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

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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

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SLIDE 1

Technische Universität München

  • J. Benk, Massive Parallel Solutions of Variable Annuity PDEs

(I5, Prof. Bungartz) www5.in.tum.de PRMIA Munich, April, 2012

Massive Parallel Solutions of Variable Annuity PDEs

Janos Benk M.Sc.

April 2012

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SLIDE 2

Technische Universität München

  • J. Benk, Massive Parallel Solutions of Variable Annuity PDEs

(I5, Prof. Bungartz) www5.in.tum.de PRMIA Munich, April, 2012

Outline

  • FITOB overview
  • Modeling framework
  • Numerical aspects
  • Usability & ThetaML
  • European Call Option
  • 5D, 6D
  • American Put Option
  • 5D, 6D
  • Guaranteed Minimum Withdrawal Benefit (GMWB)
slide-3
SLIDE 3

Technische Universität München

  • J. Benk, Massive Parallel Solutions of Variable Annuity PDEs

(I5, Prof. Bungartz) www5.in.tum.de PRMIA Munich, April, 2012

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

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SLIDE 4

Technische Universität München

  • J. Benk, Massive Parallel Solutions of Variable Annuity PDEs

(I5, Prof. Bungartz) www5.in.tum.de PRMIA Munich, April, 2012

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)

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SLIDE 5

Technische Universität München

  • J. Benk, Massive Parallel Solutions of Variable Annuity PDEs

(I5, Prof. Bungartz) www5.in.tum.de PRMIA Munich, April, 2012

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.#$
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SLIDE 6

Technische Universität München

  • J. Benk, Massive Parallel Solutions of Variable Annuity PDEs

(I5, Prof. Bungartz) www5.in.tum.de PRMIA Munich, April, 2012

European Options

  • BS-PDE based pricing (recapitulation)

Time T BS-PDE

payoff actual price

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SLIDE 7

Technische Universität München

  • J. Benk, Massive Parallel Solutions of Variable Annuity PDEs

(I5, Prof. Bungartz) www5.in.tum.de PRMIA Munich, April, 2012

European Options

  • 5D example (N=5)

0.551117 6 3.203e-3 9.953e-4

  • 7.829e-4

0.550686 5 3.422e-1 4.119e-2

  • 3.942e-3

0.548944 4 2.906e-1 5.069e-2

  • 3.801e-2

0.530167 3

  • rel. error

price level $%&'$() * + ( ! * +(,(## "

  • Results

∞

L

2

L

slide-8
SLIDE 8

Technische Universität München

  • J. Benk, Massive Parallel Solutions of Variable Annuity PDEs

(I5, Prof. Bungartz) www5.in.tum.de PRMIA Munich, April, 2012

European Options

  • 6D example (N=6)

0.634389 6 1.979e-3 7.499e-4

  • 4.047e-4

0.634132 5 6.435e-1 5.244e-2

  • 1.428e-2

0.625327 4 8.979e-2 2.290e-2

  • 3.095e-2

0.614752 3

  • rel. error

price level $%&'$-) * + ( ( ! * +(-,-## "

  • Results

∞

L

2

L

slide-9
SLIDE 9

Technische Universität München

  • J. Benk, Massive Parallel Solutions of Variable Annuity PDEs

(I5, Prof. Bungartz) www5.in.tum.de PRMIA Munich, April, 2012

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

0,29 0,53 0,59 0,57 0,67 0,69 1,07 1 Efficiency 240 259 465 966 1638 3180 4080 8760 Total runtime (seconds) 128 64 32 16 8 4 2 1 Number of processors

2.5 hours 4 minutes

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SLIDE 10

Technische Universität München

  • J. Benk, Massive Parallel Solutions of Variable Annuity PDEs

(I5, Prof. Bungartz) www5.in.tum.de PRMIA Munich, April, 2012

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)

0,46 0,63 0,84 0,68 1 Efficiency 3337 4881 7299 18060 24660 Total runtime (seconds) 128 64 32 16 8 Number of processors

Runtimes 7 hours 55 minutes

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SLIDE 11

Technische Universität München

  • J. Benk, Massive Parallel Solutions of Variable Annuity PDEs

(I5, Prof. Bungartz) www5.in.tum.de PRMIA Munich, April, 2012

American Options

  • 5D example (N=5)

0.332522 6 4.515e-2 1.660e-3

  • 3.008e-3

0.331521 5 4.113e-1 5.301e-2 6.137e-3 0.334562 4 2.842e-1 6.396e-2

  • 8.597e-2

0.303934 3

  • rel. error

price level $&.() * + ( "/ (#,,,*, +,( " ! (#,,,*, +,(# "

  • Results

∞

L

2

L

slide-12
SLIDE 12

Technische Universität München

  • J. Benk, Massive Parallel Solutions of Variable Annuity PDEs

(I5, Prof. Bungartz) www5.in.tum.de PRMIA Munich, April, 2012

American Options

  • 6D example (N=6)

0.37209 6 2.972e-2 1.854e-3

  • 2.436e-2

0.37118 5 6.843e-1 6.671e-2

  • 4.292e-2

0.35612 4 2.843e-1 3.064e-2

  • 6.369e-2

0.34839 3

  • rel. error

price level

  • Results

∞

L

2

L

$&.-) * + (

"/

  • #,,,*,

+,(,- " !

  • #,,,*,

+,(,-# "

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SLIDE 13

Technische Universität München

  • J. Benk, Massive Parallel Solutions of Variable Annuity PDEs

(I5, Prof. Bungartz) www5.in.tum.de PRMIA Munich, April, 2012

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)

0,76 0,92 0,82 0,97 1 Efficiency 361 601 1342 2282 4405 Total runtime (seconds) 128 64 32 16 8 Number of processors

Runtimes 1.25 hours 6 minutes

slide-14
SLIDE 14

Technische Universität München

  • J. Benk, Massive Parallel Solutions of Variable Annuity PDEs

(I5, Prof. Bungartz) www5.in.tum.de PRMIA Munich, April, 2012

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)

0,51 0,62 0,92 1,05 1 Efficiency 6201 7594 10357 18144 37941 Total runtime (seconds) 192 128 64 32 16 Number of processors

Runtimes 10.5 hours 1 hour 40minutes

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SLIDE 15

Technische Universität München

  • J. Benk, Massive Parallel Solutions of Variable Annuity PDEs

(I5, Prof. Bungartz) www5.in.tum.de PRMIA Munich, April, 2012

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

Time issuer holder

1.0

10

0.15

11

0.15

12

0.15

20

portfolio value

…

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SLIDE 16

Technische Universität München

  • J. Benk, Massive Parallel Solutions of Variable Annuity PDEs

(I5, Prof. Bungartz) www5.in.tum.de PRMIA Munich, April, 2012

Guaranteed Minimum Withdrawal Benefit (GMWB)

  • ThetaML script
  • todo

012() .#( !

  • 3#

! " # "

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SLIDE 17

Technische Universität München

  • J. Benk, Massive Parallel Solutions of Variable Annuity PDEs

(I5, Prof. Bungartz) www5.in.tum.de PRMIA Munich, April, 2012

Guaranteed Minimum Withdrawal Benefit (GMWB)

  • todo

012() .#( !

  • .

/4 , . 5 #67,. '8),'8) "

  • !

" # "

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SLIDE 18

Technische Universität München

  • J. Benk, Massive Parallel Solutions of Variable Annuity PDEs

(I5, Prof. Bungartz) www5.in.tum.de PRMIA Munich, April, 2012

Guaranteed Minimum Withdrawal Benefit (GMWB)

  • Heston model of the asset

012() .#( !

  • .

/4 , . 5 #67,. '8),'8) "

  • !

" # "

  • CIR model of the interest rate
  • parameter and initial values
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SLIDE 19

Technische Universität München

  • J. Benk, Massive Parallel Solutions of Variable Annuity PDEs

(I5, Prof. Bungartz) www5.in.tum.de PRMIA Munich, April, 2012

Guaranteed Minimum Withdrawal Benefit (GMWB)

  • Pricing results:

1.2618 8 1.554e-2 1.086e-2

  • 8.233e-3

1.2514 7 4.727e-2 2.819e-2

  • 1.880e-2

1.2381 6 1.275e-1 7.534e-2 4.895e-2 1.3245 5 1.276e-0 7.404e-1 4.832e-1 1.8716 4 1.318e-0 9.006e-1 6.546e-1 2.0878 3

  • rel. error

price level

∞

L

2

L

  • Runtimes for level = 8

0,33 0,536 0,63 1 Efficiency 40479 50871 85253 216000 Total runtime (seconds) 64 32 16 4 Number of processors

11 hours

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SLIDE 20

Technische Universität München

  • J. Benk, Massive Parallel Solutions of Variable Annuity PDEs

(I5, Prof. Bungartz) www5.in.tum.de PRMIA Munich, April, 2012

Thank you! Questions?

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SLIDE 21

Technische Universität München

  • J. Benk, Massive Parallel Solutions of Variable Annuity PDEs

(I5, Prof. Bungartz) www5.in.tum.de PRMIA Munich, April, 2012

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Literature: