Consider a portfolio (89,700, ¹P) with physical and options positions in two underliers whose values are represented by key vector where

Active holdings = (800  –300  –100   250) are in four assets, where

All options expire at time 1.

a. Specify a primary mapping = ().

b. Value at the following nine realizations for (the first equals , and the rest are arranged about an ellipse centered at . They were constructed as described in Section 9.3.):

c. Apply the method of least squares to your results from item (b) to construct a quadratic remapping

Weight the realization (1.200, 1.600) five times as heavily as the rest.

d. Construct a scatter plot to assess how well approximates .

e. Specify a crude Monte Carlo estimator for ⁰std(). Use sample size m = 1000. Estimate ⁰std().

f. Specify a control variate Monte Carlo estimator for ⁰std(). Use sample size m = 1000 and the fact that ⁰std() = 38,150. Estimate ⁰std().

g. Specify a stratified Monte Carlo estimator for ⁰std(). Use sample size m = 1000 and stratification size w = 16. Use the 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().

-14,730	0.01000	86,196	0.41572
-2,114	0.01792	98,811	0.54767
10,501	0.03123	111,427	0.68368
23,117	0.05280	124,043	0.80736
35,733	0.08635	136,659	0.90303
48,349	0.13621	149,274	0.96263
60,964	0.20653	161,890	0.99000
73,580	0.29997
Exhibit 10.17 Selected values of the conditional CDF of

h. Estimate ⁰std() 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.

i. 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.

j. Specify a crude Monte Carlo estimator for the 95% VaR of portfolio (89,700, ). Use sample size m = 1000. Estimate the VaR.

k. Specify a control variate Monte Carlo estimator for the 95% VaR of portfolio (89,700, ). Use sample size m = 1000 and the fact that the .05-quantile of is 21,770. Estimate the VaR.

l. Specify a stratified Monte Carlo estimator for the 95% VaR of portfolio (89,700, ). Use sample size m = 1000 and the fact that the .05-quantile of is 21,770. Estimate the VaR.

m. Estimate the 95% VaR of portfolio (89,700, ) 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.

n. 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.

a.

b., c. Computations are performed in a spreadsheet. We obtain portfolio remapping

where

d.

e. Our crude Monte Carlo estimator for ⁰std() is

It is implemented in a spreadsheet.

f. Our control variate Monte Carlo estimator for ⁰std() is

where H is defined as in [s5] above, and

The control variate Monte Carlo estimator is implemented in a spreadsheet.

g. As described on p. 376, we stratify into w = 16 disjoint subintervals based upon the conditional CDF of . For this purpose, we use the values of the CDF of indicated in Exhibit 10.17. Results are

Based upon stratification

define stratification

where

for all j.

Define 16 random vectors . That is, equals conditional on being in . Define = () for all j. Given samples for the , each of respective size , we define our stratified sampling Monte Carlo estimator for ⁰std(¹P) as

where probabilities

can be calculated from the values of the CDF of indicated in Exhibit 10.17. Sample sizes are selected as suggested on p. 377. Generating sample realizations for this estimator is not easy with a spreadsheet. The estimator is easy to code as a simple program.

h. Each of the three estimators is applied ten times to obtain a total of 30 estimates for ⁰std(¹P). Results are indicated below.

i. Based upon our results from part (i), we estimate that the crude, control variate and stratified sampling estimators have standard errors of 1.79%, 0.43%, and 0.40% (results are obtained by dividing stdev results by mean results from the above table for each estimator). These results were obtained using a sample size of 1000 for each estimator. Since standard error is inversely proportional to the square root of sample size, we estimate that the three estimators will require respective sample sizes of 3209, 187, and 156 to have a 1% standard error.

j. Based upon a sample for , we define a sample for with . Our crude Monte Carlo estimator estimates the 95% VaR as 89,700 less the sample .05-quantile of . This is implemented as a spreadsheet.

k. Based upon a sample for , we define a sample for with and another sample for with . Let H be the sample .05-quantile estimator for the , and let be the sample .05-quantile estimator for the . Our control variate estimator for 95% VaR then is

This is implemented as a spreadsheet.

l. As described on pp. 378–179, we stratify into 2 disjoint subintervals :

Based upon stratification

define stratification

where

for each j.

Define two random vectors . That is, equals conditional on being in . Define samples and . Define sample with for 1 k 50 and for 51 k 1000. VaR is estimated as 89,700 less the sample .05-quantile of the . Generating sample realizations for this estimator is not easy with a spreadsheet. The estimator is easy to code as a simple program.

m. Each of the three estimators for 95% VaR is applied ten times to obtain a total of 30 estimates for ⁰std(¹P). Results are indicated below.

n. Based upon our results from part (m), we estimate that the crude, control variate and stratified sampling estimators have standard errors of 3.38%, 1.24%, and 0.55% (results are obtained by dividing stdev results by mean results from the above table for each estimator). These results were obtained using a sample size of 1000 for each estimator. Since standard error is inversely proportional to the square root of sample size, we estimate that the three estimators will require respective sample sizes of 11,417, 1539, and 300 to have a 1% standard error.