Consider someone that writes a call, and wishes to delta-hedge against it to remain delta neutral. For this to be profitable, the price they sell this option for should be greater than or equal to the cost of delta-hedging it. (From this question answer: "An option price is equal to the cost of delta-hedging")
I understand the option writer has to use capital to delta-hedge, which would imply some risk-free cost to borrow. However, I do not understand how volatility increases the overall expense of delta-hedging.
The option writer could do the following, continuously: For each theoretical price point, compute what delta would be, and place a limit buy or sell order at that theoretical price that would result in their position matching that delta. This would result in them having a ladder of orders up and down the order book, which would slowly change over time. Notice: this is entirely agnostic of the actual volatility.
What am I missing here? Given that options' IVs correlate with historical volatilies there must be some real cost or risk associated with delta-hedging more volatile stocks. However, that cost and/or risk eludes me.