Instantaneous Variance of a Correlated Asset Spread

Summary

The document clarifies why the variance of changes in a spread between two correlated geometric Brownian motion assets includes the asset price levels. For the spread defined as the first asset price minus the second, its instantaneous change is the difference between their price changes. The variance of that difference is the sum of the individual instantaneous variances minus twice their covariance.

Each asset’s instantaneous variance scales with the square of its current price, while the covariance scales with the product of the two prices and their correlation. This yields the price-weighted expression in the cited paper. The unweighted expression applies to relative or normalized returns under suitable assumptions, rather than dollar price changes in the spread. The explanation is local and conditional; it does not provide a full finite-horizon distribution for the spread, which is generally not itself a geometric Brownian motion.

Key ideas

• For a price spread, the instantaneous change is the difference between the two asset price changes.
• The variance of the spread change equals the sum of the asset variances minus twice their covariance.
• Under correlated geometric Brownian motion, each asset’s instantaneous variance depends on its current price squared.
• The covariance term depends on correlation and the product of the current asset prices.
• An unweighted volatility expression describes normalized returns rather than dollar changes in the price spread.

Tags

# Variance of a spread for options on spreads


# Variance of a spread for options on spreads












I was reading the paper: https://people.umass.edu/nkapadia/docs/Negative_Vega.pdf

In the equation $(5)$, he is defining the variance of the spread as:

$$\sigma_1^2S_1^2 + \sigma_2^2S_2^2 - 2\sigma_1 \sigma_2 S_1 S_2 \rho$$

whereas I have always seen it defined as:

$$\sigma_1^2 + \sigma_2^2 - 2\sigma_1\sigma_2\rho$$

This is for 2 correlated GBM and the spread is $S_1 - S_2$.

What am I missing?

## Answer by Magic is in the chain (score 2, accepted)

https://quant.stackexchange.com/a/49659

I think the variance of the instantaneous shifts in the spread is meant:

$V \left[ dX \right]=V \left[ dS_1-dS_2 \right]$

And the individual variances (in the conditional and local sense) are:

$V \left[ dS_1 \right]= \sigma_1^2 S_1^2dt$

$V \left[ dS_2 \right]= \sigma_2^2 S_2^2dt$

And the covariance term is, assuming the two Brownians are correlated:

$C\left[ dS_1 , dS_2\right]=\rho \sigma_1 \sigma_2 S_1 S_2dt$

Now if plug these into your formula, you get the equation 5.

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