# Control variate for pricing a best of assets option : $\mathop{{}\mathbb{E}}[ \max ( F^1_T,F^2_T, …,F^N_T )]$

I want to use Monte Carlo to price a best of assets derivative :

$$\mathop{{}\mathbb{E}}[ \max ( F^1_T,F^2_T, ...,F^N_T )]$$

where the $$F^i_T$$ is the forward of the ith asset observed at expiry time $$T$$ of the option.

What would be a good control variate to use for variance reduction?

I know that I have to look for a function (not necessairly a traded instrument) involving the underlyings that is :

• highly correletated with the payoff above

• has a known expectation

However i don't have enough experience with choosing control variates. Any suggestions, ideas?

thank you

If the $$(F^i_T)_i$$ are lognormal, I'd choose their geometric average $$\left(\prod_{i=1}^N F^i_T\right)^{\frac{1}{N}}$$ because it's lognormal as well and hence the expectation is easy to compute.
If they are normal, I'd choose the arithmetic average $$\frac{1}{N}\sum_{i=1}^N F^i_T$$, since it's gaussian as well.