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Greeks

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Greeks

(Spot) Delta.

$\displaystyle \frac{\partial v}{\partial x}$

$\textstyle =$

$\displaystyle \phi e^{-r_f\tau} {\cal N}(\phi d_+)$

(4)

Forward Delta.

$\displaystyle \frac{\partial v}{\partial f}$

$\textstyle =$

$\displaystyle \phi e^{-r_d\tau} {\cal N}(\phi d_+)$

(5)

Driftless Delta.

$\displaystyle \phi {\cal N}(\phi d_+)$

(6)

Gamma.

$\displaystyle \frac{\partial^2 v}{\partial x^2}$

$\textstyle =$

$\displaystyle e^{-r_f\tau} \frac{n(d_+)}{x\sigma\sqrt{\tau}}$

(7)

Speed.

$\displaystyle \frac{\partial^3 v}{\partial x^3}$

$\textstyle =$

$\displaystyle -e^{-r_f\tau} \frac{n(d_+)}{x^2\sigma\sqrt{\tau}}\left(\frac{d_+}{\sigma\sqrt{\tau}}+1\right)$

(8)

Theta.

$\displaystyle \frac{\partial v}{\partial t}$	$\textstyle =$	$\displaystyle -e^{-r_f\tau} \frac{n(d_+)x\sigma}{2\sqrt{\tau}}$	(9)
	$\textstyle +$	$\displaystyle \phi[r_fxe^{-r_f\tau}{\cal N}(\phi d_+)-r_dKe^{-r_d\tau}{\cal N}(\phi d_-)]$	(10)

Vega.

$\displaystyle \frac{\partial v}{\partial\sigma}$

$\textstyle =$

$\displaystyle xe^{-r_f\tau}\sqrt{\tau}n(d_+)$

(11)

Vomma.

$\displaystyle \frac{\partial^2 v}{\partial\sigma^2} =xe^{-r_f\tau}\sqrt{\tau}n(d_+)\frac{d_+d_-}{\sigma}$

(12)

Cross.

$\displaystyle \frac{\partial^2 v}{\partial\sigma\partial x} =-e^{-r_f\tau}n(d_+)\frac{d_-}{\sigma}$

(13)

Rho.

$\displaystyle \frac{\partial v}{\partial r_d}$	$\textstyle =$	$\displaystyle \phi K\tau e^{-r_d\tau} {\cal N}(\phi d_-)$	(14)
$\displaystyle \frac{\partial v}{\partial r_f}$	$\textstyle =$	$\displaystyle -\phi x\tau e^{-r_f\tau} {\cal N}(\phi d_+)$	(15)

Dual Delta.