# Variance reduction techniques - control variates technique

In control variate technique we have to calculate $$b=\frac{\text{cov}\{{X,Y}\}}{\text{var}\{{X}\}}$$ where $$X$$ is a payoff from standard call option and $$Y$$ is a payoff from for example barrier option. Why we have to estimate $$b$$ before we use this method and we cant use payoffs which we use during pricing? Or maybe I can first calculate the payoffs for these options, then based on them, calculate the option price, and finally calculate $$b$$ using the same payoffs and change the price of the barrier option accordingly?

EDIT: My current problem: I try to calculate value of Up and out call option using MC simulation. I use three methods: 1) standard Monte Carlo, 2) anthitetic variates MC, 3) control variates MC using standard call option. The correct price in BS model is $$1.3341$$. My three methods (with the same parameters and number of simulations equal $$100000$$) gives me these results:

1. MC: $$1.3621$$

2. Antithetic MC: $$1.3763$$

3. Control variates MC: $$1.3703$$

Suppose you want to price a derivative $$X$$ (e.g. a barrier option). You simulate $$M$$ sample paths and compute $$M$$ potential discounted payoffs, $$f_X$$. The standard Monte Carlo estimate for the price of $$X$$ is simply the (arithmetic) average, $$\text{Price}=\bar{f_X}=\frac{1}{M}\sum_{i=1}^Mf_X(i).$$

The idea of control variates is that you use your $$M$$ paths to price another derivative, $$Y$$, which is (very) similar to $$X$$. For example, if you price a barrier option, then you can use a vanilla option as control variate. Importantly, for this control variate (the derivative $$Y$$), you need to know a closed-form solution, call it $$f_Y^*$$.

Now, you have got $$M$$ sample payoffs for $$X$$ (denoted by $$f_X$$) and $$M$$ sample payoffs for $$Y$$ (denoted by $$f_Y$$) as well as the closed-form solution for $$Y$$ (denoted by $$f_Y^*$$). Based on your sample payoffs, let's calculate $$\hat\beta = \frac{\mathbb{C}\text{ov}(f_X,f_Y)}{\mathbb{V}\text{ar}[f_Y]}.$$ This looks like a regression coefficient (market beta) or a minimum variance hedge. The new control variate estimate for the price of your derivative is then $$\text{Price}=\bar{f_X}+\hat\beta(f_Y^*-\bar{f_Y}).$$ The term $$f_Y^*-\bar{f_Y}$$ is the bias of your random numbers. Scaling the bias correction by $$\hat\beta$$ ensures that the new variance is at smaller (or equal to) the sample variance of $$f_X$$. The idea is that $$\hat\beta$$ emerges as optimal (i.e. variance minimising) solution of the problem $$\min_\beta\; \mathbb{V}\text{ar}[f_X+\beta(f_Y^*-f_Y)]=\mathbb{V}\text{ar}[f_X]+\beta^2\mathbb{V}\text{ar}[f_Y]-2\beta\mathbb{C}\text{ov}(f_X,f_Y).$$

Clearly, the more $$f_X$$ and $$f_Y$$ correlate, the better the method. Imagine you price a down-and-out put option. This payoff actually correlates negatively with a vanilla put payoff. Not including $$\hat\beta$$ could make your price estimate worse!

Check out Boyle, Broadie and Glasserman (1997) for more details. For example, you can use several instruments simultaneously as control variates and the optimal $$\hat\beta$$ coefficients will, of course, look similar to those from a multi linear regression.

• Thanks for great answer! I edit my question because I dont know why antithetic and control variates doesnt work good for barrier option. Jan 24, 2021 at 20:26
• @JohhnWhite Instead of just focussing on the point estimate, you should compute the confidence intervals. They should decrease as you employ more variance reduction methods (hence the name). Jan 25, 2021 at 9:07