2.3 — Impermanent Loss

Impermanent Loss — Full Derivation

Impermanent loss (IL) is the cost a liquidity provider pays for holding assets inside a constant-product pool rather than simply holding them in a wallet. It arises mechanically from the invariant $xy=k$: as the price moves, arbitrageurs rebalance the pool, and the LP ends up with more of the depreciating asset and less of the appreciating one.

“Impermanent” is a misnomer adopted by the community — the loss is perfectly real and permanent if the price does not revert. A better name would be rebalancing loss or adverse selection cost.

Setup

At time 0, the pool holds reserves $(x_0,y_0)$ with invariant $k=x_0{y}_0$. The marginal price of X in terms of Y is:

$$P_0=\frac{y_0}{x_0}$$

At time 1, external markets move the price to $P_1$. Arbitrageurs trade against the pool until its marginal price equals $P_1$.

New Reserves After Price Change

Let $r=P_1/P_0$ be the price ratio. We need reserves $(x_1,y_1)$ satisfying:

1. $x_1{y}_1=k$ (invariant preserved)
2. $y_1/x_1=P_1=r{P}_0=r\cdot y_0/x_0$

From (1): $y_1=k/x_1$. Substituting into (2):

$$\frac{k}{x_1^2}=r\cdot \frac{y_0}{x_0}=r\cdot \frac{k}{x_0^2}$$ $$x_1^2=\frac{x_0^2}{r}⟹x_1=\frac{x_0}{\sqrt{r}}$$ $$y_1=\frac{k}{x_1}=\frac{x_0{y}_0}{x_0/\sqrt{r}}=y_0\sqrt{r}$$

More symmetrically, using $P_0$ and $P_1$:

$$\boxed{x_1=x_0\sqrt{\frac{P_0}{P_1}}}\boxed{y_1=y_0\sqrt{\frac{P_1}{P_0}}}$$

When price goes up ($r>1$), $x_1<x_0$ and $y_1>y_0$: the pool has sold X (the appreciating asset) and accumulated Y. This is the mechanism of adverse selection — the pool systematically sells winners and buys losers.

LP Portfolio Value

The LP’s portfolio value (in terms of Y) at time 1 is:

$$V_{\text{LP}}=x_1{P}_1+y_1=\frac{x_0}{\sqrt{r}}\cdot r{P}_0+y_0\sqrt{r}$$

Since $x_0{P}_0=y_0$ (by the definition of $P_0=y_0/x_0$, rearranged as $x_0=y_0/P_0$, so $x_0{P}_0=y_0$):

$$V_{\text{LP}}=y_0\sqrt{r}+y_0\sqrt{r}=2y_0\sqrt{r}$$ $$\boxed{V_{\text{LP}}=2y_0\sqrt{r}}$$

HODL Portfolio Value

If the LP had simply held the initial tokens in a wallet:

$$V_{\text{HODL}}=x_0{P}_1+y_0=x_{0}r{P}_0+y_0=y_{0}r+y_0=y_0(r+1)$$ $$\boxed{V_{\text{HODL}}=y_0(r+1)}$$

The Impermanent Loss Formula

$$\text{IL}(r)=\frac{V_{\text{LP}}-V_{\text{HODL}}}{V_{\text{HODL}}}=\frac{2\sqrt{r}-(r+1)}{r+1}$$ $$\boxed{\text{IL}(r)=\frac{2\sqrt{r}}{r+1}-1}$$

where $r=P_1/P_0$.

Properties

IL(1) = 0. When price does not change, there is no loss:

$$\text{IL}(1)=\frac{2}{2}-1=0$$

IL is always non-positive. By the AM-GM inequality:

$$\frac{r+1}{2}\ge \sqrt{r}⟹2\sqrt{r}\le r+1⟹\frac{2\sqrt{r}}{r+1}\le 1$$

with equality only at $r=1$.

IL is symmetric in log-price. Let $s=\ln r$. Then:

$$\text{IL}(e^s)=\frac{2e^{s/2}}{e^s+1}-1=\frac{2}{e^{s/2}+e^{-s/2}}-1=\mathrm{sech}(s/2)-1$$

Since $\mathrm{sech}$ is an even function, $\text{IL}(e^s)=\text{IL}(e^{-s})$. A 2x price increase and a 2x price decrease produce the same IL. The loss depends on the magnitude of the log-price move, not its direction.

Second-order expansion. For small moves $s\approx 0$:

$$\text{IL}\approx -\frac{s^2}{8}=-\frac{(\ln r{)}^2}{8}$$

The loss is quadratic in the log-price change — reminiscent of gamma exposure in options.

Reference Table

Price ratio $r$	$\ln r$	IL
1.00	0.000	0.00%
1.25	0.223	-0.60%
1.50	0.405	-2.02%
2.00	0.693	-5.72%
3.00	1.099	-13.40%
5.00	1.609	-25.46%
10.00	2.303	-42.50%
0.50	-0.693	-5.72%
0.10	-2.303	-42.50%

The symmetry between $r$ and $1/r$ is visible in the table.

Connection to Options: The LP as Short Volatility

The log-price symmetry and quadratic dependence on $|\ln r|$ are not coincidental. An LP position in a constant-product pool is economically similar to a short straddle — a combined short call + short put.

More precisely, several researchers (notably Guillermo Angeris and others) have shown that an LP position can be replicated by a short variance position (or, in discrete terms, a portfolio that is short gamma). The core intuition:

• A short straddle profits if the underlying stays near the strike and loses if it moves far in either direction.
• An LP position profits (via fees) if the price stays near $P_0$ and loses (via IL) if it moves far in either direction.
• Both payoffs are concave in the price and convex in the loss.

The fee income acts as the premium collected from selling the straddle. If realized volatility is low enough that fees exceed IL, the LP profits — exactly as a straddle seller profits when realized vol is below implied vol.

This analogy is developed quantitatively in lp-profitability.

IL Is an Adverse Selection Cost

There is another way to see IL that connects directly to Kyle’s model and market microstructure. Every unit of IL is the result of an arbitrageur trading against the pool at a stale price. The arbitrageur is the informed trader; the LP is the uninformed market maker. The LP’s loss is the arbitrageur’s profit, which is exactly the information rent in Kyle’s framework.

In Kyle’s model, the market maker’s expected loss to informed flow is proportional to $\sigma$ (fundamental volatility) and $\lambda$ (price impact). In the AMM setting, IL is proportional to $(\ln r{)}^2/8\approx {\sigma }^2/8$ for small moves — also quadratic in volatility. The parallel is tight.

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Companion notebook: notebooks/defi/03-impermanent-loss.py — Altair visualization of IL as a function of $r$; overlay of the $\mathrm{sech}$ representation; comparison with short straddle payoff.

Questions to sit with:

1. IL is quadratic in log-price for small moves and $\mathrm{sech}$-shaped globally. A short straddle is locally quadratic (gamma) but linear in the tails. Where does the analogy break down?
2. If you could hedge the IL of an LP position using options on the underlying token, what strike and expiry would you choose?
3. IL depends on $r=P_1/P_0$ but is path-independent — it does not matter how the price got to $P_1$. Why? What feature of the constant-product invariant guarantees this?