Multi-Asset Minimum-Variance Hedging with a Covariance Matrix
Summary
The document explains how to hedge a fixed position in one asset using positions in two correlated hedging assets. Write the total portfolio variance as a function of the hedge holdings, then set its partial derivatives with respect to those holdings to zero. This produces a linear system based on the hedging assets’ variances and their covariance with each other, alongside their covariances with the exposed asset. Solving the system gives the holdings that minimize portfolio variance, provided the hedging assets are not perfectly correlated or anticorrelated.
The minimized quantity is total portfolio variance, so the approach adopts variance as its measure of risk; it does not specify a broader risk model. Its usefulness depends on the covariance matrix being known or estimated well. A fuller risk evaluation therefore requires a method for forecasting that matrix and backtesting whether the predicted post-hedge risk matches realized portfolio behavior. The exposition establishes the optimization setup, but does not address estimation error, transaction costs, or constraints on hedge positions.
Key ideas
- Minimize total portfolio variance by differentiating it with respect to the hedge positions and setting the derivatives to zero.
- With two hedge assets, the resulting conditions form a linear system using their covariance matrix.
- The method assumes the covariance structure is known and the hedge assets are not perfectly correlated or anticorrelated.
- The optimized risk measure is variance, rather than a broader measure such as drawdown or tail loss.
- Covariance forecasts should be evaluated and backtested against realized post-hedge risk.
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Full text
# Minimum variance hedge with more than one asset
# Minimum variance hedge with more than one asset
My portfolio comprises of 3 assets A,B,C that are correlated and the variance-covariance structure is known. At any given point in time, my position in Asset A say is given to me.
I need to construct a variance minimizing hedge using both B and C, given this position in A.
Basically I am approaching the problem by directly constructing the Variance, as a function of say x and y, the positions in B and C, and minimizing the function by setting the partials w.r.t x and y to 0. This gives me 2 equations in x and y and I can solve them for x and y.
My question is that, is this approach sound? What notion of risk is this really minimizing? I did not explicitly construct a risk model here?
## Answer by Chris Taylor (score 2)
https://quant.stackexchange.com/a/4339
If the variances are known to be $\sigma_0$, $\sigma_1$ and $\sigma_2$ and the correlations are $\rho_{01}$, $\rho_{02}$ and $\rho_{12}$ then you can do exactly as you suggest - write down the variance of the total portfolio as a function of your holdings $x_0$, $x_1$ and $x_2$ and set the partial derivatives with respect to $x_1$ and $x_2$ to zero. You end up with the following matrix equation
$$ \left[ \begin{matrix} \sigma_1^2 && \rho_{12}\sigma_1\sigma_2 \\ \rho_{12}\sigma_1\sigma_2 && \sigma_2^2 \end{matrix} \right] \left[ \begin{matrix} x_1\\x_2 \end{matrix} \right] = \left[ \begin{matrix} \rho_{01}\sigma_1 \\ \rho_{o2}\sigma_2 \end{matrix} \right] \sigma_0 x_0$$
which you can solve by inverting the matrix, as long as $\rho_{12}\neq \pm 1$ (ie your hedging assets aren't perfectly correlated or anticorrelated).
As to whether this is sound - well, it depends what you mean by "sound". You are minimizing the total variance of your portfolio, conditional on you knowing the covariance matrix. That's about all you can say. Your "risk model" is a single dimension - you are saying that the only notion of risk that you care about is the total variance.
A full risk evaluation would need a procedure for determining the covariance matrix, and some level of backtesting to determine if your forecast risk after the hedge is a reflection of the true risk you would see if you were to hold this portfolio.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.