Protocol Maths

POOL STRUCTURE MATHS

Bin limits

Nitro Finance uses $128.128$ binary fixed-point representation, where the left 128 bits represent integers and the right 128 bits represent fractional parts. The upper limit of the price is constrained by $2^{128}$, a significant value that ensures a broad range for asset pricing.

how many bins could we possibly have?

• Bin prices follow a geometric sequence, denoted as $(1+s)^i$. We now need to find the maximum integer for $i$ such that the entire value is less than the upper limit of price, $2^{128}$This can be expressed as the formula $((1+s)^i<2^{128})$.

Taking the smallest possible value of bin step($s$) which is 1 basis point, we can solve this as follows:

• $(1+0.0001)^i<2^{128}$n

• $log_2 (1.0001)^i < log_2 (2^{128})$

• $i⋅log2 (1.0001)<128$

• $i<\dfrac{128}{log2 (1.0001)} ≈ 887,273$,

• $i=887,273$

The above equation shows how many bins are needed to cover the entire range when $i$ is a positive integer, so we account for when it is negative by doubling it which equals to $2\times887,272=1,774,544$ bins.

Bin Indexing

the Nitro Finance system uses the uint24 data type. It's sufficient to cover the range of possible bins,

uint24 is $2^{24}-1 =16,777,215$

• Range: 0 to 16,777,215

• Minimum Value: 0

• Maximum Value: 16,777,215

Every bin is assigned a unique identifier (bin ID) based on a mathematical relationship between the bin's price and a fixed reference point within the system.

• $\text{Reference Point} = \dfrac{(16,777,215 - 0) + 1}{2} = \dfrac{16,777,216}{2} = 8,388,608$

$\text{bin Id} = \dfrac{log\text{(Price)}}{log(1 + \frac{\text{bin Step}}{10000})} + 8388608$

• bin Step(s) is in basis points, converted to a decimal.

• Price is the specific price level for calculating the bin ID.

• 8388608 is the midpoint of the range.

Bin pricing

The price of each bin is a function of the pair’s bin step and the bin’s index, The bin step parameter determines the constant percentage increase or decrease in price between each incremental bin.

• $8388608=$ reference point

• $Price_{\text{bin}} = (1 + \text{bin step})^{(\text{bin index} - 8388608)}$

LIQUIDITY STRUCTURE MATHS

Bin liquidity

The total liquidity in a bin: $P.(X+Z_x)+(Y+Z_y)=L_T$

Available liquidity in a bin: $P.(X)+(Y)=L_A$

$P$= price in terms of Y $X$= base asset $Y$= quote asset $Zy$= collateralized quote asset $Zx$= collateralized base asset

Bin composition

The Liquidity Composition factor (Lc) determines the ratio of the base asset to the quote asset. $Lc=\dfrac{X}{Y}$

The debt composition factor (Dc) determines the debt ratio.

$Dc=\dfrac{Zy + (P.Z_x)}{L_T}$

• Dc = 0: There is no debt in the bin

• Dc > 1: There is debt in the bin

BASIC INTERACTION MATHS

Adding liquidity

Adding liquidity to a bin will conserve the (Lc) of the bin. If a certain quantity of X (base asset) is determined, it is possible to compute the associated amount of Y (quote asset) to be added, and vice versa.

Liquidity added (L) = ${P.∆x} + {∆y}$

Bin Shares Received (B):

When adding, you will receive bin shares representing the liquidity added. $B = \dfrac{L \times TBS}{L_T}$

Removing liquidity

Removing liquidity results in the burning of Bin shares(B) which results in the receipt of assets X, and Y from the pool, and the minting of B (bin shares) associated with debt.

$X, Y, B = \dfrac{B \times (Rx, Ry, Dc \times TBS)}{TBS}$

$L$ = liquidity added $B$ = bin shares $TBS$ = total bin shares $L_T$= total liquidity $Dc$ = debt composition $Rx$= X reserve. hhh $Ry$ = Y reserve

Swaps

• Selling Base Asset (X) for Quote Asset (Y): $∆y=P\times∆x$

• Buying Base Asset (X) with Quote Asset (Y): $∆x=\dfrac{∆y}{P}$

DEBT INTERACTION MATHS

Borrowing:

Below Active Bin:

• Collateralize base asset (X), borrow quote asset (Y).

• $∆y=P\times∆Z_x$

Above Active Bin:

• Collateralize quote asset (Y), borrow base asset (X).

• $∆x=\dfrac{∆Z_y}{P}$

Repaying:

Below Active Bin:

• Return borrowed Y, retrieve collateralized X.

• $∆Z_x=\dfrac{∆y}{P}$

• Active Bin ID ≤ Borrow Bin ID ⇒ Repayment ≠ Possible

Above Active Bin:

• Return borrowed X, retrieve collateralized Y.

• $∆Z_y=P\times∆x$

• Active Bin ID ≥ Borrow Bin ID ⇒ Repayment ≠ Possible

Debt rollover

Extends the lifespan of a debt to its default duration $(T_{d})$

• $T_d$ = Default debt lifespan (7 days in this case)

• $T_{rem}$= Remaining time to expiry

• $T_{roll}$= New lifespan of the debt after rollover

Debt rollover is represented as:

• $T_{rollover} = T_{default}$

regardless of $T_{rem}$=Time remaining, the lifespan is reset to $T_d$= default time once a rollover occurs.

Blacklist (Auto debt recovery)

When the remaining time of a debt reaches zero, the debt is blacklisted and becomes liquidatable.

$T_{rem}$= 0 ⇒ Blacklist⇒ Liquidatable

Blacklisted debts automatically absorb the collateral into the liquidity pool.

Buffer Range:

Buffer Range: $±bins$, indicating the range on either side of the active bin functioning as a safeguard to preserve liquidity for swaps and prevent liquidity depletion due to borrowing activities

$±bins=\dfrac{B_P}{s}$

• Buffer percent $(B_p)$: A predetermined percentage.

• Bin Step (s): The rate of price change between each bin.

Implications:

• Borrow and Debt Rollover:

$\text{Allowed} = \begin{cases} 0, & \text{if within buffer range (\(\pm \text{bins}\))} \\ 1, & \text{otherwise} \end{cases}$

• Repayment and Liquidation:

$\text{Always Allowed} = 1$

FEE MATHS

Swap Fee (F):

The Swap Fee applies to all token exchanges within the liquidity pool.

$\text F=\text{Base factor} \times \text{Bin step}$

• Base Factor: A multiplier set by the protocol to adjust the fee rate as needed.

• Bin Step: The percentage price difference between consecutive bins.

Borrow Fee (Bf)

The Borrow Fee is incurred when initiating a borrowing position.

$Bf = \text F \times {Borrow factor}$

• Borrow Factor: A multiplier set by the protocol to control borrowing costs independently of swap fees.

• Streaming of Borrow Fee: $\dfrac{\text{Bin fee reserve}}{\text{seconds in 7-days of debt}}= λ$

  • Where $λ$ is the per-second fee allocation from the bin's fee reserve.

Repay Fee (Rf):

The Repay Fee compensates LPs for the opportunity cost of inactive liquidity due to debt.

$(\text{Bin activation index- Debt activation index}) \times \text{F}$

Bin Activation Index:

• The number of times a bin was activated

Debt Activation Index:

• Set when borrowing, matching the Bin Activation Index of the bin at that time.

Debt Rollover Fee:

The Debt Rollover Fee applies when a borrower extends their debt's lifespan.

$DRF=\text F +\text Rf$

• F: The Swap Fee.

• Rf​: The Repay Fee.