10.5.2 Example: Control Variates

10.5.2  Example: Control Variates

Results using control-variate estimator [10.46] tend to be excellent. In this example, we consider an extreme situation where a portfolio’s value is dominated by several expiring options, which render a quadratic remapping an especially poor approximation for the primary portfolio mapping. Even in this extreme case, the control variate dramatically reduces standard error for a variety of PMMRs. The one exception is a 99% value-at-risk metric, for which the control variate increases standard error modestly.

Assume a 1-week value-at-risk horizon. A portfolio (0p,1P) holds futures and options in several underliers. Its price behaviour is dominated by several options that expire at the end of the value-at-risk horizon. Let 1P = θ(1R), where 1R is conditionally joint-normal. Using the method of least squares, we construct a quadratic remapping  = (1R) for 1P. We set  = 0p. Exhibit 10.8 is a scatter plot of realizations of 1P and  .

Exhibit 10.8: Scatter plot of 1P and realizations.

Because of the expiring options, the approximation  ≈ 1P is not good enough to justify use of a stand-alone quadratic transformation. Instead, we use a Monte Carlo transformation with control-variate estimator [10.46] based upon  . We do so for four PMMRs:

  • standard deviation of loss;
  • 90% value-at-risk;
  • 95% value-at-risk; and
  • 99% value-at-risk.

For each PMMR ψ, we value corresponding PMMR si tilde for  using the methods of Section 10.3. Next, we apply methods of Section 5.8 to generate a realization {1r[1], 1r[2], … , 1r[20000]} of a sample for 1R and calculate corresponding values 1p[k] = θ(1r[k]) and  = (1r[k]). For each PMMR ψ, we employ a sample estimator H, and set

[10.47]

[10.48]

Our crude Monte Carlo estimate for ψ is simply h. Substituting h , , and si tilde into [10.46], and setting c = 1, we obtain our control-variate estimate for ψ:

[10.49]

Exhibit 10.9 indicates control variate and crude Monte Carlo estimates for each of the four PMMRs. It compares them with estimates obtained using a stand-alone quadratic transformation. It also indicates exact results for PMMR:

Exhibit 10.9: PMMR values estimated using control variates, crude Monte Carlo and a quadratic transformation. Results in the last column are exact.

Corresponding errors are indicated in Exhibit 10.10, and Exhibit 10.11 illustrates for each metric how the control variate and crude Monte Carlo estimators converged to their respective results.

Exhibit 10.10: Realized absolute errors calculated from the results of Exhibit 10.9.
Exhibit 10.11: Convergence of the control variate and crude Monte Carlo estimators is illustrated for the four PMMRs. Each graph plots preliminary estimates based upon partial sample sizes m ranging from just the first 100 sample points through the total 20000 sample points used to construct the PMMR estimates. Arrows on either side of each graph indicate the true PMMR value being estimated in each case.

Of more interest are the standard errors for each estimator. These are estimated by repeating each Monte Carlo analysis 500 times and taking sample standard deviations of the results.5 Estimated standard errors are indicated in Exhibit 10.12.

Exhibit 10.12: Estimated standard errors.

From these, we can calculate the sample size m required to achieve a 1% standard error. Results are indicated in Exhibit 10.13.

Exhibit 10.13: Sample sizes m required to achieve a 1% standard error.

For all metrics except 99%value-at-risk, use of a control variate reduces the sample size m required to achieve a 1% standard error. Typical results are far better than these. For this example, we selected a portfolio for which a quadratic remapping is an especially poor approximation for the primary portfolio mapping.