# is there an analytical proof that vega-neutral also provides (gamma & theta) neutral?

I've read an answer here that say if your security has vega, then it has gamma and theta.

is there an analytical proof that vega-neutral also provides (gamma & theta) neutral?

if you have a portfolio of calls and puts with the same maturity then your portfolio is gamma neutral if and only if it is vega neutral.

The reasons is that the BS gamma divided by the BS vega is a function of $S$ and $T$ that does not vary with $K.$ So if you construct a linear combination that has zero gamma then the vega is zero too, and vice versa.

• Do you have the analytical explanation (formula) for vega neutral is also gamma neutral? – Tidy Star Sep 10 '15 at 11:32
• P.S: I already know the analytical explanation for gamma neutral also give theta neutral. I get that analytical explanation from the Timothy falcon book (basic black scholes). – Tidy Star Sep 10 '15 at 11:34
• @TidyStar: For the Black-Scholes model, look up the expressions for vega and gamma and see how similar they are and you will understand. – Hans Apr 28 '16 at 1:39

I don't think your hypothesis is correct. If you have a very short dated ATM option, then your option will have close to infinite gamma but close to 0 vega. So this short dated ATM option is vega neutral but definitely not gamma neutral.

• I think you're wrong. It's not about gamma value vs vega value. I'm talking about vega-neutral position is also provide gamma-neutral (and I'm asking the analytical/scientific explanation behind it). – Tidy Star Sep 10 '15 at 11:26
• What is the difference between a position with vega value of 0 and being vega neutral? – mbison Sep 10 '15 at 17:11
• I think there's only one valid implied volatility at the same time for an option (with same maturity) unless it has volatility skew/smile which violate Black-Scholes assumption about the market being in the normal distribution. If this happens, the BS model calculation (including the greeks) will also be invalid. – Tidy Star Sep 10 '15 at 21:09
• I don't know, may be you calculate it wrong. I get the same result if implied volatility is same. The ratio (division) between vega value with different strike are exactly same with the ratio of gamma value with different strike. – Tidy Star Sep 10 '15 at 21:28
• Yes 2 different maturities, therefore 2 different options. Agreed so far. Not sure why you are saying you can't hedge the greeks of one option with the greeks of another option. Especially in a black scholes world where the entire vol surface is flat (say 20% for example). Why would there be any conversion of vol be needed? You can add the vega of 2 different options (with same underlying). Maybe we are talking about different things? What I mean with vega is the change in price for a parallel shift of entire flat surface. – mbison Sep 10 '15 at 22:15