Determining Strike Prices

Methodology

The methodology presented is for stablecoins pegged to $1. In order to determine the strike prices, we analyzed a variety of datasets, each spanning over the past few years. Using the datasets for the stablecoin insurance vaults we are implementing, we calculated a variety of metrics to determine the appropriate strike prices and maturities for the low, medium, and high risk payouts.

Outlined is the process we took:

1. Calculate the standard deviations for USDC, MIM, DAI, FRAX, and FEI

2. Calculate the daily deviations from the mean ($1), excluding all outliers

3. Calculate the frequency of breaking the variance threshold at various indicators (10bp, 20bp, 30bp, etc)

4. Determine appropriate strike prices which align incentives from all interacting parties. This includes providing insureds appropriate protection, ensuring counterparties earn generous revenues, and protecting the protocol from mass liquidation events.

Determine strike prices

In the current framework, we assign three strikes to each stablecoin:

• $K_1$is the "riskiest" strike, which is expected to be breached every 3 months.
• $K_2$ is a "medium risk" strike, which is defined as being breached every 18 months.
• $K_3$ is a "low risk" strike, which denotes black swan events. These are the lowest yielding but provide protection against unexpected events over a stablecoins lifetime.

We assume price deviations from $1 are independent and identically distributed (i.i.d) random variables to determine the strike prices. Refer to our technical whitepaper to read more about the assumptions and statistical components.

Strike Price Formula:

$$X_i=X(s_i)=|s_1-1|*10^4$$

where $s_i$ is the stablecoin price at a given time $t_i$

It is well known that in times of mass decollateralization spirals, these variables become correlated, and the i.i.d assumption does not hold. To alleviate this restriction and ensure our data is accurately distributed, we assume that the discrete-time series of stablecoin prices are sampled from a "continuous" (block-by-block) series $\hat{S}$.

$$s_i = \{\hat{s_j} \in \hat{S}|\smash{\displaystyle\max_{\hat{s_j}\in \hat{S}}} X_i \}$$

where we assume that any correlation spirals happen within each interval, $\Delta t=t_i-t_{i-1}$

Each strike $K_{k \in \{1,2,3\}}$ has an associated rate $r_k$, defined as the probability that the strike is breached within a given $\Delta t$. The rate is calculated using an indicator function from the discrete-time series $S=\cup s_i$, as

$$r_k =\sum_{i=1}^{n}\frac{\mathbb{1}_{X_i>K_k}}{n}$$

and can be used in a binomial distribution to find the probability $P_k$ of a particular strike being breached within a given month:

$$P_k= (1-r_k)^{d\times f}$$

where $f$ is the sampling frequency and $d$ is the number of days in a given epoch.

The equation above is solved for each $r_k$ on the interval $(0,1) \in \mathbb{R}$ given the desired values of $P_k$. This is done by using a variety of root-finding algorithms. Once $r_k$ is determined, the set of all $X_i$ can be iterated through for varying strikes until an appropriate $K$ is found. For the cases, $K_1$and $K_2$, each $P_k$, is $\frac{1}{3}$ and $\frac{1}{18}$, respectively.