The Delta of a Variance Swap

Paper: arXiv 2609.08959

Authors: Sébastien 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.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 5.0/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: This paper presents a rigorous mathematical analysis of variance swap delta, identifying a counter-intuitive zero delta for common smile models. It proposes a novel correction and includes some empirical validation, balancing strong theoretical contributions with practical relevance.

Research Flowchart

  flowchart TD
    A[Research Goal: Define and Analyze Variance Swap Delta] --> B{Methodology: Carr-Madan Spanning Formulas};
    B --> C[Inputs: Implied Volatility Smile Curve (dependence on underlying price)];
    C --> D{Computational Process: Sensitivity Analysis of Variance Price to Underlying};
    D --> E[Key Finding 1: Total Delta is Zero for Pure Moneyness Smiles];
    E --> F[Key Finding 2: Propose Smile Modification to Address Empirical Observation];