Mercurio, Delta, Gamma, Vega-neutral

semit · March 3rd, 2007, 6:55 pm

In his paper "Consistent Pricing of FX Options" (http://www.fabiomercurio.it/consistentfxsmile.pdf)Mercurio writes on p.4:"Assuming a delta-hedged position and given that, in the BS world, portfolios of plain-vanilla options (with the same maturity) that are Vega neutral are also Gamma neutral,..."Is it actually true?If yes, how can it be proved?

richbrad · March 5th, 2007, 12:42 pm

If you assume a BS WorldThen if you are vega neutral, then you are gamma neutral by definition when you have plain vanilla options. This is very simple to see if you take the BS forms of a call option, with its vega and gamma.BS Vega and Gamma:http://en.wikipedia.org/wiki/Greeks_%28 ... reeksEvery input in the formula is non-zero (before expiry) except for phi(d_1). Hence if you are gamma neutral => phi(d_1) = 0 => vega neutral. The converse is also true.

semit · March 5th, 2007, 7:19 pm

Thank you for the post.Yes, that was also my view of it. But some authors -- e.g. Hull "Option, Futures and other Derivatives" state, it does not always hold, that a vega-neutral portfolio is also gamma-neutral.The only way for explaining it I see looking at the formula for Vega and Gamma:Vega=S*sqrt(tau)*phi(d1) is zero iff phi(d1)=0, but Gamma=(S*sigma*sqrt(tau))^(-1)*phi(d1) could equal 0 if:1. phi(d1)=0 or2.S is extremely big e.g in the case of index options (since phi(d1) is bounded from above by 1)

richbrad · March 6th, 2007, 9:40 am

It is because the BS assumptions are not true!! This is the "real world"