# Calculating the Minimum Variance Hedge Ratio [closed]

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
$$\Delta{S} - h\Delta{F} = a + (b-h)\Delta{F} + e$$

1. 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?
2. Why in $$a + b\Delta{F} + e$$ b was replaced by (b - h) when I subtracted $$h\Delta{F}$$ from $$\Delta{S}$$ ?
3. What is the main idea of my calculations?
• @nbbo2, this is an answer (you address all 3 questions adequately), so consider posting it as such instead of just as a pair of comments. Or do you perhaps believe the question should be closed and therefore only post comments? Apr 20, 2023 at 7:40
• @nbbo2 thanks, not it clear. Please consider posting a full answer so I can mark it as a solution. Apr 20, 2023 at 8:58

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.