IDEAS home Printed from https://ideas.repec.org/a/inm/orijoc/v32y2020i2p390-407.html
   My bibliography  Save this article

On the Variance of Single-Run Unbiased Stochastic Derivative Estimators

Author

Listed:
  • Zhenyu Cui

    (School of Business, Stevens Institute of Technology, Hoboken, New Jersey 07030)

  • Michael C. Fu

    (The Robert H. Smith School of Business, Institute for Systems Research, University of Maryland, College Park, Maryland 20742)

  • Jian-Qiang Hu

    (Department of Management Science, School of Management, Fudan University, Shanghai 200433, China)

  • Yanchu Liu

    (Department of Finance, Lingnan (University) College, Sun Yat-sen University, Guangzhou, Guangdong 510275, China)

  • Yijie Peng

    (Department of Management Science and Information Systems, Guanghua School of Management, Peking University, Beijing 100871, China)

  • Lingjiong Zhu

    (Department of Mathematics, Florida State University, Tallahassee, Florida 32306)

Abstract

We analyze the variance of single-run unbiased stochastic derivative estimators. The distribution of a specific conditional expectation characterizes an intrinsic distributional property of the derivative estimators in a given class, which, in turn, separates two of the most popular single-run unbiased derivative estimators, infinitesimal perturbation analysis and the likelihood ratio method, into disjoint classes. In addition, a necessary and sufficient condition for the estimators to achieve the lowest variance in a certain class is provided, as well as insights into finding an estimator with lower variance. We offer a sufficient condition to substantiate the rule of thumb that the infinitesimal perturbation analysis estimator has a smaller variance than does the likelihood ratio method estimator and to provide a counterexample when the sufficient condition is not satisfied.

Suggested Citation

  • Zhenyu Cui & Michael C. Fu & Jian-Qiang Hu & Yanchu Liu & Yijie Peng & Lingjiong Zhu, 2020. "On the Variance of Single-Run Unbiased Stochastic Derivative Estimators," INFORMS Journal on Computing, INFORMS, vol. 32(2), pages 390-407, April.
    • Handle: RePEc:inm:orijoc:v:32:y:2020:i:2:p:390-407
    DOI: 10.1287/ijoc.2019.0897
    as

    Download full text from publisher

    File URL: https://doi.org/10.1287/ijoc.2019.0897
    Download Restriction: no
    File URL: https://libkey.io/10.1287/ijoc.2019.0897?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    1. L. Jeff Hong & Sandeep Juneja & Jun Luo, 2014. "Estimating Sensitivities of Portfolio Credit Risk Using Monte Carlo," INFORMS Journal on Computing, INFORMS, vol. 26(4), pages 848-865, November.
    2. Nan Chen & Yanchu Liu, 2014. "American Option Sensitivities Estimation via a Generalized Infinitesimal Perturbation Analysis Approach," Operations Research, INFORMS, vol. 62(3), pages 616-632, June.
    3. Rajan Suri & Michael A. Zazanis, 1988. "Perturbation Analysis Gives Strongly Consistent Sensitivity Estimates for the M/G/1 Queue," Management Science, INFORMS, vol. 34(1), pages 39-64, January.
    4. Michael A. Crane & Donald L. Iglehart, 1975. "Simulating Stable Stochastic Systems: III. Regenerative Processes and Discrete-Event Simulations," Operations Research, INFORMS, vol. 23(1), pages 33-45, February.
    5. Guangwu Liu & Liu Jeff Hong, 2009. "Kernel estimation of quantile sensitivities," Naval Research Logistics (NRL), John Wiley & Sons, vol. 56(6), pages 511-525, September.
    6. Eric Fournié & Jean-Michel Lasry & Pierre-Louis Lions & Jérôme Lebuchoux, 2001. "Applications of Malliavin calculus to Monte-Carlo methods in finance. II," Finance and Stochastics, Springer, vol. 5(2), pages 201-236.
    7. Mark Broadie & Paul Glasserman, 1996. "Estimating Security Price Derivatives Using Simulation," Management Science, INFORMS, vol. 42(2), pages 269-285, February.
    8. Heidergott, Bernd, 1999. "Optimisation of a single-component maintenance system: A smoothed perturbation analysis approach," European Journal of Operational Research, Elsevier, vol. 119(1), pages 181-190, November.
    9. Cui, Zhenyu & Lee, Chihoon & Liu, Yanchu, 2018. "Single-transform formulas for pricing Asian options in a general approximation framework under Markov processes," European Journal of Operational Research, Elsevier, vol. 266(3), pages 1134-1139.
    10. L. Jeff Hong, 2009. "Estimating Quantile Sensitivities," Operations Research, INFORMS, vol. 57(1), pages 118-130, February.
    11. Michael C. Fu, 1994. "Sample Path Derivatives for (s, S) Inventory Systems," Operations Research, INFORMS, vol. 42(2), pages 351-364, April.
    12. Chen, Nan & Glasserman, Paul, 2007. "Malliavin Greeks without Malliavin calculus," Stochastic Processes and their Applications, Elsevier, vol. 117(11), pages 1689-1723, November.
    13. Michael C. Fu, 2008. "What you should know about simulation and derivatives," Naval Research Logistics (NRL), John Wiley & Sons, vol. 55(8), pages 723-736, December.
    14. Heidergott, Bernd & Vazquez-Abad, Felisa J. & Volk-Makarewicz, Warren, 2008. "Sensitivity estimation for Gaussian systems," European Journal of Operational Research, Elsevier, vol. 187(1), pages 193-207, May.
    15. Bernd Heidergott & Warren Volk-Makarewicz, 2016. "A Measure-Valued Differentiation Approach to Sensitivities of Quantiles," Mathematics of Operations Research, INFORMS, vol. 41(1), pages 293-317, February.
    16. Bernd Heidergott & Haralambie Leahu, 2010. "Weak Differentiability of Product Measures," Mathematics of Operations Research, INFORMS, vol. 35(1), pages 27-51, February.
    17. Shaolong Tong & Guangwu Liu, 2016. "Importance Sampling for Option Greeks with Discontinuous Payoffs," INFORMS Journal on Computing, INFORMS, vol. 28(2), pages 223-235, May.
    18. Michael C. Fu & Jian-Qiang Hu, 1993. "Second Derivative Sample Path Estimators for the GI/G/m Queue," Management Science, INFORMS, vol. 39(3), pages 359-383, March.
    19. Michael C. Fu & L. Jeff Hong & Jian-Qiang Hu, 2009. "Conditional Monte Carlo Estimation of Quantile Sensitivities," Management Science, INFORMS, vol. 55(12), pages 2019-2027, December.
    20. Michael C. Fu, 2015. "Stochastic Gradient Estimation," International Series in Operations Research & Management Science, in: Michael C Fu (ed.), Handbook of Simulation Optimization, edition 127, chapter 0, pages 105-147, Springer.
    21. Michael C. Fu & Jian-Qiang Hu, 1999. "Efficient Design and Sensitivity Analysis of Control Charts Using Monte Carlo Simulation," Management Science, INFORMS, vol. 45(3), pages 395-413, March.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Lei, Lei & Peng, Yijie & Fu, Michael C. & Hu, Jian-Qiang, 2023. "Copula sensitivity analysis for portfolio credit derivatives," European Journal of Operational Research, Elsevier, vol. 308(1), pages 455-466.
    2. Pierre L’Ecuyer & Florian Puchhammer & Amal Ben Abdellah, 2022. "Monte Carlo and Quasi–Monte Carlo Density Estimation via Conditioning," INFORMS Journal on Computing, INFORMS, vol. 34(3), pages 1729-1748, May.
    3. Yijie Peng & Li Xiao & Bernd Heidergott & L. Jeff Hong & Henry Lam, 2022. "A New Likelihood Ratio Method for Training Artificial Neural Networks," INFORMS Journal on Computing, INFORMS, vol. 34(1), pages 638-655, January.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Michael C. Fu, 2008. "What you should know about simulation and derivatives," Naval Research Logistics (NRL), John Wiley & Sons, vol. 55(8), pages 723-736, December.
    2. Yongqiang Wang & Michael C. Fu & Steven I. Marcus, 2012. "A New Stochastic Derivative Estimator for Discontinuous Payoff Functions with Application to Financial Derivatives," Operations Research, INFORMS, vol. 60(2), pages 447-460, April.
    3. Zhaolin Hu & Dali Zhang, 2018. "Utility‐based shortfall risk: Efficient computations via Monte Carlo," Naval Research Logistics (NRL), John Wiley & Sons, vol. 65(5), pages 378-392, August.
    4. Bernd Heidergott & Warren Volk-Makarewicz, 2016. "A Measure-Valued Differentiation Approach to Sensitivities of Quantiles," Mathematics of Operations Research, INFORMS, vol. 41(1), pages 293-317, February.
    5. L. Jeff Hong & Sandeep Juneja & Jun Luo, 2014. "Estimating Sensitivities of Portfolio Credit Risk Using Monte Carlo," INFORMS Journal on Computing, INFORMS, vol. 26(4), pages 848-865, November.
    6. Yijie Peng & Michael C. Fu & Bernd Heidergott & Henry Lam, 2020. "Maximum Likelihood Estimation by Monte Carlo Simulation: Toward Data-Driven Stochastic Modeling," Operations Research, INFORMS, vol. 68(6), pages 1896-1912, November.
    7. Guangwu Liu & L. Jeff Hong, 2011. "Kernel Estimation of the Greeks for Options with Discontinuous Payoffs," Operations Research, INFORMS, vol. 59(1), pages 96-108, February.
    8. Peter W. Glynn & Yijie Peng & Michael C. Fu & Jian-Qiang Hu, 2021. "Computing Sensitivities for Distortion Risk Measures," INFORMS Journal on Computing, INFORMS, vol. 33(4), pages 1520-1532, October.
    9. He, Zhijian, 2022. "Sensitivity estimation of conditional value at risk using randomized quasi-Monte Carlo," European Journal of Operational Research, Elsevier, vol. 298(1), pages 229-242.
    10. Yijie Peng & Li Xiao & Bernd Heidergott & L. Jeff Hong & Henry Lam, 2022. "A New Likelihood Ratio Method for Training Artificial Neural Networks," INFORMS Journal on Computing, INFORMS, vol. 34(1), pages 638-655, January.
    11. Silvana M. Pesenti & Pietro Millossovich & Andreas Tsanakas, 2023. "Differential Quantile-Based Sensitivity in Discontinuous Models," Papers 2310.06151, arXiv.org, revised Oct 2024.
    12. Guangxin Jiang & Michael C. Fu, 2015. "Technical Note—On Estimating Quantile Sensitivities via Infinitesimal Perturbation Analysis," Operations Research, INFORMS, vol. 63(2), pages 435-441, April.
    13. Koike, Takaaki & Saporito, Yuri & Targino, Rodrigo, 2022. "Avoiding zero probability events when computing Value at Risk contributions," Insurance: Mathematics and Economics, Elsevier, vol. 106(C), pages 173-192.
    14. L. Jeff Hong & Guangwu Liu, 2010. "Pathwise Estimation of Probability Sensitivities Through Terminating or Steady-State Simulations," Operations Research, INFORMS, vol. 58(2), pages 357-370, April.
    15. Jiaqiao Hu & Yijie Peng & Gongbo Zhang & Qi Zhang, 2022. "A Stochastic Approximation Method for Simulation-Based Quantile Optimization," INFORMS Journal on Computing, INFORMS, vol. 34(6), pages 2889-2907, November.
    16. L. Jeff Hong & Yi Yang & Liwei Zhang, 2011. "Sequential Convex Approximations to Joint Chance Constrained Programs: A Monte Carlo Approach," Operations Research, INFORMS, vol. 59(3), pages 617-630, June.
    17. Guangwu Liu, 2015. "Simulating Risk Contributions of Credit Portfolios," Operations Research, INFORMS, vol. 63(1), pages 104-121, February.
    18. Yijie Peng & Chun-Hung Chen & Michael C. Fu & Jian-Qiang Hu & Ilya O. Ryzhov, 2021. "Efficient Sampling Allocation Procedures for Optimal Quantile Selection," INFORMS Journal on Computing, INFORMS, vol. 33(1), pages 230-245, January.
    19. Guangwu Liu & Liu Jeff Hong, 2009. "Kernel estimation of quantile sensitivities," Naval Research Logistics (NRL), John Wiley & Sons, vol. 56(6), pages 511-525, September.
    20. Makam, Vaishno Devi & Millossovich, Pietro & Tsanakas, Andreas, 2021. "Sensitivity analysis with χ2-divergences," Insurance: Mathematics and Economics, Elsevier, vol. 100(C), pages 372-383.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:inm:orijoc:v:32:y:2020:i:2:p:390-407. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Chris Asher (email available below). General contact details of provider: https://edirc.repec.org/data/inforea.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.