 Solution

Let $\tau =T-t$. The product rule and Itô's formula givewhere the Black-Scholes equation cancels the drift. Under the usual pricing growth and integrability conditions this local martingale is a martingale. Its terminal value is $f(S_T)$, since the exponential factor becomes one at $T$. Taking a conditional expectation givesThe dynamics already use the risk-neutral measure. This is the Feynman-Kac formula for the given pricing problem; a differential equation alone, without the relevant growth condition, would not justify upgrading a local martingale or uniqueness.