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Numerical Excellence in Finance


                                                     John Holden
                                  Banking on Monte Carlo and GPUs
                                                     Paris, FRANCE
                                                 28th January 2010




Experts in numerical algorithms
and HPC services
Agenda




 NAG – An Introduction
 NAG – Numerical Libraries




                Numerical Excellence in Finance – January 2010   2
Introduction to NAG
 Founded in 1970 as a co-operative project in UK
 Operates as a commercial, not-for-profit organization
 Worldwide operations
        Oxford & Manchester, UK
        Chicago, US
        Tokyo, Japan
        Taipei, Taiwan
 Over 3,000 customer sites worldwide
 Staff of ~100, over 50% technical, over 25 PhDs
 £7m+ financial turnover



 January 10              Numerical Excellence in Finance – January 2010   3
Portfolio
 Numerical Libraries
      Highly flexible for use in many computing languages, programming
       environments, hardware platforms and for high performance
       computing methods

 Connector Products for Excel, MATLAB and Maple and
      Giving users of the Excel and mathematical software packages
       MATLAB and Maple access to NAG’s library of highly optimized and
       often superior numerical routines and allowing easy integration

 NAG Fortran Compiler and GUI based Windows
 Compiler: Fortran Builder
 Visualization and graphics software
      Build data visualization applications with NAG’s IRIS Explorer

 Consultancy services
                         Numerical Excellence in Finance – January 2010   4
Why bother?
 Numerical computation is difficult to do accurately
 Problems of
      Overflow / underflow
           How does the computation behave for large / small numbers?
      Condition
           How is it affected by small changes in the input?
      Stability
           How sensitive is the computation to rounding errors?
 Importance of
      error analysis
      information about error bounds on solution


                          Numerical Excellence in Finance – January 2010   5
NAG development philosophy

 First priority: accuracy
 Second priority: performance
     How fast do you want the wrong answer?
 Algorithms chosen for
     usefulness
     robustness
     accuracy
     stability
     speed



                   Numerical Excellence in Finance – January 2010   6
Why Use NAG Maths Libraries?
 Global reputation for quality – accuracy,
    reliability and robustness…
   Extensively tested, supported and maintained
    code
   Reduce development time
   Concentrate on your key areas
   Components
       Fit into your environment
       Simple interfaces to your favourite packages
 Regular performance improvements!


                      Numerical Excellence in Finance – January 2010   7
NAG Library and Toolbox Contents
 NAG provides high-level maths and stats components
      Nonlinear equation solvers
      Summation of series and transformations, FFTs
      Quadrature
      ODEs, PDEs and integral equations
      Approximation and curve and surface fitting
      Optimization and operations research
      Dense linear algebra, including LAPACK
      Sparse linear systems and eigenproblems
      Special functions
      Random Number Generators
      ...



                      Numerical Excellence in Finance – January 2010   8
Use of NAG Software in Finance
 Portfolio analysis / Index tracking / Risk management
      Optimisation, linear algebra, copulas…
 Derivative pricing
      PDEs, RNGs, multivariate normal, …
 Fixed Income/ Asset management / Portfolio
  Immunization
      Operations research
 Data analysis
      Time series, GARCH, principal component analysis, data smoothing,
       …
 Monte Carlo simulation
      RNGs
 ……

                       Numerical Excellence in Finance – January 2010      9
Don’t take our word for it….
Financial Maths Professors speed up their optimization




                 Numerical Excellence in Finance – January 2010   10
Don’t take our word for it…...
Senior Quant @ Tier 1 bank
 “concerning the ‘nearest correlation’ algo.
 I have to say, it is very fast, it uses all the power
 of my pc and the result is very satisfactory.”

www.walkingrandomly.com
“I really like the NAG toolbox for MATLAB for the
 following reasons (among others):
 It can speed up MATLAB calculations – see my
 article on MATLAB's interp1 function for
 example, and it has some functionality that can't
 currently be found in MATLAB.”
                 Numerical Excellence in Finance – January 2010   11
                                                                       11
NAG Libraries Ease of Integration
   C++ (various)                        Excel
   C# / .NET                            MATLAB
   CUDA                                        SciLab, Octave
   Visual Basic                           Mathematica
   Java                                   Maple
   Borland Delphi                         PowerBuilder
   Python                                 R and S-Plus
   F#                                     SAS
   …                                      …
   and more                               …
                                            and more



                     Numerical Excellence in Finance – January 2010   12
NAG and Excel..
 Our libraries are
  easily accessible
  from Excel
      Calling DLLs using VBA
      NAG provide VB
       Declaration
       Statements and
       Examples




                        Numerical Excellence in Finance – January 2010   13
NAG and .NET
NAG solutions for .NET
1. Call NAG C (or Fortran) DLL from C#

2. NAG Library for .NET (beta)
  “a more natural solution”
    DLL with C# wrappers
    Integrated help
3. NAG Library for .NET (Work-in-Progress)
     as above pure C# functions



                   Numerical Excellence in Finance – January 2010   14
NAG Toolbox for MATLAB

 Contains essentially all NAG functionality
     not a subset
 Currently runs under Windows (32/64bit) or
  Linux (32/64-bit).
 Installed under the usual MATLAB toolbox
  directory
 Can be used with MATLAB compiler




                     Numerical Excellence in Finance – January 2010   15
Case study – e04uc vs fmincon
A problem from a customer from a European
 bank.
The problem involves 48 variables and has 9
 linear constraints. (No nonlinear constraints.)
No derivatives supplied.

fmincon required 1890 evaluations of the
 objective function and tool 87.6 seconds

e04uc required only 1129 evaluations and took
 49.4 seconds

                Numerical Excellence in Finance – January 2010   16
Subset of Eigenvalues and Eigenvectors
Speedup




 Sunday, 31 January 2010   Numerical Excellence in Finance – January 2010
                                  NAG Toolbox for MATLAB                    17
                                                                                 17
Best Advice – Use the Decision Trees




         Numerical Excellence in Finance – January 2010   18
NAG Library and Toolbox – recent additions
 Global optimization                More Copulas
 ANOVA – Analysis of                Extreme Value Theory
  Variance                            Statistics
 Nearest Correlation Matrix         Fast quantile selection
                                      routine
 Partial Least Squares              Wavelets
  Regression Analysis
                                     Adoption of LAPACK 3.1
 Prediction intervals for           New RNGs
  fitted models                           Scrambled Seq for QMC
 Option Pricing                          Mersenne Twister

                                          Sobol Sequence generator
 Generalised Mixed Effect                 (50,000 dimensions)
  Regression


                  Numerical Excellence in Finance – January 2010   19
                                                                        19
Other NAG software
 NAG’s High Performance libraries
     NAG SMP Library (for multi/many core/processor)
     NAG Parallel Library (for clusters architectures - MPI)
 NAG Fortran Compiler
     Windows version with GUI & Debugger
     Automatic Differentiation (AD) enabled
          In collaboration with RWTH Aachen University
 Routines for SIMD architectures (GPUs etc)
     Early successes with Monte Carlo components on NVIDIA
      hardware
          In collaboration with Professor Mike Giles


                      Numerical Excellence in Finance – January 2010   20
Summary
 NAG for Quality, World-leading Numerical
  Software Components
     accurate, reliable, robust
     extensively tested, supported and maintained code
     updated for new architectures and new algorithms




                    Numerical Excellence in Finance – January 2010   21
NAG Key Contacts
www.nag.com

Technical Support and Help
  support@nag.co.uk

Sales in France
  francois.cassier@nag.com
  Direct Dial +33 6 87 88 12 94

NAGNews http://www.nag.co.uk/NAGNews/Index.asp




                   Numerical Excellence in Finance – January 2010   22

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Numerical Excellence In Finance N A G Jan2010

  • 1. Numerical Excellence in Finance John Holden Banking on Monte Carlo and GPUs Paris, FRANCE 28th January 2010 Experts in numerical algorithms and HPC services
  • 2. Agenda  NAG – An Introduction  NAG – Numerical Libraries Numerical Excellence in Finance – January 2010 2
  • 3. Introduction to NAG  Founded in 1970 as a co-operative project in UK  Operates as a commercial, not-for-profit organization  Worldwide operations  Oxford & Manchester, UK  Chicago, US  Tokyo, Japan  Taipei, Taiwan  Over 3,000 customer sites worldwide  Staff of ~100, over 50% technical, over 25 PhDs  £7m+ financial turnover January 10 Numerical Excellence in Finance – January 2010 3
  • 4. Portfolio  Numerical Libraries  Highly flexible for use in many computing languages, programming environments, hardware platforms and for high performance computing methods  Connector Products for Excel, MATLAB and Maple and  Giving users of the Excel and mathematical software packages MATLAB and Maple access to NAG’s library of highly optimized and often superior numerical routines and allowing easy integration  NAG Fortran Compiler and GUI based Windows Compiler: Fortran Builder  Visualization and graphics software  Build data visualization applications with NAG’s IRIS Explorer  Consultancy services Numerical Excellence in Finance – January 2010 4
  • 5. Why bother?  Numerical computation is difficult to do accurately  Problems of  Overflow / underflow  How does the computation behave for large / small numbers?  Condition  How is it affected by small changes in the input?  Stability  How sensitive is the computation to rounding errors?  Importance of  error analysis  information about error bounds on solution Numerical Excellence in Finance – January 2010 5
  • 6. NAG development philosophy  First priority: accuracy  Second priority: performance  How fast do you want the wrong answer?  Algorithms chosen for  usefulness  robustness  accuracy  stability  speed Numerical Excellence in Finance – January 2010 6
  • 7. Why Use NAG Maths Libraries?  Global reputation for quality – accuracy, reliability and robustness…  Extensively tested, supported and maintained code  Reduce development time  Concentrate on your key areas  Components  Fit into your environment  Simple interfaces to your favourite packages  Regular performance improvements! Numerical Excellence in Finance – January 2010 7
  • 8. NAG Library and Toolbox Contents  NAG provides high-level maths and stats components  Nonlinear equation solvers  Summation of series and transformations, FFTs  Quadrature  ODEs, PDEs and integral equations  Approximation and curve and surface fitting  Optimization and operations research  Dense linear algebra, including LAPACK  Sparse linear systems and eigenproblems  Special functions  Random Number Generators  ... Numerical Excellence in Finance – January 2010 8
  • 9. Use of NAG Software in Finance  Portfolio analysis / Index tracking / Risk management  Optimisation, linear algebra, copulas…  Derivative pricing  PDEs, RNGs, multivariate normal, …  Fixed Income/ Asset management / Portfolio Immunization  Operations research  Data analysis  Time series, GARCH, principal component analysis, data smoothing, …  Monte Carlo simulation  RNGs  …… Numerical Excellence in Finance – January 2010 9
  • 10. Don’t take our word for it…. Financial Maths Professors speed up their optimization Numerical Excellence in Finance – January 2010 10
  • 11. Don’t take our word for it…... Senior Quant @ Tier 1 bank “concerning the ‘nearest correlation’ algo. I have to say, it is very fast, it uses all the power of my pc and the result is very satisfactory.” www.walkingrandomly.com “I really like the NAG toolbox for MATLAB for the following reasons (among others): It can speed up MATLAB calculations – see my article on MATLAB's interp1 function for example, and it has some functionality that can't currently be found in MATLAB.” Numerical Excellence in Finance – January 2010 11 11
  • 12. NAG Libraries Ease of Integration  C++ (various)  Excel  C# / .NET  MATLAB  CUDA  SciLab, Octave  Visual Basic  Mathematica  Java  Maple  Borland Delphi  PowerBuilder  Python  R and S-Plus  F#  SAS  …  …  and more  …  and more Numerical Excellence in Finance – January 2010 12
  • 13. NAG and Excel..  Our libraries are easily accessible from Excel  Calling DLLs using VBA  NAG provide VB Declaration Statements and Examples Numerical Excellence in Finance – January 2010 13
  • 14. NAG and .NET NAG solutions for .NET 1. Call NAG C (or Fortran) DLL from C# 2. NAG Library for .NET (beta) “a more natural solution”  DLL with C# wrappers  Integrated help 3. NAG Library for .NET (Work-in-Progress)  as above pure C# functions Numerical Excellence in Finance – January 2010 14
  • 15. NAG Toolbox for MATLAB  Contains essentially all NAG functionality  not a subset  Currently runs under Windows (32/64bit) or Linux (32/64-bit).  Installed under the usual MATLAB toolbox directory  Can be used with MATLAB compiler Numerical Excellence in Finance – January 2010 15
  • 16. Case study – e04uc vs fmincon A problem from a customer from a European bank. The problem involves 48 variables and has 9 linear constraints. (No nonlinear constraints.) No derivatives supplied. fmincon required 1890 evaluations of the objective function and tool 87.6 seconds e04uc required only 1129 evaluations and took 49.4 seconds Numerical Excellence in Finance – January 2010 16
  • 17. Subset of Eigenvalues and Eigenvectors Speedup Sunday, 31 January 2010 Numerical Excellence in Finance – January 2010 NAG Toolbox for MATLAB 17 17
  • 18. Best Advice – Use the Decision Trees Numerical Excellence in Finance – January 2010 18
  • 19. NAG Library and Toolbox – recent additions  Global optimization  More Copulas  ANOVA – Analysis of  Extreme Value Theory Variance Statistics  Nearest Correlation Matrix  Fast quantile selection routine  Partial Least Squares  Wavelets Regression Analysis  Adoption of LAPACK 3.1  Prediction intervals for  New RNGs fitted models  Scrambled Seq for QMC  Option Pricing  Mersenne Twister  Sobol Sequence generator  Generalised Mixed Effect (50,000 dimensions) Regression Numerical Excellence in Finance – January 2010 19 19
  • 20. Other NAG software  NAG’s High Performance libraries  NAG SMP Library (for multi/many core/processor)  NAG Parallel Library (for clusters architectures - MPI)  NAG Fortran Compiler  Windows version with GUI & Debugger  Automatic Differentiation (AD) enabled  In collaboration with RWTH Aachen University  Routines for SIMD architectures (GPUs etc)  Early successes with Monte Carlo components on NVIDIA hardware  In collaboration with Professor Mike Giles Numerical Excellence in Finance – January 2010 20
  • 21. Summary  NAG for Quality, World-leading Numerical Software Components  accurate, reliable, robust  extensively tested, supported and maintained code  updated for new architectures and new algorithms Numerical Excellence in Finance – January 2010 21
  • 22. NAG Key Contacts www.nag.com Technical Support and Help support@nag.co.uk Sales in France francois.cassier@nag.com Direct Dial +33 6 87 88 12 94 NAGNews http://www.nag.co.uk/NAGNews/Index.asp Numerical Excellence in Finance – January 2010 22