**Additional Info**

Assumptions [pertinent but not exhaustive]

- The inputs of spot price, risk-free rate, strike price, and time to maturity are known with high confidence

- The large assumption:
*our estimate of**volatility**is correct and does not vary over the life of the contract*

Definitions

- N(d1) = delta or hedge ratio of the option
- If you look closely at d1 you will see that is the equivalent of a z-score:
- The numerator is the distance of the stock price from the strike
- The denominator normalizes that distance by vol time

credit: brilliant.org

- N(d2) = probability of the option expiring in-the-money

d2 is also a z-score but look closely —it is always less than d1

This means:

**Delta is always greater than the probability of expiring in the money because delta is a hedge ratio.****As a hedge ratio, it must account not just for probability but payoff****. If you have high volatility or lots of time to expiry, the right tail of a lognormal distribution is longer. So the number of shares you need to****hedge****the potential unbounded price of the call is larger This means the divergence between d1 and d2 depends strictly on volatility and time to maturity.**Learn more