I.
- Homework I Solutions.
- Homework II.
- Homework II Solutions.
- Homework III.
- Homework III Solutions.
- Homework IV.
- Homework IV Solutions.
- Homework V.
- Homework V Solutions.
- Homework VI.
- Homework VI Solutions.
Exam Schedule:
- Winter 2008 final exam: Thursday April 17, 4.00-6.00 pm in
DENNISON ROOM 455.
The final is a closed book exam. You may bring up to ten one sided A4 pages of notes to the exam. You may also use a calculator but not its memory function.
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