The Butterfly Spread: vega

By roma, 11 January, 2021

The butterfly spread reacts differently to changes in implied volatility, depending on the price level on the volatility smile graph.

It is very common to read the description of a butterfly spread as a strategy neutral to variations in volatility, or as vega neutral strategy, by the simplistic reasoning that there are as many options bought as options sold.

We will see that things are much more interesting. The butterfly spread can quickly be seen as an excellent technique for managing implied volatility, without the need for readjustment.

Vega υ of the butterfly spread

In order to properly observe the variation of the butterfly's price according to changes in volatility, one can choose to assess its rate of variation in relation to the movements in implied volatility, its vega (read: Option’s Vega).

Once again, like delta and gamma, the calculation of the position's vega is simple since it is a matter of adding up the individual vegas of the options, paying attention to whether they are bought or sold.

Graphical representation of vega

The representation of the vega υ of the butterfly spread 48/50/52 with expiration in 5 days. All options have an implied volatility of 20% and a risk-free interest rate of 2%.
The butterfly spread vega 5 days

Butterfly spread 48/50/52 with expiration in 1 month.
The butterfly spread vega 30 days

Remarks

The vega, like the gamma (read Butterfly spread: Gamma), changes sign to positive at the extremes. Vega becomes negative in the center, at the level of the sold strike.

We thus obtain a short vega position in the center (in the middle strike), middle strike options earn money in case of a decrease of the implied volatility when the underlying is at the level of the sold strike (in our example 50). Positive vega υ at the extremes (options at the wings) generates profit from an increase of the implied volatility of outer strikes.

Once again, the butterfly spread behaves in relation to changes in volatility like the option whose strike price is closest to the spot level.