Volatility Related

To make fees truly responsive to market conditions, Cetus DLMM tracks both recent trading activity and price movements across bins. This section explains how the Volatility Accumulator (v_a), Volatility Reference (v_r), and Index Reference (i_r) work together to adjust variable fees dynamically, ensuring fair pricing and protecting against manipulation.

Volatility Accumulator (v_a)

The Volatility Accumulator measures cumulative price movements across bins during swaps. Each bin crossed represents a fixed price step, and Va adds up these crossings to reflect volatility beyond a single trade. For bin n, v_a is calculated as:

$$v_a(n)=v_rprev +∣i_r−(activeBinID+n)∣$$

Where:

• activeBinID is the ID of the currently active bin before the swap

• v_rprev is the volatility reference from previous trades

• i_r is the index reference used to stabilize v_a during rapid trades

Volatility Reference (v_r)

v_r updates according to the elapsed time t_elapsed since the last swap:

• Filter period (t_f): If t_elapsed < t_f , v_r remains unchanged to account for high-frequency trading.

• Decay period (t_d): If t_elapsed ≥ t_d , v_r resets to zero due to inactivity.

• Intermediate decay: For t_f ≤ t_elapsed < t_d , v_r decays proportionally with a decay factor ρ : v_r = ρ⋅v_a prev

This approach ensures that frequent trading increases accumulated volatility, while inactivity gradually reduces it.

Index Reference (i_r)

A variable called Index Reference (i_r) is also introduced to stabilize the calculation of volatility and prevent fee manipulation through rapid, repeated small trades. By keeping track of a reference bin, i_r ensures that high-frequency trading does not artificially inflate the Volatility Accumulator (v_a), helping variable fees reflect real market activity rather than strategic splitting of swaps.

• If t_elapsed ≥ t_f (normal activity), then i_r = activeBinID

• If t_elapsed < t_f (high-frequency trades), then i_r = i_r prev

Example

activeBinID = 200

t_f = 2 s, t_d = 6 s, ρ = 0.6

v_r at start = 0

Swap 1: crosses +2 bins (200 → 202)

i_r = 200

v_a progression:

• v_a(0) = 0 + |200 − (200+0)| = 0

• v_a(1) = 0 + |200 − (200+1)| = 1

• v_a(2) = 0 + |200 − (200+2)| = 2

v_a at end = 2

Swap 2: occurs 3 seconds later, crosses +5 bins (202 → 207)

v_r decays: v_r = 0.6 * 2 = 1.2

i_r = 202

v_a progression:

• v_a (0) = 1.2 + |202 − (202+0)| = 1.2

• v_a (1) = 1.2 + |202 − (202+1)| = 2.2

• v_a (2) = 1.2 + |202 − (202+2)| = 3.2

• v_a (3) = 1.2 + |202 − (202+3)| = 4.2

• v_a (4) = 1.2 + |202 − (202+4)| = 5.2

• v_a (5) = 1.2 + |202 − (202+5)| = 6.2

v_a at end = 6.2

Swap 3: occurs 1 second later, crosses -3 bins (207 → 204)

v_r = 1.2, i_r = 202 (remain unchanged)

v_a progression:

• v_a (0) = 1.2 + |202 − (207+0)| = 6.2

• v_a (-1) = 1.2 + |202 − (207-1)| = 5.2

• v_a (-2) = 1.2 + |202 − (207-2)| = 4.2

• v_a (-3) = 1.2 + |202 − (207-3)| = 3.2

v_a at end = 3.2