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Asymptotically-informed neural networks for Black-Scholes implied volatility computation

Samira Amiriyan, Youness Boutaib

arXiv:2609.05491v1q-fin.CPcs.LGstat.ML

TL;DR

Implied-volatility inversion is difficult in asymptotic regimes, where price perturbations can strongly affect the result. The paper builds neural architectures that learn partitions of the price–log-moneyness domain and combine local approximations with trainable gates. These models outperform standard feed-forward networks across broad parameter domains and provide initial guesses that reach near machine precision after two Householder iterations.

  • Problem

    No explicit closed-form inverse of the Black–Scholes formula is known, and inversion is especially sensitive in asymptotic regimes.

    Source: §1 · pp. 2–3

  • Method

    The architectures learn a trainable partition of the price–log-moneyness domain and combine specialised local implied-volatility approximations through gating functions.

    Source: §1 · p. 3

  • Results

    The architectures outperform standard feed-forward networks across broad parameter domains, reducing test MSRE by one to four orders of magnitude and reaching near-machine-precision accuracy after two Householder iterations.

    Source: §5 · p. 24

  • Takeaways & Limitations

    The neural networks can serve as accurate initial guesses for classical root-finding rather than replacing it.

    Source: §5 · p. 24

    The PolyC and GaussC gates are identical across log-moneyness levels, although implied-volatility-surface geometry varies substantially for small A.

    Source: §3.3 · p. 11

Abstract

from arXiv · show

The computation of Black-Scholes implied volatility is a fundamental task in quantitative finance, underpinning option valuation, model calibration and risk management. Although implied volatility is routinely used in practice, the inversion of the Black-Scholes pricing formula remains a challenging numerical problem, particularly in asymptotic regimes corresponding to extreme option prices, strikes or maturities, where the inverse map becomes highly sensitive to perturbations of the price. In this paper, we introduce a new family of asymptotically-informed neural-network architectures for implied-volatility computation. Exploiting the distinct behaviours of the Black-Scholes pricing function in different volatility regimes, we propose a family of architectures that learn a trainable partition of the price-log-moneyness domain through a system of gating functions and combines specialised local approximations of the implied-volatility function within each region. Extensive numerical experiments demonstrate that the proposed models consistently outperform standard feed-forward neural networks across a wide range of parameter domains, often by several orders of magnitude in relative accuracy while maintaining excellent generalisation properties. Furthermore, the neural-network outputs provide highly accurate initial guesses for a third-order Householder scheme, allowing near machine-precision implied-volatility computations after only two refinement iterations.

1. Introduction

Implied-volatility inversion becomes especially sensitive in asymptotic regimes, while existing neural approaches treat it largely as generic regression. The paper introduces architectures that use Black–Scholes structure to learn domain partitions and combine specialised local approximations.

1. Introduction

  • In asymptotic regimes, small or large option prices can make implied volatility highly sensitive to price perturbations, so small pricing errors may produce large volatility errors.These regimes include options that are deeply in or out of the money and very short or long maturities.Source: §1 · p. 2
  • Existing neural methods largely frame inversion as generic regression, while prior architectures can reach a precision bottleneck and larger networks may worsen generalisation.The paper argues for incorporating the analytical structure of the implied-volatility function into network design.Source: §1 · p. 3
  • The method learns a trainable partition of the price–log-moneyness domain and combines specialised local approximations through gating functions.Its structure is motivated by distinct asymptotic and central regimes of the implied-volatility function.Source: §1 · p. 3
  • The proposed architectures consistently outperform standard feed-forward neural networks across broad parameter domains, often by several orders of magnitude in relative accuracy.The networks also provide accurate initial guesses for Householder refinement.Source: §1 · p. 3
  • The network outputs provide accurate initial guesses for a third-order Householder scheme, enabling high-precision implied-volatility computations after few iterations.The paper presents the neural network as an initializer for numerical refinement.Source: §1 · p. 3
  • The approach scales by enlarging the training data or network, while inference reduces essentially to matrix multiplications and pointwise nonlinearities.The paper contrasts this with analytical approximations, which generally require new formulas and more complex expressions for separate regimes.Source: §1 · p. 4

2. Review of the analysis of the Black and Scholes formula

The section formulates implied volatility as an inverse map from log-moneyness and normalized call price to total volatility. It uses the pricing function’s distinct regimes to motivate a three-region partition and specialized approximations.

2.1. Mathematical statement of the problem.

  • The inversion map Binv associates log-moneyness A and normalized call price C with the corresponding total volatility B∗.The target is the unique B∗>0 solving the pricing equation for the observed price.Source: §2.1 · p. 4
  • A symmetry relation permits restricting analysis to A ≥ 0, corresponding to out-of-the-money calls.Source: §2.1 · p. 4
  • The formulation recalls prior normalizations, including the one used in, and assumes S0 = 1 and r = 0 without loss of generality.Source: §2.1 · p. 5
  • Normalized call prices lie in (0, 1) for fixed A, giving neural-network training a bounded, parameter-independent input range.The section identifies this range as convenient for training.Source: §2.1 · p. 5

2.2. Partitioning the price-log-moneyness space.

  • Earlier approaches use tangent-based boundaries in or affine boundary functions in [22], whose approximation deteriorates for large |A|; uses regime-wise rational functions while [22] uses global Chebyshev polynomials.Source: §2.2 · pp. 5–6
  • For fixed log-moneyness, pricing is approximately linear near its inflection point but nearly flat at very small and large total volatility.The inverse becomes particularly steep near the two asymptotic regimes, where small price perturbations can cause large implied-volatility changes.Source: §2.2 · pp. 5–6
  • The partition uses the tangent at the pricing curve’s inflection point to separate lower, central, and upper regimes; its slope is 1/√(2π), independent of A.Figure 1 illustrates the tangent-based partition for A = 3, while Figure 2 shows the induced domain regions.Source: §2.2 · pp. 6–7, Figure 1 · p. 6, Figure 2 · p. 7
  • The proposed architecture approximates each regime separately and combines the local models through smooth transitions, adapting to different local behaviors across the domain.Source: §2.2 · pp. 7–8

3. Architectural design

The architecture combines local implied-volatility approximations with gating functions that learn how to blend them across distinct regimes. The design incorporates qualitative asymptotic structure without requiring fixed regime boundaries.

3.1. General principle.

  • Trainable gating functions let the domain partition emerge during training while respecting asymptotic constraints, rather than hard-coding regime boundaries.The representation is described as a problem-specific mixture-of-experts architecture.Source: §3.1 · p. 8
  • Inter sets f2 to 1−f0−f1, whereas Free sets f2 to 1, leaving g2 as an unweighted baseline.Source: §3.1 · p. 8
  • The weighted approximation uses g2 as a global baseline, with f0g0 and f1g1 providing corrections for the lower and upper asymptotic regions.Source: §3.1 · p. 8
  • The design uses qualitative regime structure rather than explicit asymptotic formulas, and the authors report higher accuracy with smaller networks than problem-agnostic architectures.The paper characterizes the approach as fully data-driven while embedding analytical features in the model design.Source: §3.1 · p. 9

3.2. Choices for the local implied volatility functions.

  • Gen uses unconstrained scalar-output feed-forward networks for local models, while Exp exponentiates network outputs to encode positivity; FreeExp uses a negative exponential for the lower-regime correction.Source: §3.2 · p. 9
  • AsptDef and AsptAdv performed worse than the other tested architectures; their asymptotic formulas apply only on small domain subsets and may be inaccurate over training and testing ranges.Source: §3.2 · p. 10

3.3. Choices for the gating functions.

  • PolyAC and GaussAC let gating depend on both A and C, while PolyC and GaussC use C alone, require fewer parameters, and share gates across log-moneyness despite geometry varying with A.Source: §3.3 · p. 11
  • HardAC fully activates the corresponding local approximation beyond theoretical regime boundaries; polynomial gates transition more slowly, while Gaussian gates tend to be sharper.Source: §3.3 · p. 11

4. Numerical experiments

The experiments are framed as a study of architectural variants, including comparisons with standard neural networks and a root-finding approach. The setup is not an exhaustive hyperparameter search.

4. Numerical experiments.

  • The experimental framework evaluates architectural variants against standard feed-forward networks and a classical root-finding approach.Source: §4 · p. 11
  • The experiments explore selected learning rates, layer widths, and epoch counts, but not the full range of design choices; the best architecture is expected to depend on training-data ranges.Source: §4 · p. 11
  • Training data are generated on a regular grid in (A, B), using a stable, accurate pricing implementation designed to avoid cancellation, underflow, and overflow.Source: §4.1 · p. 11

4.1. Preparing the datasets.

The implementation combines local implied-volatility functions with gating functions and supplies selected architectures with auxiliary variables intended to support stability and asymptotic learning.

4.1. Preparing the datasets.

  • Selected architectures receive precomputed auxiliary variables intended to improve numerical stability and facilitate learning asymptotic behavior.Cinv, Alog, and Clog appear in some gating functions; zu and zl encode position relative to the asymptotic partition in HardAC models.Source: §4.1 · pp. 11–12
  • Twenty percent of each dataset is reserved for testing, with an additional 15% of the full dataset reserved for validation when adaptive learning-rate schedules are used.Source: §4.1 · p. 12
  • Each model combines a choice of local implied-volatility functions with a choice of gating functions through Equation (6).Source: §4.1 · p. 12

4.2. Neural networks with asymptotic regimes.

The proposed architectures combine gating functions with learned components, while varying how gates and outputs are parameterised. The section also sets up a complexity-matched comparison with standard feed-forward networks.

4.2. Neural networks with asymptotic regimes.

  • The Sig parametrisation uses the sigmoid activation and avoids explicit reciprocals and repeated evaluations of large powers, which the authors describe as appealing for optimisation.The passage contrasts Sig with a mathematically equivalent Inv parametrisation.Source: §4.2 · p. 12
  • A three-hidden-layer ReLU network takes (A, C) as inputs and outputs the three learned quantities g0, g1, and g2.Source: §4.2 · p. 12
  • PolyACInvExpInter combines PolyAC gates using Inv, positive outputs from an exponentiated neural network, and the Inter output formula.The architecture name encodes these design choices.Source: §4.2 · p. 12
  • The comparison uses SimpleGen and SimpleExp networks with three ReLU hidden layers of 2Ng neurons each, matching the proposed models’ complexity.Source: §4.3 · p. 12

4.3. Standard neural networks.

The standard-network baselines differ in their output layers, and the section identifies MSE and MSRE as commonly used loss functions.

4.3. Standard neural networks.

  • SimpleGen has an unrestricted affine output, whereas SimpleExp exponentiates its affine output to enforce positive implied-volatility predictions.Source: §4.3 · p. 13
  • The section identifies mean squared error (MSE) and mean squared relative error (MSRE) as commonly used approximation losses.Source: §4.4 · p. 13

4.4. Learning and comparison metrics.

The section motivates MSRE as the default training loss and describes the evaluation metrics, repeated experiments, and training setup used for model comparison.

4.4. Learning and comparison metrics.

  • MSRE is the default loss because it gives relative errors comparable importance and typically improves accuracy at low volatility, a regime the authors say is frequently encountered.MSE tends to prioritize large volatility values, so strong absolute accuracy can coexist with poorer low-volatility performance.Source: §4.4 · p. 13
  • All models are evaluated using MSE, MSRE, maximum absolute residual error, and maximum relative residual error.Source: §4.4 · p. 13
  • Each experiment was repeated independently 40 times, with training-versus-testing scores used to assess robustness, training variability, and generalisation to unseen data.Source: §4.4 · p. 13
  • Training used Adam variations with an initial learning rate of 10^-3 for 150 epochs; the mini-batch configuration includes batch size 128 and validation-based learning-rate reduction.The third strategy consistently produced the best results and is the only one reported in the paper’s remainder.Source: §4.4 · pp. 13–14

4.5. The Householder root finding algorithm.

The neural-network approximation supplies an initial guess for a third-order Householder root-finding method, with an alternative objective used to address numerical instability at extreme volatilities.

4.5. The Householder root finding algorithm.

  • The network approximation is used as an initial guess for Householder refinement, combining the speed of the approximation with the accuracy of an iterative solver.The authors describe this approach as standard in implied-volatility literature and use it without additional innovation.Source: §4.5 · p. 14
  • The third-order Householder method iteratively solves for volatility so the Black-Scholes price matches the observed option price.Source: §4.5 · p. 14
  • At extremely small or large volatility, direct Householder application to the pricing residual may be unstable, so the method uses alternative objective functions proposed in.Source: §4.5 · p. 14

4.6. Numerical results.

The experiments compare selected asymptotically informed architectures with a standard feed-forward network across large-, medium-, and small-range datasets, and test their use as initial guesses for Householder refinement. The proposed architectures generally improve accuracy and generalisation, with the size and preferred form of the improvement varying by parameter range.

4.6.1. A large-range dataset.

  • Roughly four orders of magnitude lower MSRE: the best asymptotic architectures outperform the standard feed-forward network on the large-range dataset.Training and testing results are nearly identical across reported metrics.Source: §4.6.1 · p. 15, Table 2 · p. 16
  • Across almost the entire admissible domain, asymptotic architectures maintain uniformly small error while the standard network’s elevated error grows at low volatility.The reported improvement is global rather than confined to isolated regions.Source: Figure 4 · p. 16, §4.6.1 · p. 16
  • After two Householder iterations, the proposed pipeline is highly accurate and computationally efficient, while Brent’s error landscape is less structured.The authors present the networks as accelerators that place the iterative solver in its convergence regime.Source: Figure 7 · p. 18, §4.6.1 · p. 18

4.6.2. A medium-range dataset.

  • Joint A-and-C gating is preferred on the medium-range dataset; Gaussian gates dominate nearly all reported metrics and remain stable across runs.The authors relate this preference to asymptotic boundaries that vary with A as A approaches zero; polynomial gates remain competitive.Source: §4.6.2 · pp. 18–19
  • More than three orders of magnitude: GaussACInvExpInter reduces test MSRE from 3.51 × 10−2 to 2.24 × 10−5 on the medium-range dataset.Training and testing errors are closely aligned.Source: §4.6.2 · p. 18, Table 4 · p. 19
  • After two Householder iterations, medium-range asymptotic architectures reach average logarithmic relative errors around −37, while Brent’s error landscape is more irregular.The refined accuracy is similar across asymptotic architectures.Source: Table 5 · p. 20, §4.6.2 · pp. 20–21, Figure 11 · p. 21

4.6.3. A small-range dataset.

  • Near A = 0, the error profile is almost horizontal and polynomial gates perform strongly; after refinement, all models converge rapidly, while Brent performs poorly across much of the domain.The neural-network-plus-Householder pipeline remains stable and highly accurate over the full small-range grid.Source: §4.6.3 · pp. 21–23, Figure 13 · p. 22, Figure 15 · p. 23
  • Approximately 39-fold: PolyACepsInvExpInter reduces test MSRE from 9.04×10−6 to 2.33×10−7 on the small-range dataset.The performance gap is smaller than on wider datasets, but the improvement is systematic.Source: Table 6 · p. 21, §4.6.3 · p. 21

5. Conclusion and possible improvements

The proposed regime-aware networks combine analytical structure, trainable gating, and local approximations, and consistently outperform standard feed-forward networks. Their outputs also provide accurate initial guesses for two Householder refinements, while richer asymptotic local approximations and extensions beyond Black–Scholes remain future directions.

Method

  • The proposed family embeds analytical structure through decomposition (6), trainable gating, and specialised local approximations, then combines network speed with high-order iterative accuracy.The architecture is motivated by distinct asymptotic and central regimes in implied-volatility inversion.Source: §5 · pp. 23–24

Results

  • The architectures consistently outperform standard feed-forward networks across broad parameter domains, improving mean and worst-case relative errors by several orders despite using smaller networks.Training and testing performance remain nearly identical throughout the experiments.Source: §5 · p. 24
  • Across datasets, the networks reduce test MSRE by one to four orders of magnitude versus standard feed-forward networks, reaching near-machine precision after two Householder iterations.The networks are framed as accurate initial guesses that place Householder directly in its rapid-convergence region, rather than as replacements for root-finding.Source: §5 · p. 24
  • Gaussian gating generally offers the best balance of accuracy, robustness, and consistency; the authors recommend Gaussian AC with interpolation (8) as a default.GaussAC is especially effective when the log-moneyness range is large enough for asymptotic-region geometry to vary significantly with A.Source: §5 · p. 24
  • Problem-specific inductive biases let the proposed models achieve substantially better accuracy with significantly fewer parameters than feed-forward networks.The comparison exploits known analytical structure rather than relying only on universal approximation with sufficiently large networks.Source: §5 · p. 24

Possible improvements

  • The current generic local approximations do not fully exploit known asymptotic shapes; future work could encode the limits C → 0+ and C → 1− explicitly.The authors identify enriching g0 and g1 as one possible improvement.Source: §5 · p. 24
  • The regime-aware architecture philosophy may extend to inverse problems in local-, stochastic-, and rough-volatility models, though this paper studies only Black–Scholes inversion.The authors describe extensions to these settings as promising future work.Source: §5 · p. 24

Appendix A. Accurate implementation of the Black-Scholes pricing formula

The appendix reformulates Black–Scholes pricing with the Gaussian Mills ratio to manage numerical loss of precision, then uses Taylor or asymptotic expansions in the regimes where each is appropriate. Its implementation selects among these approximations by h and t, with direct evaluation retained otherwise.

Appendix A. Accurate implementation of the Black-Scholes pricing formula

  • The direct Black–Scholes formula can suffer underflow, overflow, and catastrophic subtractive cancellation, especially at small total volatility.The Mills-ratio reformulation isolates the numerical challenge in a single difference term.Source: Appendix A · p. 26
  • The Mills-ratio representation isolates precision loss in Z(h−t)−Z(h+t), enabling Taylor treatment when t is small and h is moderate.For large h, Z and its derivatives become very small, making an asymptotic expansion more appropriate.Source: Appendix A · p. 26

A.1. A Taylor expansion of the call price in the small-t regime

  • For small t, the Taylor expansion replaces direct evaluation of the cancellation-prone difference; J_NZ(h,t) denotes its polynomial approximation.The appendix describes recursively computing derivatives of Z and evaluating Z with erfcx, but the supplied passages do not provide further usable implementation details.Source: §A.1 · pp. 26–27
  • With N=16 and t<0.1, the stated error bound is at most 10^-20 for h<3.8023; the appendix cautions that the bound may not be sharp at high h.The implementation uses the Taylor approximation for h<35 and switches to the asymptotic expansion beyond that threshold.Source: §A.1 · p. 27

A.2. An asymptotic expansion of the call price in the large-h regime

  • For large h, the appendix develops an asymptotic expansion and an error-based procedure for choosing its truncation orders.The expansion is used in the regime h>35, with t<h as specified in the proposition and summary of the approximation scheme.Source: Appendix A · p. 26, §A.2 · pp. 28–29, §A.3 · p. 29

A.3. Summary of the used approximations

  • The implementation uses the Taylor approximation for t<0.1 and h<35, the asymptotic approximation for h≥35 and t<0.1h, and direct Mills-ratio evaluation otherwise.For the asymptotic branch, the stated settings are N1=17, N2=20, and τmax=0.1.Source: §A.3 · pp. 29–30
  • Figures 16 and 17 compare direct and Mills-ratio-based price evaluations; oscillations from cancellation diminish as t increases, and larger h makes them disappear earlier.Figure 16 varies h at fixed t, while Figure 17 varies t at fixed h.Source: Figure 16 · p. 30, Figure 17 · p. 31

Appendix B. Householder root-finding method for the implied volatility

The appendix derives expressions for third-order Householder refinement of implied volatility, using alternative residuals where the direct objective becomes unstable at extreme volatility. It also reuses the accurate price-difference calculations from Appendix A.

Householder method

  • The third-order Householder method refines implied volatility and generalises Newton’s first-order and Halley’s second-order methods.The neural networks serve both as stand-alone approximators and as initial guesses for this iterative refinement.Source: Appendix B · p. 31

Low-volatility regime

  • At low volatility, the direct residual becomes numerically unstable as price and its derivatives approach zero, so the method switches to an alternative residual.The alternative is used when the given price satisfies the appendix’s specified condition.Source: Appendix B · p. 32

Computational expressions

  • The derivative calculations reuse ΔZ(h,t) and its regime-specific approximations from Appendix A.The appendix states that it derives explicit expressions for the quantities used in Black–Scholes implied-volatility inversion.Source: Appendix B · pp. 31, 33

High-volatility regime

  • At high volatility, the direct objective also becomes unstable as the price approaches one and its derivatives vanish, prompting another alternative residual.The appendix also notes that direct evaluations of Z(u) and Z(−v) may underflow for large B.Source: Appendix B · p. 33
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