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Conditional Monte Carlo Estimation of Quantile Sensitivities

Robert H. Smith School of Business and Institute for Systems Research, University of Maryland, College Park, Maryland 20742
Department of Industrial Engineering and Logistics Management, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China
Department of Management Science, School of Management, Fudan University, 200433 Shanghai, China
mfu{at}rhsmith.umd.edu
hongl{at}ust.hk
hujq{at}fudan.edu.cn

Estimating quantile sensitivities is important in many optimization applications, from hedging in financial engineering to service-level constraints in inventory control to more general chance constraints in stochastic programming. Recently, Hong (Hong, L. J. 2009. Estimating quantile sensitivities. Oper. Res. 57 118–130) derived a batched infinitesimal perturbation analysis estimator for quantile sensitivities, and Liu and Hong (Liu, G., L. J. Hong. 2009. Kernel estimation of quantile sensitivities. Naval Res. Logist. 56 511–525) derived a kernel estimator. Both of these estimators are consistent with convergence rates bounded by n^-1/3 and n^-2/5, respectively. In this paper, we use conditional Monte Carlo to derive a consistent quantile sensitivity estimator that improves upon these convergence rates and requires no batching or binning. We illustrate the new estimator using a simple but realistic portfolio credit risk example, for which the previous work is inapplicable.

Key Words: quantiles; value at risk; credit risk; Monte Carlo simulation; gradient estimation
History: Received: November 20, 2008; accepted: August 4, 2009.