IV is not the unknown variable in BSM. IV is directly observed. It is the market price. The unknown variable is the instantaneous (actual) volatility of the underlying. The instantaneous vol is the input that must be input into BSM to correctly replicate the option. Think of it this way....if you could trade infinitely quickly with no costs or slippage and no discontinuities in the asset price (big if's haha) then each with each passing (infinitely small) interval of time, vol would be 15%....then 11%....then 0.1%...then 4%.... Each time, you'd calculate your delta hedge at that instantaneous vol and replicate perfectly over the life of the contract.
Surely you jest! You state "IV is not the unknown ..." and you state "instantaneous volatility is the unknown ..." What do you thing "IV" is? Seems you may have had a long trying week!
IV is the implied volatility over the life of the option. Longthewings is right, it is a form of the cost of an option. Instantaneous volatility is realized volatility. That is an unknown.
Please post supporting documentation that IV is NOT implied volatility and is the odd description you state? I'm aware of individual implied volatility (What I referenced by term IV, and what I though the author referenced), and option series implied volatility, and <n>DTE implied volatility (i.e. VIX for SPX), but your reference to something else "over the life of an option" seems nonsensical.
IV is implied volatility. It is the volatility that you expect to realize over the life of the option. "instantaneous volatility" is the "realized volatility" (or some call here "Stat volatility." That is the actual volatility that you will see over the life of the option. That is unknown. When you trade volatility you are exchanging a fixed cost (which is your current implied volatility) in the form of theta for a variable cost (the actual realized volatility) in the form of gamma. Just like an interest rate swap. Implied volatility is another way of looking at the price of an option. It is directly related to the premium and current spot price. In fact, vol traders on bank floors will only talk in volatility terms: " I paid 32 vol for Netflix Dec." When I was a bank trader, I didn't even know what the premiums I was paying for. When I got a quote, I would immediately convert it into vol terms.