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When Vega Neutrality Implies Gamma Neutrality for Options

Article Quant Q&A · Author: Tidy Star
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Summary

The document examines whether a portfolio that is neutral to vega is also neutral to gamma and theta. One response gives a specific result for portfolios of calls and puts sharing the same maturity: under the Black–Scholes framework, gamma neutrality and vega neutrality are equivalent because the ratio of an option’s gamma to its vega does not depend on strike. This makes the result applicable to same-expiry combinations, rather than arbitrary option portfolios.

A second response cautions that vega and gamma need not track each other in general, citing a very short-dated at-the-money option as a case with very high gamma and very low vega. The discussion therefore limits the equivalence claim to stated assumptions; it does not establish a general relationship for mixed maturities, other pricing models, or theta. In particular, the title’s question about theta neutrality is not resolved by the responses.

Key ideas

  • For same-maturity calls and puts in Black–Scholes, gamma neutrality is equivalent to vega neutrality.
  • The equivalence follows because the gamma-to-vega ratio is independent of strike when maturity is shared.
  • A short-dated at-the-money option can have very high gamma and very low vega.
  • The discussion does not establish that vega neutrality implies theta neutrality.

Tags

OptionsDerivatives pricingRisk managementVolatility
Full text
# is there an analytical proof that vega-neutral also provides (gamma & theta) neutral?


# is there an analytical proof that vega-neutral also provides (gamma & theta) neutral?












I've read an answer here that say if your security has vega, then it has gamma and theta.

is there an analytical proof that vega-neutral also provides (gamma & theta) neutral?

## Answer by Mark Joshi (score 5)

https://quant.stackexchange.com/a/20696

if you have a portfolio of calls and puts with the same maturity then your portfolio is gamma neutral if and only if it is vega neutral.

The reasons is that the BS gamma divided by the BS vega is a function of $S$ and $T$ that does not vary with $K.$ So if you construct a linear combination that has zero gamma then the vega is zero too, and vice versa.

## Answer by mbison (score 1)

https://quant.stackexchange.com/a/20694

I don't think your hypothesis is correct. If you have a very short dated ATM option, then your option will have close to infinite gamma but close to 0 vega. So this short dated ATM option is vega neutral but definitely not gamma neutral.

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.