Monte Carlo: Antithetic and QuasiRandom

Please see: "Antithetic and QuasiRandom.xls"

As I have came to understand, the Antithetic and QuasiRandom is used to reduce the variances.

Antithetic technique:

• Each time random variable r is drawn, its complement 1-r calculated and used to drive a parallel run of the simulation. This tends to lead to negatively correlated output values; hence lower variance
• Example: this is from Monte Carlo - Antithetic and Quasi-Random.xls and it's quite direct to the point.
• Table: The rand generates the random variable, and the randns generate the distribution. S1 solves $S_T$ of the simulation. S2 solves for antithesis of $S_T$ of the simulation. Then we get the payoffs for each S1 and S2. Then get the average payoff. Then we do it many many times. 
• MC Value: Get the average of the avg pay and discount it to get the present value of the option. 

QuasiRandom technique:

• We use Low Discrepancy Sequences to determine the random variables
• Faure Sequence
• Halton Sequence
• Sobol Sequence
• Inverse normal distribution
• Box-Muller transform
• Example: This is from Monte Carlo - Antithetic and Quasi-Random.xls 
• Table: it uses Faure Sequence to determine the random variable, then Box-Miller transformation is used to determine the inverse normal distribution. Then we get $S_T$ for the share price and get the payoff.
• MC Value: Get the average of the payoff and discount it to get the present value of the option. 

As can be seen in the graph below, Monte Carlo simulation, both Quasi and Antithetic Variate, converges to Black Scholes formula.

Source: Monte Carlo - Antithetic and Quasi-Random.xls