13th December 2006
The Numerical Algorithms Group (NAG) and the finance publication Wilmott hosted a city seminar for finance industry professionals on 13th December 2006. It was an extremely well attended event, with speakers from industry and academia, and open to those working within the finance industry.
Peter W Duck - University of Manchester
“Singular perturbation problems arising in mathematical finance: fluid dynamics concepts in option pricing”
Singular perturbation theory is a widely used tool in a number of areas of physical applied mathematics, including fluid dynamics. The basic requirement for this technique (which will be outlined in the presentation) is for a differential system in which the highest-order derivative is multiplied by a small parameter. In the case of the Black-Scholes equation, the highest-order derivative (i.e. the gamma) is multiplied by the square of the volatility. Given that numerical values of the volatility are invariably small, conditions are ripe for the use of singular perturbation methods. It is shown how these lead to trivial solutions in much of parameter space, with different regions separated by thin smoothing regions (akin to ‘shear layers’ in the terminology of fluid dynamics). The procedure is first described for single-underlying option pricing problems (including barrier and early-exercise options), and then for multi-underlying options. Comparisons with exact solutions reveal that excellent approximations can be achieved, whilst at the same time the procedure gives added insight into the solution-space topology. Finally it is illustrated how the technique has the potential to provide the basis for effective evaluations of implied volatilities (and correlation coefficients). Full details of the presentation can be found here. This lecture is available as Audio/Visual or as MP3 Audio. Visit www.wilmott.com/detail.cfm?articleID=259. You are required to log in to view this.
Robert Tong - NAG Ltd
“Software issues in wavelet analysis of financial data”
Wavelet Multiresolution Analysis (WMA) is a technique that is widely applied in many areas of data processing, including financial time series. By revealing the structure of the data at multiple scales, the results of WMA can be used to challenge or confirm underlying assumptions of market models. In many cases, the software used to compute the analysis will have been provided by a third party.
The requirements this imposes on the design and implementation of such software are examined, together with possible limitations which need to be taken into account by the user.
An important consideration is the desirability for consistent output to be produced by different implementations of the wavelet algorithms, to enable valid comparisons to be made. The needs of users to choose an appropriate wavelet basis and produce results relevant to their application must be translated into efficient software in terms of scalability and the pre- and post-processing of data sets. These issues are illustrated with reference to applications such as Foreign Exchange pricing. Full details of the presentation can be found here.
To view this lecture in visual/audio, visit http://www.wilmott.com/detail.cfm?articleID=261. You are required to log in to view this.
Nicholas J Higham - University of Manchester
“Can you count on your correlation matrix?”
Correlation matrices play a fundamental role in portfolio selection. However they are often constructed from incomplete or inaccurate data and consequently need manipulating so as to make them satisfy the mathematical properties required of a correlation matrix. I will give a brief survey of the properties of correlation matrices and explain some simple tests that reveal when a matrix is not a correlation matrix. Then I will describe how to find the nearest correlation matrix to a given matrix, emphasizing the computational aspects. Finally, current research concerning a certain specially structured correlation matrix will be mentioned. Full details of the presentation can be found here.
To view this lecture in visual/audio, visit http://www.wilmott.com/detail.cfm?articleID=265. You are required to log in to view this.
As well as the above presentations, technical experts from NAG were available during registration and after the talks to answer questions relating to the use of NAG's numerical libraries as well as discuss possible future algorithmic content relevant to the finance industry.
Topics this panel has covered in previous meetings include:
- Using NAG libraries within Excel
- Using NAG libraries from other environments such as C++, C#, Java
- Using NAG libraries within proprietary packages such as Maple, MATLAB, S PLUS and R
- Routine selection for a particular problem type: for example, what NAG optimisation routine is best to solve my portfolio problem
- New algorithmic content