10.5.5 Selective Valuation of Realizations
The computationally most expensive task in estimating value-at-risk with the Monte Carlo method is performing m valuations 1p[k] = θ(1r[k]). As we have seen, variance reduction can dramatically reduce the number of valuations that must be performed. Cárdenas et al. (1999) propose a complementary technique.
For estimating a standard deviation of 1P, the precise value of every realization 1p[k] is important. Every one contributes to sample standard deviation. For estimating a quantile of 1P, the precise value of only one realization 1p[k] is important—the one equal to the quantile being estimated. Unfortunately, we only find out which one that is after we have valued all the 1p[k]! We can avoid having to value every 1p[k] by employing a quadratic remapping 
= 
(1R) to identify realizations 1r[k] for which 1p[k] clearly exceeds the quantile. Since those values 1p[k] are unimportant, we may approximate them with values 
= (1r[k]). We now formalize the technique.
Consider a portfolio (0p, 1P) with portfolio mapping 1P = θ(1R), where 1R 
Nn(1|0μ, 1|0Σ). We construct a quadratic remapping = (1R). Set 
= 0p. We wish to estimate the portfolio’s q-quantile of loss, which we denote ψ. The corresponding q-quantile of loss for the remapped portfolio is denoted 
. It is calculated using the methods of Section 10.3.
We stratify 
into w disjoint subintervals ϑj based upon the conditional PDF of as follows:
[10.76]
[10.77]
[10.78]
︙
This is illustrated for a hypothetical conditional PDF for in Exhibit 10.16.
Based upon stratification
[10.79]
define a stratification
[10.80]
where
[10.81]
The 1 – q quantile of is in Ω1 by construction. Based upon the approximation ≈ 1P, we expect the 1 – q quantile of 1P to be in Ω1 as well; but we cannot be sure. Generate a realization {1r[1], 1r[2], … ,1r[m]} and calculate corresponding values = (1r[k]). Based upon these, sort the 1r[k] into regions Ω j. For only those in regions Ω1 and Ω2, calculate 1p[k] = θ(1r[k]). For the rest, approximate 1p[k] with . Based upon the (exact or approximate) values 1p[k], estimate the value-at-risk of (0p,1P).
The purpose of the region Ω2 is to play a buffer role to protect against the possibility that the approximation ≈ 1P is poor. If the approximation is good, all realized losses 1l [k] = 0p – 1p[k] for that region should be less than the estimated value-at-risk. If this is the case, then you are done. If not, improve your value-at-risk estimate as follows.
For all realizations 1r[k] in region Ω3, calculate exact portfolio values 1p[k] = θ(1r[k]). Based upon all values 1p[k] (which are now exact in regions Ω1, Ω2, and Ω3 but approximate in the other regions) estimate value-at-risk again. Letting Ω3 play a buffer role, apply the same test as before. If all realized losses for Ω3 are less than the new value-at-risk estimate, you are done. Otherwise, repeat the same procedure again, but with Ω4 playing the buffer role. Continue in this manner until an acceptable value-at-risk estimate is obtained.
Technically, this is not a variance reduction technique, but by dramatically reducing the number of portfolio valuations 1p[k] = θ(1r[k]) that must be performed, it has the same effect.