Asset and Capital Ledger

ASSET LEDGER

We define the Asset Ledger (AL) by firstly considering the deadband around the reference ratio:

LR3    [rw,3(β)dbL,3,rw,3(β)+dbU,3]   within swap deadbandLR_3 \;\in\; [\, r_{w,3}(\text{β}) - db_{L,3} \quad,\quad r_{w,3}(\text{β}) + db_{U,3} \,] \quad \text{: \; within swap deadband}

From before (§ Liquidity Risk Factor) LR3 is defined as:

wkw_k

Liquidity Ratio

Measure, 𝜶

Reference Ratio

w3w_3

LR3LR_3

(AP+AR)/B(AP + AR)/B

rw,3,t(βt)=  pbL,3(βt)+pbU,3        (  ±db,3)r_{w,\,3,\,t}(\,\text{β}_t\,) = -\;pb_{L,\,3} (\,\text{β}_t\,) + pb_{U,\,3} \;\;\;\; (\;\pm\, db_{*,\,3}\,)

When in equilibrium, the liquidity ratio is equal to its reference: \begin{align*} \quad\quad\quad\quad LR_3 - [\, r_{w,3}(\text{β}) \,] &= 0 \\[4pt] \quad\quad\quad\quad \frac{AP + AR}{B} - [\, r_{w,3}(\text{β}) \,] &= 0 \\[4pt] \end{align*}

Define diff_ETH as any ETH deposit or withdrawal from Taker’s Protect or Close, or credits or debits of ETH relating to a cross-side swap (see § Combining Differences). AL:AP+AR+diffETHB    [rw,3(β)    (  ±db,3)]AL \quad:\quad \frac{AP + AR + diff_{\text{ETH}}}{B} \;-\; [\, r_{w,3}(\text{β}) \;\; (\;\pm\, db_{*,\,3}\,) \,]

AL is positive when the liquidity ratio exceeds the reference ratio plus the deadband:

AL=(AP+AR+diffETH)    B[pbL,3(β)+pbU,3+dbU,3]:    AP+ARB>[pbL,3(β)+pbU,3+dbU,3]\begin{align*} AL &= \left(AP + AR + diff_{\text{ETH}}\right) \;-\; B \cdot \left[ -\, pb_{L,\,3}(\text{β}) + pb_{U,\,3} + db_{U,\,3} \right]\\[4pt] &:\;\; \frac{AP + AR}{B} > \left[ -\, pb_{L,\,3}(\text{β}) + pb_{U,\,3} + db_{U,\,3} \right] \end{align*}

AL is negative when the liquidity ratio is lower than the reference ratio minus the deadband:

AL=(AP+AR+diffETH)    B[pbL,3(β)+pbU,3dbL,3]:    AP+ARB<[pbL,3(β)+pbU,3dbL,3]\begin{align*} AL &= \left(AP + AR + diff_{\text{ETH}}\right) \;-\; B \cdot \left[ -\, pb_{L,\,3}(\text{β}) + pb_{U,\,3} - db_{L,\,3} \right]\\[4pt] &:\;\; \frac{AP + AR}{B} < \left[ -\, pb_{L,\,3}(\text{β}) + pb_{U,\,3} - db_{L,\,3} \right] \end{align*}

Otherwise, we set AL to zero:

AL=0    :    elsewhere\begin{align*} AL &= 0 \\[6pt] &\;\;:\;\; elsewhere \end{align*}

CAPITAL LEDGER

We similarly define the Capital Ledger by firstly applying the deadband around the reference ratio:

LR13    [rw,13(β)dbL,13,      rw,13(β)+dbU,13]   within swap deadbandLR_{13} \;\in\; [\, r_{w,\,13}(\text{β}) - db_{L,\,13} \quad,\;\;\; r_{w,\,13}(\text{β}) + db_{U,\,13} \,] \quad \text{: \; within swap deadband}

From § Liquidity Risk Factor, LR13 is defined as:

wkw_k

Liquidity Ratio

Measure, 𝜶

Reference Ratio

w13w_{13}

LR13LR_{13}

(CPβL+D+E+CRL)(\frac{CP}{\text{β}\cdot L + D + E} + \frac{CR}{L})

rw,13,t(βt)=  pbL,13+pbU,13(βt)        (  ±db,13)r_{w,\,13,\,t}(\,\text{β}_t\,) = \;pb_{L,\,13} + pb_{U,\,13}(\,\text{β}_t\,) \;\;\;\; (\;\pm\, db_{*,\,13}\,)

When in equilibrium, the liquidity ratio is equal to its reference:

LR13[rw,13(β)]=0(CPβL+D+E+CRL)[rw,13(β)]=0\begin{align*} \quad\quad\quad\quad LR_{13} - [\, r_{w,\,13}(\text{β}) \,] &= 0 \\[4pt] \quad\quad\quad\quad (\frac{CP}{\text{β}\cdot L + D + E} + \frac{CR}{L}) - [\, r_{w,\,13}(\text{β}) \,] &= 0 \\[4pt] \end{align*}

We then define diffUSD as any USD deposit or withdrawal from Claim, Withdraw, or credits or debits of USD relating to a cross-side swap (see subsection Combining Differences).

CL:(  CP+diffUSD  βL+D+E+CRL)    [rw,13(β)    (  ±db,13)]CL \quad:\quad (\frac{\;CP + diff_{\text{USD}\;}}{\text{β}\cdot L + D + E} + \frac{CR}{L}) \;-\; [\, r_{w,\,13}(\text{β}) \;\; (\;\pm\, db_{*,\,13}\,) \,]

CL is positive when the liquidity ratio exceeds the reference ratio plus the deadband:

CL=(CP+CR+diffUSD)(LR13    [pbL,13+pbU,13(β)+dbU,13])/LR13    :LR13>[rw,13(β)+dbU,13]\begin{align*} CL &= ( CP + CR + diff_{\text{USD}} ) ( LR_{13} \;-\; [ pb_{L,\,13} + pb_{U,\,13}(\text{β}) + db_{U,\,13} ]) / LR_{13}\\[8pt] &\;\;: \quad LR_{13} > [ r_{w,\,13}(\text{β}) + db_{U,\,13}] \end{align*}

CL is negative when the liquidity ratio is lower than the reference ratio minus the deadband:

\begin{align*} CL &= -1 * (CP + CR + diff_{\text{USD}}) ([ pb_{L,\,13} + pb_{U,\,13}(\text{β}) - db_{L,\,13} ] \;-\; LR_{13}) / LR_{13}\\[8pt] &\;\;: \quad LR_{13} < [ r_{w,\,13}(\text{β}) - db_{L,\,13}] \end{align*}

Otherwise, we set CL to zero:

CL=0  :    elsewhere\begin{align*} CL &= 0 \\[6pt] &\;:\;\; elsewhere \end{align*}

Last updated