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to determine the volatility of a basket of stocks, I often use the following formula:

$\sigma_{basket}=\sum_{i}\sum_{j}w_i w_j \sigma_i \sigma_j \rho_{ij}$

where the $\sigma$ are the constituents' volatilities and the $\rho_{i,j}$ are the pair correlation coefficients of logreturns. Now, to simulate future prices, one might choose to start out with the initial basket value and then use $\sigma_{basket}$ to determine the distribution of future prices. The issue I have with this is the following: The sum of lognormally distributed random variables is not a lognormally distributed random variable. So while I assume that my naive approach may yield reasonable results, it is probably not entirely correct.

Does anyone have a view on how good/bad of an approximation it is / where this breaks down / what workarounds there are. I am vaguely aware that there is a result by Brigo on this very topic, but I have not found it.

Can anyone help with this please ? Much appreciated ...

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  • $\begingroup$ Are you looking for references where you can decide for yourself which approximation you'll take? [The basket vol as calculated above is that for a geometric basket and will thus always underestimate the price of an arithmetic basket.] $\endgroup$ – ilovevolatility May 13 at 5:17
  • $\begingroup$ Ideally I am looking for a result which allows me to see how large the error is $\endgroup$ – ZRH May 13 at 5:18
  • $\begingroup$ Have a look at the following paper to start with. It gives also some error analysis: pdfs.semanticscholar.org/a012/… $\endgroup$ – ilovevolatility May 13 at 5:19
  • $\begingroup$ thanks!! appreciate $\endgroup$ – ZRH May 13 at 5:21
  • $\begingroup$ Just to give feedback - I tested extensively, and the Lévy method works quite very accurately in the parameter ranges I am looking at $\endgroup$ – ZRH May 14 at 19:22

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