Asset and Capital Ledger

ASSET LEDGER

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

$$LR_3 \;\in\; [\, r_{w,3}(\text{β}) - db_{L,3} \quad,\quad r_{w,3}(\text{β}) + db_{U,3} \,] \quad \text{: \; within swap deadband}$$

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

$w_k$	Liquidity Ratio	Measure, 𝜶	Reference Ratio
$w_3$	$LR_3$	$(AP + AR)/B$	$r_{w,\,3,\,t}(\,\text{β}_t\,) = -\;pb_{L,\,3} (\,\text{β}_t\,) + pb_{U,\,3} \;\;\;\; (\;\pm\, db_{*,\,3}\,)$

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). $AL \quad:\quad \frac{AP + AR + diff_{\text{ETH}}}{B} \;-\; [\, r_{w,3}(\text{β}) \;\; (\;\pm\, db_{*,\,3}\,) \,]$

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

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

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

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

Otherwise, we set AL to zero:

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

CAPITAL LEDGER

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

$$LR_{13} \;\in\; [\, r_{w,\,13}(\text{β}) - db_{L,\,13} \quad,\;\;\; r_{w,\,13}(\text{β}) + db_{U,\,13} \,] \quad \text{: \; within swap deadband}$$

From § Liquidity Risk Factor, LR13 is defined as:

$w_k$	Liquidity Ratio	Measure, 𝜶	Reference Ratio
$w_{13}$	$LR_{13}$	$(\frac{CP}{\text{β}\cdot L + D + E} + \frac{CR}{L})$	$r_{w,\,13,\,t}(\,\text{β}_t\,) = \;pb_{L,\,13} + pb_{U,\,13}(\,\text{β}_t\,) \;\;\;\; (\;\pm\, db_{*,\,13}\,)$

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

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

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).

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

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

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

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:

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