While deriving the delta hedge error if we hedge with implied vol, and the true vol is different, we say that the PnL of the call option is:
$$dC=C_tdt+C_SdS+0.5C_{ss}<QV>dt - (1)$$
Where $<QV>$ is the 'realised quadratic variation' of the stock price, and not the incorrect implied vol. While I understand this from the mathematical perspective (change in a function depends on the actual change/dynamics of the independent variable), and I also understand that this call price must 'drift' at less than the risk free rate (therefore create an arbitrage with the correct call price). However, I don't see how I 'realize' this PnL.
Consider the case where I have bought a call in a market where there is no option liquidity. I come back tomorrow, I mark to model, and therefore my PnL should be given by the difference in model price today and tomorrow, which is just the above equation but with implied vol used as the quadratic variation. How do I know what is the correct value to mark my call value tomorrow? Is there a market mechanism that will force the value of my call to be given by the above equation? Does this mean I will have to remark the volatility in my model everyday, to be consistent with the PnL?
Edit: I'm trying to ask the same question in a different way. Let $<QV>$ be the actual quadratic variation and $<MV>$ be the implied quadratic variation of the stock price. Then:
$$dC(t,S_t;MV)=C_tdt+C_SdS+0.5C_{ss}<MV>dt$$ where derivatives are taken at the implied vol.
$$dC(t,S_t;QV)=C_tdt+C_SdS+0.5C_{ss}<QV>dt$$ where derivatives are taken at the true vol.
However, in equation 1, the derivatives are at the implied vol, whereas quadratic variation is at the true vol. I am not sure what function $C$ is in equation (1). It's certainly not the functions in LHS of (2) and (3). Can somebody explain what call price function is involved in equation (1)?