$Solving Constant Product for price$

Question

$Solving Constant Product for price$

The constant product is used in decentralized finance. It is used to calculate how much someone gets for currency $Y$ when giving funds of currency $X$. $X$ and $Y$ indicate the size of funds in the corresponding currencies, $z$ stands for the amount of funds of currency $X$ given into the pool, $a$ stands for the amount of funds received in currency $Y$. $p$ is the price paid for the transaction.

$k = X * Y$
$a = Y - k / (X + z)$
$p = z / a$

Now my question is: How much does someone have to give to exactly receive $p$?

Example:
$X = 100, Y_1 = 50, Y_2 = 50, z = 10$
$k = 100 *(50+50)$
$a = (50+50) - 10000 / (100+10) = 9.09...$
$p = 9.09... / 10 = 1.09...$

Math Finance

Maxbry

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Mathe

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When you say, "How much does someone have to give?" you mean "z", right?
- Maxbry
  
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  Yes, correct.

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Mathe

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When you say, "How much does someone have to give?" you mean "z", right?

Maxbry

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Yes, correct.

Answer 1

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We have that $ k = XY$ and $a =Y -\dfrac{k}{X+z}$, so $a = Y -\dfrac{XY}{X+z}$.

Now, $p = \dfrac{z}{a}=\dfrac{z}{Y -\dfrac{XY}{X+z}}=\dfrac{z(X+z)}{Y(X+z)-XY}$

We now solve for $z$ in the previous equation. This wil give us the right $z$ quantity that will guarantee a result of $p$.

$$p(Y(X+z)-XY)=z(X+z)$$$$zpY =z^2+zX $$$$z^2 +z(X-pY)=0$$ $$z=pY-X$$ We can double check this answer by replacing it in the formula for $p$:

$\dfrac{z}{a}=\dfrac{z}{Y -\dfrac{XY}{X+z}}=\dfrac{z(X+z)}{Y(X+z)-XY}=\dfrac{(pY-X)(X+pY-X)}{Y(X+pY-X)-XY} = \dfrac{(pY-X)pY}{Y(pY-X)}=p$

Mathe

$Solving Constant Product for price$

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