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I would like to calculate a basket European option with Black Scholes local volatility model.

I want to simplified the basket option into a single underlying European option. Should we get the local volatility surface with single underlying instruments as usual and get volatility by the following formula?

Assume the correlation $\rho$ is constant.

(2-equal-weighted underlying for demonstration purposes) $\sigma_{basket}^2(S^a_{t},S^b_{t},t) = (\frac{1}{2}\sigma_{a}(S^a_{t},t))^2 + 2\rho(\frac{1}{2}\sigma_{a}(S^a_{t},t))(\frac{1}{2}\sigma_{b}(S^b_{t},t))+ (\frac{1}{2}\sigma_{b}(S^b_{t},t))^2$

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Your equation holds if you replace $\sigma$ with $\text{var}$. But if you use it as such, even in the BS case ($\sigma(S,t) = \sigma$) then you have a problem, because you are manipulating a lognormal volatility and the (weighted) sum of lognormals is not lognormal. Now nothing prevents you from simulating the future price processes of assets $a$ and $b$ tying them by a Gaussian copula with parameter $\rho$, pricing vanillas on the basket and deriving an IVS for the basket. Strip this one to get the basket's LV.

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