Exploiting Mixed-Precision Arithmetics in a Multilevel Monte Carlo Approach on FPGAs

  • Steffen Omland
  • Mario Hefter
  • Klaus Ritter
  • Christian Brugger
  • Christian De Schryver
  • Norbert Wehn
  • Anton Kostiuk

Abstract

Nowadays, high-speed computations are mandatory for financial and insurance institutes to survive in competition and to fulfill the regulatory reporting requirements that have just toughened over the last years. A majority of these computations are carried out on huge computing clusters, which are an ever increasing cost burden for the financial industry. There, state-of-the-art CPU and GPU architectures execute arithmetic operations with predefined precisions only, that may not meet the actual requirements for a specific application. Reconfigurable architectures like Field Programmable Gate Arrays (FPGAs) have a huge potential to accelerate financial simulations while consuming only very low energy by exploiting dedicated precisions in optimal ways. In this work we present a novel methodology to speed up Multilevel Monte Carlo (MLMC) simulations on reconfigurable architectures. The idea is to aggressively lower the precisions for different parts of the algorithm without loosing any accuracy at the end. For this, we have developed a novel heuristic for selecting an appropriate precision at each stage of the simulation that can be executed with low costs at runtime. Further, we introduce a cost model for reconfigurable architectures and minimize the cost of our algorithm without changing the overall error. We consider the showcase of pricing Asian options in the Heston model. For this setup we improve one of the most advanced simulation methods by a factor of 3–9× on the same platform.

References

  1. 1.
    Alfonsi, A.: On the discretization schemes for the CIR (and Bessel squared) processes. Monte Carlo Methods Appl. 11(4), 355–384 (2005)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Alfonsi, A.: High order discretization schemes for the CIR process: application to affine term structure and Heston models. Math. Comput. 79(269), 209–237 (2010)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Alfonsi, A.: Strong convergence of some drift implicit Euler scheme. Application to the CIR process (2012). arXiv preprint arXiv:1206.3855Google Scholar
  4. 4.
    Andersen, L.B.G.: Efficient simulation of the Heston stochastic volatility model. J. Comput. Financ. 11(3), 1–42 (2008)Google Scholar
  5. 5.
    Bakshi, G., Cao, C., Chen, Z.: Empirical performance of alternative option pricing models. J. Financ. 52(5), 2003–2049 (1997)CrossRefGoogle Scholar
  6. 6.
    Baldeaux, J., Gnewuch, M.: Optimal randomized multilevel algorithms for infinite-dimensional integration on function spaces with ANOVA-type decomposition. SIAM J. Numer. Anal. 52(3), 1128–1155 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Berkaoui, A., Bossy, M., Diop, A.: Euler scheme for SDEs with non-Lipschitz diffusion coefficient: strong convergence. ESAIM: Probab. Stat. 12(1), 1–11 (2008)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Politi. Econ. 81(3), 637–654 (1973)CrossRefGoogle Scholar
  9. 9.
    Broadie, M., Kaya, Ö.: Exact simulation of stochastic volatility and other affine jump diffusion processes. Oper. Res. 54(2), 217–231 (2006)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Brugger, C., de Schryver, C., Wehn, N., Omland, S., Hefter, M., Ritter, K., Kostiuk, A., Korn, R.: Mixed precision multilevel Monte Carlo on hybrid computing systems. In: Proceedings of the 2014 IEEE Conference on Computational Intelligence for Financial Engineering Economics (CIFEr), London, pp. 215–222 (2014)Google Scholar
  11. 11.
    Chow, G., Kwok, K.W., Luk, W., Leong, P.: Mixed precision comparison in reconfigurable systems. In: 2011 IEEE 19th Annual International Symposium on Field-Programmable Custom Computing Machines (FCCM), Salt Lake City, pp. 17–24 (2011)Google Scholar
  12. 12.
    Chow, G.C.T., Tse, A.H.T., Jin, Q., Luk, W., Leong, P.H., Thomas, D.B.: A mixed precision Monte Carlo methodology for reconfigurable accelerator systems. In: Proceedings of the ACM/SIGDA International Symposium on Field Programmable Gate Arrays, FPGA’12, New York, pp. 57–66. ACM (2012)Google Scholar
  13. 13.
    Clark, I.J.: Foreign Exchange Option Pricing: A Practitioners Guide, 1st edn. Wiley, Chichester (2011)Google Scholar
  14. 14.
    Cox, J.C., Ingersoll, J., Jonathan, E., Ross, S.A.: A theory of the term structure of interest rates. Econometrica 53(2), 385–407 (1985)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Creutzig, J., Dereich, S., Müller-Gronbach, T., Ritter, K.: Infinite-dimensional quadrature and approximation of distributions. Found. Comput. Math. 9(4), 391–429 (2009)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    de Schryver, C., Shcherbakov, I., Kienle, F., Wehn, N., Marxen, H., Kostiuk, A., Korn, R.: An energy efficient FPGA accelerator for Monte Carlo option pricing with the Heston model. In: Proceedings of the 2011 International Conference on Reconfigurable Computing and FPGAs (ReConFig), Cancun, pp. 468–474 (2011)Google Scholar
  17. 17.
    Deelstra, G., Delbaen, F.: Convergence of discretized stochastic (interest rate) processes with stochastic dift term. Appl. Stoch. Models Data Anal. 14(1), 77–84 (1998)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Deelstra, G., Delbaen, F.: An efficient discretization scheme for one dimensional SDEs with a diffusion coefficient function of the form | x | α, α ∈ [1∕2, 1). INRIA Rapport de recherche (5396) (2007)Google Scholar
  19. 19.
    Dereich, S., Neuenkirch, A., Szpruch, L.: An Euler-type method for the strong approximation of the Cox–Ingersoll–Ross process. Proc. R. Soc. A: Math. Phys. Eng. Sci. 468(2140), 1105–1115 (2012)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Dick, J., Gnewuch, M.: Infinite-dimensional integration in weighted Hilbert spaces: anchored decompositions, optimal deterministic algorithms, and higher-order convergence. Found. Comput. Math. 14(5), 1027–1077 (2014)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Dick, J., Gnewuch, M.: Optimal randomized changing dimension algorithms for infinite-dimensional integration on function spaces with ANOVA-type decomposition. J. Approx. Theory 184(0), 111–145 (2014)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Duffie, D., Pan, J., Singleton, K.: Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68(6), 1343–1376 (2000)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Duranton, M., Black-Schaffer, D., Bosschere, K.D., Mabe, J.: The HiPEAC Vision for Advanced Computing in Horizon 2020 (2013). Available at http://www.hipeac.net/system/files/hipeac_roadmap1_0.pdf Google Scholar
  24. 24.
    Dyrting, S.: Evaluating the noncentral chi-square distribution for the Cox-Ingersoll-Ross process. Comput. Econ. 24(1), 35–50 (2004)CrossRefMATHGoogle Scholar
  25. 25.
    Engelmann, F.K.B., Oeltz, D.: Calibration of the Heston stochastic local volatility model: a finite volume scheme (2011). Available at SSRN: http://dx.doi.org/10.2139/ssrn.1823769
  26. 26.
    Giles, M.B.: Multilevel Monte Carlo path simulation. Oper. Res.-Baltim. 56(3), 607–617 (2008)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Giles, M.B.: Multilevel Monte Carlo methods (2013). arXiv preprint arXiv:1304.5472Google Scholar
  28. 28.
    Giles, M.B.: Multilevel Monte Carlo methods (2015, in preparation). Preprint available at http://people.maths.ox.ac.uk/gilesm/files/acta_draft.pdf
  29. 29.
    Glasserman, P.: Monte Carlo Methods in Financial Engineering. Stochastic Modelling and Applied Probability, 8th edn. Springer, New York (2003)Google Scholar
  30. 30.
    Glasserman, P., Kim, K.-K.: Gamma expansion of the Heston stochastic volatility model. Financ. Stoch. 15(2), 267–296 (2011)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Haastrecht, A.V., Pelsser, A.: Efficient, almost exact simulation of the Heston stochastic volatility model. Int. J. Theor. Appl. Financ. 13(1), 1–43 (2010)CrossRefMATHGoogle Scholar
  32. 32.
    Heinrich, S.: Monte Carlo complexity of global solution of integral equations. J. Complex. 14(2), 151–175 (1998)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Heston, S.L.: A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financ. Stud. 6(2), 327 (1993)CrossRefGoogle Scholar
  34. 34.
    Hickernell, F.J., Müller-Gronbach, T., Niu, B., Ritter, K.: Multi-level Monte Carlo algorithms for infinite-dimensional integration on \(\mathbb{R}^{\mathbb{N}}\). J. Complex. 26(3), 229–254 (2010)CrossRefMATHGoogle Scholar
  35. 35.
    Higham, N.J.: Accuracy and Stability of Numerical Algorithms, 2nd edn. Society for Industrial and Applied Mathematics, Philadelphia (2002). ISBN:978-0898715217CrossRefMATHGoogle Scholar
  36. 36.
    Higham, D., Mao, X.: Convergence of Monte Carlo simulations involving the mean-reverting square root process. J. Comput. Financ. 8(3), 35–61 (2005)Google Scholar
  37. 37.
    Hutzenthaler, A.J.M., Noll, M.: Strong convergence rates and temporal regularity for Cox-Ingersoll-Ross processes and Bessel Processes with accessible boundaries (2014). Online at arXiv: http://arxiv.org/abs/1403.6385
  38. 38.
    Kahl, C., Jäckel, P.: Fast strong approximation Monte-Carlo schemes for stochastic volatility models. J. Quant. Financ. 6(6), 513–536 (2006)CrossRefMATHGoogle Scholar
  39. 39.
    Kahl, C., Schurz, H.: Balanced Milstein methods ordinary for SDEs. Monte Carlo Methods Appl. 12(2), 143–170 (2006)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations. Stochastic Modelling and Applied Probability, vol. 12. Springer, New York (2010)Google Scholar
  41. 41.
    Korn, R., Korn, E., Kroisandt, G.: Monte Carlo Methods and Models in Finance and Insurance. CRC, Boca Raton (2010)CrossRefMATHGoogle Scholar
  42. 42.
    Kuo, F.Y., Sloan, I.H., Wasilkowski, G.W., Wozniakowski, H.: Liberating the dimension. J. Complex. 26(5), 422–454 (2010)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Lord, R., Koekkoek, R., van Dijk, D.: A comparison of biased simulation schemes for stochastic volatility models. Quant. Financ. 10(2), 177–194 (2010)CrossRefMATHGoogle Scholar
  44. 44.
    Malham, S., Wiese, A.: Chi-square simulation of the CIR process and the Heston model. Int. J. Theor. Appl. Financ. 16(3), 1350014-1-1350014-38 (2013)Google Scholar
  45. 45.
    Marxen, H.: Aspects of the application of multilevel Monte Carlo methods in the Heston model and in a Lvy process framework. PhD thesis, University of Kaiserslautern (2012)Google Scholar
  46. 46.
    Müller-Gronbach, T., Ritter, K.: Variable subspace sampling and multi-level algorithms. In: Ecuyer, P.L., Owen, A.B. (eds.) Monte Carlo and Quasi-Monte Carlo Methods 2008, pp. 131–156. Springer, Berlin/Heidelberg (2009)CrossRefGoogle Scholar
  47. 47.
    Ninomiya, S., Victoir, N.: Weak approximation of stochastic differential equations and application to derivative pricing. Appl. Math. Financ. 15(2), 107–121 (2008)MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Niu, B., Hickernell, F.J., Müller-Gronbach, T., Ritter, K.: Deterministic multi-level algorithms for infinite-dimensional integration on \(\mathbb{R}^{\mathbb{N}}\). J. Complex. 27(3–4), 331–351 (2011)CrossRefMATHGoogle Scholar
  49. 49.
    Novak, E.: The real number model in numerical analysis. J. Complex. 11(1), 57–73 (1995)CrossRefMATHGoogle Scholar
  50. 50.
    Pell, O., Averbukh, V.: Maximum performance computing with dataflow engines. Comput. Sci. Eng. 14(4), 98–103 (2012)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Steffen Omland
    • 1
  • Mario Hefter
    • 1
  • Klaus Ritter
    • 1
  • Christian Brugger
    • 2
  • Christian De Schryver
    • 2
  • Norbert Wehn
    • 2
  • Anton Kostiuk
    • 3
  1. 1.Computational Stochastics Research GroupUniversity of KaiserslauternKaiserslauternGermany
  2. 2.Microelectronic Systems Design Research GroupUniversity of KaiserslauternKaiserslauternGermany
  3. 3.Stochastic Control and Financial Mathematics GroupUniversity of KaiserslauternKaiserslauternGermany

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