Exercise: Verifying the PDE for the Black-Scholes call

Substitute the Black-Scholes call price $C(S, t) = S\Phi(d_1) - Ke^{-r(T-t)}\Phi(d_2)$ directly into the PDE

C_t + \tfrac12\sigma^2 S^2 C_{SS} + rSC_S - rC = 0

and verify that it holds identically.

Tasks

Compute $C_S$ (delta). Show that $C_S = \Phi(d_1)$ . Useful identity: $S\varphi(d_1)\cdot \frac{\partial d_1}{\partial S} = Ke^{-r(T-t)}\varphi(d_2)\cdot\frac{\partial d_2}{\partial S}$ .
Compute $C_{SS}$ (gamma). Show that $C_{SS} = \varphi(d_1)/(S\sigma\sqrt{T-t})$ .
Compute $C_t$ (theta, noting the negative sign). Show that

C_t = -\frac{S\varphi(d_1)\sigma}{2\sqrt{T-t}} - rKe^{-r(T-t)}\Phi(d_2).

Substitute all three into the PDE and simplify. The terms should cancel to zero.

Hint

The identity $S\varphi(d_1) = Ke^{-r(T-t)}\varphi(d_2)$ — which follows from evaluating both sides explicitly and using the relation between $d_1, d_2$ — is the key algebraic shortcut that makes the verification tractable.