Trade Impact on price quoted by the AMM

The objective of an options market maker is to identify an Implied Volatility value $(\sigma)$ at which supply meets demand. At this level, the Automated Market Maker (AMM) can generate trading fees without assuming any risk, as it is purchasing and selling options in equal quantities. The AMM is created to efficiently adjust to supply and demand to attain this IV level. Consequently, the variables applied in the pricing model are solely determined by the market. Traders' buy and sell activities are utilized to bring the price quoted by the pool closer to the actual market price, all in real-time.

Standard Size

The price impact of a trade is proportional to its size. The AMM captures this effect through the notion of Standard Size ($\varphi$).

$$\varphi = \varepsilon n$$

Where:

$\varepsilon$ is a constant which is dynamically determined by the protocol.

$n$ is the number of contracts traded.

The Standard Size is inversely proportional to vega; as the vega of the contract increases, the Standard Size decreases. This means that a lower Standard Size makes the implied volatility more responsive to a given trade, while a higher Standard Size makes it less responsive.

The Standard Size decreases as the vega of the contract increases, indicating an inverse proportionality between them. As a result, when the Standard Size is lower, the implied volatility becomes more responsive to a given trade, whereas a higher Standard Size makes it less responsive.

Implied Volatility Impact

Let $IV$ be the Implied Volatility Matrix of Otto Exchange. The matrix holds the implied volatility for every European Option used to determine Eonian Option pricing on the platform. As Eonian Option strike prices are customizable, the matrix has an infinite number of columns, and it has an infinite number of rows because each Eonian Option comprises an infinite number of options that expire at one-day intervals.

$$IV = \begin{bmatrix} \sigma_{1,1} & \dots & \sigma_{1,\infty}\\ \vdots & \ddots & \vdots\\ \sigma_{\infty,1} & \dots & \sigma_{\infty,\infty} \end{bmatrix}$$

A row in this matrix represents the implied volatilities of European Options with the same maturity but different strike prices.

A column in this matrix represents the implied volatilities of European Options with the same strike price but different maturities.

Exemple 1: The $\sigma_{5, 1}$ coefficient represents the implied volatility of the pool's lowest strike option with a 5-day maturity and the $\sigma_{5, \infty}$ coefficient represents the pool's highest strike option with a 5-day maturity.

If the pool sells 1 Standard Size: $\sigma_{new} = \sigma_{old}$ + 1%

If the pool buys 1 Standard Size: $\sigma_{new} = \sigma_{old}$ - 1%

Skew Impact

The Black-Scholes model doesn't consider the impact of strike on implied volatility, leading to the skew or volatility smile seen in options markets.

To fix this problem, the AMM computes the Skew Ratio ($SR$).

The Skew Ratio is given by the following formula:

$$(SR)_{i,j} = \frac{\sigma_{i, j}}{\sigma_{j}}$$

Where $\sigma_{j}$ is the baseline implied volatility for an option located on the $j$ column, provided by the protocol at regular intervals. We compute it by extracting market data.

For every Standard Size bought or sold by a trader at a strike $K$, the mechanism increases or decreases the Skew Ratio by a constant $c_r$ initialized at $0.0075$.

If the pool sells 1 Standard Size: $(SR)_{new} = (SR)_{old} + c_r$

If the pool buys 1 Standard Size: $(SR)_{new} = (SR)_{old} - c_r$

Therefore, the implied volatility used to compute the price of the European Option with strike price index $j$ and expiration $i$ days is:

$$\sigma_{i,j} = (SR)_{i,j} \times \sigma_j$$

When a trader buys a single Eonian Option with a certain strike price from the pool, they're actually buying an infinite number of European Options with different maturities. The trader's trade affects the entire column of the strike price they bought, and also affects every element on the line where the implied volatility of the European options they bought is. So, the trader's trade affects the entire implied volatility matrix.

One way to represent the impact of a trader buying a single Eonian Option with strike price index 3 on the Implied Volatility Matrix is as follows.

Impact of a trader buying a single Eonian Option with strike price index 3 on the Implied Volatility Matrix