Control Variate Method
$\overline{Y}$ = Price of option Y using MonteCarlo Valuation
We use option $X$ [btw]The exact/true price of option X[/btw] as control variate to determine $Y^*$ [btw] The Control variate estimate of option Y[/btw]
 $Y^* = \overline{Y} + \text{Adjustment} = \overline{Y} + \beta(X \overline{X})$
 $Var[Y^*] = Var [ \overline{Y} + \beta(X \overline{X})] \equiv Var [ \overline{Y} – \beta \overline{X}]$ = $Var[\overline{Y} ] + \beta^2Var[\overline{X}]2\beta Cov[\overline{Y},\overline{X}]$
Value of $\beta$ that minimizes $Var[Y^*]$:
$Var[Y^*]$ is minimized when $\beta = \frac{Cov[\overline{Y},\overline{X}]}{Var[\overline{X}]}$
= $\frac{\frac{\sum_{i=1}^n{(Y_i\overline{Y})(X_i\overline{X})}}{n}}{\frac{\sum_{i=1}^n{(X_i\overline{X})}}{n}}$
= $\frac{\sum_{i=1}^n{(Y_i\overline{Y})(X_i\overline{X})}}{\sum_{i=1}^n{(X_i\overline{X})}}$
When $\beta$ is set to the above value, $Var[Y^*] = Var[\overline{Y}](1 \rho_{\overline{Y}\overline{X}}^2)$
[icon name=”bullseye” class=””] Tip: Use the 2var stat feature in the BA II Plus calculator to calculate $\beta$. Enter the values for X and Y and perform 2var stat. The value of $a$ in the result set is $\beta$ that minimizes Var[Y]
Antithetic Variate Method
For every $u_i$ use $1u_i$
For every $z_i$ use $ – z_i$
Stratified Sampling

Break sample space into smaller and equal sized spaces

Scale the uniform numbers to fit into the smaller sampling spaces
Tip: Use the TI30XS ‘data’ key to calculate the new random numbers. Enter the given random numbers in L1, enter the pattern $(0,1,2,3,0,1,2,3)$ or $(0,1,2,3,4,5,6,7,8,9)$ in L2 and set L3 = $\frac{1}{n}[L1+L2]$