Risks and Outcomes of Delta-Hedging a Short Out-of-the-Money Call
Summary
The document discusses the possible outcomes of selling an out-of-the-money European call while delta hedging with the underlying stock. It contrasts a gradual rise toward the strike, where the call may expire worthless and the hedge can gain value, with a sharp overnight rally, where the short call's loss can exceed gains on the initial stock hedge. Because the hedge is only locally effective, a large move can create losses as the option's delta changes.
It also identifies practical risks beyond the stock-price path: implied volatility can rise and hurt the marked value of a short option, and margin demands or financing costs can pressure the position. The numerical illustration is explicitly approximate and depends on the pricing model and hedge adjustments. The discussion does not provide a full profit-and-loss framework, account for the premium in its examples, or quantify the effects of rebalancing frequency, transaction costs, or gap risk.
Key ideas
- A delta hedge offsets the option's immediate sensitivity to stock moves but changes as the underlying moves.
- A sudden rally can leave a short call exposed to large losses, potentially without an upper bound.
- A gradual move toward the strike may allow the call to expire worthless while the stock hedge gains.
- Rising implied volatility can worsen the mark-to-market of a short option.
- Rebalancing, gap risk, margin availability, and financing costs affect the position's realized outcome.
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Full text
# Best/worst case scenario after selling OTM call option # Best/worst case scenario after selling OTM call option 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. ## Answer by dm63 (score 3) https://quant.stackexchange.com/a/65749 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. ## Answer by AlRacoon (score 3) https://quant.stackexchange.com/a/65761 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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