# Control Variate Barrier Basket Option

I need to improve the speed of convergence of PRNG Monte Carlo. I'm opening a new thread for that purpose and I have question / need confirmation about the algorithm.

I'm pricing options with Heston, QE scheme.

• For an European Down and In Put on a basket, I was thinking about using a vanilla put as a control variate (CV).

I read that the correlation between the initial and modified (CV one) random variables had to be highly correlated to produce significant reduction. In the case my barrier is -30% from the Spot, I'm not sure this will be very correlated... What do you think ? By the way, there's no closed form formula for the Heston basket. How should I proceed then ?

• For an American D&I put ? What's the CV ? Same thing, no closed form formula.

Edit: To be a little more precise

Let's say I'm dealing with a WorstOf basket: $$\text{Pay off}(T)=\max(0,K-\min(S^T_1,S^T_2))$$ What I was thinking as a CV is the following: $$CV_1 = \max(0,K-E(S^T_1)-S^T_2)$$ $$CV_2 = \max(0,K-E(S^T_2)-S^T_1)$$ Then this would be equivalent to pricing 2 vanilla put with new strikes $K_i=K-E(S^T_i)$ where $E(S^T_i)$ is given by the analytical Heston price.

Therefore the new payoff would be:

$$\text{New Payoff} = e^{-rt}\dfrac{1}{n}\sum\limits_{j=1}^{n}\text{Old Payoff} - CV_1^{(j)} - CV_2^{(j)} + E(S^T_1) + E(S^T_2)$$

An another idea could be as below but I don't know how to apply the methodology (pricing with a deterministic volatility as close as Heston vol) http://www.iaeng.org/IJAM/issues_v45/issue_1/IJAM_45_1_07.pdf