Rebalancing a Delta Hedge After a Stock Price Crash

Summary

The document explains how to adjust a hedge after a stock price falls when holding a long call that was initially delta hedged. A long call has positive delta, so the initial hedge involves shorting shares. When the stock drops, the call’s delta decreases; the existing short stock position then hedges more exposure than the option requires.

The adjustment is to buy back some of the short stock. The same conclusion follows from gamma: a long vanilla option has positive gamma, so its delta falls as the underlying price falls. The explanation is qualitative and sign-based, with no numerical example or discussion of transaction costs, changing volatility, or other portfolio positions. It applies to the stated long-option hedge setup; other positions may require different trades.

Key ideas

• A long call has positive delta, so delta hedging it initially requires shorting stock.
• After a stock price decline, a long call’s delta decreases.
• The original short stock hedge then becomes too large relative to the option’s new delta.
• Buy back some short stock to restore the delta hedge.
• Positive gamma explains why a long option’s delta falls when the underlying price falls.

Tags

# Adjusting your delta hedge when the stock crashes and were originally delta hedged


# Adjusting your delta hedge when the stock crashes and were originally delta hedged












You are long a call option on a stock and you are delta hedged. The stock crashes in price. How do you adjust your delta, do you buy or sell stock?

Could answers please be quantitative (i am getting signs the wrong way round probably) as well as qualitative.

Note: I do not understand the solution in Heard on the Street Question 2.7.

## Answer by siou0107 (score 2, accepted)

https://quant.stackexchange.com/a/50455

To hedge your long call option (which as a delta between 0 and 1), you had to short sell some stock. If the stock price crashes the option you are long of is less in the money (or further out of the money). Therefore you are overhedged by your short stock position and need to buy back some stock.

You can also think in terms of gamma. Your long vanilla option is long gamma (gamma positive): its delta increases as the spot rises and decreases as the spot falls. Since your delta is lower on your long option (less positive on a call or more negative on a put), 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.