10.5.3 Stratified Sampling to Estimate Standard Deviation of Loss
Stratified sampling can also dramatically reduce the standard error of a Monte Carlo estimator for various PMMRs. A quadratic remapping guides us both in specifying a stratification and in selecting sample sizes mi for each subregion of the stratification. Our approach depends upon the PMMR to be estimated. Below, we consider standard deviation of loss and then value-at-risk.>
Consider portfolio (0p, 1P) with 1P = θ(1R) and 1R 
Nn(1|0μ,1|0Σ). Construct a quadratic remapping 
= 
(1R), and set 
= 0p. To estimate 0std(1L), we note
[10.50]
so it is sufficient to estimate 0var(1P). We stratify 
into w disjoint subintervals ϑj based upon the conditional PDF of as follows. Since is a quadratic polynomial of a joint-normal random vector, we may apply the methods of Section 10.3 to calculate its .01 and .99 quantiles, 
(.01) and (.99). Set
[10.51]
[10.52]
Define intervening subintervals ϑj, each of length
[10.53]
so
[10.54]
A stratification of size w = 8 is illustrated based upon a hypothetical conditional PDF for in Exhibit 10.14. In practice, stratification sizes of between 15 and 30 may be appropriate.
of size w = 8 constructed as described in the text.Based upon stratification
[10.55]
define a stratification
[10.56]
where
[10.57]
Specifically, Ω j is the set of realizations 1r for 1R such that corresponding portfolio values 1p = (1r) are in ϑj. In mathematical parlance, each set Ω j is the preimage under of the set ϑj.
Define w random vectors 1R j = 1R |1R ∈ Ωj. That is, 1R j equals 1R conditional on 1R being in Ωj. Define 1P j = θ(1R j) for all j. Then 1P is a mixture, in the sense of Section 3.11, of the 1P j. Applying [3.128] and [3.129],
[10.58]
[10.59]
Given samples {
, 
, … , 
} for the 1R j of respective sizes mj, we define an estimator for 0std(1P):
[10.60]
where
[10.61]
Since is conditionally a quadratic polynomial of a joint-normal random vector, we can apply the methods of Section 10.3 to value its conditional CDF, so values pj can be calculated. With exceptions of m1 and mw, all mj are set equal to each other. Specifically,
[10.62]
where the mj sum to m. The formula for m1 and mw is reasonable based upon empirical analyses.
Generate realizations 
simultaneously for all j by generating realizations 1r[k] for 1R, and allocating each to one of the 1R j according to which set ϑj the corresponding realization (1r[k]) falls in. Stop when you have sufficient realizations for each 1R j. For some 1R j, you will have more than enough realizations, but extras can be discarded.