08 Oracle and TWAP

Overview

On-chain, we need a price that is:

• Resistant to manipulation (not easily affected by flash loans)
• Reflects the true price over a period of time
• Does not rely on off-chain records

So time-weighted average prices are introduced:

$$\text{TWAP (Time Weighted Average Price)}$$

V2's approach is to accumulate prices:

$$a(t) = a(t-1) + price \cdot \Delta t$$

Query range price:

$$p(t_1,t_2) = \frac{a(t_2) - a(t_1)}{t_2 - t_1}$$

The key improvement in V3 is that it no longer records price directly, but records:

$$tick = \log_{1.0001}(price)$$

Therefore:

$$price = 1.0001^{tick}$$

The essential change is to convert $price \cdot \Delta t$ into $tick \cdot \Delta t$.

using:

$$\log(p_1 \cdot p_2) = \log p_1 + \log p_2$$

1. tickCumulative (core accumulator)

Definition:

$$tickCumulative(t) = \sum tick \cdot \Delta t$$

Update method:

$$tickCumulative = tickCumulative + tick_{current} \cdot (t_{now} - t_{last})$$

2. TWAP calculation

Given two points in time:

$$t_1,\quad t_2$$

Step 1: Average tick

$$\bar{tick} = \frac{tickCumulative(t_2) - tickCumulative(t_1)}{t_2 - t_1}$$

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Step 2: Restore price

$$p(t_1,t_2) = 1.0001^{\bar{tick}}$$

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3. Why use tick(log)

Geometric mean

$$\log(p_1) + \log(p_2) = \log(p_1 \cdot p_2)$$

Corresponding price relationship

$$price = 1.0001^{tick}$$

4. Observation (historical snapshot)

V3 stores the state of multiple time points on-chain, and each record contains:

• timestamp
• tickCumulative
• secondsPerLiquidityCumulative

Each observation represents a snapshot of the accumulator at a point in time.

5. secondsPerLiquidity Oracle

definition:

$$secondsPerLiquidityCumulative = \sum \frac{\Delta t}{liquidity}$$

Meaning:

Indicates the time during which one unit of liquidity participates in market making.

• The greater the liquidity → the less time it takes to distribute units
• The smaller the liquidity → the more time units will be allocated

6. Oracle query

Query two points in time:

$$t_1,\quad t_2$$

calculate:

$$TWAP = \frac{tickCumulative(t_2) - tickCumulative(t_1)}{t_2 - t_1}$$

A unified perspective with the fee system

Fee system:

$$f(i_{lower}, i_{upper}) = f_g - f_b - f_a$$

Oracle system:

$$p(t_1,t_2) = \frac{\Delta tickCumulative}{\Delta t}$$

Therefore, V2 and V3 are essentially unified, and both are expressed as:

$$value = cumulative(end) - cumulative(start)$$

The difference is only in the dimensions:

• fee: make difference in tick space
• oracle: make differences in the time dimension

The essence of Oracle in V3 is:

• Time integration of tick (log price)
• Restore the average price through difference when querying

Finally got:

$$TWAP = \frac{\Delta tickCumulative}{\Delta t}$$

Its core design is completely consistent with the fee system:

• Accumulator (cumulative)
• Interval difference (delta)

Just applied in different dimensions:

• fee → space (price range)
• oracle → time (time)