# Long Gamma vs Vega

What is the difference between being long gamma and being long Vega? I understand that gamma is the vol of delta and that vega is the vol of the underlying. However, I have also found that being long gamma and long vega basically means being long options. In that case, what is the difference between the two?

Cheers.

• Gamma increases at T->0 and S->K, Vega increases as T becomes large. So they vary with maturity in different ways. – Alex C Feb 21 '18 at 6:09
• So the difference between the two is a function of time? – Noodle22 Feb 21 '18 at 6:50
• I am not saying that. I only wanted to "prove" to you that they are not the same thing. – Alex C Feb 21 '18 at 16:29

Long gamma is being long realized volatility. Long vega is being long implied volatility. Long gamma positions benefit when realized volatility goes up or the actual underlying has volatility. Long vega positions benefit when the price of volatility goes up.

Being long plain vanilla options, one is long both gamma and long vega. However, this is not so if one starts to combine options in strategies. One can construct positions where one is long gamma and short vega.

A simple example would be a simple calendar spread--if one is long an at-the-money call with short maturity, one is long gamma and long vega. If one shorts an at-the-money longer dated maturity call on the same underlying, one is short gamma and short vega. However, the short longer dated call will be less long gamma than the shorter dated one; and short more vega than the shorter dated one. The combined position will be long gamma and short vega. The position will benefit if realized volatility goes up before the shorter dated call expires, and if implied volatility goes down.

• Okay, that makes a lot of sense! Thanks for the example, that really made it easier to understand. – Noodle22 Feb 21 '18 at 22:01

Vega (denoted by $\nu$ in what follows) is the first order sensitivity of the option price with respect to volatility $\sigma$. Gamma (denoted by $\Gamma$ in what follows), is the second order sensitivity of the option price with respect to the underlying spot price $S$.

Because for a semi-martingale $(S_t)_{t \geq 0}$ there is a direct link between the variance of the random variable $S_t$ for any fixed $t$ and its quadratic variation over $[0,t]$, it is only logical that there exists a link between Vega and Gamma.

Under BS assumptions, one can show that for an option evaluated at $t$ with time to maturity $\tau = T-t$ $$\nu(\tau) = \Gamma(\tau) \, \sigma S_t^2 \tau$$ see Appendix A of Chapter 5 of Bergomi's book "Stochastic Volatility Modeling" for a demonstartion and this Wiki page to see that it indeed holds under BS.

• Thank you so much, this was a very comprehensive explanation. – Noodle22 Feb 21 '18 at 22:00