The partial derivative of a call option with respect to $t$

Question

In Black-Scholes related computations, why do we not treat the stock price $S$ as a function of $t$ when taking partial derivatives with respect to $t$? For example, if $$c(t,T)=SN(d_1)-Ke^{-r(T-t)}N(d_2)$$ is the price of a call option and we want to find $\partial C/\partial t$, we never include the term $\partial S/\partial t$ and don't consider $S$ as a function of $t$, but as a separate variable. How can this be justified?

Answer 1

Your call formula can be written as a function of $S$, where $S$ is the initial stock price

$$c(t,T, S, K)=SN(d_1)-Ke^{-r(T-t)}N(d_2)$$

In this case, $S$ has no dependence on $t$ (this formula is not stochastic!) so you can take the $t$-derivative as your usual partial derivative.