Delta Hedging a Short Call and the Cost of Gamma Rebalancing
Summary
The note examines a short at-the-money call hedged by trading the underlying as the option’s delta changes. At initiation, a half-sized share position leaves the seller less protected against an immediate rise above the strike than a full share position would. Yet delta evolves with price, so the hedge must be rebalanced as the market moves. The trader raises exposure as the call becomes more sensitive to the underlying and reduces it as sensitivity falls.
The discussion highlights that repeated moves around the strike can make rebalancing costly, while a static full-size hedge can also perform poorly when the price crosses back and forth. The premium is compensation for taking option risk, but the note does not quantify whether it covers hedging costs or establish an optimal policy. Outcomes depend on the path, volatility, transaction costs, time remaining, and hedge frequency. Selling farther out of the money changes initial delta and premium, but does not by itself settle the trade-off.
Key ideas
- A short call’s delta changes as the underlying price and time to expiry change.
- A delta hedge requires adjusting the underlying position as option sensitivity evolves.
- Repeated price moves near the strike can create costs through hedge rebalancing.
- A static full hedge and a delta-based hedge each have path-dependent drawbacks.
- The premium’s adequacy depends on risks and costs not quantified in the note.
Tags
Full text
# Short Call Hedge. Options and gamma trading # Short Call Hedge. Options and gamma trading Let’s say a trader sells a Short Call with strike 100 (for making profit with the premium) at-the-money (for highest extrinsic value there). For hedging until expiration, he buys the underlying share at price 100. If the price goes up, ideally and theoretically, the profit from the share is equal to the loss from the Short Call. He adjusts his position of the share according to Delta. So in the first place, he buys a position of 0.5 at 100. This means, practically, that immediately above the strike, his shares make less profit than they should for hedging the full options position. However they will compensate by making already some profit below the strike. Yet this will be true only, if the price starts collecting the profit right from Delta=0 up to Delta=1. If he buys position 0.5 at the strike, and then the price only wanders around above the strike price, even though he is going to increase his position to Delta 1 eventually, the share will never earn the full profit it needs to fully compensate the short call loss (losses from the many small ups and downs not even considered yet). Would others see it the same way? And what to conclude from that? - He could ignore Delta and always hedge with full position size (he is small enough and the market big enough for that). But if the price moves around the strike for long, and he has to fully go long above and fully flat below, his earned premium will be gone soon. - Or he just has to bear this cost; it’s covered by the premium anyway. And most likely the price will go a bit underneath the strike anyway, and collect at least a bit of profit there. - Or he would need to sell a Call not ATM but OTM, so he can start his hedge from where Delta is near 0. But doesn’t sound good either, as lower extrinsic value there.
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.