# Is the vega of a portfolio of a long 0.5 delta and short two 0.25 delta calls positive or negative?

More specifically what I am trying to find out is whether the following relationship is always true or not. Same underlying for the calls, assume the most simplistic assumptions (interest rate = dividends yield, time to maturity 1 year, etc.)

Vega of Call$$_{0.25\Delta}$$ > $$\frac{1}{2}$$ * Vega of Call$$_{0.5\Delta}$$

Should be simple but could not come up with a model independent intiutive explanation.

• When you say model independent, does it mean delta and vega are model independent? Could you explain what model independent definition of delta of 0.25 do you have in mind please? – Magic is in the chain Nov 12 '19 at 22:34
• I was concerned that was not clear. What I meant was instead of comparing the BS formulas, I am looking for an intuitive explanation if that makes sense. From a no arbitrage point of view maybe. – mebiles Nov 12 '19 at 23:15

## 1 Answer Construct the delta-neutral position in the question: buy the 0.5 and sell two of the delta 0.25. Then consider the position's payout as a function of the underlying just before expiry. Its maximum then lies close to the strike of the delta 0.25 where the position's delta then will be zero.

Now, as time to expiry increases (we go backward in time!) the value function smears out as illustrated in the diagram (red to purple). Still though, the point were the delta is zero will be at the value function's maximum. However handwavy this is, it goes well with intuition. Going further backwards in time we will hit the current point where the long call is delta 0.5 and the two short calls are delta 0.25 where the underlying is at the strike of the delta 0.5.

Here we have delta = 0 still at the max and hence gamma and vega negative.