One way to derive the Black-Scholes PDE is via the Delta-hedging argument:
Suppose that $V_t = V(t, S_t)$, for some function $V: [0,T] \times \mathbb{R} \to \mathbb{R}$. We construct a portfolio by buying one unit of the derivative and shorting $\frac{\partial V}{\partial S}(t, S_t)$ units of the underlying stock. Therefore, the porfolio has value $\Pi_t= V(t, S_t) - \frac{\partial V}{\partial S}(t, S_t) S_t$ and hence by the self-financing property and Ito's formula, $$ d \Pi_t = dV_t - \frac{\partial V}{\partial S}(t, S_t) \,dS_t = \bigg(\frac{\partial V}{\partial t}(t, S_t) + \frac{1}{2} \sigma^2 S_t^2 \frac{\partial^2 V}{\partial S^2}(t, S_t) \bigg) \,dt. $$ This allows us to derive the Black-Scholes PDE: $$\frac{\partial V}{\partial t} + rS \frac{\partial V}{\partial S} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} = r V.$$
However, I notice a strange thing in this argument:
The Delta-hedging portfolio is constructed using one unit of the derivative, whose price function $V$ fluctuates according to the actual market price of that derivative, i.e. the compromised value following a bid-ask spread in trading. However, the objective of this argument is to find a PDE for $V$. Hence, this argument appears to assume that the theoretical value function (also denoted by $V$) is the same as the actual price function in the market. Have I mixed up anything in this argument? Any ideas?