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You decide to sell a European call option that is currently 10% OTM (for example the strike = 100 and the current price = 90). You have to delta hedge to keep the delta of your position at 0. What is the best and worst case scenario for you as the seller of the call?

I would think that the best case scenario is if the option remains OTM so you can pocket the premium paid at the start and the worst case scenario is if the option becomes deep ITM but I'm not sure how to actually quantify this.

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  • $\begingroup$ Reread the question carefully, I believe it mentions "delta hedged". So the answer is not so simple. What can go wrong when you are delta hedged? $\endgroup$
    – nbbo2
    Jun 28, 2021 at 0:54

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Let’s say you have sold the 100 call 10% OTM and you have a delta of 0.4 ie you bought 40% stock at 90 against the call.

I think the worst scenario is the stock skyrockets overnight. Say it goes to 200. Then you are down about 100 on the call and up 0.4*90= 36 on the stock. And if it goes even higher, the loss is unlimited.

The best case is that the stock gradually crawls from 90 to 100 by the time of expiration. Then, the option expires worthless so you pocket the premium plus gains on the stock as it went from 90 to 100. Your delta probably went from 40 to 50 during this time so the gain is approximately (50-40)*0.45=4.50.

All of this is approximate , since delta depends on the model you are using.

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  • $\begingroup$ Thank you! If you are delta hedging at the end of everyday to be delta neutral (i.e. delta = 0), how would the answer change? $\endgroup$
    – sn98
    Jun 29, 2021 at 13:40
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Delta is an instantaneous measure of risk for the stock movement. @dm63 basically describes a scenario where the gamma has hurt you because you have not rebalanced your delta hedge to account for a large move in the stock. Rebalancing your hedge is where the art of risk management comes in--how frequently should you rebalance?; how do you handle gap risk?; Will you use other options to hedge the gamma?; etc.

Additionally, you have the risk that implied vols spike. Since you are short the option, if implied volatility moves, the mark to market (MTM) on your position will negatively impact your PnL.

Also, there are risks in management of the position. If you are unable to post margin for any adverse moves in the MTM, you will be closed out of your position. Additionally, you will be negatively impacted by the costs of posting your margin.

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