#### Control Variate Method

$\overline{Y}$ = Price of option Y using MonteCarlo Valuation

We use option $X$ The exact/true price of option X as control variate to determine $Y^*$ The Control variate estimate of option Y

• $Y^* = \overline{Y} + \text{Adjustment} = \overline{Y} + \beta(X- \overline{X})$
• $Var[Y^*] = Var [ \overline{Y} + \beta(X- \overline{X})] \equiv Var [ \overline{Y} – \beta \overline{X}]$ = $Var[\overline{Y} ] + \beta^2Var[\overline{X}]-2\beta Cov[\overline{Y},\overline{X}]$

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 2-var stat feature in the BA II Plus calculator to calculate $\beta$. Enter the values for X and Y and perform 2-var stat. The value of $a$ in the result set is $\beta$ that minimizes Var[Y]

#### Antithetic Variate Method

For every $u_i$ use $1-u_i$

For every $z_i$ use $– z_i$

#### Stratified Sampling

1. Break sample space into smaller and equal sized spaces

2. Scale the uniform numbers to fit into the smaller sampling spaces

Tip: Use the TI-30XS ‘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 = $\frac{1}{n}[L1+L2]$

#### Exam FM/2

Financial MathematicsDerivative Markets