# Risk Measure

## Default Risk

We define the liability that represents the unrealized loss for the exchange at time
$t$
as
$X_t= \mathbf{q}^T(\mathbf{s_t}-\mathbf{\bar{s}})$
where
$\mathbf{q} \in \mathbb{R}^n$
is a vector of long-short imbalances (
$q_{i}>0$
implies long-heavy in the
$i$
th market),
$\mathbf{s}_t \in \mathbb{R}^n_+$
is a vector index prices at time
$t$
, and
$\mathbf{ \bar{s}}$
is a vector of average entry prices of the open positions. In other words,
$X_t$
represents the net payout from AMM to traders if all outstanding positions were to be closed at index price
$\mathbf{s_t}$
. We call the exchange default if
$X_t$
exceeds exchange-owned liquidity.
We use a monetary risk measure to quantify this default risk. Monetary risk measure is a mapping from a set of random variables to a real number, which we can think of as the amount of capital needed to cover for the risk.
We use a specific coherent risk measure called entropic value-at-risk (EVaR) for the risk calculation.

## Objective

In order to maintain solvency, the AMM charges premium / rebate as well as periodic funding to ensure that the following invariant holds with high probability at any time
$t$
$X_{t+\tau} \leq P+L+F$
where
$P$
$L$
$F$
$t+\tau$