Choosing a Minimum-Variance Hedge Ratio by Regression
Summary
The document explains the setup for hedging a spot exposure with a futures position. It models the change in spot price as a linear function of the futures price change, plus an intercept and an error term. For a position long one unit of the spot asset and short h units of futures, the change in value is the spot change minus h times the futures change. Substituting the regression model shows why the futures coefficient becomes b minus h.
The hedge ratio is selected to minimize the variance of this combined position’s value change. Under the linear model, choosing h equal to the regression slope b eliminates the component associated with futures price movements; the intercept and regression error remain. The document therefore treats the hedge ratio as an estimated quantity rather than assuming it equals one. The result depends on the approximate linear relationship and the regression estimate, and the residual risk cannot be removed by this choice of hedge ratio.
Key ideas
- A hedge combines a long spot position with a short futures position.
- The position’s value change per unit of spot exposure is ΔS − hΔF.
- Substituting the regression for ΔS yields a futures movement term with coefficient b − h.
- Setting the hedge ratio h equal to the regression slope b minimizes variance under the stated linear model.
- The intercept and regression error remain as sources of risk after hedging.
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# Calculating the Minimum Variance Hedge Ratio
# Calculating the Minimum Variance Hedge Ratio
Taken from the book: $\Delta{S}$ - Change in spot price, S, during a period of hedge. $\Delta{F}$ - Change in futures price, F, during a period of hedge. If we assume that the relationship between $\Delta{S}$ and $\Delta{F}$ is approximately linear, we can write: $\Delta{S} = a + b\Delta{F} + e$ where a and b are constants and e is an error term. Suppose that the hedge ratio is h (futures size position/exposure). EVERYTHING IS CLEAR TILL NOW Then the change in the value of the position per unit of exposure to S is $\Delta{S} - h\Delta{F} = a + (b-h)\Delta{F} + e$
- If I understand correctly, $\Delta{S}$ - $h\Delta{F}$ is change of spot price - change of futures price related to my position. Let's assume that hedge ratio is 1. Then $\Delta{S}$ - $h\Delta{F}$ is just a difference between spot price change and futures price change, why do I need it?
- Why in $a + b\Delta{F} + e$ b was replaced by (b - h) when I subtracted $h\Delta{F}$ from $\Delta{S}$ ?
- What is the main idea of my calculations?
## Answer by nbbo2 (score 1, accepted)
https://quant.stackexchange.com/a/75283
Hedging is when you are long one thing and short another thing, with the hope that the overall portfolio will be stable, it will not change much in value. Here the hedge position is: long 1 unit of S and short h units of F. Therefore the profit/loss or change in value of the position is $\Delta S−h \Delta F$. And $h$ is called the hedge ratio.
Now substitute the expression for $\Delta S$ in this, we get that the p/l is $ (a+b \Delta F+e)−h \Delta F=a+(b−h) \Delta F+e$.
The main purpose of this calculation is to find out what $h$ should be, and the conclusion will be that $h$ should be set equal to $b$, the linear regression coefficient. That will zero out the middle term, the other 2 terms $a$ and $e$ we cannot do anything about.
You say "let us assume the hedge ratio is 1", that is OK but it is a strange assumption, we are trying to calculate a value for $h$.The best we can do to minimize the variance is set h = b.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.