A lower bound for the volatility swap in the lognormal SABR model

Paper: arXiv 2306.14602

Authors: E. Alòs, F. Rolloos, K. Shiraya

Abstract

In the short time to maturity limit it is proved that for the conditionally lognormal SABR model the zero vanna implied volatility is a lower bound for the volatility swap strike. The result is valid for all values of the correlation parameter and is a sharper lower bound than the at-the-money implied volatility for correlation less than or equal to zero.

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 2.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical derivation of a new lower bound for volatility swap strikes under the lognormal SABR model in the short-maturity limit. While mathematically strong and analytically clear, it is a purely theoretical result with limited empirical validation or real-world backtesting data.

Research Flowchart

  flowchart TD
    A["Research Question: Find a sharp lower bound<br/>for volatility swap strike in SABR model"] --> B["Inputs: Conditionally Lognormal SABR Model<br/>& Short Time-to-Maturity Limit"]
    B --> C["Methodology: Perform Asymptotic Analysis<br/>for Zero Vanna Implied Volatility"]
    C --> D["Computational Step: Evaluate bounds<br/>across all correlation parameters (ρ)"]
    D --> E["Key Finding 1: Zero vanna implied vol<br/>is a proven lower bound for all ρ"]
    D --> F["Key Finding 2: Bound is sharper than ATM<br/>implied volatility for ρ ≤ 0"]