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.

- 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]

- 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]

- 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]

- 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]

- BRANDIMARTE, Paolo,
**Numerical Methods in Finance: a MatLab-based introduction**, Wiley Series in Probability and Statistics, 2002.

- SHAW, Willian,
**Modelling Financial Derivatives with**, Cambridge University Press, 1998.*Mathematica*

- BURDEN, Richard L. and FAIRES, J. Douglas,
**Numerical Analysis**, Brooks/Cole, 2001. - HEATH, Michael T.,
**Scientific Computing: an introductory survey**, McGraw-Hill, 2002.

- BJORK, Thomas,
**Arbitrage Theory in Continuous Time**, Oxford University Press, 1998.

- 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.

- OKSENDAL, Bernt,
**Stochastic Differential Equations: an introduction with applications**, Springer-Verlag, 1998. - DUFFIE, Darrel,
**Dynamic Asset Pricing Theory**, Princeton University Press, 1996.

- Assignment 1, due on Monday 27/01/03.
- Assignment 2, due on Friday 14/02/03.
- Assignment 3, due on Wednesday 19/03/03.

- Pricing American down-and-out call options by the Crank-Nicolson Method, by Mirela Cara.
- Valuing American options with jump diffusion processes, by Xuping Zhang.
- Binomial trees and trinomial trees produced by explicit finite differences, by Jae W Jun.
- Pricing American-Bermudan-Asian options via LSM, by Willian D Volterman.
- Using Microsoft Excel to build Efficient Frontiers via the Mean Variance Optimization Method, by John A McNair.