# LVR and re-centring costs

Fees are what an LP earns. This page covers the two costs fees have to beat, and the break-even volatility that ties them together.

## LVR: loss-versus-rebalancing

A pool's price only changes when someone trades. When the stock moves elsewhere, the pool is briefly stale, and arbitrageurs trade against it until it catches up. Each of those trades is a small loss for LPs. Added up, that loss is **LVR** (Milionis, Moallemi, Roughgarden and Zhang, 2022).

For liquidity spread over all prices:

```
LVR (full range) = σ² ÷ 8
```

For a range of ±w, the same loss falls on far less capital:

```
LVR(w) = σ²/8 × M(w),    M(w) = 1 ÷ (1 − (1+w)^−½)
```

M is how much more concentrated the range is than full range. At ±2% it is about 100, so a stock with 34% volatility costs a full-range LP about 1.4% a year and a ±2% LP about 146% a year while in range.

LVR accrues only while the price is inside the range.

## Re-centring

A range the price has left earns nothing. Re-centring closes it and reopens around the price. The expected number of re-centres for a price following a random walk:

```
re-centres per year = σ² ÷ ln(1+w)²
```

Each re-centre costs:

```
cost = ½ × (swap fee now + price impact) + transaction fees
```

The half is there because re-centring swaps about half the position into the other token. The swap fee includes any dynamic surcharge at the time, and price impact is measured through the pool's actual liquidity.

## Break-even volatility

Fees are fixed by recent history, while LVR and re-centring costs both grow with volatility. The volatility at which they balance:

```
σ* = √( range fee APR ÷ (M/8 + cost ÷ ln(1+w)²) )
```

- If the volatility you expect is **below σ\***, fees win.
- If it is **above σ\***, costs win.

σ\* is also called **fee-implied volatility**. Divided by the option market's implied volatility for the same risk, it becomes [vol ratio](../vol-ratio/README.md).

## Matching a range to an option

A range kept on the price has the gamma of a short at-the-money option. The option tenor with the same gamma per dollar:

```
matched tenor T = 2w² ÷ (π σ² (1+w))
```

A ±0.5% range matches an option of hours, a ±10% range one of weeks. See [How vol ratio is calculated](../vol-ratio/how-it-is-calculated.md).

## Assumptions

- Arbitrage is continuous and costless, and prices move without jumps. With trading fees arbitrage happens less often, so real LVR is somewhat lower. Opening gaps work the other way.
- Re-centring is instant, and the price leaves the range as often as a random walk would.
- Re-centring swaps go through the same pool.