10.5.1 Control Variates
Cárdenas et al. (1999) propose the following method of control variates to calculate value-at-risk. Consider a portfolio (0p, 1P) where 1P = θ(1R) for 1R Nn(1|0μ,1|0Σ). We construct a quadratic remapping = (1R) as described in Section 9.3.7. Set = 0p. We wish to estimate some PMMR for 1P, which we denote ψ. Let be the corresponding PMMR for . Since is a quadratic polynomial of a joint-normal random vector, we can value using the techniques of Section 10.3.
is a crude Monte Carlo estimator for ψ. Define a corresponding estimator
which is an estimator for . As described in Section 5.9.3, define a control variate estimator for ψ as
Variance reduction depends on how well approximates H, which depends on how well approximates 1P. Often, the approximation is nearly perfect. In those situations, the control variate estimator has essentially zero standard error. Sample sizes m as low as 100 produce accurate value-at-risk estimates, which match those obtainable using a stand-alone quadratic transformation, as described in Section 10.3. If such instances can be identified, it makes sense to forgo a Monte Carlo transformation and use the computationally less expensive quadratic transformation.
The more interesting situation is if the approximation for 1P is not good enough to justify use of a stand-alone quadratic transformation. In this case, the approximation is usually good enough to achieve variance reduction as a control variate. In rare instances, it is inadequate even for this purpose. This is sometimes the case if a portfolio holds expiring options. Even this tends only to be a problem with long-horizon high-quantile value-at-risk metrics or other PMMRs that focus on rare events, such as 99% ETL.
If is valued approximately using the Cornish-Fisher expansion, any error in that approximation will introduce an error into the control-variate Monte Carlo result.