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: 
​K1K_1K1​
is the "riskiest" strike, which is expected to be breached every 3 months.
​K2K_2K2​
 is a "medium risk" strike, which is defined as being breached every 18 months.
​K3K_3K3​
 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: 
Xi=X(si)=∣s1−1∣∗104X_i=X(s_i)=|s_1-1|*10^4Xi​=X(si​)=∣s1​−1∣∗104
                                          where sis_isi​
 is the stablecoin price at a given time tit_iti​
 
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 S^\hat{S}S^
.
si={sj^∈S^∣max⁡sj^∈S^Xi}s_i = \{\hat{s_j} \in \hat{S}|\smash{\displaystyle\max_{\hat{s_j}\in \hat{S}}} X_i  \}si​={sj​^​∈S^∣sj​^​∈S^max​Xi​}
where we assume that any correlation spirals happen within each interval, Δt=ti−ti−1\Delta t=t_i-t_{i-1}Δt=ti​−ti−1​
​
Each strike Kk∈{1,2,3}K_{k \in \{1,2,3\}}Kk∈{1,2,3}​
 has an associated rate rkr_krk​
, defined as the probability that the strike is breached within a given Δt\Delta tΔt
. The rate is calculated using an indicator function from the discrete-time series S=∪siS=\cup s_i S=∪si​
, as 
rk=∑i=1n1Xi>Kknr_k =\sum_{i=1}^{n}\frac{\mathbb{1}_{X_i>K_k}}{n}rk​=i=1∑n​n1Xi​>Kk​​​
and can be used in a binomial distribution to find the probability PkP_kPk​
 of a particular strike being breached within a given month: 
Pk=(1−rk)d×fP_k= (1-r_k)^{d\times f}Pk​=(1−rk​)d×f
where fff
 is the sampling frequency and ddd
 is the number of days in a given epoch. 
The equation above is solved for each rkr_krk​
 on the interval (0,1)∈R(0,1) \in \mathbb{R}(0,1)∈R
 given the desired values of PkP_kPk​
. This is done by using a variety of root-finding algorithms. Once rkr_krk​
 is determined, the set of all XiX_iXi​
 can be iterated through for varying strikes until an appropriate KKK
 is found. For the cases, K1K_1K1​
and K2K_2K2​
, each PkP_kPk​
, is 13\frac{1}{3}31​
 and 118\frac{1}{18}181​
, respectively. 
​
Step-By-Step User Guide
Hedge Pricing
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