Liquidity provider positions in Uni v3/v4 only collect fees when the price is in range.
How long does the price remain within a given range (in expectation)?
1/7 A short 🧵
TLDR; E[time-in-range] = halfWidth^2/σ^2, & we can use this to find the realized volatility of a pool!
2/7 We will consider positions centered at the current price at time T=0 and monitor how long it takes the price to move a certain number of ticks up or down.
In the example below, a position is minted on Jan. 1st 2022 and the price crossed ±2%, ±3%, ±5%, and ±10% in <6 days
4/7 While crossing times do seem to converge for short exit times (see 10, 20, 30bps), it turns out that the average waiting time scales as the square of the range h: waitTime ~ h^2.
I'm deriving why this relationship exists based on work by Andersen 2008 and @SinclairEuan 2014
5/7 This h^2 law is a consequence of the price *ticks* following a Brownian Motion with normally distributed steps: in a BM, the average displacement after a time t scales as σ*t^2, where σ=volatility.
This relationship can be inverted to estimate σ from average crossing times!
7/7 Key insights:
- The average time spent within a range ±h scales as (h/σ)^2
- E[first exit time] can be used to develop resilient and efficient volatility estimators
- The volatility calculation in the ETH-USDC pool can highlight volatile macro events
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