Sign up ×
Quantitative Finance Stack Exchange is a question and answer site for finance professionals and academics. It's 100% free, no registration required.

If so, is there a derivation anywhere that shows this? I was told that this could be done in a class but I don't see how it's possible.

share|improve this question

1 Answer 1

up vote 1 down vote accepted

It's quite straightforward when you plug in the formulae for the greeks into the PDE.


$\Delta =\frac{\partial c_t}{\partial S_t}=\Phi(d_1)$

$\Gamma=\frac{\partial^2 c_t}{\partial S_t^2}=\frac{\phi(d1)}{S_t\sigma\sqrt{u}}$

$\Theta=\frac{\partial c_t}{\partial t}=-rKe^{ru}\Phi(d_2)-S_t\phi(d_1)\frac{\sigma}{2\sqrt{u}}$

\begin{eqnarray} rc_t&=&\Theta+rS_t\Delta + \frac{1}{2}S_t^2\sigma^2\Gamma\\ RHS&=&-rKe^{ru}\Phi(d_2)-S_t\phi(d_1)\frac{\sigma}{2\sqrt{u}}+rS_t\Phi(d_1)+\frac{1}{2}S_t^2\sigma^2 \frac{\phi(d1)}{S_t\sigma\sqrt{u}}\\ &=&-rKe^{ru}\Phi(d_2)-\frac{S_t\phi(d_1)\sigma}{2\sqrt{u}}+rS_t\Phi(d_1)+ \frac{S_t\sigma\phi(d1)}{2\sqrt{u}}\\ &=&rS_t\Phi(d_1)-rKe^{ru}\Phi(d_2)\\ &=&rc_t\\ &=&LHS \end{eqnarray}

share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.