Greeks (Option)

• Notation is equivalent to the BSM formula.
• Derivative of CDF of Normal Distribution

Delta §

i.e. sensitivity to either underlier price or strike price

$$\begin{array}{rlr}{\Delta }_{C,S}& :=\frac{\partial C}{\partial S}:=\frac{\partial C}{\partial x}{\mid }_{x=S_t}& =e^{-q(T-t)}N(d_1)\\ {\Delta }_{C,K}& :=\frac{\partial C}{\partial K}& =e^{-r(T-t)}N(d_2)\end{array}$$

Table of Deltas for Call/Put options as execution time approaches.

$t\to T$ or $\tau {\to }_0$	At the Money Call $S_t-K=0$	In the Money Call $S_t-K>0$	Out of the Money Call $S_t-K<0$
$d_1$	$0$	$\infty$	$-\infty$
$N(d_1)$	$\frac{1}{2}$	$1$	$0$
${\Delta }_C$	$\frac{1}{2}$	$1$	$0$
${\Delta }_P$	$-\frac{1}{2}$	$0$	$-1$
For puts	At the money Put $K-S_t=0$	Out of the Money Put $K-S_t<0$	In the Money Put $K-S_t>0$

Gamma §

i.e. second derivative sensitivity to underlier price

$${\Gamma }_C:=\frac{{\partial }^{2}V}{\partial S^2}={\Gamma }_P=\frac{e^{-q\tau }{N}^{\prime }(d_{+})}{S_t\sigma \sqrt{\tau }}$$

Theta §

i.e. sensitivity to time (=time decay)

$$\begin{array}{rlrl}{\Theta }_C:=\frac{\partial V}{\partial \tau }& =-S_t{e}^{-q\tau }\sigma N^{\prime }(d_{+})& +q{S}_t{e}^{-q\tau }N(d_{+})& -rK{e}^{-r\tau }N(d_{-})\\ {\Theta }_P:=\frac{\partial V}{\partial \tau }& =-S_t{e}^{-q\tau }\sigma N^{\prime }(d_{+})& -q{S}_t{e}^{-q\tau }N(-d_{+})& +rK{e}^{-r\tau }N(-d_{-})\end{array}$$

Vega §

i.e. sensitivity to volatility

$${\mathcal{V}}_C:=\frac{\partial C}{\partial \sigma }$$