Techniques
Basic Idea
Basic idea of antithetic variables: If satisfy
then satisfies
if , then
Thus, if , then has less than half the variance of either or .
Specifically, suppose are i.i.d. copies of and are i.i.d. copies of . The estimators of are separately
with
Or we can generate copies of and copies of respectively and construct
If , then
To construct the negative correlation, we can apply the Chebyshev’s inequality: If is an increasing function of and is a decreasing function of , then .
Brief proof
Let and be two independent random variables having the same distribution, and be two functions as above, then
One General Case
Suppose we want to estimate when .
By the inverse CDF method, are i.i.d. random variables with distribution , and hence the Monte Carlo estimator is
On the other hand, the and have the same distribution of , i.e., follows distirbution and is independent of . The second estimator is
Beside, since the CDF is increasing, is also increasing. Hence if is an increasing function (e.g. Pay-off) or decreasing function, then and are increasing & decreasing or decreasing & increasing, making .
Our Antithetic variable estimator is then
Let since (iid) is independet of (iid), is also i.i.d., and thus
Examples
Example (1)
Suppose we want to estimate
The basic Monte Carlo estimate would generate independently from and return the estimate .
Since are also independent and have distribution, another estimate would be .
Note that is a decreasing function of and is an increasing function of . Therefore
Example (2)
Suppose we want to estimate , where is the price at time of maturity coming from a GBM.
Note that for
for some non-negative constants .
The basic Monte Carlo estimator is
where are i.i.d. .
Since are also i.i.d. , a second estimator is .
Note that is an increasing function of and is a decreasing function of . Therefore
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