Risk Measure

Default Risk
We define the liability that represents the unrealized loss for the exchange at time ttt
 as
Xt=qT(st−sˉ)X_t= \mathbf{q}^T(\mathbf{s_t}-\mathbf{\bar{s}})Xt​=qT(st​−sˉ)
where q∈Rn\mathbf{q} \in \mathbb{R}^nq∈Rn
 is a vector of long-short imbalances (qi>0 q_{i}>0qi​>0
 implies long-heavy in the iii
th market), st∈R+n\mathbf{s}_t \in \mathbb{R}^n_+st​∈R+n​
 is a vector index prices at time ttt
, and sˉ\mathbf{ \bar{s}}sˉ
 is a vector of average entry prices of the open positions. In other words, XtX_tXt​
 represents the net payout from AMM to traders if all outstanding positions were to be closed at index price st\mathbf{s_t}st​
. We call the exchange default if XtX_tXt​
 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 ttt
 
Xt+τ≤P+L+FX_{t+\tau} \leq P+L+FXt+τ​≤P+L+F
where PPP
 is cumulative premium, LLL
 is externally provided liquidity, and FFF
 is cumulative funding. Note that we consider the liability at some future time t+τt+\taut+τ
, which requires some forecast of index price movement.
In the following sections, we discuss how they are priced using EVaR.
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Last modified 4mo ago
Risk Measure

Default Risk

Objective

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