Estimating the Volatility of a Weighted Stock Basket

Summary

The document asks how to estimate the future price distribution of a basket of stocks. It presents a portfolio variance expression built from constituent weights, volatilities, and pairwise correlations of log returns, then raises a modeling concern: a weighted sum of lognormally distributed prices is generally not itself lognormally distributed. Applying the basket volatility to a single lognormal price process may therefore be an approximation rather than an exact model.

No answer, derivation, empirical comparison, or workaround is included, so the document does not establish when the approximation is adequate or how its errors behave. It points to a result by Brigo as a possible source for further study, but gives no details. Readers should also note that the displayed expression is a variance formula; a standard deviation would require taking its square root. The post is useful as a statement of a modeling issue, but it leaves the proposed method and its limitations unresolved.

Key ideas

• Basket variance depends on constituent weights, volatilities, and pairwise return correlations.
• A weighted sum of lognormal stock prices is generally not lognormally distributed.
• Using basket volatility in a single lognormal price simulation is an approximation whose accuracy is not assessed here.
• The document raises the modeling question but provides no derivation, evidence, or solution.

Tags

# Volatility of a stock basket


# 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 ...

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