# Volatility of a stock basket

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 ...

• 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.] May 13 '19 at 5:17
• Ideally I am looking for a result which allows me to see how large the error is
– ZRH
May 13 '19 at 5:18
• Have a look at the following paper to start with. It gives also some error analysis: pdfs.semanticscholar.org/a012/… May 13 '19 at 5:19
• thanks!! appreciate
– ZRH
May 13 '19 at 5:21
• Just to give feedback - I tested extensively, and the Lévy method works quite very accurately in the parameter ranges I am looking at
– ZRH
May 14 '19 at 19:22