Four estimators of the same call price from the same normal draws: the antithetic and control-variate tricks shrink the confidence band without adding a single simulation.
Parameters
100
10300
100
10300
1.00
0.253
3.0%
00.1
20.0%
0.050.6
5000
20020000
7
1100
Metrics
Black–Scholes price
9.4134
Running estimates and 95% bands
95% confidence band (±1.96·SE)
PlainAntitheticControl variateBothBlack–Scholes
What the tricks buy
Estimator
Standard error
Variance reduction
Plain draws needed
Saved
● Plain
0.2006
×1.0
5,000
0
● Antithetic
0.1491
×1.8
9,050
4,050
● Control variate
0.0830
×5.8
29,242
24,242
● Both
0.0419
×22.9
114,384
109,384
Both: each Z and −Z pair is averaged first (antithetic), then the discounted S_T of the pairs serves as the control variate. Draws saved = the plain draws needed for the same standard error minus the draws used.