# Derivation of Call Delta from Black Scholes Model

How is call delta mathematically derived from Black Scholes Model (without approximation) ? Please help me understand each step mathematically. And how it is approximated to say that delta is the probability of option expiring in the money?

Look here for a detailed derivation of the formula for $$\Delta$$ (be aware that this particular website uses $$r_d$$ to denote the risk-free rate and $$r_f$$ to denote the dividend yield). You can always ask for more specific help regarding a particular step in the derivation.
It is easy to see that $$\mathbb{Q}[\{S_T\geq K\}]= \Phi(d_2)$$. Just replace $$S_T=S_0\exp\left(\left( r-q-\frac{1}{2}\sigma^2\right)T +\sigma\sqrt{T}Z\right)$$ where $$Z\sim N(0,1)$$ and isolate $$Z$$ on the left-hand side. This is the risk-neutral probability of expiring ITM. Note that $$\Delta=\Phi(d_1)=\Phi(d_2+\sigma\sqrt{T})\approx \Phi(d_2)$$. This is since $$\sigma\sqrt{T}$$ is typically very small.
• That derivation is excellent but it starts with a small typo: in BSCallPrice the roles of $r_f$ the risk free rate and $r_d$ the dividend rate are reversed. Oct 6 '19 at 15:39
• I think it's not a typo because they use $r_f$ constantly and (as far as I checked) consistently to denote the dividend yield and $r_d$ as risk-free interest rate. So I believe they do this intentionally. However, I do agree with you that this notation is odd because $r_f$ should be the risk-free rate (or perhaps the rate in a foreign currcency) but not the dividend yield. Oct 6 '19 at 16:33