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.
Figures & tables
Figure 1. (A): The partition of the space (B,C) into three regions defined by the tangent to the curve CBS(A,⋅) at its inflection point, for A=3 . (B): Inverse-volatility map Binv(A,)˙ for A=3 , and the corresponding partition of the (C,B) -domain into three regions defined by the tangent to the curve CBS(A,⋅) at its inflection point.
Figure 2. Partition of the (A,C) -domain induced by the inequalities ( 5 ). The central region corresponds to the approximately linear part of the pricing function, while the upper and lower regions correspond to the two asymptotic regimes.
Table 1. Families of gating functions f0 and f1 considered in this work.
Figure 3. Evolution of the training MSRE averaged over 25 independent runs for the selected architectures on the dataset (A,B)∈[0,16]×[10−5,7.07] . The shaded regions represent one standard deviation around the mean.
Figure 4. Pointwise relative error, on a logarithmic scale, of the chosen architectures over the domain (A,B)∈[0,16]×[10−5,7.07] . The evaluation grid contains 1000×1000 points.
Figure 5. Evolution of the logarithmic relative error ( 11 ) for a fixed value A=5 over the interval B∈[10−5,2.00] . The curves correspond to the neural-network output ( B0 ), after one Householder iteration, and after two Householder iterations. B is sampled uniformly over a grid of 500 points.
Model
Avg Log Error
Std Log Error
Max Log Error
SimpleExp
−33.56
7.38
−19.70
PolyACInvGenInter
−38.85
2.61
−35.35
PolyACInvGenFree
−38.85
2.60
−35.35
GaussACInvGenInter
−38.72
2.59
−35.35
PolyCInvExpInter
−38.70
2.62
−35.35
Table 3. Average, standard deviation and the maximum of the logarithm of the relative error ( 11 ) for the chosen architectures after 2 Householder iterations with A=5 and B∈[10−5,2.00] . B is sampled uniformly over a grid of 500 points.
Figure 6. Heat map of the logarithm of the relative error ( 11 ) of the chosen neural networks after 2 iterations of the Householder root finding method over the range of values (A,B)∈[0,16]×[10−5,7.07] , with 1000 grid values for each variable.
Figure 7. Comparison of the logarithm of the relative error ( 11 ) between Brent’s method (maximum of 50 iterations) and the proposed neural-network-plus-Householder approach on the domain (A,B)∈[0,0.5]×[10−5,2] . The evaluation grid contains 200×200 points.
Figure 8. Evolution of the training MSRE averaged over 25 independent runs for the selected architectures on the dataset (A,B)∈[0,3]×[10−7,1.22] . The shaded regions represent one standard deviation around the mean.
Figure 9. Pointwise relative error, on a logarithmic scale, of the chosen architectures over the domain (A,B)∈[0,3]×[10−7,1.22] . The evaluation grid contains 1000×1000 points.
Figure 10. Evolution of the logarithmic relative error ( 11 ) for a fixed value A=1.5 over the interval B∈[10−7,1.22] . The curves correspond to the neural-network output ( B0 ), after one Householder iteration, and after two Householder iterations. B is sampled uniformly over a grid of 500 points.
Model
Avg Log Error
Std Log Error
Max Log Error
SimpleExp
−34.39
6.00
−18.78
GaussACInvExpInter
−37.26
2.42
−34.65
GaussACInvGenInter
−37.25
2.43
−34.94
PolyACepsInvExpFree
−37.33
2.42
−34.79
PolyACSigExpFree
−37.13
2.34
−34.65
Table 5. Average, standard deviation and the maximum of the logarithm of the relative error ( 11 ) for the chosen architectures after 2 Householder iterations with A=1.5 and B∈[10−7,1.22] . B is sampled uniformly over a grid of 500 points.
Figure 11. Comparison of the logarithm of the relative error ( 11 ) between Brent’s method (maximum of 50 iterations) and the proposed neural-network-plus-Householder approach on the domain (A,B)∈[0,3]×[10−7,1.22] . The evaluation grid contains 200×200 points.
Figure 12. Evolution of the training MSRE averaged over 25 independent runs for the selected architectures on the dataset (A,B)∈[0,10−5]×[10−5,0.18] . The shaded regions represent one standard deviation around the mean.
Figure 13. Pointwise relative error, on a logarithmic scale, of the chosen architectures over the domain (A,B)∈[0,10−5]×[10−5,0.18] . The evaluation grid contains 500×500 points.
Figure 14. Evolution of the logarithmic relative error ( 11 ) for a fixed value A=5.10−6 over the interval B∈[10−5,0.18] . The curves correspond to the neural-network output ( B0 ), after one Householder iteration, and after two Householder iterations. B is sampled uniformly over a grid of 500 points.
Model
Avg Log Error
Std Log Error
Max Log Error
SimpleExp
−38.70
2.92
−10.948
PolyACepsInvExpInter
−39.06
2.88
−10.925
PolyCSigExpInter
−39.07
2.92
−10.926
GaussACInvExpInter
−39.40
2.85
−10.932
HardACInvExpInter
−38.62
2.91
−10.957
GaussCInvExpInter
−38.94
2.91
−10.953
Table 7. Average, standard deviation and the maximum of the logarithm of the relative error ( 11 ) for the chosen architectures after 2 Householder iterations with A=5.10−6 and B∈[10−5,0.18] . B is sampled uniformly over a grid of 500 points.
Figure 15. Comparison of the logarithm of the relative error ( 11 ) between Brent’s method (maximum of 50 iterations) and the proposed neural-network-plus-Householder approach on the domain (A,B)∈[0,10−5]×[10−5,0.18] . The evaluation grid contains 200×200 points.
Appendix figures & tables2 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 16. The logarithm of the ratio between the direct evaluation of the Black-Scholes formula ( 1 ) (denoted here Cdefault(h,t) ) and the Mills-ratio-based implementation ( 12 )(denoted Cexpanded(h,t) ), with varying h and fixed t . (A) t=0.001 , (B) t=0.01 , (C) t=0.1 and (D) t=1 . The magnitudes of the oscillations caused by subtractive cancellation decay as t increases.
Figure 17. The logarithm of the ratio between the direct evaluation of the Black-Scholes formula ( 1 ) (denoted here Cdefault(h,t) ) and the Mills-ratio-based implementation ( 12 )(denoted Cexpanded(h,t) ), with varying t and fixed h . (A) h=0.01 , (B) h=0.1 , (C) h=1 and (D) h=10 . The magnitudes of the oscillations decay as t increases. Larger values of h lead to an earlier disappearance of the oscillations.
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