Do vega and gamma always have the same sign (ie both positive or both negative)? Under what circumstances can they have opposite signs?
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$\begingroup$ 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. $\endgroup$– nbbo2Commented Sep 21, 2016 at 19:59
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1$\begingroup$ In Black Scholes Model ? $\endgroup$– user16651Commented Sep 21, 2016 at 20:04
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2 Answers
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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.
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1$\begingroup$ By the way that's called a "short ATM Calendar Spread". $\endgroup$– nbbo2Commented Sep 23, 2016 at 16:30
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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 $$\frac{\text{Vega}}{\Gamma}=S^2\sigma{\tau}>0$$
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3$\begingroup$ 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. $\endgroup$ Commented Sep 21, 2016 at 21:11