# Risk Measure

## Default Risk

We define the liability that represents the unrealized loss for the exchange at time $t$ as

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 $t$

where $P$ is cumulative premium, $L$ is externally provided liquidity, and $F$ is cumulative funding. Note that we consider the liability at some future time $t+\tau$, which requires some forecast of index price movement.

In the following sections, we discuss how they are priced using EVaR.

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