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Why Long Options Have Positive Gamma in Black–Scholes

Article Quant Q&A · Author: Guess601
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Summary

The note explains the usual sign of option gamma: under Black–Scholes assumptions, a long call or put has positive gamma, while a short position has negative gamma. Gamma describes how an option’s delta changes as the underlying price moves. For a long put, for example, delta becomes less negative as the underlying rises, which corresponds to positive gamma.

The explanation depends on holding other factors constant in the Black–Scholes framework. The note cautions that real markets can behave differently when implied volatility changes with the underlying price. That relationship introduces mixed-derivative effects, so the simple Black–Scholes gamma alone may not describe the position’s observed sensitivity. No empirical study or quantitative evidence is provided; the answer is a conceptual explanation with a model caveat.

Key ideas

  • In Black–Scholes with other inputs held constant, long calls and puts have positive gamma.
  • A long put’s delta becomes less negative as the underlying price rises.
  • Short option positions have gamma with the opposite sign to equivalent long positions.
  • Spot-dependent implied volatility can add mixed-derivative effects beyond plain Black–Scholes gamma.

Tags

OptionsDerivatives pricingVolatility
Full text
# Long positions (call or put) have positive gamma, and short positions (call or put) have negative gamma


# Long positions (call or put) have positive gamma, and short positions (call or put) have negative gamma












I was reading a lot about the idea that long positions (call or put) have positive gamma, and short positions (call or put) have negative gamma. But I couldn't understand why. In "Bunds and Bund Futures" book, the author argues as: "The reason for this is that as the futures price rises the delta of a long call (or long put) option will become more positive (or less negative). For short call (or short put) options, the deltas will become more negative (or less positive) as the futures price increases." But why!? Why "as the futures price rises the delta for a long put option will become less negative"? I would really appreciate some feedback about this.

## Answer by CABLE (score 1)

https://quant.stackexchange.com/a/55097

It works in the Black-Scholes world assuming other factors are constant. For the Black-Scholes, you can check online the formula for Gamma. It is positive obviously if you are long. In reality this may not hold. A simple example would be the case when the implied vol is correlated with spot price. In this case, the "Gamma" would not be as simple as the plain Black-Scholes since there will be "mixed-derivative" terms when calculating second derivatives.

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.