> For the complete documentation index, see [llms.txt](https://bubble-bot.gitbook.io/bubblebot-guide/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://bubble-bot.gitbook.io/bubblebot-guide/hedge-risk/analyzing-bubble-levels-price.md).

# Analyzing Bubble Levels (Price)

Speculate on pegged assets, multifold your earnings

## Methodology

We analyzed multiple charts (datasets) from recent years to establish the appropriate price levels for stable coins, which are pegged at $1. Utilizing data from diverse sources pertaining to stable coins, we computed optimal strike prices and expiration dates for low, medium, and high-risk scenarios, incorporating a range of variables.

We conducted the following steps in our analysis:

1. Calculated standard deviations for USDC, DAI, USDT, and FEI.
2. Identified daily deviations from the mean ($1), excluding outliers.
3. Analyzed the frequency of breaches in variance thresholds at different indicators (e.g., 10bp, 20bp, 30bp, etc.).
4. Determined optimal strike prices that strike a balance among the interests of all stakeholders.

In the current system, each stable coin is associated with three strike levels:

1. **Faty Risk Strike**: Anticipating a breach every three months.
2. **Smart Risk Strike**: Expecting a breach every 18 months.
3. **Low Risk Strike**: Representing rare, black swan events.

These strikes provide a balance between yield and security against unforeseen events during the stable coin's lifespan. To calculate these striking prices, we assume that price variations from $1 follow an independent and identically distributed (i.i.d) random variable pattern. You can find more details on the assumptions and statistical aspects in our technical whitepaper.

### Strike Price Formula:

$$
X\_i=X(s\_i)=|s\_1-1|\*10^4
$$

                                          where $$s\_i$$ is the stable coin price at a given time $$t\_i$$ 

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 

$$
r\_k =\sum\_{i=1}^{n}\frac{\mathbb{1}\_{X\_i>K\_k}}{n}
$$

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.