Operator Deep Smoothing for Implied Volatility
Authors: Ruben Wiedemann
Antoine Jacquier
Lukas Gonon
Venue: ICLR 2025 Poster (2025)
Keywords: Financial Engineering
Neural Operators
Option Pricing
Function Interpolation
Nowcasting
Primary area: other topics in machine learning (i.e., none of the above)

Abstract: We devise a novel method for nowcasting implied volatility based on neural operators.
Better known as implied volatility smoothing in the financial industry, nowcasting of implied volatility means constructing a smooth surface that is consistent with the prices presently observed on a given option market.
Option price data arises highly dynamically in ever-changing spatial configurations, which poses a major limitation to foundational machine learning approaches using classical neural networks.
While large models in language and image processing deliver breakthrough results on vast corpora of raw data, in financial engineering the generalization from big historical datasets has been hindered by the need for considerable data pre-processing.
In particular, implied volatility smoothing has remained an instance-by-instance, hands-on process both for neural network-based and traditional parametric strategies.
Our general *operator deep smoothing* approach, instead, directly maps observed data to smoothed surfaces.
We adapt the graph neural operator architecture to do so with high accuracy on ten years of raw intraday S&P 500 options data, using a single model instance.
The trained operator adheres to critical no-arbitrage constraints and is robust with respect to subsampling of inputs (occurring in practice in the context of outlier removal).
We provide extensive historical benchmarks and showcase the generalization capability of our approach in a comparison with classical neural networks and SVI, an industry standard parametrization for implied volatility. 
The operator deep smoothing approach thus opens up the use of neural networks on large historical datasets in financial engineering.

Published as a conference paper at ICLR 2025
OPERATOR DEEP SMOOTHING FOR IMPLIED VOLATIL- ITY
Ruben Wiedemann
Antoine Jacquier Imperial College London
Imperial College London w.iedemann222 1eee@eimmriiaaaaaaaammeemieaaaaa ac. uk
jww.jaieeeeeieeriii... ac .uku
Lukas Gonon
University of St. Gallen
lukas. genoneunisg.co
ABSTRACT
We devise a novel method for nowcasting implied volatility based on neural oper- ators. Better known implied volatility smoothing in the financial industry, now- casting of implied volatility means constructing a smooth surface that is consistent with the prices presently observed given option market. Option price data arises highly dynamically in ever-changing spatial configurations, which poses major limitation foundational machine learning approaches using classical neural networks. While large models in language and image processing deliver breakthrough results on vast corpora of raw data, financial engineering the gen- eralization from big historical datasets has been hindered by the need for consider- able data pre- pre-proeessing particular, implied volatility smoothing has remained instance-by byiinstance, hands-on process both for neural network-based and tra- ditional parametric strategies. Our general operator deep smoothing approach, instead, directly maps observed data to smoothed surfaces. We adapt the graph neural operator architecture to do so with high accuracy on ten years of raw in- traday S&P 500 options data, using single model instance. The trained operator adheres to critical no-arbitrage constraints and is robust with respect to subsam- pling of inputs (occurring in practice in the context of outlier removal). We pro- vide extensive historical benchmarks and showcase the generalization capability of our approach in comparison with classical neural networks and SVI, indus- try standard parametrization for implied volatility. The operator deep smoothing approach thus opens up the use of neural networks on large historical datasets financial engineering.
INTRODUCTION
Options trading experienced phenomenal growth in recent years. its 2023 trading volume re- port (Cboe Global Markets, Inc. 2024), the CBOE announced the fourth consecutive year of record- breaking volumes on its options exchanges, citing an all-time high number of transactions for Eu- ropean options on the S&P 500 index. European options are financial derivative contracts that give their holder the right, but not the obligation, to either buy or sell an underlying asset at predeter- mined price (the strike) at predetermined time (the expiry). An option specifying the right buy (respectively to sell) is called a Call (respectively Put) option. Options are traded on wide range of underlyings, including stocks, indices, currencies and commodities, and can be used hedge against or speculate on the price movements of the underlying asset.
A key concept in options trading the so-called implied volatility, which transforms the nominal price of option into conceptually and numerically convenient metric. The implied volatility surface is the collection of implied volatilities as observed at a specific point in time, visualized surface over the (strike, expiry)-domain. It provides intuitive representation of the current state of the options market and is crucial for hedging and risk management. The extraction a smooth surface from quoted option prices called implied volatility smoothing and allows to infer

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(or nowcast) theoretical option prices for interpolated strike values and expiry times. It remains one of the key challenges in options trading.
Conventionally, implied volatility smoothing relies on parametric surfaces whose parameters op- timized based on the distance to observed prices while adhering to absence-of-arbbeeee which ensure the consistency of prices extrapolated from the smoothed surface. The development of such ad-hoc models for implied volatility traces back to SVI (Gatheral 2004). which models implied volatility slice-wise for each expiry and successfully captures its key features on Equity indices. A continuous interpolation scheme for SVI slices was provided in Gatheral & Jacquier (2014), yield- ing a full surface. Nowadays, sophisticated market makers employ custom parametrizations, which can be considered proprictary trading secrets and reduce SVI to a benchmark role.
conditions,
Regardless of the particular parametric surface model used, the conventional smoothing approach boils down to the continued execution of a numerical optimization routine: A smoothed surface expires as soon as quotes are updated (whenever markets move), necessitating the re-calibration of parameters. Success and duration of this routine is sensitive to initial conditions, search heuristics, and termination criteria, which exposes practitioners to considerable process uncertainties during trading hours (or online). In response, we introduce novel operator deep smoothing approach, replacing the instance-by-ine by-instance optimization with a single evaluation of a neural network. This greatly simplifies online calibration, the upfront cost of training the network offfine from historical data (in the spirit of Hernandez) 2016); Horvath et al. (2021) Liu et al. (2019)). Our unique use of neural operators (Kovachki al., 2003) is fundamentally directed by the observation that the raw inputs for volatility smoothing (the collections of observed volatilities) over time vary in size and spatial arrangement: Options expire, new expiries and strikes become available (the total number of options listed is steadily increasing, see Figure TOp, and the coordinates of existing options evolve continuously in the domain of the implied volatility surface (illustrated in Figure Tb and Figure 99) This setting excludes classical neural networks which required fixed-size inputs from direct application. Neural operators, instead, conceptualize observed data as point-wise discretizations of latent functions in implicit fiiiiteddimensionall function spaces and are well suited for the task.
Contributions We introduce operator deep smoothing, a general approach for discretization- invariant data interpolation based on neural operators, and apply it implied volatility smoothing. Our technique transcends traditional parametric smoothing and directly maps observed volatilities to smoothed surfaces. Comparable neural network based approaches are limited to certain option markets (e.g. FX markets, where options by default spread out on fixed rectilinear grids, as relied upon by [Bergeron al. (202T) for its VAË approach) or require data pre-processing (as in Cont & Vuletic (2023). which achieves fixed rectilinear grids by linear interpolation of market values, setting aside questions related to no-arbitrage constraints). Instead, our technique novelly adapts the graph neural operator (GNO) architecture (Anandkumar et al., 2020) to consistently smooth input data of any size and spatial arrangement. While neural operators have successfully been used in Physics to numerically solve partial differential equations, our application is the first in finan- cial engineering and highlights the values of their dbvcretization-ivaaiine under-explored. We employ our method on years of intraday S&P 500 options data, smoothing more than 60 million volatility datapoints using a single model instance with around 100 thousand trained parameters. We report errors that substantially improve on the SVI industry benchmark and are highly competitive with Ackerer et al. (2020). which performs smoothing by training one classi- cal neural network per volatility surface. We proceed successfully demonstrate the generalization capabilities of our model for end-of-day options data of the S&P 500 as well as three further major US indices. No data from these three indices has been used for training.
properties, so far rather
We explore the technical implications and limitations of our method in Section 5 the broader impact of our contributions:
Here we discuss
Operator Deep Smoothing for Implied Volatility Our method massively simplifies volatility smoothing, an area where effective methods mean competitive advantage and frequently remain trade secrets. We believe that our approach lowers the entry barrier for effective volatility smoothing, even among industry professionals. Practitioners and researchers who are not directly involved in options trading frequently employ rudimentary methods (SVI or linear/spline interpolation). Here, our trained operator, served as a hands-off tool, could provide "cheap" and accurate surfaces for use in downstream tasks. Ultimately, our method may be useful for all participants of option mar-
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kets. This includes the general publie, whose trading in such markets has been increasing substan-
tially (Doherty et al. 2023) and which benefits from broadly accessible investment tools.
Neural Operators for Discretization-Inann proach constitutes the first application of neural operators for interpllation/exttpllaatoo tasks and paves the way to future research on the versatility of the discretization-inet cnniiiaattioninnaiiannee invariance of neural operators industrial applications characterized by dynamic and spatially irregular data. At least in financial engineering, surfaces analogous to the implied volatility surface (e.g. nigher-dimensional equiva- lents, as, for example, the volatility cube) are ubiquitous. We expect our technique to be transferable and to streamline and robustify engineers' and researchers' algorithms and data pipelines.
Interpolation Our operator deep smoothing ap-
The aforementioned SVI was developed for internal use at Merrill Lynch in 1999 and later advocated in Gatheral (2004). Its extension to surface- SSVI in Gatheral & Jacquier (2014) has been eagerly adopted by practitioners, which have since contributed to its robust calibration and generalizations (Corbetta et al. 2019; Hendriks Martini 2017 Guo et al. 2016). was augmented in Ackerer et al. (2020) by multiplicative neural network corrector, based guided network training by means no-arbitrage soft constraints from Zheng (2018). The absence-of- arbitrage conditions for implied volatility surfaces providing safeguards for option pricing were formulated in Roper 2210)) and we provide an equivalent formulation, based on Fukasawa (2012);
Literature review
Lucic (2021). for practical purposes. In InCChaaiigne et al. (2020) static arbitrage constraints were used to perform option calibration (with an additional regularization technique), which can be considered to be bestance-y--instance smoothing of nominal price data. In [ergeron et al. (2021) classical VAE (variational autoencoder) was applied to implied volatility smoothing on FX markets, where strikes of quoted options are tied to a fixed grid of deltas ! This specificity of FX markets allows the use of a conventional feedforward neural network based decoder. Recent option calibration
approaches based on neural networks have been proposed in Baschetti et al. 2024);
Hernandez
(216)); Horvath et al. (202T): : Van Mieghem et al. (2023).
comprehensive account on neural operators is given in Kovachki et al. (2023), unifying previous research on different neural operator architectures and techniques (Anandkumar et al. 2020; Li
al., 2020). Subsequent developments investigating the expressivity of these architectures as well as their generalizations include Hao et al. (2023); Huang et al. (2024); Lanthaler et al. (2023):
ct al. (202I) Lingsch et al. (2023); Tran et al. (2021).
Outline We review financial concepts and the challenges of implied volatility smoothing in Sec- 2 In Scction we provide a review of neural operators (Section and introduce our operator deep smoothing approach for general interpolation tasks (Section 3..2p In Section we perform ex- periments for implied volatility smoothing of S&P 500 options data. Finally, Section 5 gathers technical implications and limitations as well outlooks regarding the use of neural operators for interpolation purposes.
Code We make code for the paper available at the location https:://gt thub. cwm/wwii...o.o..............o..e rwi el operator- deepr-deepsseee saoppp-mmotting---eee for- imp wmpppiedvolatiiii lity In partieular, the code repository contains a general PyTorch (Paszke et al. 2019) implementation of the graph neural operator archi- tecture for operator deep smoothing.
BACKGROUND: IMPLIED VOLATILITY
We consider a market of European options written on an underlying asset, which we observe at a given instant To. We denote the T forward price of the underlying asset by FT.
European Call options The market consists of a finite collection of European Call options each identified by its expiry T E (To, oo) and its strike K E oo). We write C(T, K) for its (undis- counted) price. In practice, these are traded for fixed expiries Ti, ......, Tii for each T., only finite range of strikes Ki.-.. Ki is available, widening as the expiry increases (Figure
Delta the derivative the price with respect to the underlying asset and standard FX strike quoting. In practice, market participants trade hoth Call and Put options, which are mathematically equivalent through the well-known Put-Call parity. The latter thus allows to speak in terms of Call options only.
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Black-Scholes The volatility of the underlying asset with a single parameter v E (0, oo). Its popularity stems from the closed-form expression it admits for the price of a European Call option with time-to-expiry
model is the simplest diffusive asset model and captures the
T = - To and log- log-moneyness & = log( F)..r):
BS( k v) = 2 (d1 (T, b)) - e + (d2 (T,
(in units of the time-T forward of the underlying). Here, & denotes the cumulative distribution function of the standard Normal distribution, while
-k k)) OVT L.V. d2(T, uVT 2 ZOV..
-k 1 UVT
Implied volatility While not able (any longer) to fit market data, the mathematical tractability of the Black-Scholes model gave rise to the concept of implied volatility: Given Call option with price C(T, K), its implied volatility v(T, is defined by C(T, K) FT..T BS(T, k, v(T; k)). By using time-to-exxe -ot.ioiiyyllog---oo. expiry/log-monyynesss coordinates, implied volatility provides universal way to consis- tently compare the relative expensiveness of options across different strikes, expiries, underlyings, and interest rate environments. Its characteristic shape helps traders make intuitive sense of the states of option markets relative to a (flat) Black-Scholes model baseline.
Implied volatility smoothing This refers fitting smooth surface ú: (0, x I (0, oo) to a collection v = {(([,, j= of observed implied volatilities. Naive strategies such as cubic interpolation are ill-fated: Surfaces generated by such interpolation rules will in general correspond to Call option prices that are exploitable by so-called arbitrage, namely costless trading strategies generating a guaranteed profit. In option markets, an arbitrage called static if set up solely from fixed positions in options and finitely many rebalancing trades in the underlying. Beyond simple interpolations, practitioners have devised ad-hoc parametrizations for implied volatility (in particular the aforementioned SVI), which are not expected to perfectly match all reference prices. Instead, the model parameters are optimized with respect to an objective function that measures market price discrepancy and includes penalization terms ruling out static arbitrage. These penalization terms are commonly formulated on the basis of the following theorem, which summarizes the shape constraints of the implied volatility surface (Gatheral & Jacquier 2014 Lucic 2021 Roper 2010).
Theorem 2.1 (Volatility validation). Let 0: (0, 00) x R- - (0, oc) be continuous and satisfying:
(i) Calendar arbitrage: For each ls E R. 6(-, /F non-decreasing and vanishes at the origin.
(ii) Strike arbitrage: For every > the slice 0, ((T,) .) is of class C² with
(22kk) 2
But(T, " 0r, 0,0 020
and lim sup ktoo
k
T'
where
But(T, ki to v1, U2) (1 (1+ (T, vo)0117)) (1 (1+
k, VT) + Ug
Then, (T, BS(T, k 8(T, k)) defines a Call price surface that is free from static arbitrage.
Condition 2.1 is equivalent to option prices increasing in expiry, while Condition 211 arises when computing the implied probability density Jr of the underlying:
fr(.) d2- .)) But(r, u, dev. 0z),
wVT
where y is the probability density of the standard Normal distribution. Since probability density needs to be non-negative, equation 3 explains why Condition 2. ((i is required.
3 NEURAL OPERATORS FOR DISCRETIZATIIO SMOOTHING
3.1 BACKGROUND: NEURAL OPERATORS
We provide full details about notations and terms, well additional context in Appendix A.
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(a) Relative trading volume per quoted interval, averaged over S&P 500 Left: Rectangular domain w.r.t time-to-epprryllg- moneyness (99.9% trading volume with maturities < year), Right: Rectangular domain w.r.t. transformed coordinates (96.6% of trading volume while more evenly populated); boundary delineated in Left.
(b) Scatter plots of example sets of option quotes v1, V2. V3-
dataset 2012-2021.
Observed on 15.10.2012, 18.05.2017, and 04.01.20211 respectively, at 10:50:00, each.
Figure 1: Spatial distribution of option quotes in tameett-exppryllog-momeye domain.
Philosophy The development of neural operators is based on the philosophy that observed data a = {ar] arises as the evaluation of a latent function a: D + RR.,, defined on some domain D D DR, at a discretization л = {-r} 12 of D. That is, a = a/x, or a₁ a(x), 1 1=1,...,,
input-output relationship of data a + u is then "really" described by an operator F: A - U between function spaces A and U. Neural operators are abstract neural network architectures F"; A + U, with implementations that integrate equation 4 consistently across the variable dis- cretization n.
Technicalities review the core concepts neural operators from Kovachki et al. (2023). Let D be bounded domain in Rd and A and UI be Banach spaces of functions mapping from D to Re and RR'a respectively. Neural operators are finitely parametrized mappings F A + U with universality for continuous target operators and with discretization-innarree
implementations F0. In the space C(AA, Lf) topologized by uniform convergence on compacts, the architecture F is called universal if { (F)) Jace dense in C(A,lf), with + the parameter set. An implementation of Fd an algorithm e0 which accepts observed data a = al and outputs a function u E l and is such that (-)--= (-I=) E C(A,U). Now, 0 is called discretization -tneiiiaationnnnaraannt if lim x(n) = F= in C(A,U), given a discrete refinemen D.
Let K be set of input functions, compact in A, and let e >> 0. In combination, universality and pesretization-invariane allow to posit the existence of parameters 0, such that for all a K,
(a) F(a) l1z S E,
(5)
irrespective of the particular discretization æ given that the data alT is scattered sufficiently densely across D. The training of neural operators is analogous to the classical dinee-dameniional setting. It happens in the context of implicit training distribution μ on the input space. A and aims at minimization of the generalization error
Ryp 0 + Ea || (alxx - F(a)llu,
(6)
through the use of gradient descent methods applied empirical estimates of equation 6 These esti-
mates constructed from training dataset D ((a(*) ) of features a a ( (0) and
labels u (( the basis of (mini) batching heuristics and are frequently transformed or augmented by additional terms through the use of apposite loss function..
3.2
OPERATOR DEEP SMOOTHING
Let
[e(z)){ be the collection of observed data, for example implied volatilities as in Sec- This notation silently adopts the neural operator philosophy, connecting the data point o(x))
tion
discrete refinement of is nested sequence (17(")) JnEN discretizations D for which for every
0, there exists N E 3 such that (Rg e): x o (N)) covers D.
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(a) Smoothed surface with input volatility quotes (bluc). 8ats 0.005, (0, v) = 1.020.
(b) Left: Slices of implied volatility scaled by vT: absence of cross- ings indicates absence of calendar arbitrage. Right: Implied probability density factor; positivity indicates absence of butterfly arbitrage. In fact, (e v) and Cim (8, v) O.
Figure 2: Operator deep smoothing of quotes v from
at 11:50:00. Compare Figure 13
with coordinates xı and hinting latent function D R giving rise the observed values. The smoothing or interpolation task consists of constructing an appropriately regular estimate é of The operator deep smoothing approach uses neural operator F trained using historical data, to generate i := F.9 (v). fundamentally lever- ages the discretization -nnvariancee produce consistent results even the "sensors" z; of the latent function a change in availability and/or location in the domain D. This the situation for volatil- ity smoothing. where the sensors xy (71, k)) the (time-to-eppiry log-moneyness) coordinates of the quoted options, which as noted in Section and illustrated in Figure Tb move with the market.
........pp. from the data v 1/sT. with {......... Tpp.
Methodology Translating the smoothing task to an operator learning problem, we find that the target operator is the (continuous) identity operator
F C(D) + C(D), U A
meaning that F = Fl and A = 4 = CD)). The training dataset the collection D = {v(() of historical data (labels and features coincide), and we train a suitable neural operator architecture ño such that
|F (v) < E, for all E D, where v denotes the postulated latent function of which we observe v/., and some given error tolerance for the smoothing task. Instead of minimizing empirical estimates of the co-nom F (v) ul/z.. however, we suggest a fitting loss based on the root mean square relative error,
- -(()))))<(<))
Cm(0, v) =
IV [(x))
ZEn
for its smoothness and invariance the scale of the data Additional engineering techniques, such as subsampling of inputs during training, are explored in our practical investigation in Section 4
Depending on the application, smoothing task may subject to constraints. For volatility smoothing, the smoothed surface 80 = F must be free static arbitrage (Theo- rem 2. This is effectively enforced by augmenting the loss function with additional penalization terms, moving away from a pure operator learning problem. This does not only promote the rel- evant properties in the neural operator output but can also help define it when faced with sparsity of data in the domain D (in this context see also Li et al. (2021)). We handle the strike arbitrage constraint 2. (ii) via
Constraints
Laui (f: vf:
=| 80, 0000000000000000000 0000 - |,
Empirical estimates of co-norms and T.2 no-mo are equivalent loss functions finiteddimensional spaces.

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where we ignore the asymptotic condition since experiments focused the bounded do- main (Figure Lap. The inclusion of & promotes strictly positive implied densities (we will use 10 10-",, ") while choose the 1-norm to induce sparsity in the constraint violation. The calendar arbitrage constraint (() can be tackled analogously with
Leal (0; v) (0+ [(T.k) - 20 (T, -E) -1,
(10)
where again we ignore the asymptotic condition since D is bounded away from zero time-to-expiry.
Interpolating graph neural operator Various neural operator architectures exist, mostly arising from the kernel integral transform framework of Kovachki al. (2023). Most prominently, these include Fourier neural operators (FNO), delivering state-of-the-art results on fixed grid data, as well as graph neural operators (GNO), able to handle arbitrary mesh geometries (both reviewed in Ap- pendix [A.I]. While highly effective with documented universality, these neural operators are not directly applicable for interpolation tasks their layers include a pointwise-applied linear transfor- mation, which limits the output to the set of the input data locations. Dropping this local transforma- tion results in an architecture proved to retain universality Kovachki al. (2023) and that least for its implementation as a GNO allows to interpolate functions. On the other hand, Lanthaler jet al. (2023) proves universality for the architecture combining the local linear transformation with simple averaging operation, suggesting the fundamental importance of the collaboration of local and non-local components for the expressivity neural operators. This was noted in Kovachki et al. 2023), for whom retaining the local components can be "beneficial in practice", and confirmed in our experiments, where a purely non-local architecture led to substantially reduced performance.
We therefore propose ability to handle irregular mesh geometries. We use a purely non-local first layer (dropping the pointwise linear transformation), and use it to produce hidden state at all required output locations, enabling subsequent layers to retain their local transformations. Since GNOs do not theoretically guarantee smooth output, augment the training with additional regularization terms such as Creg (0; v) = || with A the Laplace operator, and provide a full description in Appendix B
architecture for operator deep smoothing leveraging GNOs' unique
4
EXPERIMENTS
We detail our practical investigation of the operator deep smoothing approach for implied volatility.
4.1
MODEL TRAINING
Dataset and splits We perform our numerical experiments using 20-minute interval CBOE S&P 500 Index Option data from 2012 to 2021. The dataset amounts to a collection of 49089 implied volatility surfaces and just above 60 million individual volatility datapoints (after domain trunca- tion). We refer the reader to Appendix C. for details on the preparation of the dataset. We allocate the first nine years of data (2012 2020) to training, keeping 750 randomly drawn surfaces for validation purposes, and use the final year of the dataset (2021) for testing. This yields a training dataset Dirain containing Prrain 43442 surfaces, validation dataset D containing fhat 750 surfaces, and a test dataset Diest with Peest = 4897 surfaces.
Data transformation Motivated by Figure la we transform time-to-expiry and fog-moneyness via p / and z k/p. Intuitively, this transformation converts the "natural" scaling of implied volatility by the square root of time-to-expiry to a scaling of the input domain. From here on, we consider the domain in these coordinates, setting D = (Prnin: Panax) X (zmin 2max) = (0.01, 1) x (-1.5,0.5)
(T. -ooooddniasssss,, D becomes a cone-shaped region, that, on average, contains 96.6% of traded options (with time-to-expiry below one year) and, with respect to (p. 2) coordiiaass, is more evenly populated, improving the numerics.
Model configuration We employ the interpolating GNO as introduced in Section 3.2 and de-
scribed in detail in Appendix B The model hyperparameters (giving rise to 102529 trainable pa- rameters in total) were identified by manual experimentation and are detailed in Appendix C.2 We perform ablations for the connectivity of the graph structure (and thus for the tradeoff between ex- pressivity and computational complexity) in Appendix C.6
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(araa:
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0.01
(dm)):
SVI
OpDS
0.0015
(bid-ask spread)
0.0010
0.0005
2012 2013 2014 2015 2016 2017 2018 2019 2020 202 FehMarApcMaayyuu 20
Jul AugSep OotNovDee
T
Figure 3: Benchmark of our method (OpDS) vs. SVI, split between training period (left half; 2012 2020) and testing period (right half; 2021). Top and middle: Surface-averages (6ams) and (fmm)): shaded regions depict respective interquartile range. Bottom: Surface-average bid-ask spread (s).
Loss function We implement Vega-weighted version of Lat (equation 8) We compute Lbat (equation 9 using finite differences on a synthetic grid. We use a multiplicative formulation of Lea that is invariant the level of implied volatility. We provide precise description in Appendix and perform an ablation study for the weighting of the arbitrage losses in Appendix C.7
C.3
Model training We train the GNO for 500 epochs on Dirain using the AdamW optimizer with learning rate À = 10-4 and weight decay rate 3 = 10-5, and use a pseudo batch size 64 by accumulating gradients. We randomly subsample the inputs and randomize the grids which we compute the arbitrage losses. The training is performed in around 250 hours using an NVIDIA Quadro RTX 6000 GPU. The validation loss reported in Table
4.2 RESULTS
Evaluation metrics Let [u(x)) Jren be a collection of observed implied volatilties and é the smoothed surface. We measure absolute relative error:
() v(z)) 16(x) v(z))/u(z)
The average value of dabs over the whole surface v the mean absolute percentage error (or MAPE); denote it by (8at).. As in Corbetta et al. (2019), we moreover realize the importance of analyzing smoothing algorithm
terms of nominal price error relative to the bid-ask spread s(x)
(x, (z)) [S, Vbia (x)(())) 5 We define
6sge (6(x), u(r) 2| BS+ 6(x)) BS =(=)))/s(x))
Asp (()),((())) < indicates that the prediction ((x) for the option lies within the bid-ask spread.
Evaluation and model finetuning During production use, the GNO would be retrained regularly using the most recent available data. We emulate this procedure for our benchmark in Figure Following the evaluation of the first month test data (January 2021), the GNO re-trained for TO epochs this data (with each mini batch augmented by equal amount of data from the training dataset Drrai),, before the next month is evaluated. The process is repeated until the entire dataset Diest is assessed. This fineuuningevvaluation procedure takes circa 1.8 GPU hours per month.
(2))( and (x) the implied volatilities corresponding to Bid and Ask option prices while BS_ the Black- Scholes formula for Call (resp. Put) options positive (resp. negative) log moneyness values:
(b (7. k w)) (d w)). BS (T, k v)
k 0
( w)) (T,
& 5500
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0.0
Figure 4: Average spatial distribution of benchmark metrics and arbitrage terms over Prest The terms But(-, 0, O₂ é, i)) and 0-[ appear in Theorem 2.1 It is apparent that depicted values are strictly positive (indicating absence of arbitrage), lying above the threshold of 10-3
Analysis It is apparent from Figure 3 that operator approach substantially improves SVT' smoothing capabilities, with respect to both 6ab and 6xp о Our approach, with monthly finetuning, smooths the volatility surface with MAPE of around 0.5%, while SVI fluctuates between 1% and 2%. The various figures in Appendix D illustrate the qualitative improvements of our method over SVI. Moreover, our approach appears highly competitive with Ackerer al. (2020), which performs instance-brr banbice-b--istaanee volatility smoothing using classical neural networks and reports a MAPE of around 1 % for synthetically generated data _
We reenact its backtesting for the period Jan-Apr 2018 using our method and summarize the results in Table 5 in Appendix C.5
the historical tightening of the bid-ask spread (displayed in the bottom), driven by increased competition on the S&P 500 option market. This explains why dep small early the training dataset, both for our approach and SVI, while Jass is large: Wide spreads make dyp more lenient error metric but are accompanied by noisier prices, necessitating greater need for correction by the smoothing algorithm, in turn captured by
important fact to consider when analyzing Figure
abs Similarily, spikes in (which indicate periods of market stress) help explain spikes in dabs-
Complementary to Figure Figure resolves the error metrics as well as the terms controlling the absence of arbitrage spatially, averaged over time. dabs tends to be larger the Call side (positive mog-moneyness), in accordance with Call option's 's noisier prices (Call options experience less trading than Put options). Moreover, we discern that, average, the smoothed surfaces completely free of arbitrage (indicated by non-negativity).
Generalization To test the generalization of our approach, we procure end-of-day options data for the S&P 500 (SPX), the NASDAQ-100 (NDX), the Dow Jones Industrial Average (DJX), and the Russell 2000 (RUT) from the OptionMetrics Ivy DB US database, accessed by us through the Wharton Research Data Services (WRDS). We evaluate the trained operator on the data for the month of January 2021 (right after the training period on the CBOE S&P 500 intraday data), and report the averages of (8at)) and (dspr) well as the average arbitrage losses Leal and Lru in T Firstly, our method maintains its performance on end-of-day S&P 500 data, validating the soundness of our approach: While end-of-day data is slightly different from intraday data, method still yields small error metrics and arbitrage-free prices. Secondly, the method generalizes well to other indices. We want stress the fact that our operator has solely been trained intraday S&P 500 data. Its accurate and virtually arbitrage-free output on end-of-day data of other indices is a strong indicator of the robustness of our approach. We provide further example plots for these datasets in Appendix D.3
"We produce the SVI benchmark as described in Section
"Compare Table of Ackerer al. (2020). which note, however, performs smoothing larger strikefexpiry domain.
*We emphasize the following: First, the backtest involves dropping 50% of points for each surface, and our trained operator continues perform accurate smoothing, strong indicator of the robustness of our approach with respect suhsampling of inputs. Second, while Ackerer to perform the backtest once, our operator approach enables us to run 25 repetitions in around two minutes on a consumer grade laptop CPU, which is the average time that it takes Ackerer et al. (2020) to train one network.
(020) requires train 61 neural networks
Compare, e.g., Figure 11 and Figure

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Table 1: Average error metries and arbitrage losses for end-of-day options data for US indices in January 2021. The GNO has been trained solely on intraday S&P 500 data from before 2021.
SPX
NDX
DJX
RUT
(dahs)
0.00272
0.01057
0.01629
0.00885
(spe)
0.64423
1.53306
0.20736
1.03183
Cout 6.41e-05
2.16e-04
1.61e-03
4.48e-06
Leal
0.00000
0.00000
3.13e-08
0.00000
5 DISCUSSION
Summary We provide a novel method for implied volatility smoothing, resulting from an appli- cation our general operator approach for discretization-inves leverages a GNO to directly map given data consistently across size and spatial arrangement smoothed surfaces, transcending classical parametric smoothing techniques. the example of volatility smoothing, benefits include a massively simplified online calibration process.
data interpolation. The approach
Learning from large datasets By moving the application of neural networks from the instance- by-instance level (Ackerer et al. (020) the "operator level", we leverage the information contained in the entire training dataset for the smoothing of every single surface. In other words, our method "unlocks" large historical options datasets for volatility smoothing. We argue that substantial outperformance against against/Ackerer /Ackerer et al. (2020) in the "Extrapolation- -stting of the benchmark detailed in Table of Appendix C.5 to this circumstance.
The discretization-invaainee of the GNO entails that our method is robust with respect subsampling of inputs. In practice, subsampling of inputs occurs in the context of outlier removal. In the example of volatility smoothing, certain quotes may be determined spurious. Simply removing anomalous datapoints from the input is compatible with our method (moreover, we leverage this fact during operator training to improve generalization, compare Appendix CK
Subsampling of inputs
Compression Figure 3 makes the compression qualities of the operator approach apparent: We compute the entire historical timeseries using single GNO instance, with around 100 thousand parameters. Evaluating the SVI benchmark, on the other hand, requires 61454 model instances (one per slice), a total of 307270 parameters. A comparison with Ackerer al. (022), which for each smoothed surface trains new neural network of around 5085 parameters
striking: Smoothing of the CBOE dataset 2012-202 2021 at its 20-minute interval frequency would require more than 200 million parameters (more with rising frequency). At the same time, we expect our GNO to perform accurate smoothing over the entire training period and beyond (with regular finetuning), and our model instance remains fixed, even when moving to higher- figher-frequency data.
Limitations and perspectives Compared to ad-hoc volatility parametrizations like SVI, our neu- ral operator approach loses interpretability of parameters, which for some practitioners may be stringent requirement. This disadvantage is generally shared by neural network based engineer- ing solutions. Moreover, some situations dimensionality reduction (even without interpretability of parameters) may be a desirable additional feature that is not directly achieved by our operator approach. Combining the VAE method (Bergeron al. 2021) with our operator approach could lead to further promising potential applications of neural operators, Huang et al.] (2024) introduces for in-
neural mappings, which generalize neural operators to mixed GNO-based encoder, fit to handle put or output spaces. This motivates a discretizationninaariie raw incoming market data, and a classical decoder to extend the operator approach VAE-like architecture.
'CCmputed as the sum of 120 = 3 40 parameters for the input layer. three times 1640 = 41 x 40 parameters for the hidden layers, 41 parameters for the output layers, plus additional parameters of the SSVT prior and scaling parameter.
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REPRODUCIBILITY
Primarily, we ensure reproducibility by providing the codebase and model weights used to produce all results in this paper as part of the supplementary material. The codebase includes the data pro- cessing components, the GNO architecture, the loss functions, the error metrics, the production of the SVI benchmark, the notebooks used to train and evaluate the models (including hyperparam- eters and data splits), as well as the notebooks to produce the plots and tables in this paper. The full reproduction of results on intraday data (in particular of Figure is contingent on access to the proprietary CBOE options data, which we are not allowed to provide. In fact, we have stripped the codebase from intermediate benchmarking artifacts that would expose the proprictary data (which some notebooks for the plots rely on). The dataset can be purchased from CBOE, but is expensive. The OptionMetrics end-of-day options data for the suite of indices considered in the final paragraph of Section[4] 4 on the other hand, is more readily and freely available to researchers with subscriptions via the Wharton Research Data Services (WRDS) platform. The provided code allows directly reproduce the experimental results, in particular, Table and the plots in Appendix D.3 To do so, one would need to download the data from WRDS, persist it at prespecified location detailed in the codebase, and then run the respective notebooks, which automatically load the trained model weights.
To avoid any unclarities in our technique, the Pytorch implementation of our general graph neural operator architecture follows the mathematical definition given Appendix closely as possible. Moreover, the concrete steps undertaken as part of our experiments are detailed in Appendix C
Appendix C.I gives a summary of the processing of the options data.
The hyperparameter configuration of our model finally employed in our experiments is detailed in Appendix C2
The loss functions and their weights are explicitly defined in Appendix
ETHICS STATEMENT: DATA AVAILABILITY
The effectiveness of our method is fundamentally tied to the quality and frequency of the options data used for training. Our operator was trained using proprietary dataset with 20-minute interval frequency, which may not be accessible to many researchers due to its cost. More easily available datasets, such as OptionMetrics data (e.g. freely available from the Wharton Research Data Ser- vices), usually only provide daily snapshots of volatility surfaces (end-of-day data). We acknowl- edge that training using such pwer-ffrquency data may yield an operator that does not have the same smoothing performance as ours. This raises important ethical questions about data accessibil- ity in financial engineering research, where high-quality, high-frequency data is often locked behind paywalls. In our case, the following strategies could potentially mitigate the resulting limitations: Augmenting low-frequency datasets with synthetically generated data based on established paramet- ric models like SVI, or combining data from multiple indices to increase the effective sample size. Practitioners working with lower-frequency data might want to carefully evaluate these approaches and consider the relationship between their data sampling frequency and their intended use case.
ACKNOWLEDGEMENTS
Access to data was made possible through the Clearify Project funding by an Imperial College SME engagement grant jointly with Zeliade Systems, as well as the MSc in Mathematics and Finance, Imperial College London. AJ gratefully acknowledges financial support from the EPSRC grants EP/W032643/1 and FP/T032146/1 RW was supported by the Department of Mathematics, Imperial College, through the Roth scholarship scheme. Computational resources and support were provided by the Imperial College Research Computing Service (Harveyl (017). For the purpose of open access, the author(s) has applicd a Creative Commons Attribution (CC BY) licence (where permitted by UKRI, 'Open Government Licence" or 'Creative Commons Attribution No-derivatives (CC BY- ND) licence' may be stated instead) to any Author Accepted Manuscript version arising.
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Nikola Kovachki, Zongyi Li, Burigede Liu, Kamyar Azizzadenesheli, Kaushik Bhattacharya, An- drew Stuart, and Anima Anandkumar. Neural operator: Learning maps between function spaces with applications to PDEs. Journal of Machine Learning Research, 24(89):1-977 2023.
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A
NEURAL OPERATORS
We give a review of the kernel integral transform neural operator framework of Kovachki et al. and expand in more detail its graph neural operator.
2023
Notation and terms Let A and U be input and output space of an operator learning problem, as introduced Section 3.1 Then, A and U are Banach spaces of functions + IK and D - R respectively, where D is (bounded) domain in IR.. The mathematical analysis of neural operators Kovachki al. (2023) summarized hereafter, as well as the definition of universality and discretization-inaarii
in Section 3.1 make use of the following terms and notations.
bounded and connected open set that is topologically regular (in the sense that it is the interior of its closure). A domain D is Lipschitz if its boundary locally graph of a Lipschitz continuous function defined on an open ball of Rd-1 An open ball - for any metric space X' (X, d) is the set
In the context Kovachki al. (2023). domain
(x, = {y E X: d(y, <E).
A discrete refinement of X is nested sequence (π₂) of discretizations of X (finite subsets of X), such that for every & >> 0 there is N E N such that (B((x,)) e): I E nN} covers X.
We consider the space C(A,U) U)) of continuous operators between A and 4. C(A,U) is topologized by uniform convergence on compact sets. With respect to this topology, sequence (F₂) mee in C'A,U) converges with limit F' E C(A,U), if, for every & and every compact set in A, holds
lim | Fllco, =0 n-000
Here,
H 00 = sup [|H()),,, H HE C(A,))))
aEK
It is well known that this topology on C(A,U) is induced by the metric
DC | F Ber
p(F,G) Σ 1V (ec
F,G,E
Therefore, the notion of density in C(A,U), used define universality of neural operators in Section 3.1 is well defined.
A.1 KERNEL INTEGRAL NEURAL OPERATORS
Kernel integral transform neural operators and universality operator consists of the sequential application of:
kernel integral transform neural
1) lifting layer
Lpp: [a: D - R + [ho: D Re0 ho(x) )((x)))((()))}
P(a(x)))
given by the pointwise application of function P: Ret + R
2) The forward propagation through J neural operator layers Lo...., LJ-1i
[bo D + RoO Le, [h1: D RR 4 ب [hj-1: D + R*); each layer Lj operates
hy+1(y) - (Wshs(o) + x))(nx)))(())d by(1)) E D,
(1I)
where
W, e B5111e is weight matrix applied pointwise,
Ky C(D D, Xe)+ is a kernel function parametrizing the integral transform and subject to integrability conditions,
The bias term by is itself function from D RS#+1
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j is classical neural network activation function.
3) A projection layer
Lo: [hJ: D - RR + [u: D - Rim u(x) (h)(()))
given by the pointwise application of function Q: RR - IR
In practice, components (lifting, kernel functions, projection) are implemented as classical feed- forward neural networks (FFNs). This neural operator architecture is universal in the following sense.
Theorem (Universal Approximatio;; Theorem of Kovachki (bounded) Lipschitz domain. Assume:
2023 Let D be
A = Whas (D) for & N20 and 1 S P1 < 0c, or A= C(D).
(D) U Whisss (D) for N20 and S p2 < oc, or = C(D).
Then, subset of kernel integral neural operators, with kernel functions and bias functions taken from a suitable set of FNNs, is dense in C(A,U).
Discretization-nnariantt implementations Consider neural operator F and x { [oi)[=] be a discretization D. To make sense of a basic dnscretizationnnnaarii implementation for Fd associate with partition (D...... Dp) of D for which Aa(D:) > 0 and E D; for = 1,..., Here Às denotes the Lebesgue measure on Rdd Consider the following implementation of 1 (writ- ten in terms of a single constituent layer L = (W, k. b, o)):
(v) = (h(y)) + Σ 2. [(l.a))M)A)(DD)) + Mu)) WER D.
(12)
Kovachki et al. (2023) establishes the following.
Theorem A.2 (Discretization Invariance; Theorem Kovachki et al. (2023 ) Let Fo : A - U
be a kernel integral neural operator, where A and U both continuously embed into C(D). Then, the
implementation of Fd based on equation 12 is discretization invariint as defined in Section 3.1
equation 12 suggests the straightforward (quasi) Monte-Carle inspired implementation
(()+)(y)
(Wh())+
Aa(D) |T|
Σ xxyyx)) ( EE D.
(13)
Most effectively, low-discrepancy sequence D.
A.2 GRAPH NEURAL OPERATORS
The curse of dimensionality makes the direct implementation equation 13 prohibitively expensive in practice. Instead, Anandkumar et al. (2020) introduces gruph neural operators (or, GNOs, for short) which replace the kernel integral operation at the heart of the framework by sum ap- proximation and organizes the constituent terms using directed graph structure: The discretiza- tion я = (#1,..., Fm of the input data h/ enriched with a directed graph structure Gn = (V, (V,E) E), allowing the following implementation f of FO;
L(b+)))) = WWhw) (i
Σ (
nly, x))x))+))++ + +(y)))+b())))))) y V.V.
(14)
Here, (y) is the set of so-called in-neighbors clear that. to compute output at y, the point must be included node into the graph Gn. On the other hand, it necessary to drop the local linear transformation with W if y is not part of the input data locations (compare our discussion in 3.. It important reconcile the input and output
in the graph Gb: x E Nin(y) iff (x. y) E E.
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locations of the layers when creating the graph structure to enable an efficient implementation using message passing algorithms.
The graph structure can be meticulously adjusted to implement various complexity- tech- niques like Nyström approximation or integration domain truncation that effectively aim at sys- tematic reduction of the size of (y)) the naive implementation equation 12 is recovered for the case of a complete directed graph (with self-loops) for which Nin = т. Note that choosing Nin (v) strict subset of breaks the guaranteed smoothness in the GNO output.
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B INTERPOLATION GRAPH NEURAL OPERATOR
We detail our modifications of the GNO architecture.
Let {u< be the given data.
Graph construction Arguably part of the model architecture is the graph construction: Compile the set Yout {ur 1= of points y E D at which compute the smoothed surface ö(y). During operator training. this will be the set of input locations (to compute the fitting loss) as well as any additional locations needed to compute auxiliary loss terms (the arbitrage losses Luu (0, v)) and (0, v) in the case of volatility smoothing). For each y E Tout, we construct the set of in-neighbors from the set of input data locations:
(y) Tin
(15)
In other words, we employ Nyström approximation with nodes limited to the input data locations. This is an important prerequisite to enable the use the GNO architecture for interpolation tasks (or, more generally phrased, allows us employ kernel functions with input skip connections). We set Gy = (Tout: E), where
E = U ((r,v): E Nin
yc
Forward propagation
Given Gv, perform the first step of the forward propagation follows:
ho(xx) = Po(u(x)),
x E Tin
h1 (1) = (on 2o) (C(hoo: + bo). y/ E Taut-
For the subsequent layers j = =.....,J -
we then proceed using the classical scheme:
hy(()) (h (v));
h1++(1))= (o, 2,) (w) +b,) y E Tout
In the above:
Pj: R9 - Rd and 2, Ro - RS are layer-individual lifting and projection, in view of A.1 implemented simply as FNNs.
W, E x&xx, weight matrix (not present for j = 0), while by R is a constant bias term.
Kj is the sum approximation of the kernel integral with kernel weight function n. D2 x R x & I 17511xe xē; and kernel bias function <²y: D2 R2 1 + Rē:- (both with state and input skip connections):
K, (,(h)v)))
Σ
bz),
+ 2,
[vin()) (.Ni..))
Both w.W and R are implemented as FNNs in our case, again to satisfy the requirements of A.1 and to keep things simple.
Note that omitting the local linear transformation in the first layer allows to extract the fist hidden state b (y) for all y E Tour from the lifted input hg, which is defined solely for the input locations x E Tin Providing each layer with its own lifting and projection allows to separate the hidden channel size co....,CJ from the the dimensions čo ... ōب of the space in which the integral transformation performed. Moreover, the individual lifting and projection help re-parametrize the state before performing the integral transform (inspired by the succesful Transformer architecture), which allows to keep the size the kernel weight matrix low.
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SUPPLEMENTARY INFORMATION: DATA, MODEL, TRAINING, EVALUATION C
This section contains additional information regarding our empirical study of the operator deep smoothing method for implied volatility smoothing.
C.I DATA
Data source Our numerical experiments are based on the "Option Quotes" dataset product avail- able for purchase from the CBOF. Our version of the dataset contains relevant data features for S&P 500 Index options for the years 2012 through 2021 and is summarized on a 20-minute interval basis.
Data preparation We compute simple mids for options and underlying by averaging bid and ask quotes and use this aggregate as a reference price for all our subsequent computations. We cal- culate discount factors and forward prices from Put-Call parity using the industry standard tech- nique based on linear regression. We compute time-to-expin in units of one year well as log- moneyness as defined Section 2 We extract implied volatilities using the py-vollib- vectorieee project available in the Python Package Index at the location https: //pppi.ppp.....m
hetg/roojccto implements a vectorized version of Jäckel (2015)'s Let's s--beoertiiinall state-of-the-art method for computing implied volatility. We discard all implied volatilities money options, or, in other words, we compose our implied volatil- ity surface from Put options for non-post non-positive log moneeneesss values and Call options for positive log-moneyness values.
wy-voolib-vettorizeed
py-vollib-vectorizz
C.2 MODEL
We proceed detail the hyperparameter configuration of the modified GNO architecture introduced in Appendix B
The choice of in-neighborhoods The construction of the in-neighborhood sets for the graph neu- ral operator is crucial hyperparameter choice, fundamentally dictating the computation routes (and thus complexity) of the forward pass of the model. We already explained Appendix B that we employ Nyström approximation with subsampling from the input data nodes, to unlock the GNO for interpolation tasks. Additionally, we employ truncation. Truncation limits the spatial extent of the in-neighborhoods and is a way to incorporate information about the locality structure of the learning task at hand directly into the graph neural operator architecture. Since implied volatility smoothing requires limited global informational exchange along the time-to-expiry axis, impose the following restriction on the nn-neighoorhood sets (y):
(wn C. (N),
(16)
where
(P 2,) = -
Pyl S sp}
(17)
is the set of all available options (pt, 21) contained in the slices with time-to-expiry pl close than p to the time-to-expiry Py of y. We explain our reasoning more precisely:
The input data for volatility smoothing is not arbitrarily scattered over the (p, =)-domain, but arranged as dense z-slices that are sparseley distributed along the p-axis (compare Figure 5 as well as three more examples pictured in Figure [b)). Condition 2. 2.) of Theorem 2. imposes monotonicity of the output surface along the time-to-expiry axis. This constraint is inherently
"local": To generate compliant output surface, it is sufficient for the hidden states at given output location to receive information from their immediate neighboring slices |T| We computed the maximum distance (with respect to p-coordinates) between slices over entire dataset as A m R 0.269, which is thus established as a lower bound for 7. and finally explains our choice p = 0.3. We note that because we use three hidden GNO layers (see below) - the domain of influence of each input point ultimately is unrestricted: The compositional structure allows information to travel slice slice in steps of length ò = 0.3, which amounts to total distance of 4 x 0.3 = 1.2. This exceeds the size of the considered domain D = (Pmin- Pmax.) X (Cmin; zmax) in the direction of the p-axis. It is therefore not motivated to increase the level of p.
A collection of slices that is monotonously increasing in pairs is montonously increasing whole.
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We do not perform similar truncation in the direction of the log mee-mnnyynss axis. In particular, this allows all points in any given slice to connect with each other indiscriminately. which in view of the nonlinear shape constraint (i) of Theorem 2.1 is motivated. Moreover, such truncation would limit extrapolation distance in z-direction.
To concretely compute (y) for given y the basis of equation 16 employ the following low-diserepancy subsampling heuristic, parametrized by the hyperparameter K: First, we compute Ny)(y) and convert it to sequence ascending order. Of this sequence we take every k-th element, where is the largest step size such that the final number of nodes Nín does not exceed K. This gives (v) Note that, by sorting (y) and performing "sparse" selection, promote cow-disceppancy properties for (y), which intuitively aid the convergence properties of the kernel integrals (compare
sorting it by two-dimensional Euclidean distance y in
The hyperparameter K, finally, constitutes upper bound the size of the My) It allows control the computational complexity of the model, in a trade-off, of course, with the expressivity of the GNO. After manual experimentation, we settle on value of K 50 and perform ablation study in Appendix C.6 to validate our choice.
fl Psl 1 p
0.8
(x)(
0.4
11100
110
0100
10000
0.0
1.50
1.25
1.00
0.75
0.50
0.25
0.00
0.25
0.50
Figure 5: Graph construction: Given data v = (UZ)eEn (S&P 500 on 10.05.2012 at 16: 16:10:00)
generate in-neigboorhood for output point by subsampling from (m 2n) by first sorting and then taking sparse selection, promoting low-discrepancy.
GNO layers and kernels The below choices amount to a total number of 102529 trainable pa- rameters.
We employ three hidden layers and channel size of 16: J= 4, and C1, C1, C3 16 (Co and Cy are determined as by the sealar dimension volatility data). use GELU-activations for the hidden layers and Softplus acttivtssactivatioon for the output layer (to ensure the positivity of the smoothed surfaces): oo...., JJ = GELU, and σı Softplus.
We retain Po.... PJ-1 and Q, as single-hidden layer FNNs with 64 hidden nodes and GELU- activations for the hidden layers. The remaining lifting and projections remain unutilized. In 16.
particular, Co io...., c,
We implement the kernel weight and bias functions as two-hidden layer FNNs with 64 hidden nodes and
for the hidden layers.
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Published as a conference paper at ICLR 2025
C.3 TRAINING
Loss function To ease notation we write 60 = =F (v). We implement a Vega-weighted version of the fitting loss equation 8
( [Tv]
1/2
(n(
Σ (x)
ZETV
Here,
V(x, v(x))
wy(x; Ener V(z, v(x))
V1,
where Vax, u(x)) the Black-Scholes Vega, the sensitivity of the Black-Scholes option price with respect to its volatility parameter:
V(p. 2, = 0, k. v) = c(di(t,k,))v
(18)
For the implementation of the no-arbitrage penalization terms Lba and Lealt, we first generate dis- cretizations "r (Pi.-.., -Pm) and nı (,, of [Pminn Pmax] and [zmin= Pmax]- We resolve
the derivative terms O.v0 and 02 the synthetic rectilinear grid π TTp x Ta using (central)
finite differences. Then, we translate Lba directly from equation
Lour(O:
1
Σ ZEx
(But(æ; (x())) De. 0 (r), A (x)) - -e)
where But made consistent with the transformed coordinates and we used obvious notation for the finite differences. We use 10-3, On the other hand, we enforce the monotonicity constraint of Theorem 2.1 using
5 (pit1, (pt, (P++12))/p3)
Leal (6; Tp T2) =
Pi
-6)
Pi+1
Compared to derivative based implementation. this implementation is independent of the scale and in our empirical experiments has provided an improved signal. Since the Nyström approximation employed by the graph neural operator (as well as the choice equation smoothness the operator output, we additionally introduce 1102 1|2 and T0 ||2 as regularization terms:
[6] break the guaranteed
Ereg.p (0: V.)
C (())
Σ .
60 (x))
V
FE
We compose the final loss function weighted of all terms introduced:
XntCCii (0; v), Abur ( v. Tp Tz),
C(A:V, (p; =
Seal V, Tp Rz);
Creg-p (0; V, TT X nz),
Avog-s Creg. (0: v, Tp x T
The specific weights are
Aua
Acal
Abul
Areg-p
Avog-z
1 10
10
0.01
0.01
The particular weighting of the individual terms has initially been retrieved by manual experimen- tation, led by the findings of Ackerer et al. (2020). To additionally validate our choices, perform an ablation study Appendix C.7
Validation loss
Table 2 displays descriptive statistics of the validation losses.
20

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Published as a conference paper at ICLR 2025
Table 2: Validation loss.
std
1%
25%
50%
75%
99%
C
0.0591
0.0807
0.0351
0.0450
0.0506
0.0578
0.1489
Lm
0.0182
0.0182
0.0066
0.0121
0.0162
0.0203
0.0479
Dut
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
Leal
0,0001
0.0006
0.0000
0.0000
0.0000
0.0000
0,0001
Lieg-r
0.8567
2.4065
0.3280
0.4702
0.5788
0.7703
2.7578
Lieg-z
0.7610
0.1403
0.4815
0.6488
0.7590
0.8679
1.0812
C.4 EVALUATION
Here provide additional results supplement our performance evaluation. Table and Table display descriptive statistics of our approach (OpDS) versus SVI. OpDS* refers to benchmarking without monthly finetuning. Moreover, we include Figure and Figure 8 which have been created just like Figure B and Figure 4 but without monthly finetuning (OpDS*) ). Finally, Figure 7 shows the average spatial distribution of benchmark metries and arbitrage term over the training dataset, which complements the same averages on the test dataset shown in Figure
Table 3: Descriptive statistics for surface-MAPE's (Dabs) over Deal/Diest.
mean
50%
99%
OpDS OpDS SVI
0.009/0.00s
0.007/0.001
0.003/0.003
0.006A0.004
0.008/0.005
0.01080005
0.021/D.007
0.009/0.001 007
0.007/0.001
0.003/0.003
0.006/0.07
0.008/0
0.0100.008
0.021/0:012
0.021/0.015
0.006/0.002
0.007/0.010
0.016/0.013
0.020/0.014
0.025/0.016
0.034/0.020
Table 4: Descriptive statistics for surface- averages (6xpe) over Dval'D DDsst-
lest-
meam
sid
1%
25%
50%
75%
99%
OpDS
0.479/1.266 .265
0.662N1.347
0.193/0.609
0.274/1.025
0.330V1.240
0.550/11451
1.526/2-453
OpDS
0.479/1.86 186
0.662/0.574
0.193/0.731
0.274/1.457
0.330/1 826
0.550/2.233
1.526/3. 445
0.877/0.826
0.715/3.320
1.646/3.914
3.710/5.247
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Published as a conference paper at ICLR 2025
(ataa)
SVI
0.02
OpDS
0.01
(dm)):
SVI
OpDS
0.0015
(bid-ask spread)
0.0010
0.0005
2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 FehMarrpccaayyuuu Jul AugSep NotNovDee
To
Figure 6: Benchmark comparison between operator deep smoothing without finetuning (OpDS*) and SVI, split between training period (left half; from 2012 to 2020) and testing period (right half; 2021). Surface-averager (fabs) (top) and (Rupr) (middle); shaded region indicate interquartile range. Bottom plot: Surface-ave aureage- spread (s).
8,30
8.p
But(- 8, a,i
a. 16
1.0
0.06
0.04
0.03
0.4
0.2
1.5
-1.0
-0.5
0.0
-1.5
-1.0
-0.5
0.0
-13
110
-0.
0.0
-15
-10
-0.5
Figure 7: OpDS: Average spatial distribution of benchmark metrics and arbitrage term over train dataset.
6pg
But(-, 0,0,, 0E0)
1.0
0.05
0.01
0.6
0.4
0:01
Figure 8: OpDS*; Average spatial distribution benchmark metrics and arbitrage terms over test dataset inon-finetuned model).
C.5 COMPARISON TO CLASSICAL NEURAL NETWORKS
We reproduce the experiment underlying Table of Ackerer al. (2020) using our operator deep smoothing approach. Given observed option quotes, involves dropping 50% of datapoints, and then measuring the MAPE of the smoothed surface at retained datapoints ("Train"") as well as different settings dictating how
dropped datapoints ("Test"). "Interpolate" and "Extrapolate" exactly the datapoints which to drop are selected (for details refer to Ackerer et al.) (2020)). The experiment is performed end-of-day S&P 500 data in the period from January to April 2018 and
22

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Published as a conference paper at ICLR 2025
averaged percentiles reported. Table reproduces the relevant of Table of Ackerer et al. 2020 ("DS") as well as our results averaged over 25 repetitions ("OpDS").
Table 5: Backtesting results ((5am), i.e. MАPH) of Operator Deep Smoothing vs. Deep Smoothing ("DS"; taken from [2]): quantiles in %, Jan-Apr 2018 end-of-day SPX data.
Interpolation
Extrapolation
Train
Test
Train
Test
X
905
450
Q95
(J05
4550
(/95
Q05
(50
Q95
Q05
150
Q95
OpDS
10
0.5
0.7
1.0
0.5
0.7
1.1
0.5
0.7
1.0
0.7
0,9
1.3
DS
10
0.5
0.7
1.2
0.5
0.8
1.2
0.4
0.6
0.9
1.2
1.7
2.4
C.6 ABLATION: NYSTRÖM APPROXIMATION
We explore the impact of our hyperparameter choice K = 50 introduced in Appendix C.2 ling the size of the Nyström approximation of the integral kernels. We perform an ablation study by resuming training of our trained GNO for the additional values for K 3, 5, 10, 20, 30, 40 well as = 60, 70. We focus on the data D2018 of the period Jan-Apr 2018, and perform two additional training runs starting from our final GNO-checkpoinl (trained for 500 epochs on Duain) follows:
control-
20 epochs each for K = 40, 30, 20, 1,,5, 3. in this order. We plan to understand how low K can be for our method to still produce meaningful results.
20 epochs each for K 60, 70, in this order. We plan understand how much additional information the GNO can extract by increasing the value of K.
Descriptive statistics for losses and evaluation metrics over Dmns itself are printed Table 6 and Table We can read from Table increasing K does not significantly improve performance for either on or ōer The mean values for both metrics remain relatively stable for K 50, suggesting diminishing returns with larger K (or, in view of the slightly increasing tendency, the need for additional training). On the other hand, reducing K below its original value of 50 leads to gradual degradation in performance. It is expected that very small K K-values, especially K 10, result in substantially poorer performance, but it is noteworthy that the progression is quite graceful. Table paints similar picture for the fitting loss Lit, while the auxiliary loss terms slightly increase as K incrases. We argue that the increased expressivity awarded by larger values of K leads to slightly more irregular surfaces, and thus slightly increased arbitrage loss terms. Finally, we argue that our choice of K = 50 is validated, where decreasing K is reasonable strategy when computational resources are scarce and accuracy requirements are not too stringent.
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Published as a conference paper at ICLR 2025
Table 6: Loss terms over D2018 for different values of K, after training for 20 additional epochs on 20118-
mean
std
1%
25%
50%
75%
99%
C
3
0.1113
0.0934
0,0577
0.0819
0.0978
0.1174
0.3771
5
0.1008
0.1022
0,0499
0.0714
0.0849
0.1018
0.3074
10
0.0914
0.1220
0.0412
0.0635
0.0742
0.0885
0.2739
20
0.0854
0.1484
0.0371
0.0580
0.0669
0.0799
0.2802
30
0.0852
0.1834
0.0360
0.0561
0.0650
0.0776
0,2916
40
0.0890
0.2341
0.0367
0.0561
0.0647
0.0779
0,3511
50
0.0906
0.2416
0,0388
0.0564
0.0644
0,0782
0,4900
60
0.0930
0.2956
0.0371
0.0555
0.0644
0.0773
0.5752
70
0.0889
0.2293
0.0371
0.0556
0.0642
0.0773
0.4404
Li
3
0.0639
0.0346
0.0237
0.0462
0.0575
0.0739
0,2017
5
0.0484
0.0243
0.0195
0.0369
0.0466
0.0557
0.1428
10
0.0354
0.0188
0,0132
0.0283
0.0340
0.0402
0.0681
20
0.0264
0.0168
0,0102
0.0202
0.0245
0.0294
0.0638
30
0.0234
0.0167
0.0088
0.0171
0.0215
0.0258
0.0785
40
0.0220
0.0169
0.0084
0.0157
0.0200
0.0238
0.0963
50
0.0207
0.0174
0.0084
0.0147
0.0183
0.0221
0,1122
60
0.0215
0.0186
0.0083
0.0150
0.0192
0.0230
0.1236
70
0.0217
0.0181
0,0081
0.0152
0.0195
0.0234
0.1218
Lou
3
0.0000
0.0001
0.0000
0.0000
0.0000
0.0000
0.0001
5
0.0000
0.0005
0.0000
0.0000
0.0000
0.0000
0.0000
10
0.0002
0.0035
0.0000
0.0000
0.0000
0.0000
0.0001
20
0.0003
0.0079
0.0000
0.0000
0.0000
0.0000
0.0000
30
0.0005
0.0117
0.0000
0.0000
0.0000
0.0000
0.0001
40
0.0008
0.0163
0.0000
0,0000
0.0000
0.0000
0.0002
50
0.0009
0.0168
0.0000
0.0000
0.0000
0.0000
0.0010
60
0.0012
0.0216
0.0000
0.0000
0.0000
0.0000
0.0004
0.0008 70
0.0155
0.0000
0.0000
0.0000
0.0000
0.0005
Ceal
3
0.0000
0.0006
0.0000
0.0000
0.0000
0.0000
0.0001
5
0.0001
0.0006
0.0000
0.0000
0.0000
0.0000
0.0001
10
0.0001
0.0007
0.0000
0.0000
0.0000
0.0000
0.0000
20
0.0001
0.0007
0.0000
0.0000
0.0000
0.0000
0.0001
30
0.0001
0.0007
0.0000
0.0000
0.0000
0.0000
0.0003
40
0.0001
0.0007
0.0000
0.0000
0.0000
0.0000
0.0004
50
0.0001
0.0008
0.0000
0.0000
0.0000
0.0000
0.0005
60
0.0001
0.0008
0.0000
0,0000
0.0000
0.0000
0.0009
70
0.0001
0.0008
0.0000
0.0000
0.0000
0.0000
0.0009
3
1.1568
2.4882
0.3534
0.5468
0.6905
1.0271
6.9051
5
1.3544
3.0528
0.3823
0.5419
0.7105
1.1629
8.1210
10
1.4194
3.3473
0.3897
0.5763
0.8204
1.2643
8.6570
20
1.4614
3,2880
0,3676
0.6552
0.8875
1.3222
9.0101
30
1.5008
3.3752
0.3790
0.7009
0.9327
1.3734
8.8143
40
1.5716
3.5193
0.4026
0.7334
0.9835
1.4533
9.2937
50
1.5256
3.5376
0.4661
0.7785
1.0336
1,4189
9.2313
60
1.5924
3.6494
0.4175
0.7514
1.0053
1.4776
9.4071
70
1.5610
3.4815
0.3999
0.7508
0.9877
1.4551
8,8669
Lugz
3
0.7183
0.0900
0.4913
0.6589
0.7346
0.7832
0.8723
5
0.7134
0.0788
0,5375
0.6597
0.7193
0.7649
0.8887
10
0.7240
0.0773
0.5572
0.6623
0.7260
0.7773
0.8966
20
0.7413
0.1010
0.5583
0.6585
0.7384
0,8171
0.9738
30
0.7476
0.1177
0,5362
0.6563
0.7420
0.8326
1.0086
40
0.7545
0.1404
0,5280
0.6516
0.7467
0,8425
1.0482
50
0.7616
0.1523
0.5270
0.6533
0.7586
0,8520
1.0745
60
0.7593
0.1647
0.5255
0.6517
0.7542
0.8463
1.0617
70
0.7624
0.1489
0.5286
0.6538
0.7585
0.8552
1.0720
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Published as a conference paper at ICLR 2025
Table 7: Evaluation metrics over D2018 for different values of K, after training for 20 additional epochs on 22118
K
mean
std
1%
25%
50%
75%
99%
(Bats)
3
0.0414
0.0232
0.0159
0.0289
0.0376
0.0473
0.1427
0.0323
0.0140
0.0127
0.0242
0.0310
0.0379
0.0898
10
0.0230
0.0093
0,0081
0.0178
0.0221
0.0270
0.0500
20
0.0166
0.0078
0.0064
0.0127
0.0156
0.0194
0.0473
30
0.0145
0.0079
0.0053
0.0106
0.0135
0.0169
0.0529
40
0.0135
0.0083
0.0051
0.0098
0.0126
0.0152
0.0562
50
0.0127
0.0087
0.0050
0.0092
0.0116
0.0138
0.0571
60
0.0132
0.0101
0.0048
0.0093
0.0120
0.0145
0.0628
70
0.0133
0.0095
0.0048
0.0093
0.0121
0.0148
0.0622
(8mr)
3.7755 5
4.9387
0.8977
1.4838
2.0117
3.2423
24.8441
2.7120
3.3901
0.6315
1.1012
1.4551
2.4550
18.7650
10
1.8229
2.4049
0.4316
0.7320
0.9282
1.7497
12.9783
20
1.2243
1.6054
0.3171
0.4991
0.6227
1.1553
7,7893
30
0.9980
1.2846
0.2787
0.4229
0.5180
0.9376
5.6118
40
0.9307
1.2340
0.2742
0.3907
0.4768
0.8775
5.1398
50
0.8634
1.1477
0.2627
0.3639
0.4356
0.8218
4.4631
60
0.9012
1.2789
0.2647
0.3710
0.4483
0.8584
5.2339
70
0.9038
1,2559
0.2607
0.3773
0.4570
0.8498
5.2972
C.7 ABLATION: WEIGHTING OF ARBITRAGE PENALTIES
To assess the impact of weighting of the arbitrage penalties in the loss function, we perform the following experiment: We resume training of our trained GNO for 20 additional epochs on the full training dataset Drrain varying the weights A and Amr of Leal and Lbar More precisely, we equally weight both terms Leal and Lutt at the values Aar = 0, 1, 10, 100, 1000, 10000 (we include the original value Aub 10 to maintain fair baseline). We start each training run from our final GNO-checkpoint (trained for 500 epochs on Dusin with Aub = 10). The results are reported in Table & and Table and make the following observations:
The particular choices Aut affect the achieved loss terms in the expected ways. For a value of Auth > 10 all traces of the arbitrage penalties vanish from the table. the same time, however, accuracy (as measured by La. dabc. and & suffers. For choices Aub < 10. it is possible to read a non-zero average for the calendar loss from the table. For Aar = 1000 and Auo- At the same time, however, accuracy (as measured by Cat. dabs drel) suffers. Our choice Aab 10 is validated: Aua = or even Auo = 0 do not seem to unlock substantial additional accuracy of the GNO. If there is small effect, it comes cost of increased arbitrage in the smoothed surfaces, as measured by Leal and Lhut-
Acal has a counter-regularizing effect p-direction, and we suspect overfitting of the monotonic- ity constraint. Lour instead, remains stable for all values. Practitioners will be aware that the calendar arbitrage constraint is usually more demanding than the butterfly arbitrage constraint.
25

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Published as a conference paper at ICLR 2025
Table Loss terms over vaal for different values Xarb, after training for 20 additional epochs on
D Drrain
Aurh
mean
std
1
25%
50%
75%
99%
C
0
0.0577
0.0750
0.0341
0,0441
0.0499
0.0567
0.1312
1
0.0575
0.0733
0.0344
0.0441
0.0497
0.0568
0.1375
10
0.0583
0.0774
0.0351
0.0446
0.0501
0.0573
0.1452
100
0.0581
0.0599
0.0352
0.0456
0.0511
0.0580
0.1358
1000
0.0617
0.0658
0.0380
0.0480
0.0535
0.0609
0.1474
10000
0.1304
0.0802
0.0825
0.1023
0.1148
0.1381
0.2793
Cat
0
0.0180
0.0181
0.0066
0.0118
0.0160
0.0202
0.0457
0.0181
0.0182
0.0063
0.0118
0.0162
0.0202
0.0435
10
0.0182
0.0181
0.0065
0,0120
0.0164
0.0204
0.0469
100
0.0195
0.0315
0.0069
0.0123
0.0165
0.0206
0.0497
1000
0.0225
0.0479
0.0088
0.0136
0.0177
0.0218
0.0529
10000
0.0603
0.0669
0.0287
0.0401
0.0474
0.0652
0.1369
Cbut
0
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0001
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0001
10
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
100
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
1000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
10000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
Geal
0
0.0001
0.0006
0.0000
0.0000
0.0000
0.0000
0.0001
1
0.0001
0.0006
0.0000
0.0000
0.0000
0.0000
0.0001
10
0.0000
0.0006
0.0000
0.0000
0.0000
0.0000
0.0001
100
0.0000
0.0003
0.0000
0.0000
0.0000
0.0000
0.0000
1000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
10000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
Ereg-p
0
0.8019
2.1876
0.3091
0.4490
0.5489
0.7329
2.8897
1
0.7925
2.1203
0.3102
0.4410
0.5395
0.7268
2.9587
10
0.8236
2.2886
0.3301
0.4578
0.5573
0.7476
2.6687
100
0.7756
1.2831
0.3332
0.4866
0.5858
0.7758
3.4684
1000
0.8216
0.8114
0.3927
0.5394
0.6450
0.8389
3.5378
10000
2.0534
0.7765
1.2892
1.6319
1.8667
2.2504
5,5746
Lagg
0
0.7621
0.1394
0.4876
0.6498
0.7599
0.8690
1.0786
I
0.7633
0.1390
0.4876
0.6506
0.7596
0.8695
1,0829
10
0.7572
0.1379
0.4793
0.6477
0.7529
0.8619
1.0702
100
0.7596
0.1409
0.4909
0.6512
0.7554
0.8649
1.0812
1000
0.7467
0.1339
0.4903
0.6442
0.7414
0.8494
1.0558
10000
0.7507
0.1462
0.5013
0.6365
0.7422
0.8278
1.1014
26

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Table 9: Evaluation metrics over vall for different values of Xat. after training for 20 additional
epochs on Drain-
mean
std
1%
25%
50%
75%
99%
(aabs)
0
0.0108
0.0097
0.0045
0.0073
0.0098
0.0122
0.0265
1
0.0109
0.0100
0.0042
0.0071
0.0098
0.0122
0.0262
10
0,0110
0.0091
0.0044
0,0073
0.0101
0.0125
0.0260
100
0.0121
0.0217
0.0046
0.0074
0.0101
0.0125
0.0276
1000
0,0139
0.0291
0.0057
0,0086
0.0109
0.0134
0.0302
10000
0.0388
0.0360
0.0182
0.0265
0.0315
0.0419
0.0923
(8sp))
0
0.6138
1.0777
0.2218
0.3130
0.3728
0.7277
1.9947
1
0,6039
1.0790
0.2221
0.3062
0.3685
0.7026
1.9557
10
0.6186
1.0552
0.2200
0,3080
0.3775
0.7475
2.1704
100
0.7241
2.7257
0.2227
0.3137
0.3797
0.7378
2.4623
1000
0.9303
4.1817
0.2399
0.3480
0.4265
0.9299
2.8245
10000
3.1238
7.3292
0.5128
0.8566
1.3404
3.6653
14.2364
C.8
IN THE CONTEXT OF IMPLIED VOLATILITY SMOOTHING
Figure illlstrrate the fact that the coordinates options continuously evolve with respect to their hewe-tt-exxirrloogmmoooeee moneyness coordinates. It shows the trajectory of three S&P 500 Put options piring on 19.06.201 2020, with strike prices of K = 2000, 3000, 4000, over the period from 01 01.06.2019 11l.0622000 The 2020 (Covid) stock market crash and subsequent recovery and the sudden in- crease in log-moneyness of the options is clearly discermible. Moreover, over the lifetime of these options, all three options leave our smoothing domain (one temporarily), which means that from that point onward these options are dropped from the input of the GNO (and one returns). Figure 10) instead, shows that the number listed options to be processed by the GNO has continuously been increasing. It is the discretization-ine ieveriiaaioonnnvvriancee invariance of the GNO architecture that allows our method to produce consistent results over the whole timeline of Figure 10
1.0
K 2000
K 3000
0.8
K 4000
0.6
0.1
0.0
-1.25
-1.00
-0.75
-0.50
-0.25
0.00
0.25
0.50
Figure 9: S&P 500 Put option paths, expiry 19.06.2020. The path is depicted from 01.06.2019 to 01.06.2020.
27

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Published as a conference paper at ICLR 2025
3000
2500
2000
1500
1000
500
0
2012
2013
2014
2015
2016
2017
2015
2019
2020
2021
2022
Figure 10:
Development of the total number of listed Put and Call options (out-the-money and contained in our smoothing domain).
C.9 ANALYSIS OF COMPUTATIONAL COMPLEXITY
Let Tin be the (r, coooodiinss of the volatility inputs v and Tout the points at which the smoothed surface is to be computed. As part of the forward pass, the computation of the kernel integrals in each layer involves for each y E U Tout the summation of [W(x) S К many terms. In total, this amounts to the summation of O(Ain + Nous )) many vectors, where Tiin |i is the number of input datapoints and Youu = /oot is the number of queried output points. The production of each of these vectors involves the evaluation of the kernel FNNs well the matrix multiplication
of its output with the previous hidden state h (a)) (in our implementation, a 16 16 matrix veetoor multiplication). We take these as base units of computation and do not decompose their complexity further (in terms of e.g. the hidden channel size). Moreover, each layer has its own lifting net- work (the last one has projection well) and local linear transformation (the first one has not), which involve O + Noat) many FNN evaluations/matrii multiplications. The computational ef- fort therefore is O + now Row)K) (where J is the number of layers), counted in evaluations of (relatively small) feedforward neural networks. We derive the following strategies to control the -omm-to-smooteed -wrothhhddusffce....
Changing the subsample size K: We refer Appendix C.6 which contains an ablation study for K. It shows that there is potentially quite a bit of leeway to reduce K below the level of K = 50, reducing the complexity of the forward pass.
Decreasing the resolution of the output surface: Lower-resolution output can be traded for faster execution speed.
It could be motivated to investigate smaller FNN components for ker- nels, liftings, and projections with faster execution speeds.
Targeting the constants:
Since method is purely compute- based, the of more and/or faster GPUs will allow to process surfaces more quickly. This is a great advantage over instance-by. instance smoothing, whose fitting routines are sequential in nature and hard to accelerate.
Accelerate compute:
We benchmarked the execution with nin 897 and nour 2500 (i.e., with an output grid of size 50 x 50) on consumer- grade laptop CPU as well as an Nvidia Quadro RTX 6000 GPU in Ta- ble TO The numbers confirm the linear scaling of compute in the subsample size K and the great accelerability of our method using GPUs. They also show that computation times for typical grid sizes are fast enough for performing smoothing several times per second, which is more than suf- ficient for most applications. However, there exist participants in modern financial markets that operate at aub-millisecond timescales (kigh-frequeno) trading or HFT). At the same time, our cur- rent implementation prioritizes methodological validation over speed optimization (in particular the graph construction crute-forced and takes around 100 ms), and prevents us from drawing defini- tive conclusions about the method's applicability across the full spectrum of HFT applications. A
28

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comprehensive optimization study targeting HFT-sppeific requirements lies beyond the scope of this
work.
Table 10: Computation times for different subsample sizes K on CPU and GPU,
K
10
20
30
40
50
CPU (ms)
74.6
149
215
273
336
GPU (ms)
3.57
6.93
10.0
13.2
16.7
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D ADDITIONAL PLOTS
D.1 EXAMPLE PLOTS
We plot the results of operator deep smoothing (OpDS) vs. SVI on example inputs. To aid the visual clarity of our plots, we display only every third market datapoint.
0.011
0.107
0.5
SV1
0.4
0.4
OyDS
Op18
0.8
0.3
0.2
0.2
0.1
0.184
0.5
Mikt
Miss
SVI
SVI
0.1
Opts
0.4
Op128
0.2
0.1
0.356
0.586
0.5
0.3
SVI
SV1
OyDS
0.4
OpDS
0.3
0.2
-1.5
1.0
-0.5
0.0
0.5
0.855
= k/VF
0.5
Mk
0.4
0.2
0.5
Figure 11: Smoothing of quotes Dvai from 20.07.2012 at 10:50:00.
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0.074
0.151
0.5
0.50
Mlcz
Mitt
SVI
0.45
SVI
OpDS
OpDS
0.4
0.35
0.30
0.25
0.2
0.20
0.15
0.1
0.10
0.247
== 0.400
0.5
0.50
SVI
0.45
SV1
OpDS
OpDS
0.35
0.90
0.25
0.2
0.20
0.15
0.1
0.10
0.649
0.800
0.45
SVI OpDs
0.45
SVI
0.40
OpDs
0.55
0.35
0.90
0.25
0.25
0.20
0.20
0.15
0.10
0.10
1.5
-1.0
-0.5
0.0
0.5
-1.5
-1.0
-0.5
0.0
0.5
R/VF
Figure 12: Smoothing of quotes Deai from 21.10.2016 at 13:10:00.
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0.027
0.123
0.6
MK
Mis
0.5
OpDS
0.5
OpDS
0.4
0.4
0.3
0.3
0.2
0.2
7- 0.200
T 0.277
Mkt
0.6
Mks
8VI
SV1
0.5
OpDS
0.5
OpDS
0.4
0.3
0.3
0.373
0.449
0.6
SVI
0.5
OgDS
0.5
OpDS
0.4
0.4
0.2
0.2
0.699
0.918
Mkk
0.6
Mie
5V1
0.5
OpDS
0.8
OpD8
0.4
0.3
0.3
0.2
0.2
R/VF
k/VF
Figure 13: Smoothing quotes V E D Dirain from 04.01.2021 at 10:50:00.
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D.2
MONTHLY BREAKDOWN OF SPATIAL DISTRIBUTIONS BENCHMARK METRICS ON TEST DATASET
Jan 2021
Fab 2021
Mar 2021
1.0
0.4
0.2
Ape 2021
May 2021
Jon 2011
1.0
0.5
0.4
0.05
Jul
Ang
Sep
0.8
0.5
0.4
0.02
0.01
0.8
0.2
-1.5
-1.0
-0.5
0..
0.5
-1.5
-1.0
-0.5
0.0
0.3
-1.5
-1.0
-0.5
0.0
0.5
Figure 14: Average spatial distribution of dabs Drest for OpDS with monthly finetuming, per month. Blank cells indicate that no data was available for the particular region in the respective month.
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Jan 2021
Fib 2021
Mar 2021
1.0
0.4
0.2
May
1.0
0.8
0.6
0.1
0.2
Jul 2021
Aug 2021
9ep 2021
1.0
0.8
0.2
0.8
0.6
0.4
0.2
1.5
-1.0
-0.5
0.0
0.5
115
-1.0
-0.5
0.0
0.5
-1.5
-1.0
-0.5
0.0
0.5
Figure 15: Average spatial distribution of &n on Drest for OpDS with monthly finetuning, per month. Blank cells indicate that no data was available for the particular region in the respective month.
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D.3
EXAMPLE PLOTS: OPTION METRICS END-OF-DAY US INDEX OPTIONS DATA
0.022
0.118
0.6
Mist
0.6
Mkt
SVI OpDS
SVI
0.5
OpDS
0.4
0.4
0.3
0.3
0.2
0.2
k/VF
k/VF
0.195
0,271
Mikt
0.6
Mkr
OpDS
OpDS
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
x=kiv
T= 0.367
7= 0.444
0.6
Mid
0.6
SVT
OpD8
OpDS
0.5
0.5
0.4
0.4
0.3
0.3
0.2
4/VF
0.603
0.942
0.6
Miat
0.6
MK
SVI
SVI
OpDS
OpDS
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
-1.5
-1.0
-05
0.0
0.5
-1.5
-1.0
-0.5
0.0
0.5
=
k//r
k/v/
Figure 16: Smoothing SPX end-of-day data from 07.01.2021. Every third datapoint displayed.
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T 0.022
0.118
0.6
Midt SVI
AIK
SVI
0.5
OpDs
0.5
OpDS
0.4
0.4
0.3
0.3
0.2
195
0.297
03.6
Misl
0.6
Mkt
SVI
SVI
0.5
OpDS
0.5
OpDS
0.4
0.3
0.3
0.2
0.2
0.367
0.6
Mist
0.6
Mkt
SVI
SVI
0.5
OpDS
0.5
OpDS
0.4
0.4
0.3
0.3
0.2
0.
1.6933
0.6
0.6
OpDS
0.5
OpDS
0.4
0.3
0.2
0.2
k/vF
0.6
0.5
OpDS
0.4
0.9
-1.5
-1.0
0.0
0.5
k/VF
Figure 17: Smoothing of NDX end-of-day data from 07.01.2021. Every second datapoint displayed.
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0.003
0.022
Midt
0.8
SVI OpDS
SVI
OpDS
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
k/vF
L/VF
0,118
0.195
0.6
Mist
Mist
SVI
SVI
0.5
OpDS
0.5
OpDS
0.4
0.4
0.3
0.3
0.2
0.2
k/VT
k/v/
0.444
0.603
0.5
OpDS
OpDS
0.4
0.4
0.3
0.3
0.2
0.2
k/v/
1.5
1.0
0.5
0.0
0.5
0,942
k/
Mkt
SVI
0.5
OpDS
0.3
0.2
-1.5
-1.0
-0.5
0.5
k/VF
Figure 18: Smoothing of DJX end-of-day data from 07.01.2021. Every datapoint displayed.
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D.4 EXAMPLE PLOTS: SYNTHETIC SSVI DATA
SSV/ is a continuous interpolation scheme for SVI (see Section B
introduced in Gatheral & Jacquier 2014). We generate synthetic volatility surface data, which then apply our smoothing method. We consider the following formulation of SSVI (taken from Lind & Gatheral (2023)):
0r 1 2T
1 1 pp(-))) V(o(0,) p) +)++(( +1 (1-02))
02)).
(svv((, -
where ((0,)= = 1707 and
1 e 0, T + (V - -0)
1
N2T
+ (V'
8)
R1
R1
N2
F2
R1
making Ø (V, (V.V' p. 7, K1, R2) the parameter vector. We set the parameters the following values, reported as historical averages for the S&P 500 over the period 1996-2021 in|Lind & Gatheral 2023
V
V
P
1
R1
R2
0.04
0.04
0.11
-0.5
0.01
1.19
0.49
5.5
0.1
We produce synthetic market data vssvi by evaluating the SSVI instance ūssvi on the synthetic (8 x 51)-grid
min = Tp X T=
with " {0.16, 0.28, 0.4, 0.52, 0.64, 0.76, 0.88, 1} and n. (-1.5 -1.46,...6...... .......... 0.46, We
then proceed to smooth Vssvi with our method, producing êopns = 100 (Vssvi), and plot the results in Figure [9| Moreover, compute the MAPE smoothing error, both over the grid Tin, as well as over high-resolution (100 x 100)-
Tout Tp 22
where 7p and T. discretize [0.01, and [-1.5, 0.5). respectively, uniformly with one hundred nodes each. We achieve the following values:
Toul
2.01% % 2.42%
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T= 0.010
7 = 0.052
0.5
OpDS
0.5
OpDS
0.4
0.4
0.3
0.3
0.2
0.2
k/vF
k/vF
0.128
0.236
Mkt
Mktt
0.5
OpDS
0.5
OpDS
0.4
0.4
0.3
0.3
0.2
0.2
k/VF
4/VF
0.377
0.552
0.5
OpDS
0.5
OpDS
0.4
0.3
0.3
0.2
0.2
k/VF
== 0.759
T= 1.000
Mkt OpDS
Mktt
0.5
0.5
OpDS
0.4
0.4
0.3
0.3
0.2
0.2
-1.0
0.5
-1.0
0.0
0.5
2-
--K//F
k//F
Figure 19: Smoothing of synthetic SSVI data. Every second datapoint displayed.
39

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Published as a conference paper at ICLR 2025
E SVI
SVI, originally devised by Gatheral (Gatheral 2004), stands for Stochastic Volatility Inspired and low-dimensional parametrization for implied volatility slices (namely for each maturity). It captures the key features of implied volatility of Equity indices, and has become the iide benchmark for implied volatility smoothing on such markets. Its raw variant parametrizes the "slice" of implied volatility at given time-to-eppile as
b (p(k: - m) + V(k m)² o²
(1)) =
E R.
where Φ = (a, b, σr) is the parameter vector.
Calibration While stylistically accurate, SVI does not easily guarantee absence static arbitrage opportunities. and several authors have investigated this issue (Gatheral & Jacquier 2014: Martini & Mingone 2022, Mingone 2022 Martini & Mingone 2023). To produce our SVI benchmark therefore rely on the constrained SLSQP optimizer provided by the SciPy scientific computing package for Python, with the mean square error objective, a positivity constraint and the constraint equation 2 (computed in closed form), and the following parameter bounds:
E H
b E []. p=|-1,1],
1-1.5,0.5] E [10-8, 8,2)
We ignore the calendar arbitrage condition.
40


Reviews:
[{"review_id": "GlERT7RhMq", "reviewer": "Reviewer_GefV", "summary": "This paper introduces a novel approach to smoothing implied volatility using neural operators, focusing on Graph Neural Operator (GNO) architectures. Unlike traditional parametric models or classical neural networks, the proposed operator deep smoothing method directly maps observed implied volatilities to smooth, arbitrage-free surfaces, eliminating the need for instance-by-instance recalibration. By harnessing the unique capability of neural operators to handle data of varying sizes and spatial arrangements, the model effectively addresses the challenges posed by the dynamic and irregular nature of real-world financial data. Validated on a decade of intraday S&P 500 options data, the method outperforms the SVI benchmark and other machine learning techniques while generalizing effectively to unseen datasets, including end-of-day options for other indices, without retraining.", "questions": "See Weaknesses.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- **Novelty:** This paper proposes a novel application of GNO architectures to implied volatility smoothing, a task traditionally reliant on parametric models like SVI. By leveraging the discretization-invariance of neural operators, the method effectively addresses the challenges of dynamic and irregular financial data, marking a significant step forward in financial engineering.\n\n- **Practical Significance:** The operator deep smoothing approach eliminates the need for instance-by-instance recalibration, streamlining the online calibration process. This drastically reduces computational overhead, making the method highly practical for real-time applications in trading, risk management, and other financial operations.\n\n- **Robust Validation:** The method is extensively validated on a decade of intraday S&P 500 options data, demonstrating superior accuracy compared to the SVI benchmark and other machine learning techniques. Its strong generalization performance on unseen datasets, including end-of-day options for other indices, further highlights its robustness and adaptability.\n\n - **Reproducibility:** The authors provide open-source code, model weights, and detailed implementation details, including architecture, loss functions, and training setups. These resources ensure the experiments are easy to replicate and extend.", "weaknesses": "- **Limited Benchmark Comparisons:** The paper benchmarks its approach against SVI [1] and Ackerer et al. [2] but does not include comparisons with other key methods, such as SSVI [3] and VAE-based approaches [4]. Incorporating these would provide a more comprehensive evaluation. Additionally, using synthetic data, as in [2], could further strengthen the experimental validation.\n\n - **Insufficient Analysis of Computational Efficiency:** While the paper highlights the elimination of instance-by-instance recalibration, it lacks a detailed discussion on computational complexity, including runtime and memory requirements. These metrics are crucial for assessing the method’s practicality in high-frequency financial contexts.\n\n - **Abstract Treatment of Discretization-Invariance:** The explanation of discretization-invariance is largely theoretical. Providing concrete financial scenarios, such as handling abrupt market shifts, would better illustrate its practical significance and strengthen the narrative.\n\n- **Clarity of Experimental Figures**: Figures 3 and 4 have unclear legends and insufficient annotations, which reduce their interpretability. Improving their clarity and labeling would make the results more accessible and impactful.\n\n[1] A parsimonious arbitrage-free implied volatility parameterization with application to the valuation of volatility derivatives.\n\n[2] Deep Smoothing of the Implied Volatility Surface.\n\n[3] Arbitrage-free SVI volatility surfaces.\n\n[4] Variational Autoencoders: A Hands-Off Approach to Volatility.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "UGIkBPahsi", "reviewer": "Reviewer_o4xg", "summary": "This paper proposes a novel approach to nowcast implied volatility using neural operators, advancing the technique of implied volatility smoothing by constructing consistent, smooth surfaces that reflect current option market prices. Unlike traditional machine learning models, which struggle with dynamically changing spatial configurations in option price data, this work leverages a graph neural operator architecture to achieve robust and accurate predictions. Extensive benchmarks showcase that this work achieves superior generalization and accuracy over conventional neural networks and the SVI parametrization.", "questions": "please refer to the weaknesses part", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "1. This paper is well-written and easy to follow.\n\n2. This proposed method is resilient to input subsampling, aligns with no-arbitrage conditions, and eliminates the need for extensive data pre-processing.", "weaknesses": "1. The contribution is limited. It is focused on applying the Graph Neural Operator (GNO) architecture to the specific task of nowcasting implied volatility, without modifications to the operator itself.\n\n2. It's important to discuss various neural operators and clarify why the Graph Neural Operator (GNO) was chosen for this task.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "2qpTwBBaYl", "reviewer": "Reviewer_Xa1Y", "summary": "**Disclaimer**: I have no background on financial engineering so I cannot comment on the originality/significance aspect of this work. I might have missed essential points and only base my review on technical soundness and empirical validation\n\nThis paper introduces a method to nowcast implied volatility based on neural operators. This is achieved by a graph neural operator with MLPs as the lifting/projection layers and the kernel functions. The model uses a composite loss function that includes the fitting loss, no-arbitrage constraints, enforcing monotonicity and regularization terms to ensure smoothness. The method improves over the baseline SVI model.", "questions": "See Weaknesses section.", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The proposed method appears to be an interesting approach to model implied volatility surfaces for option pricing. The method through composite loss functions that include constraints such as no-arbitrage allows for direct training of a neural network architecture that directly provides smooth surfaces that directly satisfy these constraints. While I cannot comment directly how difficult/hands-on other alternative approaches are in this domain, the proposed model appears to have the flexibility to introduce other loss terms or vary the graph structure to incorporate other structural properties directly into the model. The model leads to lower errors compared to the SVI method.", "weaknesses": "The paper gives a good detail on the exact instantiation and implementation of the proposed method. I understand that there might be little prior work to base this paper on. However, it is unclear how the exact choices have been made and what would be their impact if changed. Note that the choices here might be purely made by domain knowledge, which I cannot judge and therefore might miss the specific considerations made here. However, I think the reader would benefit from more detail on the following: \n\n**In-Neighborhood sets**: The choice of in-heighborhood sets is arguably key model choice in this work. However, this work lacks detail on how the choice has been made and the subsampling heuristic has been devised. The authors argue that the choice is made because volatility smoothing requires limited global information exchange. I would then expect that the error of this method would go up if larger neighborhood sets are used. This paper would benefit from an additional analysis on changing the $\\bar{p}$ and $K$ parameters. \n\n**Other model choices**: Again, there is little information about all the other model choices and/or instantiations. In particular, the choice of the kernel functions as MLP (as opposed to other choices) would be interesting for the reader. I'm also wondering whether the MLP can output negative values? I understand that there is a softplus operation in the architecture and ensures positive values but again the choice of a MLP would benefit from further explanation/ablation. \n\n**Loss function weighting coefficients:** I would kindly ask the authors to include more detail how the weighting of the loss function has been performed. As it is stated right now, there is no information on how to select/adjust the weightings on a new pr.oblem. It would help the dissemination the method if the authors would share their methodology.\n\n**Ablations**: The claims in the paper could be strengthened by providing ablations In particular, different (wider) choices of the graph neighborhood would justify the choice of the authors. Additionally, omitting the no-arbitrage term and any of the other auxiliary terms would give more insight on the effect of the individual chosen components. \n\nI would reconsider my score if additional justification on the specific choices are included in the paper and the additional ablations on in-neighborhood and loss function weighting are performed.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)