The Greeks: Delta, Gamma, Theta, Vega

Using the Black-Scholes formula, derive closed-form expressions for the Greeks of a European call on a non-dividend-paying stock:

• Delta ${\Delta }_c=\frac{\partial c}{\partial S}$
• Gamma $\Gamma =\frac{{\partial }^{2}c}{\partial S^2}$
• Vega $\nu =\frac{\partial c}{\partial \sigma }$
• Theta ${\Theta }_c=\frac{\partial c}{\partial t}=-\frac{\partial c}{\partial T}$
• Rho ${\rho }_c=\frac{\partial c}{\partial r}$

Then verify that ${\Delta }_c=N(d_1)$ and that the BSM PDE ${\Theta }_c+rS{\Delta }_c+\frac{1}{2}{\sigma }^2{S}^2\Gamma =rc$ holds identically.