Volatility of a Product of Correlated Lognormal Stocks

Summary

The document asks how to calculate the volatility of a product of two stock prices modeled as correlated geometric Brownian motions. The log of the product equals the sum of the two log prices, so its log return is the sum of their log returns. Its variance therefore includes both individual variances and a covariance term, which depends on the correlation between the underlying Brownian shocks. This explains why simply combining the two volatilities without accounting for correlation is generally incomplete.

The replies suggest either applying Ito’s lemma to the product or working directly with log returns. The displayed variance expression in one reply appears to omit squares on the individual volatilities and the time factor on the covariance term; for constant volatilities over horizon t, the log-return variance is t times (sigma1 squared plus sigma2 squared plus twice rho sigma1 sigma2). This is log-return volatility; volatility of the non-log price level is a different quantity.

Key ideas

• The log return of a product of two prices is the sum of their log returns.
• The variance of that sum includes a covariance term determined by correlation.
• For constant volatilities, log-return variance scales with the time horizon.
• Ito’s lemma can be used to derive the dynamics of the product price directly.
• Log-return volatility and the volatility of the non-log price level are distinct quantities.

Tags

# Volatility of Multiple Stocks


# Volatility of Multiple Stocks












According to BSM, Stock Price follows log-normal distribution s.t. $$S(t)=S(0)*\exp(\sigma\sqrt t Z-(\sigma^2t)/2)$$ where Z is standard normal variable Then volatility of this stock is $\sigma \sqrt t$.

Suppose I model a new stock $S'(t)=S_1(t)*S_2(t)$ where $S_1(t)$ and $S_2(t)$ will follow log-normal distribution as mentioned above with parameters $\sigma_1,Z1$ and $\sigma_2,Z2$ and $Z_1$ and $Z_2$ are correlated with a factor rho, then how do I calculate volatility of new stock denoted by price S'(t) ?

Currently, according to How to calculate the volatility matrix with multiple stocks

I calculated volatilty = sqrt(sigma1^2+sigma^2)*sqrt(t) but I am not sure if it's correct especially for correlated Zs

## Answer by user32416 (score 1)

https://quant.stackexchange.com/a/20906

Use Ito's lemma on the function $f(x,y) = xy$ and then extract out the diffusion term.

## Answer by Phun (score 0)

https://quant.stackexchange.com/a/20908

To obtain the vola for the log returns is easy and you don't need itos lemma, since

$$ \log S'(t) = \log S_1(t) + \log S_2(t),$$ therefore $$ var(S'(t)) = var(\log S_1(t)) + var(\log S_2(t)) + 2covar(\log S_1(t),\log S_2(t))\\ = \sigma_1t+\sigma_2t+2\rho \sigma_1\sigma_2.$$

However, to get the vola for the non-log stock price you indeed need to use the ito formula.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.