It is well known that hedging with implied volatility involves a PnL:
In the Wilmott paper (http://web.math.ku.dk/~rolf/Wilmott_WhichFreeLunch.pdf), they imply that the collective PnL from such a strategy is the integral of above expression across time.
However, this seems to assume that the market implied volatility stays constant at $σ_i$. Otherwise, one would also encounter the mark-to-market PnL governed by the sensitivity of the option to implied volatility among other terms:
Why is the mark to market PnL not accounted for in the above analysis?