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 iith 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.


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.

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