# 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, va 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******r******prev*** is the volatility reference from previous trades
* ***i******r*** is the index reference used to stabilize va 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*