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

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  • 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
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    References listed on IDEAS

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    1. P. Heidelberger & P. A. W. Lewis, 1984. "Quantile Estimation in Dependent Sequences," Operations Research, INFORMS, vol. 32(1), pages 185-209, February.
    2. 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.
    3. 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.
    4. Mark Broadie & Paul Glasserman, 1996. "Estimating Security Price Derivatives Using Simulation," Management Science, INFORMS, vol. 42(2), pages 269-285, February.
    5. Chiahon Chien & David Goldsman & Benjamin Melamed, 1997. "Large-Sample Results for Batch Means," Management Science, INFORMS, vol. 43(9), pages 1288-1295, September.
    6. L. Jeff Hong, 2009. "Estimating Quantile Sensitivities," Operations Research, INFORMS, vol. 57(1), pages 118-130, February.
    7. Lee Schruben, 1983. "Confidence Interval Estimation Using Standardized Time Series," Operations Research, INFORMS, vol. 31(6), pages 1090-1108, December.
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    Citations

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    Cited by:

    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. Guangwu Liu, 2015. "Simulating Risk Contributions of Credit Portfolios," Operations Research, INFORMS, vol. 63(1), pages 104-121, February.
    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. 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.
    5. 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.
    6. 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.
    7. 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.
    8. 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.
    9. 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.

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