Paper: arXiv 2506.08067

Authors: Roberto Baviera, Michele Domenico Massaria

Abstract

We investigate the asymptotic behaviour of the implied volatility in the Bachelier setting, extending the large-strike results established for the Black-Scholes framework. Exploiting the theory of regular variation, we derive explicit expressions for the Bachelier implied volatility in the wings of the smile, directly linking them to the tail decay of the underlying returns’ distribution. Furthermore, we establish a rigorous connection between the analyticity strip of the characteristic function and the asymptotic slope of the volatility smile. Moreover, we show that if the implied variance grows linearly for large absolute moneyness degree, the underlying returns’ distribution must exhibit exponential tail decay, and the corresponding characteristic function is analytic in a horizontal strip of the complex plane. These findings characterizes valid models based solely on the observable asymptotic behaviour of the smile.

Complexity vs Empirical Score

  • Math Complexity: 8.0/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical extension of Benaim-Friz (2009) implied volatility asymptotics to the Bachelier framework using regular variation theory. While mathematically formal and well-written, it is purely theoretical and lacks any empirical implementation or market data validation.

Research Flowchart

  flowchart TD
    A[Research Goal: Asymptotic Behaviour of Bachelier Implied Volatility] --> B[Inputs: Underlying Return Distributions & Characteristic Function]
    B --> C[Methodology: Apply Regular Variation Theory]
    C --> D[Analyze Tail Decay & Analyticity Strip in Complex Plane]
    D --> E[Derive Explicit Wing Formulas for Implied Volatility]
    E --> F[Outcome: Linear Implied Variance Growth Implies Exponential Tail Decay & Analytic CF]

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1 of 115 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.