This course is an introduction to the numerical techniques used by both academics
and practioners in Financial Mathematics. It is intended for graduate students
in mathematics, statistics, business, economics or physcal sciences in general,
who have an interest in numerical methods for financial engineering. To motivate
the methods, we will review some key financial problems such as the time evolution
of asset prices, hedging and pricing financial instruments and analysing real data sets,
so in principle no extensive previous knowledge of finance
is required. These problems will then be addressed and solved by the use of
discrete models (binomial and trinomial trees), Monte Carlo methods, numerical
solutions of PDE's and SDE's and maximum likelihood estimators. Although all
the examples treated will be drawn from finance, the course can be taken by
students solely interested in the mathematical side of the techniques above.
A strong component of the course will be the implementation of the computer
routines in MatLab and occasional use of Maple for symbolic calculations.
For the finance part of the course, no formal prerequisites are necessary.
A general knowledge of financial terms such as provided by a cursory reading
of the first eight chapters of the book ''Options, Futures and Other Derivatives'', by J. Hull, should
be more than enough. As for mathematical prerequisites, students should have
taken introductory level courses on partial differential equations, probabilty
theory (discrete state space) and statistics. Previous contact with discrete
time stochastic processes is welcome but not strictly necessary. More advanced
topics like Brownian motion and stochastic differential equation will be introduced
in the course in a pragmatic fashion. Concerning computer skills, we will
provide tutorials and general help with MatLab, but students are expected
to have some previous exposure to writing programms in a high level computer
language such as C, Fortran or MatLab itself.
Here is a list of topics covered in the course. The information in
square brackets refer to book sections where the material can be found at more or less the same level
as we are going to be treating them. We use the following abbreviations for the book authors: B=Brandimarte,
Bj=Bjork, B/R=Baxter/Rennie, D=Duffie, H=Heath, O=Oksendal, WHD= Wilmott et al, S=Shaw.
Part I - Lattice Methods
- One Period Binomial Model [Bj 2.1][B/R 2.1][WHD 10.1]
- Portfolios and Arbitrage [Bj 2.1.2]
- Contingent Claims [Bj 2.1.3]
- Completeness [Bj 2.1.3]
- Risk Neutral Valuation [Bj 2.1.4]
- The Binomial Tree [Bj 2.2][B/R 2.2][WHD 10.2]
- Recombining and non-recombining trees
- The Cox-Rubenstein-Ross (CRR) case [WHD 10.2.1]
- The Jarrow-Rudd (JR) case [WHD 10.2.2]
- Pricing vanilla european options on a tree [WHD 10.4]
- Pricing american options on a tree [WHD 10.5]
Part II - Review of Continuous Time Finance
- Motivation to Stochastic Differential Equations [O 1.1][Bj 3.1]
- Brownian Motion [O 2.2][Bj 3.1]
- Information Flow [Bj 3.2]
- Stochastic Integrals [Bj 3.3]
- Conditional Expectations [Bj 3.4]
- Martingales [Bj 3.4]
- Ito's Formula [Bj 3.5]
- Examples of SDE's
- Geometric Brownian Motion [Bj 4.2]
- Linear SDE's [Bj 4.3]
- Change of Measure and Girsanov Theorem [B/R 3.4]
- Arbitrage [Bj 6.2]
- Risk Neutral Valuation [Bj 6.4]
Part III - Monte Carlo Methods
- Introduction to Monte Carlo Integration [B 4.1]
- Pseudo Random Number Generators [B 4.2.1]
- Generating random variables with a given distribution [B 4.2.2 - 4.2.4]
- Setting the number of replications [B 4.3]
- Variance Reduction Techniques [B 4.4]
- Simulation of SDE's Sample Paths [B 7.1][S 21]
- Pricing vanilla european options [B 7.2]
- Pricing path dependent options [B 7.3]
- Barrier Options [B 7.3.1]
- Asian Options [B 7.3.2]
- Lookback Options [B 7.3.3]
Part IV - Numerical Solutions of Parabolic PDE's
- Introduction and Classification of 2nd Order Partial Differential Equations
- Characteristic Curves [H 11.1]
- Examples [B 5.1]
- The PDE Approach to Finance
- The Cauchy Problem & Probability Theory [Bj 4.5]
- The Feymann-Kac Formula [Bj 4.5]
- The Black-Scholes Equation
- Reduction of the Black-Scholes Equation to a Diffusion Equation [WHD 5.4 and 8.1][S 15.3]
- Finite Difference Methods [WHD 8][S 13][B 5.2]
- Explicit [WHD 8.4][S 14.5][B 5.3.1]
- Fully Implicit [WHD 8.6][S 14.7][B 5.3.2]
- Crank-Nicholson [WHD 8.7][S 14.8][B 5.3.3]
- Examples in Option Valuation [B 8]
- Free Boundary Conditions [WHD 9][S 16]
- Linear Complementarity, PSOR [WHD 9.2]
- American Options [B 8.5]
- Convergence and Stability [B 5.4]
We are not going to be following the integrity of any particular book. If the students wish to purchase a book, the
following is the closest to what we are going to be doing, both in spirit and content:
- BRANDIMARTE, Paolo, Numerical Methods in Finance: a MatLab-based introduction, Wiley Series
in Probability and Statistics, 2002.
A highly useful compilation and (most importantly) critical view of all sorts of numerical procedures encountered in
Finance, emphasizing the role of Mathematica
for symbolic calculation, is
- SHAW, Willian, Modelling Financial Derivatives with Mathematica, Cambridge University
The Numerical Analysis developed in the course will be at the level of the following two books:
- BURDEN, Richard L. and FAIRES, J. Douglas, Numerical Analysis,
- HEATH, Michael T., Scientific Computing: an introductory survey, McGraw-Hill, 2002.
For the Financial Mathematics background needed for the course, we are going to be drawing material mostly from
- BJORK, Thomas, Arbitrage Theory in Continuous Time, Oxford University Press, 1998.
The following are other introductory level books in Financial Maths which might also help to understand the background:
- BAXTER, Martin and RENNIE, Andrew, Financial Calculus: an introduction to derivative pricing,
Cambridge University Press, 1998.
- GOODMAN, Victor and STAMPFLI, Joseph The Mathematics of Finance: modelling and hedging, The
Brooks/Cole Series in Advanced Mathematics, 2001.
- HULL, John, Options, Futures and Other Derivatives, Prentice Hall, 2000.
- WILMOTT, Paul, HOWISON, Sam and DEWYNE, Jeff, The Mathematics of Financial Derivatives: a student
introduction, Cambridge University Press, 1998.
These last two books are mathematically more demanding but contain material which will be essential for
the students seeking for a deeper view on subject:
- OKSENDAL, Bernt, Stochastic Differential Equations: an introduction with applications,
- DUFFIE, Darrel, Dynamic Asset Pricing Theory, Princeton University Press, 1996.
The final mark will be calculated based on the solutions for the assignment questions posted during the course (40%)
as well as a final project. The assignments should be solved individually, whereas the
final project can be done in groups of up three students. It will consist of the implementation of the methods studied in
the course to the solution of an specific practical problem of financial engineering. Marks will be given to the
fully operating computer code (20%) solving the problem, a short written essay explaining the techniques used, accuracy,
estimate of computing time, etc (20%) and an industrial style oral presentation conveying it to the rest of the class (20%).
A timetable for returning the projects at the end of the course, as well as for the presentations, will be decided in class.
I will be suggesting project ideas along the lectures, and students are encouraged to contact me to discuss their
These were the final projects presented by students taking the course for credit. Click on the titles to see
the pdf form of the submitted written versions when available.