Schadner's Black-Scholes inverter earns a place as an independent check: its relative error never exceeds 2.22e-15 across 42 synthetic cases. It expresses implied volatility as an integral over (0,1), with a contour fixed before the price is known. No root finding is required. The accuracy makes it a usable second inverter, though the paper gives no evidence that it is faster. Nor does it tell us whether a quote is mispriced.
What does the fixed contour buy?
The idea behind it goes back to the Delves and Lyness identity from 1967. A quantile Q(p) is a zero of P(z) minus p; for an analytic function, a contour integral returns a zero enclosed by the contour. Constructing that contour is the difficult part. It must enclose the desired zero and exclude every other one, without prior knowledge of the zero's location.
Schadner uses the density to do this. Above each real point, he chooses a height h(x) that keeps the real part of the complex density positive throughout the vertical segment. Integration upward then gives the complex cumulative distribution function (CDF) a strictly positive imaginary part above the axis. It therefore cannot equal a real probability there. Conjugation gives the corresponding result below the axis. The ribbon bounded by κ(x) = x + ih(x) and its mirror image contains only the real quantile, for every p simultaneously.
When the endpoint term vanishes, integration by parts gives a compact expression. Q(p) equals (1/π) times the imaginary part of the integral of κ'(x) Log[(P(κ(x)) minus p)/(1 minus p)]. The paper supplies elementary heights for generalized beta, generalized gamma, GIG, generalized hyperbolic and GEV laws, along with normal, Student and variance-gamma limits. For the normal case, h(x) = 1/(2(1+x²)) keeps the density's phase at or below 1/4.
An existence theorem extends the construction to any strictly positive real-analytic density after changing coordinates to (0,1). As the paper explains, existence is what the theorem guarantees; obtaining an elementary height requires further estimates for the particular density. The quantitative result, Proposition B.3, requires explicit complex-disk bounds derived from the density. The paper also notes that the proposition cannot infer those bounds from finitely many real-axis evaluations.
For the finance application, Schadner starts from his earlier result: the normalized call price equals 1 minus an inverse-Gaussian CDF evaluated at 4/v². This makes σ²T a quantile of a GIG law with parameters (2k, 1/2, k/2). Rescaling the GIG height yields κ_k(x) = x + i x²/(k² + x + x²/4). A t/(1 minus t) substitution, followed by a square root, places the integral on (0,1).
The inputs are log-moneyness k > 0, the forward-normalized price, and T. Along the contour, the integrand evaluates the Black-Scholes call at complex volatility and compares it with the observed price. The earlier paper established the connection to an inverse-Gaussian quantile. This paper supplies an integral using only forward functions.
Adaptive quadrature still has work to do
The paper uses 'explicit' to mean an exact integral of forward functions along a specified contour. The result meets that definition. Computing it still requires adaptive quadrature: the test used SciPy with absolute and relative tolerances of 2e-12. A t = s² substitution was also needed to remove a square-root singularity at zero.
There is case handling too. Negative k is treated through put-call parity, while k = 0 has a separate normal-CDF expression. Near the left endpoint, the logarithm's imaginary part can underflow. The paper says the code must preserve the upper-half-plane limit with argument +π. It also uses algebraically equivalent expressions for the logarithm to avoid cancellation when the IG CDF approaches 0 or 1.
The quantile calculations show why cancellation deserves attention. In 24-digit runs, beta and gamma at p = 0.01 cancel order-one contributions down to 10^-7 or 10^-6. Going from order one to 1e-7 loses about seven digits. For its Black-Scholes calculation, the paper uses algebraically equivalent logarithm expressions to avoid cancellation when the IG CDF is near zero or one.
Fifteen digits, without a timing comparison
The test grid combines k in {0.01, 0.05, 0.1, 0.25, 0.5, 1, 2} with total vol in {0.1, 0.2, 0.4, 0.8, 1.5, 3}. Maturities of 0.25, 1 and 4 are assigned cyclically. Prices range from 3.72e-91 to 0.866. The largest absolute vol error is 1.33e-15, and repricing the recovered vols leaves a maximum price discrepancy of 2.22e-16.
Another 68 unit-maturity evaluations remain within 2.22e-15 relative error. Three 50-digit mpmath checks agree to 1.67e-51. Across the general quantile tests, all 96 cases spanning 12 laws fall below 10^-17. Student t with a = 1/2 at p = 0.01 is the weakest case, recording 8e-18. There, the tangent reconstruction magnifies error around quantiles near ±10^3.
Schadner states the limitations plainly. The discrepancies, he writes, "do not constitute uniform quadrature error bounds". He describes quantifying quadrature error near extreme levels as "a useful next step".
We did not find any timing.
At a tolerance of 2e-12, we expect an adaptive rule to evaluate the complex price at many contour points for each inversion. The paper claims no speed advantage. Without timing results, the evidence says nothing either way about the integral's practicality as a replacement for a root finder. And all 42 inputs are exact model prices. None is a noisy quote.
On an ETF options desk
An inverter agreeing with yours to fifteen digits gives you the same surface. Relative value depends on the distance between quoted vol and a fair vol estimate. Fifteen digits of inversion accuracy tell us neither how large that distance is nor whether it survives spreads and delta-hedging costs. The paper uses no market data, provides no evidence on those questions, and makes no claim to answer them.
Independence is its useful contribution. The contour shares no code with seed-and-refine solvers, making it a credible check in the deep wings, where the tests reach prices of 3.72e-91. We are evaluating it for that purpose on end-of-day ETF options. We have no results yet.
A benchmark placing it within a small multiple of an iterative solver's time per option would change the conclusion here.