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Risk Measure

Default Risk

We define the liability that represents the unrealized loss for the exchange at time
tt
as
Xt=qT(stsˉ)X_t= \mathbf{q}^T(\mathbf{s_t}-\mathbf{\bar{s}})
where
qRn\mathbf{q} \in \mathbb{R}^n
is a vector of long-short imbalances (
qi>0 q_{i}>0
implies long-heavy in the
ii
th market),
stR+n\mathbf{s}_t \in \mathbb{R}^n_+
is a vector index prices at time
tt
, and
sˉ\mathbf{ \bar{s}}
is a vector of average entry prices of the open positions. In other words,
XtX_t
represents the net payout from AMM to traders if all outstanding positions were to be closed at index price
st\mathbf{s_t}
. We call the exchange default if
XtX_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
tt
Xt+τP+L+FX_{t+\tau} \leq P+L+F
where
PP
is cumulative premium,
LL
is externally provided liquidity, and
FF
is cumulative funding. Note that we consider the liability at some future time
t+τt+\tau
, which requires some forecast of index price movement.
In the following sections, we discuss how they are priced using EVaR.