# Section 10.5.5 Exercise

###### 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.

1. Specify a primary mapping 1P = θ(1R).
2. 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]

1. 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.

1. Construct a scatter plot to assess how well  approximates 1P.
2. Specify a crude Monte Carlo estimator for 0std(1P). Use sample size m = 1000. Estimate 0std(1P).
3. 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).
4. 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 .
1. 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.
2. 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.
3. 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.
4. 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.
5. 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.
6. 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.
7. 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.