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Why Delta-Hedged Call Writers Face Gamma Loss and Theta Gains

Article Quant Q&A · Author: barbatos233
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Summary

The document examines the profit profile of a call writer who holds stock to offset the call’s delta. Because the stock position has no gamma while the short call has negative gamma, the combined portfolio retains negative gamma. The question is whether this curvature means the hedge must lose money whenever the stock price moves. The answer confirms the negative-gamma profile but explains that stock-price movements are only one component of the position’s profit and loss.

The offsetting consideration is theta: the option writer benefits from time decay, which the answer characterizes as compensating for the expected loss associated with the option’s curvature under the model. Thus, the hedge is not guaranteed to lose overall merely because its gamma is negative; the result depends on time passage and other factors as well as the stock move. The brief exchange gives no derivation, numerical example, or treatment of transaction costs and changing volatility, so it offers only a qualitative explanation.

Key ideas

  • A delta hedge in stock does not remove the short call’s negative gamma.
  • Negative gamma makes the hedged portfolio’s local profit profile concave with respect to the stock price.
  • The call writer receives a theta benefit as time passes.
  • The exchange describes theta as offsetting the expected gamma-related loss under the model.
  • A negative-gamma profile alone does not determine total profit or loss.

Tags

OptionsDerivatives pricingRisk management
Full text
# Negative-gamma delta hedging (for a call option writer): how will the stock price affect the portfolio profit?


# Negative-gamma delta hedging (for a call option writer): how will the stock price affect the portfolio profit?












Suppose a (European) call option writer is hedging their risk by taking a long position in stocks (holding $\delta_C$ shares). The value of the portfolio is $V(S)=\delta_CS-C$. Then is the gamma of the portfolio $\Gamma_P=-\Gamma_C<0$? If so, is the plot of the profit of the portfolio against stock price a concave down graph below the $x-$axis (the reflection of the yellow curve in the following graph by the $x-$axis)? Then wouldn't it imply that such a hedging will always generate a loss (negative profit) no matter how the stock price changes (if changes in other greek variables are small enough)?

I am asking this question since I am not sure whether or not the results I derived above are correct, and I find this strategy taken by an option writer to be a bit absurd if my conclusion (i.e. the portfolio will never be able to generate gain from change in stock price) is correct, since they still get a chance to gain when stock price decreases if they do not hedge at all. My sincerest gratitude for any help (or references)!

## Answer by dm63 (score 2, accepted)

https://quant.stackexchange.com/a/70660

Yes the p/l profile is the reflection of the yellow line so it appears that the option writer always loses money. However the option writer benefits from time decay, denoted by the Greek letter theta. The theta equals the expected loss from the yellow line, basically. Consult almost any book on options or Google the black scholes equation.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.