5.9.3 Control Variates for Monte Carlo Estimators of Quantiles
Our development of control variates assumes that a crude Monte Carlo estimator can be represented as a sum:
[5.82]
In theory, this is always possible. In practice, it may be inconvenient. This is true when the Monte Carlo method is used to estimate quantiles of random variables, an application of particular relevance for value-at-risk.
Consider a crude Monte Carlo estimator for a q-quantile of a random variable f (U). If {f (u[1]), f (u[2]), … , f (u[m])} is a realization of a sample, we estimate the q-quantile of f (U) by selecting that value f (u[k]) such that q m of the values are less than or equal to it. With considerable effort, this computation may be represented as a sum, but doing so is inconvenient. Such a representation is not necessary for applying a control variate.
Suppose ξ(U) is a control variate for f (U), as defined earlier. We wish to estimate some parameter ψ of f (U). Suppose the corresponding parameter 
of ξ(U) is known. Let
[5.83]
be an estimator for ψ. Define a corresponding estimator
[5.84]
which is an estimator for . Define a control variate estimator for ψ as
[5.85]
We shall use this form of control variate estimator in Chapter 10 for estimating portfolio value-at-risk. Note the similarity between this estimator and control variate estimator [5.62], which we developed earlier. Indeed, if H represents a sum, then
[5.86]
and control variate estimator [5.81] reduces to control variate estimator [5.62]:
[5.87]
[5.88]
We shall call 
a control variate for H. If the correlation between and H is difficult to estimate, we may simply set c = 1.