Exercises
10.6
Consider a portfolio (89,700, 1P) with physical and options positions in two underliers whose values are represented by key vector 1R N2(1|0μ,1|0Σ) where:
[10.82]
Active holdings ω = (800 –300 –100 250) are in four assets, where
[10.83]
All options expire at time 1.
- Specify a primary mapping 1P = θ(1R).
- Value 1P at the following nine realizations for 1R (the first equals 1|0μ, and the rest are arranged about an ellipse centered at 1|0μ. They were constructed as described in Section 9.3.8.):
[10.84]
- Apply the method of least squares to your results from item (b) to construct a quadratic remapping
[10.85]
Weight the realization (1.200, 1.600) five times as heavily as the rest.
- Construct a scatter plot to assess how well
approximates 1P.
- Specify a crude Monte Carlo estimator for 0std(1P). Use sample size m = 1000. Estimate 0std(1P).
- Specify a control variate Monte Carlo estimator for 0std(1P). Use sample size m = 1000 and the fact that 0std(
) = 38,150. Estimate 0std(1P).
- Specify a stratified Monte Carlo estimator for 0std(1P). Use sample size m = 1000 and stratification size w = 16. Use values shown in Exhibit 10.17 for the conditional CDF of
(calculated using the methods of Section 10.3) to construct your stratification. Estimate 0std(1P).

Exhibit 10.17: Selected values of the conditional CDF of
.

- Estimate 0std(1P) 10 times using each of your estimators of items (e), (f), and (g). Based upon the results, construct a (very crude) estimate of the standard error of each of the estimators.
- Based upon the estimated standard errors from item (h), estimate for each of your estimators the sample size required to achieve a 1% standard error.
- Specify a crude Monte Carlo estimator for the 95%value-at-risk of portfolio (89,700, 1P). Use sample size m = 1000. Estimate thevalue-at-risk.
- Specify a control variate Monte Carlo estimator for the 95%value-at-risk of portfolio (89,700, 1P). Use sample size m = 1000 and the fact that the .05 quantile of
is 21,770. Estimate thevalue-at-risk.
- Specify a stratified Monte Carlo estimator for the 95%value-at-risk of portfolio (89,700, 1P). Use sample size m = 1000 and the fact that the .05 quantile of
is 21,770. Estimate thevalue-at-risk.
- Estimate the 95%value-at-risk of portfolio (89,700, 1P) 10 times using each of your estimators of items (j), (k), and (l). Based upon the results, construct a (very approximate) estimate of the standard error of each of the estimators.
- Based upon the estimated standard errors from item (m), estimate for each of your estimators the sample size required to achieve a 1% standard error.