# 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?

• For a long option position they are both positive and for a short option position they are both negative. ("option" = vanilla call or vanilla put). So yes, they always have the same sign. Sep 21, 2016 at 19:59
• In Black Scholes Model ?
– user16651
Sep 21, 2016 at 20:04

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.

• By the way that's called a "short ATM Calendar Spread". Sep 23, 2016 at 16:30

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 $$\frac{\text{Vega}}{\Gamma}=S^2\sigma{\tau}>0$$

• Note that this is true for single European vanilla options. It no longer holds for portfolios of options. Even when the volatility surface is flat, you can build a portfolio of options with different maturities to obtain gamma and vega positions of opposite signs. Similarly, when the volatility smile of a maturity is not flat, you can achieve the same with a position in different strikes of that maturity. Sep 21, 2016 at 21:11