# Determining Sugar levels (price)

## Methodology

We studied several medical histories (datasets) from recent years to determine striking prices for stable coins fixed at $1. Using data from medical insurance vaults for stable coins, we calculated optimal strike prices and expiry dates for low, medium, and high risk scenarios using a variety of variables.&#x20;

The following is an outline of the steps we took: Standard deviations for USDC, BUSD, DAI, USDT, and FEI are calculated.&#x20;

Determine the daily variations from the mean ($1) while removing any outliers.&#x20;

Determine the frequency with which the variance threshold is breached at various indicators (10bp, 20bp, 30bp, etc)&#x20;

Determine optimal strike prices that balance the motivations of all parties involved.

## Determining the Sugar strike levels (prices)

In the present system, each stable coin is assigned three strikes:

the "riskiest" strike, with a breach projected every three months.&#x20;

a "medium risk" strike, characterized as a breach every 18 months.&#x20;

a "low risk" attack that represents black swan occurrences.

These are the lowest yielding yet give security against unforeseen occurrences during the lifespan of a stable coin. To compute the striking prices, we assume that price variances from $1 are independent and identically distributed (i.i.d) random variables. More information on the assumptions and statistical components may be found in our technical whitepaper.&#x20;

_Strike Price Formula:_&#x20;

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

&#x20;                                         where $$s_i$$ is the stable coin price at a given time $$t_i$$&#x20;

It is well known that in times of mass de collateralization 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 stable coin 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&#x20;

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

and can be used in a binomial distribution to find the probability $$P_k$$ of a particular strike being breached within a given month:&#x20;

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

where $$f$$ is the sampling frequency and $$d$$ is the number of days in a given epoch.&#x20;

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.&#x20;





