Usual assumptions of the Black Scholes model apply. Given a risky asset and a risk-less asset which evolve respectively according to \begin{align}dS_t &= \mu S_t dt + \sigma S_t dW_t \\ \\ dM_t &= rM_t dt \end{align}We construct a portfolio $$V_t = \phi_t S_t + \psi_t M_t \tag{1}$$ and make it self-financing. The strategy ($\phi, \psi$) is self-financing if the only change in portfolio value comes from capital gains $$dV_t = \phi_t dS_t + \psi_t dM_t \tag{2}$$ If we then define $X$ as a claim at $T$ which only depends on $(S_t)_{0 \leq t \leq T}$ we have a replicating portfolio: $$V_T(\phi, \psi) = X_T \tag{3}$$ as a consequence, the price of the claim at any time $t$ becomes $$\Pi_t (X) = V_t$$ We seek to replicate a European option payoff with our self financing strategy (1) defined above.
The amounts $\phi_t$ and $\psi_t$ of stock and bond will generally be stochastic functions of the future stock price at $t$, or possible the entire stock-price history up till $t$. We have already seen in our derivation of the Black Scholes that the short stock delta-hedged portfolio is comprised of a stochastic $$\phi = \partial_s V (S_t , t)$$ For the portfolio to be self-financing, its value at $T$ must coincide with the claim $X$. Hence at $T$, $(1)=(3)$, $$V_T(\phi, \psi) = X_T= \phi_T S_T + \psi_T M_T$$