The probability of BTC or ETH being above X$ is calculated per maturity, as shown below:
This probability depends first on the model we use in pricing the options and on many parameters:
- \(F\): forward price of the underlying
- \(K\): strike level
- \(\tau\): time-to-maturity
- \(\sigma\): implied volatility
In the Black-Scholes framework, where the underlying follows a log-normal distribution, the risk neutral probability \(P_{K}\) of being above a strike \(K\) is:
\begin{equation*}
P_{K} = N(d_{2})
\end{equation*}
with:
\begin{equation*}
d_{2} = \frac{Log(\frac{F}{K}) - \frac{\sigma^{2}}{2}\tau}{\sigma \sqrt{\tau}}
\end{equation*}
where \(N(.)\) is the standard normal cumulative distribution function.