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How Calendar Spreads Can Separate Gamma and Vega Exposure

Article Quant Q&A · Author: Trajan
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Summary

Vega and gamma often have the same sign for a single option under standard assumptions, but a calendar spread can produce opposite exposures. The example pairs a long near-dated at-the-money option with a short longer-dated at-the-money option, yielding long gamma and short vega. The intuition is that at-the-money option vega tends to decline as expiration approaches, while gamma tends to increase.

For European options in the Black–Scholes model, the document gives formulas showing that vega divided by gamma is positive, so the two Greeks share a sign for an individual option under that model. The calendar-spread example illustrates how combining maturities can alter the portfolio’s exposures. These relationships depend on the option structure and modeling assumptions; the discussion does not cover other strikes, market conditions, or hedging effects.

Key ideas

  • For a European option in Black–Scholes, vega and gamma have the same sign.
  • At-the-money option vega generally falls as expiration approaches, while gamma rises.
  • A long near-dated and short longer-dated at-the-money calendar spread can be long gamma and short vega.
  • Portfolio Greeks can have opposite signs even when the component option Greeks share a sign.

Tags

OptionsVolatilityDerivatives pricing
Full text
# Vega and Gamma signs


# Vega and Gamma signs












Do vega and gamma always have the same sign (ie both positive or both negative)? Under what circumstances can they have opposite signs?

## Answer by fni (score 4, accepted)

https://quant.stackexchange.com/a/30245

Usually vega and gamma go in the same direction, but you can have opposite exposure in a calendar spread.

For an ATM option, vega decreases closer to maturity while gamma increases. If you implement the following:

-long a 1 month ATM option

-short a 2 months ATM option

you should be long gamma and short vega.

## Answer by user16651 (score 3)

https://quant.stackexchange.com/a/30244

In the Black Scholes model, for an European option, we have $$\text{Vega}=Ke^{-r\tau}\phi(d_2)\sqrt{\tau}$$ and $$\Gamma=Ke^{-r\tau}\phi(d_2)\frac{1}{S^2\sigma\sqrt{\tau}}$$ thus $$\frac{\text{Vega}}{\Gamma}=S^2\sigma{\tau}>0$$

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.