# When should we delta hedge?

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?

• Why did you sell the option? – will Apr 29 at 19:26