Rebalancing a Delta Hedge After a Stock Price Decline
Summary
For a long call initially hedged to delta neutrality, the hedge is typically a short stock position because the call has positive delta. If the stock price falls, the call’s delta usually falls as well, leaving the trader short more shares than needed for the new delta exposure. The adjustment is to buy back some of the short stock position.
The explanation connects this trade to the call’s positive gamma: as the underlying moves down, the long call’s delta decreases, so reducing the short hedge realizes the direction of adjustment associated with gamma hedging. The answers also mention that a sharp adverse event may raise implied volatility and affect skew, which can benefit a long option. These volatility effects are conditional rather than guaranteed, and the document gives qualitative guidance rather than a hedge ratio or discussion of transaction costs and other risks.
Key ideas
- A long call has positive delta, so delta neutrality initially requires shorting stock.
- When the stock falls, the call’s delta generally decreases and the short hedge becomes too large.
- Buying back some short stock rebalances the position toward delta neutrality.
- A long call’s positive gamma explains why its delta changes as the stock price moves.
- Changes in implied volatility and skew can also affect the option’s value during a sharp decline.
Tags
Full text
# How to adjust delta hedging if stock price decreases? # How to adjust delta hedging if stock price decreases? > Question: You are long a call option no MITCO stock. You have delta hedged your position. You hear on the radio that the CEO of MITCO has just been arrested for running a massive Ponzi scheme. The stock price plunges $10. How do you adjust your hedge (qualitatively)? That is, do you borrow and buy stock or sell stock and lend? Explain carefully. I know that delta is the rate of change of option's value with respect to stock price. Since stock price decreases, so is delta. But I do not know how to adjust the delta hedging. ## Answer by AlRacoon (score 7, accepted) https://quant.stackexchange.com/a/50117 You would be over hedged in your call position if it was delta neutral before the stock cratered. Since you are long delta on the call, you would have shorted stock to make the original position delta neutral. When the stock fell, your long call delta would have fallen, and you would buy to cover some of your short stock hedge. However, being long the call, you would have been long gamma. As such, this trade should have been a great trade for you. Your position would have been overhedged. But as you were short the stock, your short delta on your hedge would not have been changing as the stock tanked. Meanwhile, your delta on your long call would have been decreasing. As a result, your short delta would have been increasingly short as the stock was crashing--just the way you want to be positioned as the stock is tanking. You would also have been long Vega in your long call position. As the stock tanks, the implied volatility should have gone up which would have benefited your long call position slightly. Also, skew would have worked in your favor. ## Answer by siou0107 (score 7) https://quant.stackexchange.com/a/50115 You are long a vanilla option, so long gamma (positive gamma). If the stock price decreases, so does the delta of your option. Since you short-sold the stock to hedge, you now have short-sold too much since delta has decreased. As a consequence, you must buy back some stock.
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.