Vega of a Delta-Neutral Call Position with Unequal Strikes
Summary
The document considers a position that buys one at-the-money call with delta 0.5 and sells two calls with delta 0.25 on the same underlying. It offers an intuitive argument for the position’s vega sign by first considering its payoff just before expiry: the payoff profile has a maximum near the strike of the short calls, where the position’s delta is zero. As time to expiry increases, the profile is described as smoothing while retaining a maximum at the zero-delta point.
At the current underlying level, the position is presented as having zero delta at that maximum, and therefore negative gamma and negative vega. This is a qualitative explanation based on the described payoff shape, not a model-independent proof. Its applicability depends on the option setup and assumptions; the document does not provide a derivation across models or explore how dividends, rates, or other contract features might change the result.
Key ideas
- The portfolio buys one 0.5-delta call and sells two 0.25-delta calls.
- Its near-expiry payoff is described as reaching a maximum near the short calls’ strike.
- The explanation links zero delta at the payoff maximum with negative gamma and vega.
- The argument is intuitive and does not establish a model-independent result.
Tags
Full text
# Is the vega of a portfolio of a long 0.5 delta and short two 0.25 delta calls positive or negative?
# 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.
## Answer by Mats Lind (score 1)
https://quant.stackexchange.com/a/49688
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.