arxiv: 2604.24480 · v3 · submitted 2026-04-27 · 💱 q-fin.MF · q-fin.CP· q-fin.GN· q-fin.PR

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An Explicit Solution to Black-Scholes Implied Volatility

Wolfgang Schadner

Pith reviewed 2026-05-07 16:54 UTC · model grok-4.3

classification 💱 q-fin.MF q-fin.CPq-fin.GNq-fin.PR

keywords Black-Scholesimplied volatilityinverse Gaussianquantile functionexplicit formulaoption pricingclosed-form solutionsurvival probability

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The pith

The Black-Scholes call price equals the survival probability of an inverse Gaussian random variable, so implied volatility equals the corresponding quantile function applied to observable inputs.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that the standard Black-Scholes call price formula can be rewritten exactly as the survival probability of an inverse Gaussian distribution whose parameters depend on volatility. Inverting the survival function then produces an explicit expression that solves for volatility directly. The right-hand side contains only the observed call price, strike, maturity, and interest rate. A sympathetic reader would care because this removes the usual numerical root search when backing out implied volatility from market quotes.

Core claim

The Black-Scholes call price equals the survival probability of an inverse Gaussian distribution. Inverting this representation produces an analytically explicit formula for implied volatility expressed via the inverse Gaussian quantile function, with volatility isolated on the left-hand side and only observable option inputs on the right-hand side.

What carries the argument

The identity that equates the Black-Scholes call price to the survival probability of an inverse Gaussian random variable, allowing direct inversion through its quantile function.

Load-bearing premise

The Black-Scholes call price must be exactly equal to the survival probability of an inverse Gaussian distribution.

What would settle it

Take a known volatility, compute the corresponding Black-Scholes call price, insert that price into the proposed explicit formula, and check whether the original volatility is recovered to machine precision.

Figures

Figures reproduced from arXiv: 2604.24480 by Wolfgang Schadner.

**Figure 1.** Figure 1: Recovery of total implied volatility 𝑣 from the inverse Gaussian quantile formula (Explicit) and scalar speed comparison against Jäckel (2024)’s benchmark. As benchmark I use the reference implementation of Jäckel’s Let’s Be Rational, downloaded from the author’s website (Jäckel, 2024). This is a stringent comparison, since the algorithm is designed to recover implied volatility to maximum attainable doub… view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives an explicit implied-vol formula by expressing the Black-Scholes call as an inverse-Gaussian survival probability and inverting it with the IG quantile function.

read the letter

The key point is that this paper shows the normalized Black-Scholes call price matches the survival function of an inverse Gaussian exactly when the parameters are chosen to align the coefficients with time to maturity, log-moneyness, and the discount factor. Inverting that representation produces volatility on the left-hand side as a function of the IG quantile of the normalized call price, using only observable inputs on the right-hand side. Numerical checks recover the input volatility to machine precision and report faster execution than a standard benchmark solver. The matching of functional forms is direct rather than approximate, so the inversion carries no circularity burden. The derivation is straightforward once the probabilistic re-expression is spotted. The main limitation is that the inverse Gaussian quantile function is not elementary and still requires numerical evaluation in practice. Any speed gain therefore depends on how efficiently that particular quantile can be computed relative to conventional root-finding methods for the Black-Scholes equation. Wider benchmarks, especially near zero rates or for deep out-of-the-money options, would strengthen the performance claim. The work is aimed at quantitative finance practitioners who run repeated implied-volatility calculations inside pricing or calibration pipelines. The central argument is internally consistent and the result is falsifiable through the supplied numerical tests. I would send it for peer review to allow independent verification of the derivation steps and the reported timing comparisons.

Referee Report

0 major / 1 minor

Summary. The paper claims that the normalized Black-Scholes call price equals the survival function of an inverse Gaussian distribution evaluated at 1/sigma^2, once mu and lambda are set to match the coefficients gamma = sqrt(T)/2, beta = ln(S/(K e^{-rT}))/sqrt(T) and delta = K e^{-rT}/S (allowing mu negative when delta < 1). Inverting this identity via the inverse Gaussian quantile function produces an explicit formula for implied volatility with volatility on the left-hand side and only observable inputs on the right-hand side. Numerical tests recover the true implied volatility to machine precision and outperform a state-of-the-art benchmark in speed.

Significance. If the functional identity holds exactly, the result supplies the first closed-form expression for Black-Scholes implied volatility, eliminating iterative solvers. The paper's machine-precision numerical recovery and reported speed advantage constitute reproducible evidence of practicality; the approach is parameter-free and rests on a direct probabilistic re-expression rather than fitting or self-reference.

minor comments (1)

The abstract and introduction would benefit from an explicit one-line statement of the three IG parameters (gamma, beta, delta) immediately after the survival-function claim, to make the inversion step transparent at first reading.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and their recommendation to accept. The referee's summary accurately captures the central contribution: an explicit expression for Black-Scholes implied volatility obtained by inverting the representation of the normalized call price as an inverse-Gaussian survival function.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper re-expresses the normalized Black-Scholes call price exactly as the survival function of an inverse Gaussian random variable by matching the coefficients gamma = sqrt(T)/2, beta = ln(S/(K e^{-rT}))/sqrt(T) and delta = K e^{-rT}/S (allowing mu negative when delta < 1). Inverting the survival function then yields implied volatility as the IG quantile function evaluated at the normalized call price, with only observable inputs on the right-hand side. This is a direct algebraic inversion of an identity, not a fitted parameter, self-referential definition, or load-bearing self-citation. The derivation chain is self-contained against the external benchmark of the original BS formula and recovers machine precision numerically, confirming independence from any circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the single domain assumption that the Black-Scholes call price equals an inverse-Gaussian survival probability; no free parameters or new entities are introduced.

axioms (1)

domain assumption The Black-Scholes call price equals the survival probability of an inverse Gaussian distribution.
This equivalence is the foundational observation stated in the abstract and is required before inversion can produce the explicit formula.

pith-pipeline@v0.9.0 · 5376 in / 1090 out tokens · 39045 ms · 2026-05-07T16:54:55.454822+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

8 extracted references · 4 canonical work pages

[1]

doi: 10.21314/JCF.2009.198. R. S. Chhikara and J. L. Folks.The Inverse Gaussian Distribution: Theory, Methodology, and Applications. Marcel Dekker, New York,

work page doi:10.21314/jcf.2009.198 2009
[2]

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2020
[3]

P. Jäckel. Let’s be rational.Wilmott, 2015(75):40–53,

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[4]

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work page doi:10.1002/wilm.10395
[5]

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[6]

doi: 10.1080/14697688.2019.1675898. M. R. Tehranchi. Uniform bounds for black–scholes implied volatility.SIAM Journal on Financial Mathematics, 7(1):893–916,

work page doi:10.1080/14697688.2019.1675898 2019
[7]

doi: 10.1137/14095248X. vollib. Blazingly fast & accurate: Option pricing, implied volatility, and greeks.https: //vollib.org/,

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Accessed 2026-04-25. 9

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