Paper: arXiv 2604.24480

Authors: Wolfgang Schadner

Abstract

This paper observes that the Black–Scholes call price can be written as the survival probability of an inverse Gaussian distribution, equivalently as a probability in variance space. Inverting this representation yields an analytically explicit formula for implied volatility in terms of the corresponding inverse Gaussian quantile function, with volatility on the left-hand side and only observable option inputs on the right-hand side. Numerical tests recover implied volatility to machine precision and, in a controlled setting, show the formula to be faster than a state-of-the-art benchmark.

Complexity vs Empirical Score

  • Math Complexity: 7.5/10
  • Empirical Rigor: 6.5/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides a rigorous mathematical derivation linking Black-Scholes inversion to Inverse Gaussian quantile functions and backs this with a high-performance empirical benchmark against industry-standard numerical methods.

Research Flowchart

  flowchart TD
    A[Research Goal: Explicit Solution for Implied Volatility] --> B[Methodology: Link Black-Scholes to Inverse Gaussian]
    B --> C[Inputs: Option Price, Strike, Spot, Time, Rate]
    C --> D[Process: Invert Survival Probability via IG Quantile Function]
    D --> E[Outcome: Analytically Explicit IV Formula]
    E --> F[Validation: Machine Precision & Faster Benchmarking]

Cited References (30)

  1. Infinite divisibility of the hyperbolic and generalized inverse Gaussian distributions (1977)
  2. On the non-holonomic character of logarithms, powers and the $n$th prime function (2005)
  3. Ordinary Differential Equations (1926)
  4. The Calculation of Implied Variances from the Black–Scholes Model: A Note (1982) — link
  5. A Simple Formula to Compute the Implied Standard Deviation (1988) — link
  6. A Note on a Simple, Accurate Formula to Compute Implied Standard Deviations (1996) — link
  7. An Explicit Implied Volatility Formula (2017) — link
  8. Tighter Bounds for Implied Volatility (2017) — link
and 22 more
  1. An Exact and Explicit Implied Volatility Inversion Formula (2018) — link
  2. Generalized incomplete gamma functions with applications (1994) — link
  3. Asymptotics and closed form of a generalized incomplete gamma function (1996) — link
  4. Uniform Bounds for Black–Scholes Implied Volatility (2016) — link
  5. A Closed-Form Model-Free Implied Volatility Formula through Delta Families (2021) — link
  6. Let’s Be Rational (2015) — link
  7. Let’s Be Rational : Reference Implementation (2024)
  8. Operator Deep Smoothing for Implied Volatility (2024) — link
  9. The pricing of options and corporate liabilities (1973)
  10. Generalized incomplete gamma functions with applications (1994)
  11. Saddlepoint Methods for Option Pricing (2009) — link
  12. The Volatility Surface: A Practitioner’s Guide (2006)
  13. The Moment Formula for Implied Volatility at Extreme Strikes (2004) — link
  14. A property of the generalized inverse Gaussian distribution with some applications (1983)
  15. Asymptotics of Implied Volatility to Arbitrary Order (2014) — link
  16. The Inverse Gaussian Distribution: Theory, Methodology, and Applications (1989)
  17. Arbitrage-free SVI volatility surfaces (2014)
  18. Can there be an explicit formula for implied volatility? (2012)
  19. Statistical Properties of the Generalized Inverse Gaussian Distribution (1982)
  20. Blazingly Fast & Accurate: Option Pricing, Implied Volatility, and Greeks (2023)
  21. A PDE Method for Estimation of Implied Volatility (2020) — link
  22. A better approximation formula for the Black–Scholes implied volatility (2008)

0 of 30 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.