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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?

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    $\begingroup$ Why did you sell the option? $\endgroup$
    – will
    Apr 29, 2019 at 19:26

2 Answers 2

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By delta hedging you are saying that you have a view on the path and the volatility of the option you are trading, but not on its direction; in your case, that being short delta.

From a theoretical perspective, all options are priced fairly and not delta hedging simply increase the variance of your payouts.

In your example, selling a call and delta hedging, then, at time T, if the volatility of the call stays constant and the option's gamma is "consistently" hedged, then it means that you will lose an amount on your delta hedge equal in value to the premium you received for your option, so PNL = 0.

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    $\begingroup$ Because of this 0 PNL, should banks therefore sell the call slightly higher than the fair price to earn a profit? From my understanding, banks are always delta-hedging, therefore to have a positive profit, they should sell it higher than the fair price. $\endgroup$
    – Victor
    Apr 30, 2019 at 0:07
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    $\begingroup$ Yes, that is how Banks make money in Options, from the "markup" or bid/ask spread. They sell the option to a customer for slighly more than the "correct" price. $\endgroup$
    – Alex C
    Apr 30, 2019 at 1:28
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As @Alex C mentioned, this is how banks make money. They will sell/buy you an instrument with a markup and hedge their position. Over the life of the trade, they will adjust the hedge. If the costs of hedging over the life of the trade are less than the markup, they will profit. Of course the banks, as market makers, will have a portfolio of positions and they can efficiently hedge a number of trades as the risks of each individual trade will be offset by other positions on the book. Also, they will use this "inventory" of positions to buy and sell to clients such that they will not have to hedge the position over the entire life--only over their holding period.

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