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When we sell an option and we hedge it using Delta, we replicate the option payoff until maturity according to its Delta. If we replicate the option perfectly and with high frequency, we should be able to pay just the payoff at the end if it is exercised, otherwise we end with zero PnL at maturity.

How traders and banks makes their PnL if the hedge is perfect ?

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The phenomenon you describe is the cost of maintaining the delta hedge due to the actual volatility of the underlying (other costs include bid-ask spread, market impact etc.) To compensate for the costs of delta hedging and to make some money to make a market in the option, the market maker charges more for the option than the option value. Hence the "implied volatility" (the volatility implied by the price of the option, all other inputs constant) tendency to being greater than the actual volatility.

Further, the market maker hedges their net delta of their full book, rather than each and every option. Should there be offsetting deltas from being long/short and puts vs calls, etc., the market maker does not have to trade the delta of each individual option, and therefore reduces their overall delta hedging costs.

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  • $\begingroup$ Thank you. I heard about the consideration of more charge on the option value, but still not clear for me regarding the no arbitrage principle: indeed the option payoff could be replicated by a delta hedging folio so why the buyer will accept a higher volatility. On the argument of hedging all the folio: all the folio is hedged so each option is hedged. If the folio is perfectly replicated the broker won’t earn on all the folio, no ? $\endgroup$
    – Ouissem
    Commented Jan 27 at 20:19
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At all times the market maker posts 2 prices: the bid and the ask (which is higher than the bid). When someone sells to the marketmaker, they receive the bid price, when someone wants to buy they have to pay the ask price. So the market maker earns the spread between these two prices, just like a street vendor sells you an apple for a higher price than he paid at the wholesale fruit market - that is his profit.

The hedging makes it possible to eliminate the "inventory risk" i.e. the danger of price changes between the time the marketmaker buys from X and the time he sells to Y. But the profit comes (mostly) from the price difference.

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  • $\begingroup$ Thank you. what about the case where a bank sells an OTC option and hedge it in delta. The bank will track the option payoff until the maturity and if the tracking is perfect the bank will replicate the payoff exactly, so no earn. $\endgroup$
    – Ouissem
    Commented Jan 27 at 20:05
  • $\begingroup$ No, the bank collects p+x from the customer on the first day, where p is the fair price. They set x aside in a separate bank account. Then they hedge the option at p and if all goes well they make no profit ot loss on that. But remember: they still have the x that they collected at the beginning and that is their profit. The hedging is a way to protect the initial markup of the option. $\endgroup$
    – nbbo2
    Commented Jan 28 at 12:52
  • $\begingroup$ Understand. Thank you nobb2 $\endgroup$
    – Ouissem
    Commented Jan 29 at 22:01
  • $\begingroup$ OK, So the bank earns x because the bank overvaluated the option to make earn. $\endgroup$
    – Ouissem
    Commented Apr 5 at 6:57

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