Let's say I'm the seller of a European call option on a non-dividend paying stock. I pocket the premium $c_0$ of the call at $t=0$.
If I start to delta-hedge right away, this is equivalent to replicating the call and the cost of the strategy will converge to the price of the option $c_0$ if the option is priced fairly (according to the Black-Scholes formula) and if the delta-hedging is done sufficiently frequently.
So, it seems unless at $t=0$ the option is priced higher than its theoretical price, delta-hedging right away leads to an average null profit. Am I correct? So what are the precise scenarios when we should use delta-hedging?
For example, if at a time $t$, the call price $c_t$ is below the price at $t=0$ ($c_t < c_0$), the delta-hedging strategy will cost in average $c_t$. Does it mean I will lock an average positive profit?