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In the end I found that fitting a SABR smile to each tenor (borrowing a result from this answer) was sufficient to build a local vol surface that was smooth and well-behaved enough to build a variance surface worked nicely. I also fitted a Heston model to it, and the two surfaces do look fairly similar. Here is the final code and the fits generated (the long ...


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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 $b$ ...


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