So, suppose you use calls spread to approximately replicate the payoff and therefore 'hedge' the position of shorting a binary call. Suppose the binary call's time to maturity is 1 month. The underlier current price is 100 dollars. You buy a call with strike 99.5 dollars and sell a call with strike 100.5 dollars and they both have time to maturity which is 1 month.
So, my question is how would you price/charge the binary option you sell then? Suppose the theoretical price of a binary option is 0.6 dollars, which can be obtained using Black-Scholes or take the limit of calls spread to zero...which is just math. But is this the real price or the real way of pricing a binary option?
I imagine there are two parts to the pricing of a binary option in practice: the replication cost and the hedging cost. The replication cost is easy to understand and should be close to 0.6 dollars. But because it is not perfect hedging, there is some residual exposure. For instance, if the stock price ends up being 100.4 dollars, then you yourself need to cough up and pay a little. So as the seller of the binary option, you would definitely charge a little for this risk you are taking, correct? I heard trader/quant/financial engineer use the word 'bump'. I imagine this must be it.
Again, thank you very much for your time.