How do you derive the delta of an option from the Black Scholes model?

First, compute $d_1$. \begin{equation} d_1 = \frac{ln(\frac{S_t}{K})+(r+0.5 \sigma^2)(T-t)}{\sigma \sqrt{T-t}} \end{equation} Finally, look up $N(d_1)$ in the $z$-statistic table. This value is the value of delta. You'll have a value between 0 and 1, indicating the probability of an option to end up (In The Money) ITM. An ATM option has therefore a delta of 0.5, indicating a 50/50 probability that the option ends up ITM. For put-options, the values are between 0 and -1.

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