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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

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