The gamma of Butterfly spread shows accelerations and decelerations in changes of its value. We saw in Butterfly: Sensitivity to Delta Variations that the butterfly gains and loses value on both sides of its center. The increase in Butterfly spread price reverses at the level of the sold strike.
Butterfly gamma Г
The best way to fully appreciate the variation of the delta of the butterfly spread is to analyze its rate of delta variation in relation to the movements of the underlying, its gamma (see: The Gamma).
The calculation of the gamma Г of the position is quite simple since it is a matter of adding up the individual gammas of the options, whether they are bought or sold.
An interesting element with the study of the gamma Г of the butterfly spread is that it informs us about the theta of the position, the effect of time on its valuation. Indeed, we know that gamma and theta have values of equivalent size and opposite signs. It is often said that gamma is a "proxy" of the theta.
Graph – gamma Г of the butterfly
The representation of the gamma of butterfly spread with 10 days to expiration, the options all having an implied volatility of 20% and a risk-free interest rate of 0,5% gives (x-axis – strike as % of spot):
The gamma Г changes sign, it is positive at the extremes (outer wings) and negative in the center (at the level of the sold strike). We deduce that the butterfly spread "costs money" (owner of Butterfly spread has to pay theta) at the extremes and actually gains in the center (middle strike). Since middle strikes are sold trader will collect theta if spot is trading close to middle strike.
We can therefore deduce that the butterfly spread behaves like the option whose strike price is closest to the spot level.