Impermanent loss

Last modified: April 22, 2026

In the previous section, we derived the value of a CenturionDEX v3 position as a function of the current price. We now use those formulas to study impermanent loss in a systematic way.

Consider a CenturionDEX v3 position defined by a price interval $[p_a,p_b]$ and liquidity parameter $L$. Let $p_0$ denote the entry price, that is, the market price at the moment the position is opened. Let

be the initial real balances of tokens $X$ and $Y$ contributed by the liquidity provider. For any current price $p>0$, let

denote the current real balances of the position.

As before, the value of the position, expressed in units of token $Y$, is

If, instead of depositing liquidity, the provider had simply held the initial token amounts $x_0$ and $y_0$, then the value of that passive portfolio at price $p$ would be

We define the impermanent-loss fraction by

Thus, $\operatorname{IL}(p)$ measures the relative difference between the value of the LP position and the value of passively holding the original assets.

This analysis generalizes the numerical position studied earlier in the example above, where the parameters were

We now distinguish three cases, depending on the location of the entry price $p_0$ relative to the chosen interval.

Case 1: $p_a \le p_0 \le p_b$

In this case, the position is opened inside its active interval. Therefore, the initial balances are

Subcase 1.1: $p_a \le p \le p_b$

When the current price remains inside the interval, the current balances are

Hence,

while

Therefore,

In particular, $\operatorname{IL}(p_0)=0$, as expected.

Subcase 1.2: $p \le p_a$

If the price falls below the range, the position is entirely held in token $X$, so

Thus,

and

Subcase 1.3: $p \ge p_b$

If the price rises above the range, the position is entirely held in token $Y$, so

Hence,

and

Case 2: $p_0 \le p_a$

In this case, the position is opened below the chosen interval, so the provider initially deposits only token $X$. The initial balances are

Subcase 2.1: $p \le p_a$

As long as the price remains below the interval, the position stays entirely in token $X$. Therefore,

So,

and hence

There is no impermanent loss in this regime because the LP position coincides with simply holding token $X$.

Subcase 2.2: $p_a \le p \le p_b$

Once the price enters the interval, the balances become

Therefore,

while

Thus,

Subcase 2.3: $p \ge p_b$

If the price rises above the interval, then

Hence,

whereas

After simplification, we obtain

Case 3: $p_0 \ge p_b$

In this case, the position is opened above the chosen interval, so the provider initially deposits only token $Y$. The initial balances are

Subcase 3.1: $p \le p_a$

If the price falls below the interval, then the position is entirely converted into token $X$, so

Thus,

while

Therefore,

Subcase 3.2: $p_a \le p \le p_b$

Inside the interval, the balances are again

Hence,

and since $W(p)=L\left(\sqrt{p_b}-\sqrt{p_a}\right)$, we get

Subcase 3.3: $p \ge p_b$

If the price remains above the interval, then the position is still fully in token $Y$, and therefore

So,

and consequently

Again, there is no impermanent loss in this regime because the position coincides with a passive holding of token $Y$.

A final remark is in order for Case 3, especially when token $Y$ is a stablecoin such as USDC. In that setting, a CenturionDEX v3 position opened above the selected range initially consists only of token $Y$. By contrast, a CenturionDEX v2 position would require liquidity to be supplied in both tokens, so the provider would first need to swap part of the stablecoin balance into token $X$. As a result, if the price of token $X$ later increases, the CenturionDEX v3 position in Case 3 may remain entirely in stablecoin and therefore exhibit zero impermanent loss relative to its initial deposit, while the corresponding CenturionDEX v2 position will generally move away from the original 50/50 value split and incur impermanent loss relative to simply holding the two deposited assets. For this reason, comparisons between CenturionDEX v2 and CenturionDEX v3 in this regime must be interpreted carefully.