I'm looking at the chapter, "The Greek Letters" in Hull's book (Options and derivatives...) and in particular the paragraph "Dynamic Aspects of Delta Hedging". He demonstrates two examples of how dynamic hedging works (within the Black-Scholes framework), for a short position on a European call option with expiry 2 weeks.
In particular, after shorting a call option, given that delta (assume that it can be computed somehow) changes over time, the seller should buy or sell stocks (might need to borrow money from some cash account) in order to neutralise the change in the value of the option due to changes in the value of the stock (delta neutral). During this hedging lifecycle, some rebalancing/hedging costs accumulate (occurring from borrowing money to buy extra stocks while hedging at discrete times) till the expiry of the option where the seller closes his position with the buyer.
The author claims that, if there are no transaction costs and the rebalancing is frequent, then the Discounted accumulated hedging cost at expiry (which appears as the cash remaining "to return to the bank" after having delivered the payoff and sold the owned stocks) should be equal to the Black-scholes price of the call option at inception. Anyone can (mathematically) prove why is that or provide reference?
Question 2: (for sanity)
How does a trading desk/seller knows what the delta of the option is every time they hedge the option? Do they approximate numerically using Black-Scholes? If yes, which approximation scheme of the derivative is used?