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Cross-Asset Delta Hedging Under Correlated Geometric Brownian Motion

Article Quant Q&A · Author: Mattiatore
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Summary

The document explains how an option on one asset can be delta-hedged using a different, correlated asset. It defines the hedge sensitivity of the second asset to changes in the first and gives a formula under a bivariate geometric Brownian motion assumption: the sensitivity scales with the assets’ correlation, their volatilities, and their relative prices. The option’s hedge amount in the second asset is then obtained by dividing its delta by this sensitivity.

It also describes estimating the relationship empirically by regressing daily price changes in the second asset on those in the first; the regression slope corresponds to correlation times the ratio of standard deviations. The approach depends on a usable estimate of correlation and volatility, or on a suitable price-change regression. The formula is model-dependent, and the hedge becomes unstable or undefined when the estimated cross-asset sensitivity is near zero.

Key ideas

  • Under a bivariate geometric Brownian motion, cross-asset price sensitivity depends on correlation, relative volatility, and relative prices.
  • The second asset’s hedge amount can be derived by dividing the option delta by its sensitivity to that asset.
  • A regression of one asset’s daily price changes on another’s can estimate the cross-asset sensitivity.
  • A near-zero correlation or regression slope makes the resulting hedge ratio unstable.

Tags

OptionsDerivatives pricingRisk managementStatistics
Full text
# Delta Hedging using another correlated asset


# Delta Hedging using another correlated asset












My question is about the following (from Maxime de Bellefroid, Ch. 5 The Greeks):

- From my understanding $\Delta_2$ is the sensitive of the option (on the first instrument with underlying $S_1$) with respect to the price of the second instrument $S_2$, I guess this value can only be estimated and there are no closed formulas?

- How is the value of $\frac{\partial S_2}{\partial S_1}$ calculated?

## Answer by nbbo2 (score 6)

https://quant.stackexchange.com/a/74512

Look carefully. $\frac{\partial S_2}{\partial S_1}$ is explicitly given as $\frac{\partial S_2}{\partial S_1} = \frac{\rho_{12}\sigma_2}{\sigma_1}\frac{S_2}{S_1}$

If you know the correlation and the standard deviations of daily returns for asset 1 and asset 2 you can use this formula. But if not you can just do a least squares regression of the daily price changes (not returns) of asset 2 on the daily price changes of asset 1 and essentially come up with the same result (remeber the formula for the slope on a bivariate regression, it is the correlation times the ratio of the two standard deviations).

Then the closed formula for finding $\Delta_2$ is $\Delta_2 = \Delta / \frac{\partial S_2}{\partial S_1}$, a simple division (let us hope the denominator is not zero).

## Answer by Bruce Chang (score 4)

https://quant.stackexchange.com/a/78791

This example is from Exotic Options and Hybrids: A Guide to Structuring, Pricing and Trading page 68 equation 5.3.

- We are trying to find out $\Delta_2$, which is the amount of $S_2$ we need to perform delta hedge. We can use this chain rule equation to solve for $\Delta_2$. We only need to know $\Delta$ and $\frac{\partial S_2}{\partial S_1}$. The former is known and the latter depends on your assumption of asset price.

- Here it assumes $S_1, S_2$ follow a bivariate geometric Brownian motion:

$$ \begin{align} dS_1&=S_1\mu_1 dt+S_1\sigma_1dW_1 \end{align} $$ $$ \begin{align} dS_2&=S_2\mu_2 dt+S_2\sigma_2dW_2 \end{align} $$ with $$dW_1dW_2=\rho_{1,2} dt$$ This gives $\frac{\partial S_2}{\partial S_1} = \rho_{1,2} \frac{\sigma_2S_2}{\sigma_1S_1}$

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.