Price derivatives. Implement the solver. Prove it under interview conditions.

Under risk-neutral measure $\mathbb{Q}$, the asset price follows geometric Brownian motion:

$dS_t = r\,S_t\,dt + \sigma S_t\,dW_t^{\mathbb{Q}}$

Applying Itô's lemma to $V(S_t,\,t)$, and invoking the no-arbitrage condition:

$\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV = 0$

Terminal condition for a European call: $V(S_T,\,T) = \max(S_T - K,\;0)$

The closed-form solution is:

$V = S\,\mathcal{N}(d_1) - K e^{-rT}\mathcal{N}(d_2)$

$d_{1,2} = \frac{\ln(S/K) + \bigl(r \pm \tfrac{1}{2}\sigma^2\bigr)T}{\sigma\sqrt{T}}$