Skip to main content
SpringerLink
Log in

A Multistep Scheme to Solve Backward Stochastic Differential Equations for Option Pricing on GPUs

  • Conference paper
  • First Online:
Advances in High Performance Computing (HPC 2019)

Part of the book series: Studies in Computational Intelligence ((SCI,volume 902))

Abstract

The goal of this work is to parallelize the multistep method for the numerical approximation of the Backward Stochastic Differential Equations (BSDEs) in order to achieve both, a high accuracy and a reduction of the computation time as well. In the multistep scheme the computations at each grid point are independent and this fact motivates us to select massively parallel GPU computing using CUDA. In our investigations we identify performance bottlenecks and apply appropriate optimization techniques for reducing the computation time, using a uniform domain. Finally, a Black-Scholes BSDE example is provided to demonstrate the achieved acceleration on GPUs.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Bender, C., Steiner, J.: Least-squares Monte-Carlo for backward SDEs. In: Numerical methods in finance, pp. 257–289. Springer, Heidelberg (2012)

    Google Scholar 

  2. Bouchard, B., Touzi, N.: Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations. Stochast. Process. Appl. 111(2), 175–206 (2004)

    MathSciNet  MATH  Google Scholar 

  3. Crisan, D., Manolarakis, K.: Solving backward stochastic differential equations using the cubature method: application to nonlinear pricing. SIAM J. Fin. Math. 3(1), 534–571 (2012)

    MathSciNet  MATH  Google Scholar 

  4. Dai, B., Peng, Y., Gong, B.: Parallel option pricing with BSDE method on GPU. In: 2010 Ninth International Conference on Grid and Cloud Computing, pp. 191–195. IEEE (2010)

    Google Scholar 

  5. El Karoui, N., Peng, S., Quenez, M.C.: Backward stochastic differential equations in finance. Math. Finan. 7(1), 1–71 (1997)

    MathSciNet  MATH  Google Scholar 

  6. Gobet, E., Lemor, J.P., Warin, X.: A regression-based Monte Carlo method to solve backward stochastic differential equations. Annal. Appl. Probab. 15(3), 2172–2202 (2005)

    MathSciNet  MATH  Google Scholar 

  7. Gobet, E., López-Salas, J.G., Turkedjiev, P., Vázquez, C.: Stratified regression Monte-Carlo scheme for semilinear PDEs and BSDEs with large scale parallelization on GPUs. SIAM J. Sci. Comput. 38(6), C652–C677 (2016)

    MathSciNet  MATH  Google Scholar 

  8. Hendricks, C.: High-Order Methods for Parabolic Equations in Multiple Space Dimensions for Option Pricing Problems, Dissertation, University of Wuppertal (2017)

    Google Scholar 

  9. Lemor, J.P., Gobet, E., Warin, X.: Rate of convergence of an empirical regression method for solving generalized backward stochastic differential equations. Bernoulli 12(5), 889–916 (2006)

    MathSciNet  MATH  Google Scholar 

  10. Pardoux, E., Peng, S.: Adapted solution of a backward stochastic differential equation. Syst. Control Lett. 14(1), 55–61 (1990)

    MathSciNet  MATH  Google Scholar 

  11. Peng, S.: Probabilistic interpretation for systems of quasilinear parabolic partial differential equations. Stochast. Stochast. Rep. 37(1–2), 61–74 (1991)

    MathSciNet  MATH  Google Scholar 

  12. Peng, Y., Gong, B., Liu, H., Dai, B.: Option pricing on the GPU with backward stochastic differential equation. In: 2011 Fourth International Symposium on Parallel Architectures, Algorithms and Programming, pp. 19–23. IEEE (2011)

    Google Scholar 

  13. Peng, Y., Liu, H., Yang, S., Gong, B.: Parallel algorithm for BSDEs based high dimensional American option pricing on the GPU. J. Comp. Inform. Syst. 10(2), 763–771 (2014)

    Google Scholar 

  14. Teng, L.: A Review of tree-based approaches to solve forward-backward stochastic differential equations. arXiv preprint arXiv:1809.00325 (2018)

  15. Teng, L., Lapitckii, A., Günther, M.: A Multi-step Scheme based on Cubic Spline for solving Backward Stochastic Differential Equations. arXiv preprint arXiv:1809.00324 (2018)

  16. Zhao, W., Zhang, G., Ju, L.: A stable multistep scheme for solving backward stochastic differential equations. SIAM J. Numer. Anal. 48(4), 1369–1394 (2010)

    MathSciNet  MATH  Google Scholar 

  17. Zhao, W., Chen, L., Peng, S.: A new kind of accurate numerical method for backward stochastic differential equations. SIAM J. Sci. Comput. 28(4), 1563–1581 (2006)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors were partially supported by the bilateral German-Portuguese Project FRACTAL – FRActional models and CompuTationAL Finance, the bilateral German-Hungarian Project CSITI – Coupled Systems and Innovative Time Integrators and the bilateral German-Slovakian Project ENANEFA – Efficient Numerical Approximation of Nonlinear Equations in Financial Applications all financed by the DAAD.

Author information

Authors and Affiliations

  1. Applied Mathematics and Numerical Analysis, University of Wuppertal, Gaussstrasse 20, 42119, Wuppertal, Germany

    Lorenc Kapllani, Long Teng & Matthias Ehrhardt

Authors
  1. Lorenc Kapllani
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Long Teng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Matthias Ehrhardt
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Matthias Ehrhardt .

Editor information

Editors and Affiliations

  1. Institute of Information and Communication Technology, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Ivan Dimov

  2. Institute of Information and Communication Technology, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Stefka Fidanova

Rights and permissions

Reprints and permissions

Copyright information

© 2021 The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Kapllani, L., Teng, L., Ehrhardt, M. (2021). A Multistep Scheme to Solve Backward Stochastic Differential Equations for Option Pricing on GPUs. In: Dimov, I., Fidanova, S. (eds) Advances in High Performance Computing. HPC 2019. Studies in Computational Intelligence, vol 902. Springer, Cham. https://doi.org/10.1007/978-3-030-55347-0_17

Download citation

Publish with us

Policies and ethics

Search

Navigation