How Vega Changes in a Vega-Neutral Calendar Call Spread

Summary

The document poses a risk-sensitivity question about a calendar call position: long one long-dated at-the-money call and short a quantity of short-dated at-the-money calls chosen to make initial portfolio vega zero. It expresses the later portfolio vega as the long option’s current vega minus the original hedge ratio times the short option’s current vega. The key point is that initial neutrality does not by itself determine later vega; both option vegas can change as time passes and market conditions evolve.

The author notes that the long-dated option initially has greater vega, so the short position’s size exceeds one contract, but does not resolve the sign of the later exposure. The question specifies that spot is again at the strike before the short option expires, yet supplies no pricing assumptions or calculations that establish whether vega is positive, negative, or zero. This is a useful setup for studying changing Greeks, but it is not a worked solution or a complete trading rule.

Key ideas

• Initial vega neutrality sets the short-call quantity using the two options’ vegas at inception.
• The hedge ratio can exceed one when the long-dated option has greater initial vega.
• Later portfolio vega depends on both options’ vegas at that later time.
• Returning spot to the strike alone is not shown to determine the sign of portfolio vega.

Tags

# Vega for long long-term ATM call and short short-term ATM call


# Vega for long long-term ATM call and short short-term ATM call












> 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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.