Risk Premium Calculation

Risk Premium is calculated based on the Delta, Vega, and Theta Risk that the OLP is exposed to, taking into account "how much" and "in what direction" the trader's new trade has impacted the OLP's risk, and how much Greeks risk should be reflected in each time frame.

The formula used by Moby to calculate Risk Premium is as follows:

Calculation based on Risk Premium calculation methodology updated as of 02 April 2024.

$$\bold{Risk\,Premium}=(RP_{\delta}+RP_{\nu}+RP_{\theta})*UR\,Mul * Market\,Factor\\ \ \\ \bold{RP_{Greeks}} = UG_{Greeks}*Direction_{Greeks}*Weight_{Greeks}$$

Calculation

The sequence of Risk Premium calculation can be divided into four steps:

1. Determine UG: Calculate Unit Greeks based on the size and risk of the options position held by OLP before and after the trader's transaction

2. Determine Direction: Calculate the direction of risk by comparing the magnitude and symbol of the OLP's risk before and after the trader's trade

3. Determine Weight: Determine the weight of the Greeks based on the type of Greeks and the timeframe of the OLP

4. Determine UR Mul: Calculate the UR Mul that causes the Risk Premium to increase rapidly when the Utility Ratio of OLP exceeds a certain threshold

5. Determine Market Factor: Calculate a set of factors such that the Risk Premium may have a tendency similar to the market spread (from CEXs). Those factors are determined based on each option position's spec, including Moneynes, DTE, Underlying Asset, Call/Put, etc

Parameters

The parameters used to calculate the risk premium and the calculation method for each parameter are shown below:

$$G_0 = \{Delta_0, Vega_0, Theta_0\}$$

G_0: Overall Delta, Vega, and Theta values for OLP before new trades are reflected.

$$G_1 = \{(Delta_0 - Delta_{\text{NT}}), (Vega_0 - Vega_{\text{NT}}), (Theta_0 - Theta_{\text{NT}})\}$$

G_1: Overall Delta, Vega, and Theta values of the OLP after the new trades are reflected. (OLP takes the opposite position of the new trade, so the Greeks of the new trade (Greeks_NT) are subtracted from the Greeks of the old OLP (Greeks_0)).

$$G_1 = \{(Delta_0 - Delta_{\text{NT}}), (Vega_0 - Vega_{\text{NT}}), (Theta_0 - Theta_{\text{NT}})\}$$

OLPDV, or OLP Deposit Value, is the sum of all tokens' value(wBTC, wETH, USDC) held by OLP minus the sum of all tokens' value paid by traders as collateral. This makes OLPDV stand for the pure liquidity provision from LP users.

$$OLPDV = \sum_{i=1}^{n} (\text{OLP Owned Tokens Value} - \text{Collateral value from Traders})$$

When a trader opens a long position, OLP simultaneously opens a short position (=underwriting), providing liquidity for the trading pair. Currently, UR (Utility Ratio) is calculated based on the ratio of collateral in OLP used for options underwriting among the assets deposited in OLP.

UG(Unit Greeks)

UG (Unit Greeks) is the unit of Greeks used to assess the level of impact of a trader's requested trade on OLP risk, calculated based on the existing OLP's Greeks and the OLP's new Greeks reflecting the requested trade.

$$\bold{UG_{Greeks}} = \begin{cases} \sqrt{(F*0.01)*|G_{1}|*scalFac}*min(max(\frac{G_{1}}{G_{0}}, 1), 2) &\text{} (\delta) \\ \sqrt{|G_{1}|*scalFac/(F*0.01)}*min(max(\frac{G_{1}}{G_{0}}, 1), 2) &\text{} (\nu,\theta) \end{cases}$$

• G0: The overall Delta, Vega, and Theta exposure of the OLP before new trades are reflected

• G1: The overall Delta, Vega, and Theta values of the OLP after reflecting new trades

• scalFac: If OLPDV increases, the size of positions that traders can trade increases, and it is possible that the size of OLP's average Greek exposure will rise accordingly. In this case, even if the degree of influence on OLP's Greeks exposure is similar, the Risk Premium may be overcalculated. To prevent these issue, scalFac is used in the calculation process for Unit Greeks

• Futures Price Scaling: Unlike Delta, the formula for calculating Vega and Theta includes the underlying token's price. In the case of Delta, scaling is performed by multiplying (Futures Price*0.01) to obtain the Unit Greeks of option positions based on different underlying assets. Thus, Unit Delta can be calculated based on each underlying asset's 1% price movement

Direction

D (Direction) is a unit that evaluates the direction in which a new trade affects OLP risk, calculated by comparing the sign and absolute value of OLP's risk before and after the new trade is reflected

$$D(i_0, i_1) = \begin{cases} 1 & \text{if } i_0 \times i_1 < 0 \\ 1 & \text{if } i_0 \times i_1 \geq 0, \ |i_0| \geq |i_1| \\ 0.1 & \text{if } i_0 \times i_1 \geq 0, \ |i_0| < |i_1| \end{cases}$$

• Positive Direction: A trade in the direction of increasing the risk of the OLP, applying 100% of the Unit Greeks

  • If the sign of the risk before and after the new trade is the same, and the absolute value of the risk after the new trade is larger

  • If the sign of the risk before and after the new trade is reversed (the OLP is exposed to risk in a new direction)

• Negative Direction: A trade in the direction that reduces the risk of the OLP, applying 10% of the Unit Greeks

  • If the sign of the risk before and after the new trade is the same, and the absolute value of the risk after the new trade is smaller

Greeks Weight

W (Greeks Weight) is a weight that considers the impact of each greek on Risk Premium, and uses different values for each type of Greeks and term. Individual weights are calculated by considering the market spread by term and the risk-managing capability of Risk Premium. In the case of term, Short/Mid/Long-Term is categorized based on DTE.

$$W(i, \text{DTE}) = \begin{cases} w_{i,s} & \text{if DTE = Short Term} \\ w_{i,m} & \text{if DTE = Mid Term} \\ w_{i,l} & \text{if DTE = Long Term} \end{cases}$$

$$\text{where,}$$

$$w_{i,t} = \begin{pmatrix} w_{\text{Delta},s} & w_{\text{Vega},s} & w_{\text{Theta},s} \\ w_{\text{Delta},m} & w_{\text{Vega},m} & w_{\text{Theta},m} \\ w_{\text{Delta},l} & w_{\text{Vega},l} & w_{\text{Theta},l} \end{pmatrix}$$

$$i = \text{Type of Greeks} \\{DTE} = \text{Date To Expiry of New Trade}$$

Short-Term: 0DTE, 1DTE

Mid-Term: near-week, near-month

Long-Term: DTE greater than near-month

UR Mul

UR Mul encourages traders to avoid trading at high UR levels by increasing the Risk Premium as UR increases. If UR is lower than a certain threshold (UR_Thr), UR Mul is maintained as UR Mul_Initial, but if it exceeds the threshold, UR Mul increases rapidly in the form of a quadratic function.

$$\bold{UR\,Mul} = \begin{cases} \ Mul_{Initial} &\text{if } UR_1<UR_{Thr} \\ \ ((UR_1-UR_{Thr})(UR_1+1)+Mul_{Initial}) * min(max(UR_1/UR_0, 1), 2) &\text{if } UR_1\ge UR_{Thr} \end{cases}$$

$$UR_0 = \text{Utility Ratio before reflecting new trades}\\ UR_1 = \text{Utility Ratio after reflecting new trades}$$

If UR_Thr is set to 40% and UR Mul_Initial is set to 0.5, UR Mul is fixed to 0.5 until UR exceeds 40%. If UR exceeds 40%, UR Mul causes the Risk Premium to increase rapidly at the rate of quadratic form

For example, when UR reaches about 70%, UR Mul increases to 1, which means that the Risk Premium is calculated twice compared to when the UR is 40%.

The slope of the parameter and quadratic function can be dynamically changed during the risk management process of the platform. Plus, UR Mul adopts the size of UR_1 (Utility Ratio after reflecting the new trades) and aggravates how the new trades impact UR_1 (UR_1 / UR_0) so that incentivize the trades that contribute to the Utility Ratio management.

Market Factor

In determining Moby's Risk Premium, we tried to determine the Proper Level based on the three criteria below.

• How much is the Risk Premium discounted compared to the Market Spread?

• How effectively does Risk Premium manage OLP’s Greeks Risk?

• How similar are Risk Premiums to Market Spreads?

Market Fator is a factor set applied to meet the third standard. It is not an internal factor of the platform, such as OLP's Greeks Risk, but rather the spec of the position the trader wants to open or the market situation (Moneynes, DTE, Underlying Asset, Call/Put, etc.). The Market factor currently used in the Risk Premium calculation process includes:

• RP Mul: The Multiplier that allows Risk Premium to be calculated differently depending on Moneynes and DTE of a specific options position

• Underlying Ratio: Factor to reflect the tendency that the Market Spread appears differently depending on the Underlying Asset

• Call/Put Ratio: Factor to reflect the tendency that Market spread to appear differently depending on the Call/Put

• Market Volatility Ratio: Factor to reflect the tendency that Market Spread can be changed significantly when the Underlying Asset's Real Volatility increases rapidly

Additional elaboration regarding RP Mul

Traders who frequently trade options on CEX may have experienced the phenomenon that the Market Spread varies greatly depending on the strike price and the remaining DTE of options.

In general, if the option has a long DTE, the market spread is tighter, which can be fully confirmed through data analytics such as Laevitas. In fact, the data below shows that the longer the DTE, the tighter the spread is, not only in the 50delta option but also in the 25delta options.

So, with the DTE fixed, how much does Moneyness or how far away the Strike Price is from the ATM affect the Market Spread? We conducted polynomial regression analysis on the Market Spread based on the Monyness of the Historical Market Spread of Options CEXs. As a result, we confirmed that it is possible to approximate the Market Spread for each individual DTE with a quadratic equation for Moneyness as shown below.

X-axis: Moneyness(|Delta|), Y-axis: Market Spread of each Moneyness / Market Spread of ATM