# Anti-thetic sampling and second moment matching

Background:

This is in reference to ch 7 problem 10 of Mark Joshi's concepts of mathematical finance.

Question:

A normal random generator produces the following draws:

$$0.68, -0.31, -0.49, -0.19, -0.72, -0.16, -1.01, -1.60, 0.88, -0.97$$ What would these draws become after antithetic sampling and second moment matching.

Solution from Joshi - The sum-square of these and their negatives is $13.3482$. Divide by $20$ to get $0.6674$, whose square root is $\pm 0.81695$. Divide the twenty numbers by this quantity to get

0.83, -0.83,

-0.38, 0.38,

-0.60, 0.60,

-0.23, 0.23,

-0.88, 0.88,

-0.20, 0.20,

-1.24, 1.24,

-1.96, 1.96,

1.08, -1.08,

-1.19, 1.19.

I am confused by this solution as there is no formula for doing this calculation in the book. If someone could point out the formula or how he goes about getting this solution would be appreciated.

• Anthithetic means given 10 numbers {x,y,z,...} you generate 20 numbers consisting of positive/negative pairs {x,-x,y,-y,z,-z,...} that's easy. But also you want the standard deviation to be exactly 1. Because the standard deviation of these 20 is a little two low (0.8) you divide by 0.8 to scale the numbers up somewhat to exactly hit $\sigma=1$ – noob2 Feb 1 '18 at 20:46
• This is quite clearly explained in the section "Variance Reduction". Quoting: "For example, if the variance of our sample is $V$ and we rescale every random number by $V^{-1/2}$, we obtain a sample with variance 1.". – LocalVolatility Feb 1 '18 at 23:22