Risk Management for the Liquidity Providers

Options are leveraged, volatile products that incur large risks along multiple dimensions. This is why current AMM designs for other markets, whilst instructive, cannot be directly mapped to options. Existing options AMMs attempt to deal with this complexity by increasing fees to levels which almost certainly protect LPs, but result in extremely high prices and little trader interest. This tension is resolved in traditional finance through risk management techniques, allowing market makers to quote competitive prices whilst ensuring that they maintain high risk-adjusted returns.

A successful options AMM should emulate this process, hedging risk to enable competitive pricing. Delta hedging smooths the exposure of the AMM to a large directional move in the underlying asset. This is combined with an asymmetric spread which incentivizes trades that hedge the AMM by charging a tee that is snewed according to its current vega risk exposure. First, let's define the risk metrics of an Eonian Option: the Greeks.

Greeks Computation

Delta $\frac{\partial E}{\partial S}$: the change in the price of an option given a 1 USD move in the underlying asset.

Vega $\frac{\partial E}{\partial \sigma}$: the change in the price of an option given a 1% change in the Implied Volatility.

For Eonian Options:

Let $u = \sqrt{1+\frac{8}{\sigma^2T}}$

If $S > K$:

$$\delta_{i} = -\frac{u-1}{2u}(\frac{S}{K})^{-\frac{u+1}{2}}$$

$$\nu_{i} = \frac{K(1+\frac{u}{2})ln\frac{S}{K}}{u\sigma(1+\frac{T\sigma^2}{8})}(\frac{S}{K})^{-\frac{u-1}{2}}$$

If $S<K$:

$$\delta_{i} = \frac{u+1}{2u}(\frac{S}{K})^{\frac{u-1}{2}}$$

$$\nu_{i} = \frac{K(1-\frac{u}{2})ln\frac{S}{K}}{u\sigma(1+\frac{T\sigma^2}{8})}(\frac{S}{K})^{\frac{u+1}{2}}$$

Delta Risk

Calculating the Delta Risk

Without active risk management, LPs are subject to Impermanent Loss and a greatly increased probability of ruin.

Therefore, we calculate the total delta exposure of the AMM.

$$\Delta = \sum_{i=1}^{n} \delta_{i,j} \times \rho_{i,j}$$

Where:

$\rho_{i,j}$ is the number of contracts the AMM is long (if positive) or short (if negative) for a given strike.

$n$ is the number of different strike prices represented in the AMM portfolio.

Managing the Delta Risk

The total delta exposure of the AMM is then used to delta hedge properly its position, by buying, selling or short selling the underlying asset. These hedging actions are undertaken by the protocol risk team.

Vega Risk

Calculating the Vega Risk

Need reviewing

Managing the Vega Risk

The AMM manages its vega risk by incorporating its vega exposure into the fee charged to a trader. The fee is calculated as follows:

$$f = A \times E + B \times H \times VU + C \times S_t$$

Where $A$, $B$, $C$ are coefficients that will be chosen by the governance. $H$ is set to 0 if the trade brings the vega of the AMM closer to 0 and 1 otherwise.

Therefore, the fee have 3 components:

• A flat fee based on the option price

• A dynamic vega risk component

• A flat fee for exchange cost