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You are long a long-term ATM call and short a short-term ATM call. The ratio is adjusted to make the total vega zero. If before expiry of the short-term option, spot is again at the strike price. Is vega positive, negative or zero?

I tried a reasoning without entering into formulas:

Let's denote $v_{LT}(t) = \frac{\partial c_{LT}(t)}{\partial \sigma}$ the vega of the long-term call at time $t$, $v_{ST}(t) = \frac{\partial c_{ST}(t)}{\partial \sigma}$ the vega of the short-term call at time $t$.

At $t=0$, the portfolio is vega-neutral, we therefore have shorted $\frac{v_{LT}(0)}{v_{ST}(0)}$ of the short-term call, the vega of the portfolio is:

\begin{align} v_{LT}(0) - \frac{v_{LT}(0)}{v_{ST}(0)} v_{ST}(0) = 0 \end{align}

Also, we know that a long-term option is more sensitive to change in volatility, therefore: $v_{LT}(0) > v_{ST}(0)$, which means $\alpha= \frac{v_{LT}(0)}{v_{ST}(0)} > 1$.

Let $t=t^*$ the time when spot is again at strike price. The vega of the portfolio is:

\begin{align} v_{LT}(t^*) - \frac{v_{LT}(0)}{v_{ST}(0)}v_{ST}(t^*) \end{align}

However, here I still don't know what the vega of the portfolio is. Anyone has an idea how to get the vega of the portfolio at $t=t^*$? I know I didn't use the ATM property but I don't know how to use it here.

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    $\begingroup$ hint : the Vega of an ATM option, all else being equal , is proportional to square root of expiration. $\endgroup$ – dm63 Apr 4 at 10:21

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