massive parallel solutions of variable annuity pdes
play

Massive Parallel Solutions of Variable Annuity PDEs Janos Benk - PowerPoint PPT Presentation

Sep 19, 2023 •150 likes •375 views

Technische Universitt Mnchen Massive Parallel Solutions of Variable Annuity PDEs Janos Benk M.Sc. April 2012 J. Benk, Massive Parallel Solutions of Variable Annuity PDEs (I5, Prof. Bungartz) www5.in.tum.de PRMIA Munich, April, 2012

  1. Technische Universität München Massive Parallel Solutions of Variable Annuity PDEs Janos Benk M.Sc. April 2012 J. Benk, Massive Parallel Solutions of Variable Annuity PDEs (I5, Prof. Bungartz) www5.in.tum.de PRMIA Munich, April, 2012

  2. Technische Universität München Outline • FITOB overview • Modeling framework • Numerical aspects • Usability & ThetaML • European Call Option • 5D, 6D • American Put Option • 5D, 6D • Guaranteed Minimum Withdrawal Benefit (GMWB) J. Benk, Massive Parallel Solutions of Variable Annuity PDEs (I5, Prof. Bungartz) www5.in.tum.de PRMIA Munich, April, 2012

  3. Technische Universität München FITOB overview • FITOB: developed research software at our Chair • Modeling framework: by given a set of tradable and non-tradable assets, defined by stochastical differential equations (SDE): • The resulting Black-Sholes partial differential equation (BS-PDE) • Models defined by: , , , and J. Benk, Massive Parallel Solutions of Variable Annuity PDEs (I5, Prof. Bungartz) www5.in.tum.de PRMIA Munich, April, 2012

  4. Technische Universität München FITOB overview • Numerical aspects: – automatic computational domain estimation – Combination Technique for higher dimension • distributed memory parallelization – efficient Multigrid solvers • finite difference • adaptive time stepping • shared memory parallel • constraints enforcements (American, Barrier features) – hybrid parallelization – no derivative or SDE specific methods (e.g., log transformation) J. Benk, Massive Parallel Solutions of Variable Annuity PDEs (I5, Prof. Bungartz) www5.in.tum.de PRMIA Munich, April, 2012

  5. Technische Universität München FITOB overview ����� ������ User interaction: ������ �� ������ �� • Modeling Language ThetaML: – general and unique product description ��������������� – Presented previously ���� ��� ���������������� ���������������� • XML configuration file: � ��! ���� ������ – SDE models �"�� – start values (and intervals) ������������#���� �"�� – solver parameters – could be coupled to GUI ����������������������������������� ��������������������� � �������������������������� ��!��"����#�$��%����%���#�$� &�'�����#(��� � �)*�+,-)..,)/0.���������#�$��� J. Benk, Massive Parallel Solutions of Variable Annuity PDEs (I5, Prof. Bungartz) www5.in.tum.de PRMIA Munich, April, 2012

  6. Technische Universität München European Options • BS-PDE based pricing (recapitulation) payoff actual price BS-PDE Time 0 T J. Benk, Massive Parallel Solutions of Variable Annuity PDEs (I5, Prof. Bungartz) www5.in.tum.de PRMIA Munich, April, 2012

  7. Technische Universität München European Options ����� $%&'�$��()� ������ ��� • 5D example (N=5) ������ ��� ������ �*� ������ �+� ������ �(� ������ �� � ��! �� �������������*� �+��(,(#�����#���� �"�� • Results L L level price rel. error ∞ 2 3 0.530167 -3.801e-2 5.069e-2 2.906e-1 4 0.548944 -3.942e-3 4.119e-2 3.422e-1 5 0.550686 -7.829e-4 9.953e-4 3.203e-3 6 0.551117 J. Benk, Massive Parallel Solutions of Variable Annuity PDEs (I5, Prof. Bungartz) www5.in.tum.de PRMIA Munich, April, 2012

  8. Technische Universität München European Options ����� $%&'�$��-)� ������ ��� • 6D example (N=6) ������ ��� ������ �*� ������ �+� ������ �(� ������ �(� ������ �� � ��! �� �������������*� �+��(��-,-#�����#���� • Results �"�� L L level price rel. error ∞ 2 3 0.614752 -3.095e-2 2.290e-2 8.979e-2 4 0.625327 -1.428e-2 5.244e-2 6.435e-1 5 0.634132 -4.047e-4 7.499e-4 1.979e-3 6 0.634389 J. Benk, Massive Parallel Solutions of Variable Annuity PDEs (I5, Prof. Bungartz) www5.in.tum.de PRMIA Munich, April, 2012

  9. Technische Universität München European Options Parallel results for 5D and level = 6 • with regular grid number of points ~ 1.07e+9 • with combination technique number of points ~ 1.0e+7 (56 X 2.5e+5) Runtimes Number of processors 1 2 4 8 16 32 64 128 Total runtime (seconds) 8760 4080 3180 1638 966 465 259 240 Efficiency 1 1,07 0,69 0,67 0,57 0,59 0,53 0,29 4 minutes 2.5 hours J. Benk, Massive Parallel Solutions of Variable Annuity PDEs (I5, Prof. Bungartz) www5.in.tum.de PRMIA Munich, April, 2012

  10. Technische Universität München European Options Parallel results for 6D and level = 6 • with regular grid number of points ~ 6.8e+10 • with combination technique number of points ~ 1.2e+8 (84 X 2e+6) Runtimes Number of processors 8 16 32 64 128 Total runtime (seconds) 24660 18060 7299 4881 3337 Efficiency 1 0,68 0,84 0,63 0,46 7 hours 55 minutes J. Benk, Massive Parallel Solutions of Variable Annuity PDEs (I5, Prof. Bungartz) www5.in.tum.de PRMIA Munich, April, 2012

  11. Technische Universität München American Options ����� ��$&�.��()� • 5D example (N=5) ������ ��� ������ ��� ������ �*� ������ �+� ������ �(� ������ �� �������"/�� ��������(#�,��,��,�*, �+,�(���� • Results �"�� level price rel. error L L ∞ 2 � ��! �� 3 0.303934 -8.597e-2 6.396e-2 2.842e-1 ������(#�,��,��,�*, �+,�(���#���� 4 0.334562 6.137e-3 5.301e-2 4.113e-1 �"�� 5 0.331521 -3.008e-3 1.660e-3 4.515e-2 6 0.332522 J. Benk, Massive Parallel Solutions of Variable Annuity PDEs (I5, Prof. Bungartz) www5.in.tum.de PRMIA Munich, April, 2012

  12. Technische Universität München American Options ����� ��$&�.��-)� ������ ��� • 6D example (N=6) ������ ��� ������ �*� ������ �+� ������ �(� ������ �-� ������ �� �������"/�� ��������-#�,��,��,�*, �+,�(,�-���� • Results �"�� level price rel. error L L ∞ 2 � ��! �� 3 0.34839 -6.369e-2 3.064e-2 2.843e-1 ������-#�,��,��,�*, 4 0.35612 -4.292e-2 6.671e-2 6.843e-1 �+,�(,�-���#���� �"�� 5 0.37118 -2.436e-2 1.854e-3 2.972e-2 6 0.37209 J. Benk, Massive Parallel Solutions of Variable Annuity PDEs (I5, Prof. Bungartz) www5.in.tum.de PRMIA Munich, April, 2012

  13. Technische Universität München American Options Parallel results for 5D and level = 6 • with regular grid number of points ~ 1.07e+9 • with combination technique number of points ~ 1.0e+7 (56 X 2.5e+5) Runtimes Number of processors 8 16 32 64 128 Total runtime (seconds) 4405 2282 1342 601 361 Efficiency 1 0,97 0,82 0,92 0,76 1.25 hours 6 minutes J. Benk, Massive Parallel Solutions of Variable Annuity PDEs (I5, Prof. Bungartz) www5.in.tum.de PRMIA Munich, April, 2012

  14. Technische Universität München American Options Parallel results for 6D and level = 6 • with regular grid number of points ~ 6.8e+10 • with combination technique number of points ~ 1.2e+8 (84 X 2e+6) Runtimes Number of processors 16 32 64 128 192 Total runtime (seconds) 37941 18144 10357 7594 6201 Efficiency 1 1,05 0,92 0,62 0,51 10.5 hours 1 hour 40minutes J. Benk, Massive Parallel Solutions of Variable Annuity PDEs (I5, Prof. Bungartz) www5.in.tum.de PRMIA Munich, April, 2012

  15. Technische Universität München Guaranteed Minimum Withdrawal Benefit (GMWB) • Initial investment of 1.0 into an asset (forming a portfolio) • We model the payment period as 10 years long • In the payment period the portfolio structure stays the same (one asset) • Withdrawal rights at each payment dates • In case of no-withdrawal a fee of 2% is paid issuer 0.15 0.15 0.15 portfolio value Time 0 10 11 12 … 20 holder 1.0 J. Benk, Massive Parallel Solutions of Variable Annuity PDEs (I5, Prof. Bungartz) www5.in.tum.de PRMIA Munich, April, 2012

  16. Technische Universität München Guaranteed Minimum Withdrawal Benefit (GMWB) ����� 0�12()� • ThetaML script ������ �� ������ �� • todo .�����#�(� � ��! ��� ���� ���������� 3# � ��! �� �"�� ����������#���� �"�� J. Benk, Massive Parallel Solutions of Variable Annuity PDEs (I5, Prof. Bungartz) www5.in.tum.de PRMIA Munich, April, 2012

Recommend

Michael Zmistowski District Director Chairman

Michael Zmistowski District Director Chairman

Re#rement Solu#ons with fixed annuity, variable annuity, target date, balanced and stable value funds PANELISTS MODERATOR Michael Baney, CIMA Michael Zmistowski

619 views • 15 slides
Remez inequality and propagation of smallness for solutions of second order elliptic PDEs Part

Remez inequality and propagation of smallness for solutions of second order elliptic PDEs Part

Remez inequality and propagation of smallness for solutions of second order elliptic PDEs Part II. Logarithmic convexity for harmonic functions and solutions of elliptic PDEs Eugenia Malinnikova NTNU March 2018 E. Malinnikova Propagation of

415 views • 16 slides
Remez inequality and propagation of smallness for solutions of second order elliptic PDEs Part

Remez inequality and propagation of smallness for solutions of second order elliptic PDEs Part

Remez inequality and propagation of smallness for solutions of second order elliptic PDEs Part III. Eigenfunctions of Laplace-Beltrami operator Eugenia Malinnikova NTNU March 2018 E. Malinnikova Propagation of smallness for elliptic PDEs

290 views • 18 slides
High-frequency instabilities of small-amplitude solutions of Hamiltonian PDEs Bernard Deconinck

High-frequency instabilities of small-amplitude solutions of Hamiltonian PDEs Bernard Deconinck

High-frequency instabilities of small-amplitude solutions of Hamiltonian PDEs Bernard Deconinck Department of Applied Mathematics University of Washington bernard@amath.washington.edu Hamiltonian PDEs: Analysis, Computations and Applications

767 views • 54 slides
Families of periodic solutions for some Hamiltonian PDEs ( with G. Arioli ) (1) The problem etc.

Families of periodic solutions for some Hamiltonian PDEs ( with G. Arioli ) (1) The problem etc.

1 Families of periodic solutions for some Hamiltonian PDEs ( with G. Arioli ) (1) The problem etc. (2) Main results (3) Numerical results (4) Proofs ICERM, April 2016 (old-fashioned plain T X) E 1.1 The problem etc. 2 We consider

529 views • 14 slides
Reduced-Order Modeling for PDEs with bifurcating solutions: applications in CFD Annalisa Quaini

Reduced-Order Modeling for PDEs with bifurcating solutions: applications in CFD Annalisa Quaini

Introduction Model and Applications Methodology Numerical results Conclusions Reduced-Order Modeling for PDEs with bifurcating solutions: applications in CFD Annalisa Quaini Department of Mathematics, University of Houston Joint work with:

539 views • 30 slides
Annuity Adjustments Caleb CJJohnson Presenter 1 Effective Rates and Annuity Adjustments

Annuity Adjustments Caleb CJJohnson Presenter 1 Effective Rates and Annuity Adjustments

WRS Effective Rates and Annuity Adjustments Caleb CJJohnson Presenter 1 Effective Rates and Annuity Adjustments How to Participate: Ask questions once the Click to maximize presentation has ended. dashboard You will be muted for the

525 views • 18 slides
Semi-analytical solutions for transport PDEs in heterogeneous media Dr Elliot Carr

Semi-analytical solutions for transport PDEs in heterogeneous media Dr Elliot Carr

Monash Workshop on Numerical Di ff erential Equations and Applications Melbourne, Australia, 10-14 February 2020 Semi-analytical solutions for transport PDEs in heterogeneous media Dr Elliot Carr elliot.carr@qut.edu.au @ElliotJCarr

408 views • 40 slides
Massive Parallel GPU-accelerated Simulation of the Milky Way Galaxy Simon Portegies Zwart 1608

Massive Parallel GPU-accelerated Simulation of the Milky Way Galaxy Simon Portegies Zwart 1608

Massive Parallel GPU-accelerated Simulation of the Milky Way Galaxy Simon Portegies Zwart 1608 Lippershey For the last 400 years telescopes became larger CAStLe group Computational Astrophysics and Cosmology Open Access Springer Journal

517 views • 34 slides
Viscosity Solutions of Path-Dependent PDEs Zhenjie Ren CMAP, Ecole Polytechnique The 3rd young

Viscosity Solutions of Path-Dependent PDEs Zhenjie Ren CMAP, Ecole Polytechnique The 3rd young

Viscosity Solutions of Path-Dependent PDEs Zhenjie Ren CMAP, Ecole Polytechnique The 3rd young researchers meeting in Probability, Numerics and Finance June 29, 2016 Zhenjie Ren PPDE Le Mans, 29/06/2016 1 / 21 Motivation Table of Contents

1.83k views • 77 slides
VS-Net: Variable Splitting Network for Accelerated Parallel MRI Reconstruction Jinming Duan 1

VS-Net: Variable Splitting Network for Accelerated Parallel MRI Reconstruction Jinming Duan 1

VS-Net: Variable Splitting Network for Accelerated Parallel MRI Reconstruction Jinming Duan 1 , 2 , Jo Schlemper 2 , 3 , Chen Qin 2 , Cheng Ouyang 2 , Wenjia Bai 2 , Carlo Biffi 2 , Ghalib Bello 2 , Ben Statton 2 , Declan P ORegan 2 ,

512 views • 39 slides
Massive Graviton Geons: self-gravitating massive gravitational waves Katsuki Aoki, Waseda

Massive Graviton Geons: self-gravitating massive gravitational waves Katsuki Aoki, Waseda

Massive Graviton Geons: self-gravitating massive gravitational waves Katsuki Aoki, Waseda University KA, K. Maeda, Y. Misonoh, and H. Okawa, PRD 97, 044005 (2018), [arXiv: 1710.05606]. 2018/03/03 Introduction Vacuum solutions to the Einstein

395 views • 27 slides
Building Solutions to Nonlinear Elliptic and Parabolic Partial Differential Equations Adam

Building Solutions to Nonlinear Elliptic and Parabolic Partial Differential Equations Adam

Building Solutions to Nonlinear Elliptic and Parabolic Partial Differential Equations Adam Oberman University of Texas, Austin http://www.math.utexas.edu/~oberman Fields Institute Colloquium January 21, 2004 Early History of PDEs Early PDEs

1.17k views • 70 slides
Parallel, adaptive multigrid methods for parabolic PDEs and applications Feng Wei Yang

Parallel, adaptive multigrid methods for parabolic PDEs and applications Feng Wei Yang

Parallel, adaptive multigrid methods for parabolic PDEs and applications Feng Wei Yang Department of Mathematics University of Sussex F.W.Yang@sussex.ac.uk 25 January 2016 Feng Wei Yang Seminar at Warwick 25 January 2016 1 / 46 Objectives

1.03k views • 46 slides
Massive-parallel Input & Output with structured extensible data (HDF5 successful

Massive-parallel Input & Output with structured extensible data (HDF5 successful

INSTITUT FR WELTRAUMFORSCHUNG Massive-parallel Input & Output with structured extensible data (HDF5 successful implementation) (Pencil Code User Meeting, Espoo/Finland, 14 th of August 2019) Philippe-A. Bourdin Philippe.Bourdin@oeaw.ac.at

133 views • 12 slides
On a nonlinear model for tumor growth: Global existence of weak solutions Hamiltonian PDEs:

On a nonlinear model for tumor growth: Global existence of weak solutions Hamiltonian PDEs:

Tumor Growth On a nonlinear model for tumor growth: Global existence of weak solutions Hamiltonian PDEs: Analysis, Computations and Applications, Fields Institute Konstantina Trivisa January 10-12, 2014 Tumor Growth Collaborators 1 Donatella

599 views • 49 slides
Parallel Community Detection for Massive Graphs E. Jason Riedy, Henning Meyerhenke, David Ediger,

Parallel Community Detection for Massive Graphs E. Jason Riedy, Henning Meyerhenke, David Ediger,

Parallel Community Detection for Massive Graphs E. Jason Riedy, Henning Meyerhenke, David Ediger, and David A. Bader 14 February 2012 Exascale data analysis Health care Finding outbreaks, population epidemiology Social networks Advertising,

636 views • 36 slides
Outline Parallel / Distributed Computers Air Traffic Network Example Modeling and

Outline Parallel / Distributed Computers Air Traffic Network Example Modeling and

Outline Parallel / Distributed Computers Air Traffic Network Example Modeling and Simulation Parallel Discrete Event Simulation Logical processes & time stamped messages Local causality constraint and the synchronization PDES

154 views • 4 slides
GFP-X: A Parallel Approach To Massive Graph Comparison Using Spark Stephen Bonner, John Brennan,

GFP-X: A Parallel Approach To Massive Graph Comparison Using Spark Stephen Bonner, John Brennan,

GFP-X: A Parallel Approach To Massive Graph Comparison Using Spark Stephen Bonner, John Brennan, Georgios Theodoropoulos, Ibad Kureshi and Andrew Stephen McGough School of Engineering and Computing Sciences Durham University, Durham, UK

305 views • 17 slides
AM 205: lecture 16 Last time: hyperbolic PDEs Today: parabolic and elliptic PDEs,

AM 205: lecture 16 Last time: hyperbolic PDEs Today: parabolic and elliptic PDEs,

AM 205: lecture 16 Last time: hyperbolic PDEs Today: parabolic and elliptic PDEs, introduction to optimization The -Method: Accuracy The truncation error analysis is fairly involved, hence we just give the result: u n +1 u n

657 views • 50 slides
Viscosity Solutions of Fully Nonlinear Path Dependent PDEs Nizar TOUZI Ecole Polytechnique

Viscosity Solutions of Fully Nonlinear Path Dependent PDEs Nizar TOUZI Ecole Polytechnique

Motivation and examples Preliminaries Fully nonlinear path dependent PDE Wellposedness Viscosity Solutions of Fully Nonlinear Path Dependent PDEs Nizar TOUZI Ecole Polytechnique France Joint work with Ibrahim EKREN and Jianfeng ZHANG Happy

675 views • 40 slides
Fully Nonlinear Elliptic Path Dependent PDEs The viscosity solutions to the Dirichlet problem REN

Fully Nonlinear Elliptic Path Dependent PDEs The viscosity solutions to the Dirichlet problem REN

Fully Nonlinear Elliptic Path Dependent PDEs The viscosity solutions to the Dirichlet problem REN Zhenjie Ecole Polytechnique, Paris Joint work with Nizar Touzi September, 2013 Journ ee DIM RDM-IdF 2013 REN Zhenjie Fully Nonlinear

589 views • 30 slides
CHARITABLE LEAD ANNUITY TRUSTS (CLAT) Prepared by John R. Anzivino, CPA February 2012 SEE

CHARITABLE LEAD ANNUITY TRUSTS (CLAT) Prepared by John R. Anzivino, CPA February 2012 SEE

CHARITABLE LEAD ANNUITY TRUSTS (CLAT) Prepared by John R. Anzivino, CPA February 2012 SEE BEYOND THE NUMBERS Characteristics of a Charitable Lead Annuity Trust Grantor Transfer Assets Annuity Charitable Lead Payments to Charity Charity

350 views • 10 slides
The Massive Parallel Sequencing era: "Global sequencing" Richard Christen CNRS UMR

The Massive Parallel Sequencing era: "Global sequencing" Richard Christen CNRS UMR

The Massive Parallel Sequencing era: "Global sequencing" Richard Christen CNRS UMR 6543 & Universit de Nice christen@unice.fr http://bioinfo.unice.fr 1 At the end of 2007, three next-generation sequencing platforms appeared:

573 views • 42 slides

More recommend