Recall - GBM & More

• Stock prices in traditional finance are modeled using GBM

• When $p > 0.5$, the expected growth rate of the price of the asset is

• Here the term $-\frac{\sigma^2}{2}$ is known as the volatility drag.
• In the same case of $p > 0.5$, if we have $\frac{2\sqrt{\mu}}{\sqrt{3}} < \sigma < 2\sqrt{\mu},$ we have

Basis Cash

Basis Cash

• We plot the distribution of $\text{log}\left(\frac{S_i}{S_{i-1}}\right)$ for price data $\{S_i\}$.
• Ideally for stablecoins, we should have a delta function at $X = 0$.
• For all the price data, we see a Gaussian distribution centered at 0.
• For all-time price data, we see a spread around $[-0.2, 0.2]$.
• For more recent price data (which is stabilized around $\$1$, we see the variance gets smaller.

Ampleforth

• Prices before June 26th are more stable ($\sim \$1$).
• Volume trade increase led to price fluctuations.
• Here too, we expect the distribution of prices to be Gaussian.

Ampleforth

Empty Set D$\phi$llar

Empty Set D$\phi$llar

Drift & Volatility Analysis

• We analyse $\mu, \sigma$ of BAC prices for a data set of size 542.
• Plot of $(\mu, \sigma)$ values for a given window size:

Drift & Volatility Analysis

• As the window size increases, $\mu, \sigma$ tends to stabilize
• $\mu$ and $\sigma$ roughly follow a similar profile

LP Wealth Factor

• We plot the factor $f = \frac{\sigma^2}{2\mu}$, for max LP wealth growth, we want $f \in (0.66, 2)$.
• We observe an irregular pattern here, need more investigation on this.