Skip to main content
Book cover

Advances in Modeling and Simulation pp 193–210Cite as

Combined Derivative Estimators

  • 188 Accesses

Abstract

We discuss combinations of simulation-based derivative estimators using infinitesimal perturbation analysis (IPA) and the likelihood ratio method (LRM). We first provide a historical perspective on combinations of IPA and LRM and then turn to connections with the generalized likelihood ratio (GLR) method. We re-derive a GLR estimator for barrier options through a combination of IPA and LRM. We then consider the behavior of a GLR estimator for a discrete-time approximation to a diffusion process as the time step shrinks. We show that an average of low-rank GLR estimators has a continuous-time limit, even though each individual estimator blows up. The limit matches an estimator previously derived through Malliavin calculus and also through a combination of IPA and LRM.

Keywords

  • Sensitivity analysis
  • Simulation
  • Likelihood ratio method

This is a preview of subscription content, access via your institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-031-10193-9_10
  • Chapter length: 18 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   149.00
Price excludes VAT (USA)
  • ISBN: 978-3-031-10193-9
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Hardcover Book
USD   199.99
Price excludes VAT (USA)
Learn about institutional subscriptions

References

  1. Avramidis, A., L’Ecuyer, P.: Efficient Monte Carlo and quasi-Monte Carlo option pricing under the variance gamma model. Manage. Sci. 52(12), 1930–1944 (2006)

    CrossRef  MATH  Google Scholar 

  2. Broadie, M., Glasserman, P.: Estimating security price derivatives using simulation. Manage. Sci. 42, 269–285 (1996)

    CrossRef  MATH  Google Scholar 

  3. Capriotti, L., Giles, M.B.: Algorithmic differentiation: adjoint Greeks made easy. Risk 25 (2012)

    Google Scholar 

  4. Chen, N., Glasserman, P.: Malliavin Greeks without Malliavin calculus. Stoch. Process. their Appl. 117, 1689–1723 (2007)

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. Fournié, E., Lasry, J.-M., Lebuchoux, J., Lions, Touzi, N.: Applications of Malliavin calculus to Monte Carlo methods in finance. Financ. Stoch. 3 391–412 (1999)

    Google Scholar 

  6. Giles, M.B.: Vibrato Monte Carlo. In: Monte Carlo and Quasi-Monte Carlo Methods 2008. Springer (2009)

    Google Scholar 

  7. Giles, M.B., Glasserman, P.: Smoking adjoints: fast Monte Carlo Greeks. Risk 19, 88–92 (2006)

    Google Scholar 

  8. Glasserman, P.: Smoothing complements and randomized score functions. Ann. Oper. Res. 39, 1–25 (1993)

    MathSciNet  Google Scholar 

  9. Glasserman, P.: Monte Carlo Methods in Financial Engineering. Springer, New York (2004)

    MATH  Google Scholar 

  10. Glasserman, P., Yao, D.: Some guidelines and guarantees for common random numbers. Manage. Sci. 38, 884–908 (1992)

    CrossRef  MATH  Google Scholar 

  11. Glynn, P.W.: Stochastic approximation for Monte Carlo optimization. In: Proceedings of the 1986 Winter Simulation Conference, pp. 356–365 (1986)

    Google Scholar 

  12. Glynn, P.W.: Optimization of stochastic systems via simulation. In: Proceedings of the 1989 Winter Simulation Conference 90–105 (1989)

    Google Scholar 

  13. Glynn, P.W., L’Ecuyer, P.: Likelihood ratio gradient estimation for stochastic recursions. Adv. Appl. Probab. 27, 1019–1053 (1995)

    CrossRef  MathSciNet  MATH  Google Scholar 

  14. Gobet, E., Munos, R.: Sensitivity analysis using Ito-Malliavin calculus and martingales, and applications to stochastic optimal control. SIAM J. Control. Optim. 43, 1676–1713 (2005)

    CrossRef  MathSciNet  MATH  Google Scholar 

  15. Griewal, A., Walther, A.: Evaluating Derivatives. SIAM, Philadelphia (2008)

    Google Scholar 

  16. Ho, Y.C., Cao, X.R.: Perturbation analysis and optimization of queueing networks. J. Optim. Theory Appl. 40, 559–582 (1983)

    CrossRef  MathSciNet  MATH  Google Scholar 

  17. L’Ecuyer, P.: A unified view of the IPA, SF, and LR gradient estimation techniques. Manage. Sci. 36, 1364–1383 (1990)

    CrossRef  MATH  Google Scholar 

  18. L’Ecuyer, P.: On the interchange of derivative and expectation for likelihood ratio derivative estimators. Manage. Sci. 40, 738–747 (1995)

    CrossRef  MATH  Google Scholar 

  19. L’Ecuyer, P.: Quasi-Monte Carlo methods with applications in finance. Finance Stoch. 13(3), 307–349 (2009)

    CrossRef  MathSciNet  MATH  Google Scholar 

  20. L’Ecuyer, P., Glynn, P.W.: Stochastic optimization by simulation: convergence proofs for the GI/G/1 queue in steady-state. Manage. Sci. 40(11), 1562–1578 (1994)

    CrossRef  MATH  Google Scholar 

  21. L’Ecuyer, P., Perron, G.: On the convergence rates of IPA and FDC derivative estimators. Oper. Res. 42(4), 643–656 (1994)

    CrossRef  MathSciNet  MATH  Google Scholar 

  22. Lemieux, C., L’Ecuyer, P.: Efficiency improvement by lattice rules for pricing Asian options. In: Proceedings of the 1998 Winter Simulation Conference, pp. 579–586 (1998)

    Google Scholar 

  23. Peng, Y., Fu, M.C., Hu, J.Q., Heidergott, B.: A new unbiased stochastic derivative estimator for discontinuous sample performances with structural parameters. Oper. Res. 66(2), 487–499 (2018)

    CrossRef  MathSciNet  MATH  Google Scholar 

  24. Peng, Y., Fu, M., Hu, J., L’Ecuyer, P., Tuffin, B.: Generalized likelihood ratio method for stochastic models with uniform random numbers as inputs. Unpublished manuscript (2020)

    Google Scholar 

  25. Peng, Y., Fu, M., Hu, J., L’Ecuyer, P., Tuffin, B.: Variance reduction for generalized likelihood ratio method by conditional Monte Carlo and randomized quasi-Monte Carlo. Unpublished manuscript (2021)

    Google Scholar 

  26. Peng, Y., Fu, M., Hu, J., L’Ecuyer, P., Tuffin, B.: Variance reduction for generalized likelihood ratio method in quantile sensitivity estimation. In: Proceedings of the 2021 Winter Simulation Conference (2021)

    Google Scholar 

  27. Reiman, M.I., Weiss, A.: Sensitivity analysis for simulation via likelihood ratios. Oper. Res. 37, 830–844 (1989)

    CrossRef  MathSciNet  MATH  Google Scholar 

  28. Rubinstein, R.: Sensitivity analysis and performance extrapolation for computer simulation models. Oper. Res. 37, 72–81 (1989)

    CrossRef  Google Scholar 

  29. Rubinstein, R.: Sensitivity analysis of discrete event systems by the “push out’’ method. Ann. Oper. Res. 39, 229–250 (1992)

    CrossRef  MathSciNet  MATH  Google Scholar 

  30. Wang, Y., Fu, M.C., Marcus, S.I.: A new stochastic derivative estimator for discontinuous functions with application to financial derivatives. Oper. Res. 60, 447–460 (2012)

    CrossRef  MathSciNet  MATH  Google Scholar 

  31. Zazanis, M., Suri, R.: Perturbation analysis gives strongly consistent sensitivity estimates for the M/G/1 queue. Manage. Sci. 34, 39–64 (1988)

    MathSciNet  MATH  Google Scholar 

  32. Zazanis, M., Suri, R.: Convergence rates of finite-difference sensitivity estimates for stochastic systems. Oper. Res. 41, 694–703 (1993)

    CrossRef  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

I thank the editors for organizing this Festschrift and the reviewers for their helpful comments.

Author information

Authors and Affiliations

  1. Columbia Business School, New York, NY, USA

    Paul Glasserman

Authors
  1. Paul Glasserman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Paul Glasserman .

Editor information

Editors and Affiliations

  1. Randwick, NSW, Australia

    Zdravko Botev

  2. NVIDIA, Berlin, Germany

    Alexander Keller

  3. University of Waterloo, Waterloo, ON, Canada

    Christiane Lemieux

  4. Rennes Bretagne-Atlantique, INRIA, Rennes Cedex, France

    Bruno Tuffin

Rights and permissions

Reprints and Permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Verify currency and authenticity via CrossMark

Cite this chapter

Glasserman, P. (2022). Combined Derivative Estimators. In: Botev, Z., Keller, A., Lemieux, C., Tuffin, B. (eds) Advances in Modeling and Simulation. Springer, Cham. https://doi.org/10.1007/978-3-031-10193-9_10

Download citation