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The Delta of a Variance Swap

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  • S'ebastien Bossu
  • Sebastian Gaitan-Escarpeta

Abstract

We define the variance swap delta as the sensitivity of the price of variance to a change in underlying price. We use Carr-Madan spanning formulas to analyze this sensitivity when the implied volatility smile curve may depend on the underlying price. We show that the variance swap total delta is zero for the class of smile curves that are pure functions of (log) moneyness, which goes against the empirical observation that variance is up when the market is down. We propose a simple modification of the smile to correct this issue.

Suggested Citation

  • S'ebastien Bossu & Sebastian Gaitan-Escarpeta, 2026. "The Delta of a Variance Swap," Papers 2609.08959, arXiv.org.
    • Handle: RePEc:arx:papers:2609.08959
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    References listed on IDEAS

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    1. Andrew Papanicolaou, 2018. "Consistent Time-Homogeneous Modeling of SPX and VIX Derivatives," Papers 1812.05859, arXiv.org, revised Mar 2022.
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