IDEAS home Printed from https://ideas.repec.org/a/inm/oropre/v58y2010i2p357-370.html
   My bibliography  Save this article

Pathwise Estimation of Probability Sensitivities Through Terminating or Steady-State Simulations

Author

Listed:
  • L. Jeff Hong

    (Department of Industrial Engineering and Logistics Management, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China)

  • Guangwu Liu

    (Department of Management Sciences, City University of Hong Kong, Kowloon, Hong Kong, China)

Abstract

A probability is the expectation of an indicator function. However, the standard pathwise sensitivity estimation approach, which interchanges the differentiation and expectation, cannot be directly applied because the indicator function is discontinuous. In this paper, we design a pathwise sensitivity estimator for probability functions based on a result of Hong [Hong, L. J. 2009. Estimating quantile sensitivities. Oper. Res. 57 (1) 118--130]. We show that the estimator is consistent and follows a central limit theorem for simulation outputs from both terminating and steady-state simulations, and the optimal rate of convergence of the estimator is n -2/5 where n is the sample size. We further demonstrate how to use importance sampling to accelerate the rate of convergence of the estimator to n -1/2 , which is the typical rate of convergence for statistical estimation. We illustrate the performances of our estimators and compare them to other well-known estimators through several examples.

Suggested Citation

  • 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.
    • Handle: RePEc:inm:oropre:v:58:y:2010:i:2:p:357-370
    DOI: 10.1287/opre.1090.0739
    as

    Download full text from publisher

    File URL: http://dx.doi.org/10.1287/opre.1090.0739
    Download Restriction: no
    File URL: https://libkey.io/10.1287/opre.1090.0739?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. 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.
    2. 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.
    3. Mark Broadie & Paul Glasserman, 1996. "Estimating Security Price Derivatives Using Simulation," Management Science, INFORMS, vol. 42(2), pages 269-285, February.
    4. Chiahon Chien & David Goldsman & Benjamin Melamed, 1997. "Large-Sample Results for Batch Means," Management Science, INFORMS, vol. 43(9), pages 1288-1295, September.
    5. L. Jeff Hong, 2009. "Estimating Quantile Sensitivities," Operations Research, INFORMS, vol. 57(1), pages 118-130, February.
    6. P. Heidelberger & P. A. W. Lewis, 1984. "Quantile Estimation in Dependent Sequences," Operations Research, INFORMS, vol. 32(1), pages 185-209, February.
    7. Lee Schruben, 1983. "Confidence Interval Estimation Using Standardized Time Series," Operations Research, INFORMS, vol. 31(6), pages 1090-1108, December.
    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. 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.
    2. Bernd Heidergott & Warren Volk-Makarewicz, 2013. "A Measure-Valued Differentiation Approach to Sensitivity Analysis of Quantiles," Tinbergen Institute Discussion Papers 13-082/III, Tinbergen Institute.
    3. 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.
    4. Borgonovo, Emanuele & Plischke, Elmar & Rabitti, Giovanni, 2024. "The many Shapley values for explainable artificial intelligence: A sensitivity analysis perspective," European Journal of Operational Research, Elsevier, vol. 318(3), pages 911-926.
    5. Joshi, Mark S. & Zhu, Dan, 2016. "An exact method for the sensitivity analysis of systems simulated by rejection techniques," European Journal of Operational Research, Elsevier, vol. 254(3), pages 875-888.
    6. Guangwu Liu, 2015. "Simulating Risk Contributions of Credit Portfolios," Operations Research, INFORMS, vol. 63(1), pages 104-121, February.
    7. 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.
    8. 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.
    9. 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.
    10. 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.

    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. 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.
    2. Silvana M. Pesenti & Pietro Millossovich & Andreas Tsanakas, 2023. "Differential Quantile-Based Sensitivity in Discontinuous Models," Papers 2310.06151, arXiv.org, revised Oct 2024.
    3. 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.
    4. 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.
    5. 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.
    6. 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.
    7. 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.
    8. 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.
    9. 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.
    10. Makam, Vaishno Devi & Millossovich, Pietro & Tsanakas, Andreas, 2021. "Sensitivity analysis with χ2-divergences," Insurance: Mathematics and Economics, Elsevier, vol. 100(C), pages 372-383.
    11. Pesenti, Silvana M. & Tsanakas, Andreas & Millossovich, Pietro, 2018. "Euler allocations in the presence of non-linear reinsurance: Comment on Major (2018)," Insurance: Mathematics and Economics, Elsevier, vol. 83(C), pages 29-31.
    12. Xi Chen & Kyoung-Kuk Kim, 2016. "Efficient VaR and CVaR Measurement via Stochastic Kriging," INFORMS Journal on Computing, INFORMS, vol. 28(4), pages 629-644, November.
    13. Huang, Zhenzhen & Kwok, Yue Kuen & Xu, Ziqing, 2024. "Efficient algorithms for calculating risk measures and risk contributions in copula credit risk models," Insurance: Mathematics and Economics, Elsevier, vol. 115(C), pages 132-150.
    14. Pesenti, Silvana M. & Millossovich, Pietro & Tsanakas, Andreas, 2019. "Reverse sensitivity testing: What does it take to break the model?," European Journal of Operational Research, Elsevier, vol. 274(2), pages 654-670.
    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. Katja Schilling & Daniel Bauer & Marcus C. Christiansen & Alexander Kling, 2020. "Decomposing Dynamic Risks into Risk Components," Management Science, INFORMS, vol. 66(12), pages 5738-5756, December.
    17. J P C Kleijnen & W C M van Beers, 2013. "Monotonicity-preserving bootstrapped Kriging metamodels for expensive simulations," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 64(5), pages 708-717, May.
    18. Weihuan Huang, 2023. "Estimating Systemic Risk within Financial Networks: A Two-Step Nonparametric Method," Papers 2310.18658, arXiv.org.
    19. 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.
    20. M. Merz & R. Richman & T. Tsanakas & M. V. Wuthrich, 2021. "Interpreting Deep Learning Models with Marginal Attribution by Conditioning on Quantiles," Papers 2103.11706, arXiv.org.

    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:oropre:v:58:y:2010:i:2:p:357-370. 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.