# Risk Measure

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.

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

$P$

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.

Last modified 4mo ago