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Suppose I want to open a long position (that which increases in value as the commodity goes up in value) in some commodity whose price at the time of opening the trade is, say, $p$. The position is opened on margin, and the margin is 3 times of the collateral. That is, if I have $p$ dollars in my margin account I can open a position of size $3p$, and the liquidation price will be $p$. Suppose also that the collateral for the commodity I'm trading in the long position is also denominated in the said commodity, which means that as the commodity goes down, the liquidation price (when the position has to be closed) goes up.

Now, I want to obtain the graph of the liquidation price change.

Let $y$ be the liquidation price and $x$ be the price of the commodity. Suppose I have 1 unit of the commodity and I open a long position for 1 unit. I think that in this case I come up with the following differential equation:

$$3 \frac{dy}{dx}=-x, y(p) = \frac{p}{3},$$

which doesn't give me the correct graph.

Can someone please help me with this? I'm missing some important point here and thus the DE I'm coming up with is not correct.

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