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

Question

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

Answer 1

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.