HyperIV: Real-time Implied Volatility Smoothing
Authors: Yongxin Yang
Wenqi Chen
Chao Shu
Timothy Hospedales
Venue: ICML 2025 poster (2025)
Keywords: Implied volatility surface
quantitative finance
hypernetworks
Primary area: applications->everything_else
TL;DR: Construct an arbitrage-free implied volatility surface from only 9 option contracts in 2 milliseconds.

Abstract: We propose HyperIV, a novel approach for real-time implied volatility smoothing that eliminates the need for traditional calibration procedures. Our method employs a hypernetwork to generate parameters for a compact neural network that constructs complete volatility surfaces within 2 milliseconds, using only 9 market observations. Moreover, the generated surfaces are guaranteed to be free of static arbitrage. Extensive experiments across 8 index options demonstrate that HyperIV achieves superior accuracy compared to existing methods while maintaining computational efficiency. The model also exhibits strong cross-asset generalization capabilities, indicating broader applicability across different market instruments. These key features -- rapid adaptation to market conditions, guaranteed absence of arbitrage, and minimal data requirements -- make HyperIV particularly valuable for real-time trading applications. We make code available at https://github.com/qmfin/hyperiv.

HyperIV: Real-time Implied Volatility Smoothing
Yongxin Yang' Wenqi Chen
Chao Shu
Timothy Hospedales
Abstract
(i) the surface must be arbitrage-free, thus simply interpolat- ing extrapolating from potentially arbitrageable quotes does not work; (ii) the model requires frequent recalibration to reflect market movements, requiring computational effi- ciency; (iii) at higher frequencies, the limited availability of high-quality option prices is often the bottleneck, especially for fitting the tails of volatility curves. These practical con- straints have historically favoured simple parametric models (Gatheral, 2004; Gatheral & Jacquier, 2014; Hendriks & Martini, 2019; Mingone, 2022; Zaugg al., 2024) over machine learning approaches (Ackerer et al., 2020; Zheng et al., 2021; Bergeron et al., 2021; Ning et al., 2023; Gonon et al., 2024).
We propose HyperIV, a novel approach for real- time implied volatility smoothing that eliminates the need for traditional calibration procedures. Our method employs a hypernetwork gener- ate parameters for compact neural network that constructs complete volatility surfaces within 2 milliseconds, using only 9 market observations. Moreover, the generated surfaces are guaranteed to be free of static arbitrage. Extensive exper- iments across 8 index options demonstrate that HyperIV achieves superior accuracy compared existing methods while maintaining computa- tional efficiency. The model also exhibits strong cross-asset generalization capabilities, indicating broader applicability across different market in- struments. These key features rapid adapta- tion to market conditions, guaranteed absence of arbitrage, and minimal data requirements make HyperIV particularly valuable for real-time trading applications. We make code available at https: /ggthtt...b..... www. htt.////nnrhiii.../////mmm
In this paper, we introduce more challenging and practi- cally demanding problem: constructing implied volatility surfaces using extremely sparse observations (fewer than 10 contracts) at high frequency (e.g.. one-minute interval). This setting better reflects real-world trading conditions, where only a small set of options is actively traded with reliable prices at any given moment, and where surfaces must be generated within milliseconds for time-sensitive applications. To address this challenge, we propose Hy- perIV, a hypernetwork based method that constructs implied volatility surfaces for new market conditions in 2 millisec- onds, after training on historical data. We validate HyperlV through extensive experiments on both one-minute inter- val and end-of-day data across diverse assets, including major U.S. indices (S&P 500, NASDAQ 100) and interna- tional indices (MSCI World, MSCI Emerging Markets), Our evaluation encompasses over 150,000 surfaces and 50 mil- lion data points, demonstrating the model's 's robustness and generalization capabilities. The results show that HyperlV achieves superior accuracy compared to existing methods while maintaining computational efficiency and arbitrage- free properties.
1. Introduction
The implied volatility surface is a fundamental concept in options pricing. Market participants prefer to quote prices in terms of implied volatilities, as these values are more stable over time and provide an exact transformation to option prices. These implied volatilities serve as crucial inputs for more sophisticated models, which are then used to value exotic derivatives, determine margin requirements, and pro- vide liquidity in options markets. At its core, constructing an implied volatility surface involves converting discrete option price quotes into a continuous, smooth surface that spans all possible combinations of strike prices and maturi- ties. This process is commonly known as implied volatility smoothing. The construction faces several key challenges:
1.1. Literature Review
Implied volatility smoothing has been extensively studied, with approaches generally falling into two categorics (Home- scu, 20. The first category comprises indirect methods that initially fit a model for option prices, typically driven by underlying asset dynamics such as (local) stochastic volatility models (Heston, 1993; Hagan et al., 2002), Levy processes (Madan et al., 1998; Kou, 2002; Carr et al., 2002).
Queen Mary University of London ² University of Edinburgh. Correspondence to: Yongxin Yang <yongxin. yang qmul.ac.u ac.>.>
Proceedings of the 4nnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 20255 by the author(s).

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HyperIV: Real-time Implied Volatility Smoothing
or rough volatility models (Christian Bayer & Gatheral, 2016; El Euch & Rosenbaum, 2018; Gatheral al., 2018). These methods then convert the fitted price surface to implied volatility surface through the inverse Black-Scholes formula, using root-finding algorithms. The second category consists of direct methods that employ specific closed-form representations, either through carefully designed functions (Gatheral, 2004; Gatheral & Jacquier, 2014; Hendriks & Martini, 2019; Mingone, 2022; Zaugg al., 2024) or uni- versal function approximators like neural networks (Ackerer et al., 2020; Zheng et al., 2021; Bergeron et al., 2021; Ning et al., 2023; Gonon et al., 2024). Direct methods have gained greater adoption due to their easier calibration and faster exe- cution. For simpler models, such as the Stochastic Volatility Inspired (SVI) approach (Gatheral, 2004) and its extensions (Gatheral & Jacquier, 2014; Guo et al., 2016; Hendriks & Martini, 2019; Martini & Mingone, 2022), ensuring absence of arbitrage can be achieved through constraints on the pa- rameter space (Fukasawa, 2010; Homescu, 2011; Cohort et al., 2019). However, these methods may fail when market conditions fall outside the scope of their chosen functions. For instance, very short-term options markets can exhibit W-shaped volatility curves (Glasserman & Pirjol, 2023), which some models struggle to capture. Neural network- based methods offer greater flexibility in addressing such challenges. Ackerer et al. (2020) proposed a hybrid ap- proach where a neural network corrects simpler methods (c.g., SVD). However, enforcing no-arbitrage conditions on neural networks is more complex, typically requiring care- ful architecture design, auxiliary loss functions, and data augmentation techniques (Zheng et al., 2021). Additionally, these machine learning methods generally assume dense ob- servations and require longer calibration time compared to simpler models. For constructing complete implied volatil- ity surfaces from extremely sparse data, the Longitude al- gorithm (Zetocha, 2022) addresses strike interpolation and extrapolation for individual maturities by employing the Neri-Schneider entropy maximisation algorithm (Neri & Schneider, 2012), and then handles maturity extrapolation through time translation of log-normal cumulative distribu- tion functions. This concept of transforming distributions across maturities was further developed in (Zetocha, 2023) using an optimal vransport-inspire framework for generat- ing arbitrage-free transformations of an implied volatility surface. Building on these approaches, (Cao et al., 2024) further investigated the theoretical foundations, particularly the 'One-X property" and its role in establishing conditions to eliminate calendar arbitrage. They also explored the use of Log Gaussian Mixtures (as opposed to simpler Log Gaus- sian distributions) for better capturing complex volatility features like W-shapes.
dure mirrors model training in machine learning, typically involving iterative optimisation from randomly initialised parameters. Deep learning approaches have emerged to accelerate this calibration procedure. Hernandez (2016) pi- oneered a direct mapping from prices/implied volatilities on a rectangular mesh grid of moneyness and maturity to model parameters. an inverse approach, Liu et al. (2019); Benth et al. (2021); Blanka Horvath & Tomas (2021) map model parameters to prices/implied volatilities, necessitat- ing lightweight second-stage optimisation on the neural network inputs (equivalently, model parameters). A key limitation of these approaches is their reliance on fixed mon- eyness and maturity grids, which rarely align with actual market observations. More recent work by Bayer et al. (2019); Yang & Hospedales (2023); Baschetti et al. (2024) addresses this by adopting point-wise approach that ac- cepts continuous inputs for moneyness and maturity rather than using fixed grid positions. Our method builds upon these calibration acceleration techniques, eliminating the need for repeated calibration after the initial model training. While similar to (Hemandezz 2016), our approach differs in two key aspects: first, we employ hypernetwork architec- ture (Ha et al., 2016) where one neural network generates parameters for more compact network that models implied volatility surface more accurately and flexibly; second, we handle a variable-sized set of option contracts with continu- ous moneyness and maturity values, rather than requiring fixed-grid data. Alternative calibration-fre free methods have been proposed by Bergeron et al. (2021) and Gonon et al. (2024). However, these upproaches have not been validated in sparse data scenarios, and adapting them to this new setting needs non-trivial modifications, as we discuss in Section 4.2.
Our technical approach leverages hypernetworks (Ha et al., 2016), which utilise one neural network to generate the parameters of another. This architecture has demonstrated remarkable success across various domains, particularly in implicit neural representations (Sitzmann et al., 2020) where networks learn continuous mappings from coordinates to signals. Recent work by Mundinger et al. (2024) showcases hypernetworks' capability in representing PDE solution op- erators, suggesting their potential for financial applications. The connection between implied volatility smoothing and implicit neural representations emerges naturally: both do- mains involve mapping from low- -iw-demensional coordinates pooneyness-mattrrre pairs or spatial coordinates) to func- tion values (volatilities or signal intensities).
1.2. Contributions
We make three main contributions. First, we introduce a new problem setting that echoes few-shot learning in machine learning, constructing complete implied volatility surfaces from fewer than 10 data points. Second, we develop a
Implied volatility models contain free parameters that re- quire calibration before deployment. This calibration proce-
2

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HyperIV: Real-time Implied Volatility Smoothing
hypernerwork- baase method that generates arbitrage-free implied volatility surfaces within 2 milliseconds. Third, we perform comprehensive experiments on over 150,000 sur- faces and 50 million contracts, demonstrating our method's ' superior performance compared to several baselines adapted to the same setting.
ter a is typically calibrated by minimising the squared differences between the model predicted prices and the observed market prices for finite set of option con- tracts {(k1, t1, F1, T1, (kN, IN. PN, TN, dN, Vx)), i.e, through the optimisation:
F1, $1. V1),, (k2, 22, F2, r2, 82, V2).....
N
o = argmin
Σ N
Fu.r, - vi)² (2)
2. Preliminaries
Options are financial contracts that provide the holder with the right, but not the obligation, to buy (call option) or sell (put option) an underlying asset at specific price (strike price) on a designated future date (maturity date). The underlying assets for options can include indices, stocks, currencies, commodities, and other financial derivatives (e.g., futures or even options themselves). These instruments are primarily used to hedge against price movements of the underlying asset. The two main categories of options are European and American options. European options permit exercise only on the maturity date, while American options allow exercise at any time before maturity. These options are primarily traded on exchanges, such as the Chicago Board Options Exchange (CBOE). There also exist exotic options (e.g., Barrier Options), which typically traded in the tver-the-counter (OTC) market. In this study, we focus on European options with underlying assets consisting of indices, given their widespread market presence.
The assumption of constant o is unlikely to hold in real markets, thus limiting the effectiveness of the Black-Scholes model. However, if we reduce Eq. 2 to a single option contract, we can determine the unique o that matches the option price exactly (up to machine precision);
02 = argmin (V(ki, r2, Vi)²
(3)
The solution exists and is unique because the derivative of V with respect to a (known as Vega in option Greeks) is strictly positive. This or serves a proxy for option price Vi and is referred to as implied volatility. Using o; instead of Vi offers three key advantages: (i) it eliminates the need to treat put and call options separately, as the implied volatil- ity remains identical for both types with matching strike and maturity: (ii) implied volatility exhibits less dependence on the underlying asset price and provides more numerically stable range for modelling purposes; (iii) both the shape and level of the implied volatility surface demonstrate remark- able stability, with stylised features such as the volatility smile remaining consistent across different time periods and option contracts.
Notations We introduce the notations used throughout this paper. For given option contract, we use K to denote its strike price, for its annualised time to maturity, S for the current price of the underlying asset, and F for the forward price of the underlying asset, calculated as F = (S- - D)ert, where r represents the risk-free interest rate and D denotes the dividend. While r, D and F are time-dependent, we omit this dependency for notational simplicity. The vari- ables (K, 1, S,r, r, D) are directly observable from the market. However, there exists ambiguity in the option's price due to the presence of both bid and ask prices. In this work, we define the option price V as the mid-point between the best bid and best ask price. Finally, we define the log forward- moneyness as k- = ))
Implied volatility smoothing refers to the process of con- structing smooth surface that maps pairs of moneyness (k) and maturity (f) to implied volatility (σ)), using data from finite number of option contracts. This process typically involves a parametric function dg and requires solving the following optimisation problem:
1 N
min
Σ 1=1
(k) o;)²
(4)
In practice, implied volatility errors of the same magni- tude can lead to significantly different pricing errors across different moneyness values. To address this, practitioners often use Vegghweighted loss functions. An alternative ap- proach is to minimise errors directly in the price space by transforming the implied volatilities back to option prices:
The
model was the first widely used method for pricing European options, characterised by a single pa- rameter- volatility a R4 under the risk-neutral measure. For European option, the model estimates the theoretical price through the following closed-form formula:
V (l,,t, =e (54b (6d. (k,)) - be (ōd_ (k, fla))) (1)
1 N
- VV)² (5)
min
where (k, = ov. (- fo2t). Here 8
represents a call option, 8 = -1 represents a put op-
The parametric function (-)+) cannot be arbitrarily speci- fied, as it must meet certain constraints to prevent arbitrage opportunities, which we will examine in detail in Sec. 3.3.
tion, and (-) denotes the cumulative distribution func-
tion of the standard normal distribution. The parame-
3

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HyperIV: Real-time Implied Volatility Smoothing
3. Methodology
F
K
3827,08
0.0192
0.0399
3755
17.00
-0,0190
-0.2519
3.1. Problem Setting
2
3827.08
0,0192
0.0399
3825
42.40
-0,0005
0.2056
-0.1567
3827,08
0.0192
3900
15.00
0.0189
0.2471
Eq. 4 is generally not considered a challenging problem under conventional settings for two main reasons: (i) the parameter space of 0g is relatively modest, ranging from just few parameters (Gatheral & Jacquier, 2014) to several hundreds (Zheng al., 2021) or thousands (Ackerer et al., 2020) a scale that is considered small in the deep learning era: (ii) most studies use end-of-day (EOD) data for their experiments. For widely traded options, such as the S&P500 index option, EOD data typically contains tens thousands of contracts, providing sufficient data for model training. Additionally, computational time is less critical since the model needs to be trained/calibrated only once per day.
4
3836.34
0.0822
0.0411
3675
39.45
-0.0430
0.2335
-0.2498
3836.34
0,0822
3840
92.15
0.2148
3836.34
0,0822
0.0411
4000
26.30
0,0418
0.1936
0.2343
3853.73
0.2384
3575
74.40
-0,0751
-0.2503
8
3853.73
0,2384
0.0439
3855
163.40
0,0003
0.2209
0.5203
3853.73
0.2384
4125
50.40
0.1917
Table 1. A snapshot of the reference set 2
non-negative, achieved by using a softplus activation func- tion in the last layer; (ii) it must satisfy the arbitrage-free constraints, which are detailed in See. 3.3. We found that a simple multi-layer perceptron (MLP) with two hidden lay- ers, each containing 16 neurons, performs well after initial experimentation. Notably, this network has only 337 param- eters. The architectures of (o(-) and (-) are described in detail in Appendix A.
In this work, we present more challenging scenario - fitting the implied volatility surface using one-minute interval data with only a small set of observed contracts (typically fewer than 10). This setting better reflects real-world option trad- ing conditions, where within any given minute, only a few hundred contracts might be available, and even fewer will have reliable prices (indicated by reasonably small bid/ask spreads). The model must generate the surface within mil- liseconds, making traditional neural network training or fine-tuning approaches less practical. Specifically, we pro- pose a model of the following form:
The trainable parameters reside solely within the hyper- network go (-).), as the parameters for the implied volatility surface network generated on-the-fly rather than being trained. The objective function is formulated as:
M
NOO
min
1 M
1
Σ 1=1
(k 1000),1)) - o!! -3)
(8)
NO
where w(() go (2 In this expression, j iterates over the number of intervals in historical period, and i iterates over the number of option contracts within a given interval, while Z represents small set of option contracts for that interval. While the reference set Z could be constructed through any consistent rules, we select 9 contracts that are closest to fat-the- money, 25 A Call, 25 A Put} x (7 days, month, 3 months) in this work. The rationale behind this selection is that these options correspond to the most liquid contracts in our datasets, ensuring price reliability'. A snapshot of the reference set can be found in Table 1.
oo (k f) = fe(k, z|2z
(6)
where Z = {(k), 01), (k2, f2, ...). represents ref- erence set containing fewer than 10 elements. The model parameter 0 can be trained using historical data and remains fixed during deployment.
3.2. A Hypernetwork Approach
We propose to model fo ( using a hypernetwork (Ha et al., 2016). Specifically, a hypernetwork, denoted as go(-), takes a set as input and outputs a flattened vector. This flattened vector is split, and each segment reshaped appro- priately that a compact network that models the implied volatility surface, hws (-), can utilise them its parameters. Finally, this compact network produces the estimated im- plied volatility for any pair of (k, t) within the considered domain. This procedure is represented as follows:
After training, the model can generate a complete implied volatility surface for any new reference set 2 through a single forward pass of the hypernetwork. This allows real- time implied volatility smoothing for new market conditions without further optimisation.
3.3. Arbitrage-free Constraints
w go 2)) a h (k. t)
The implied volatility surface network (k; Ae must comply with arbitrage-free constraints, as discussed earlier. A fundamental requirement in option pricing theory is that the model must be free of static arbitrage trading strategies
Here, (-) can be any neural network that handles sets (Za- heer et al., 2017), meaning the order of option contracts in Z does not affect the output. In this work, we choose to use Transformer encoder (Vaswani et al., 2017) without positional embedding, in the spirit of (Lee et al., 2019), and apply mean pooling over the sample axis. The design re- quirements for (-) are as follows: (i) its output must be
'Here, A refers to the derivative of the option price with respect to the underlying asset's price. A 25 A implies that a $1 increase in the underlying asset will lead approximate $0.25 increase in the call option price or $0.25 decrease in the put option price.
4

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HyperIV: Real-time Implied Volatility Smoothing
that guarantee a positive profit with non-zero probability while having zero probability of loss (Carr et al., 2003). There are two cases: calendar spread arbitrage and butterfly arbitrage. The absence of calendar spread arbitrage implies the monotonicity of option prices with respect to maturity, while the absence of butterfly arbitrage implies the corre- sponding density is non-negative.
Index
Interval
Start
End
Interval #
Options
SPX
I-min
2023-01-03
2023-08-31
65,151
9,606,502
NDX
1-min
2023-01-03
2023-08-31
64,963
7,315,928
SPX
1-day
2013-01-02
2023-08-31
2,650
17.728,464
NDX
1-day
2013-01-02
2023 2023-08-31
2,681
10,432,080
RUT
1-day
2013-01-02
2023-08-31
2,685
6,155,780
VIX
I-day
2013-01-02
2023-08-31
1,652
388,182
MXWLD
1-day
2015506-00
2023 01-31
2,023
1,524,585
MXEF
1-day
2015-06-08
2023-08-31
2,029
1,040,392
Calendar spread arbitrage We denote the non-discounted call option price from the Black- Scholes model as V V F ($(d (k; 4f0)) e b (k, (k))) An im- plied volatility surface is free of calendar spread arbitrage if, for all & E R and E R_,, the derivative of V with respect to t is non-negative. Defining tl) g(d (k.t) o(R,t)/'s
Table 2. Dataset Statistics
(2 L) max ())
(16)
(2(() = (Trapz(p(:()))) - 1)²
(17)
where o(-) represents the probability density function of the standard normal distribution, we obtain:
where Trapz(-) denotes the numerical integral using the trapezoidal rule. Empirically, we have found that it is not necessary to tune the weight of Eq. 14 when added to Eq.8, as this auxiliary loss will approach zero in the end. Since Eq. 14 depends on a specific reference set Z, we apply it to each reference set within batch individually.
av_ K bl: t)) Ot 2
t) (kkk, 21 (k,
(9)
Øt
This leads to the following constraint:
(k, t) 2 O E and / R+ (10)
okk, 21 at
4. Experiments
Butterfly arbitrage An implied volatility surface is free of butterfly arbitrage if the corresponding density is valid, meaning it is non-negative and integrates to one for all time slices:
4.1. Dataset
We conduct our experiments using two different time inter- vals: one-minute and one-day (end-of-day). For the one- minute interval data, we consider index options on S&P 500 (SPX) and NASDAQ 100 (NDX). For the one-day interval data, we additionally include Russell 2000 (RUT), CBOE Volatility Index (VIX), MSCI World Index (MXWLD), and MSCI Emerging Markets Index (MXEF). We filter out in- the-money options, options with prices lower than 10.1, and options with maturities longer than 2 years. The details regarding data volume after pre-processing can be found in Table 2.
t(k, 20 O V R and E R+
(11)
fo t(k,t)dk=1 =1 E
(12)
where p(k,t) t) is obtained by twice differentiating V with re- spect to strike price K and applying the -hange-of-variable K- + k:
σ2 aK
p(k, 1) = 0K2 Ok
= b(k,
(13)
Train-Test Split For the one-minute interval data, we train the model using data before 2023-08-01 and test the sub- sequent intervals. For the one-day data, we train the model using data before 2023-01-01 and test on the remaining intervals. While each asset-interval has its own dedicated model, totalling 8 models, we also examine the generaliza- tion across assets in Sec. 4.5.
where the function g(k,t) t) is given by: g(k,t)
(1 o[ket] 0h
Bo(k.e) 2
to(k..) (R.
2 Ok
(k,, t) 88 (ko()))))
Based on Eq. 10, 11, and 12, we
okz
introduce an auxiliary loss function:
Σ REX
(((k)))) + Ea(1)
min I
KC
LET
4.2. Baseline Methods
(14)
Here, T represents an ordered set of evenly spaced time-to- maturity samples over [0 and IC denotes an ordered set of evenly spaced log moneyness samples over [-1.5, 1.5). The notation| represents the cardinality of a set. The individual terms of the auxiliary loss function are defined as:
While our problem setting is new, several existing methods can be adapted address Some deep learning approaches like those of (Ackerer et al., 2020; Zheng et al., 2021) re- quire calibration/traning for each time interval from scratch, making them impractical for our scenario due to their param- eter complexity, since we have only 9 observed data points. Therefore, we focus on the following adaptable methods:
& (ke,t) = max (0. (o.oo(k) - 2t (k,t)
(15)
0t
SSVI Hernandez (2016) proposed to map option prices
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HyperIV: Real-time Implied Volatility Smoothing
Model
SSVI
VAE
GNO
HyperlV
Model Param. #
332,289
102,529
243,153
1125
$1.25
Imp. Vol. Surf. Param.
337
21.00
21.00
Training Time (6)
2.87
5.08
2.82
50.75
Training Memory (MiB)
9,743
77,815
821
80.50
0050
Testing Time (ms)
13.11 0.47
0.34
7.86
2.09
20.25
25
Testing Memory (MiB)
673
2,275
532
Time to madona2 20 03
0.0 0.5
money hooss
1.
pim 0.5
mattra) 03, (e moneshees
30
1.
a
1.0
00
Table Model parameters and computational costs
to parameters of an established mathematical finance model (single-factor Hull-White model in their original work) using neural network. In theory, it can be adapted to predict the parameters of an implied volatility model (e.g., SSVI), however, since calibrating SSVI with 9 option contracts takes comparable time to the forward pass of neural network, we opt to use SSVI directly. Specifically, we employ SSVI's power-law parameterisation to model the total implied variance: wekk tls. Set (1+ pn(s2t)->k + +//++//// ((n(s2t)= p)² +1 p²) and the implied volatility is computed as o okk 1) =
Figure I. Implied volatility smoothing from sparse data. Red stars denote the 9 reference option contracts, blue dots represent test contracts held out from training, and the grey surface illustrates the volatility surface constructed by HyperlV using the reference data.
Refeetss Cornett Mode Preictin
0.6
Rehreres
Pron
RefermnnCeeee Irdetia
Greed
Redd Pocktics
Imos ent What Ry
D.A
02
0.2
1ng moneyntes
log moneyness
Log moneyness
(k.1)
VAE Bergeron et al. (2021) introduced a method that maps implied volatilities from fixed grid of deltas and maturities to latent variables. These latent variables are then combined with moneyness monteyess-maturity pairs through decoder to predict the corresponding implied volatilities. This approach can be viewed within the framework of Eq. 6 where Z is not a set of tuples {(k), o)) (k2, 122, o2).... but rather a vector of volatilities 2 = [011 22. J. since the coordinates (k;, t.) remain consistent across intervals. In our implemen- tation, we upscaled their network architecture match the parameter count of other baselines, which led to signifi- cant performance improvements. Unlike the original study which focused on foreign exchange markets where fixed- grid options are readily available, index options require us to create virtual options since they do not trade on a fixed grid. We provide the details of this virtual grid creation in Appendix B.
Figure 2. The slices of implied volatility surface
diction for any given (k. t) pair depends exclusively on the reference set Z, with complete independence from other queried points on the volatility surface.
The computational characteristics of SSVI, VAE, GNO, and HyperlV are summarised in Table 3. The training time refers to the runtime for one SGD step for a batch of 128 inter- vals, with each interval containing 16 option contracts. The total training duration spans 500 epochs, with each epoch processing all intervals in batches of 128. The testing time refers to the runtime for constructing the implied volatility surface using new, unseen reference set and evaluating over a uniform grid of 100 moneyness (k) and 100 maturity (t) values, totalling 10,000 points. All training and testing procedures are executed on NVIDIA A 100 GPU with 80G VRAM, with the exception of SSVI, which runs on an Intel Xeon CPU.
GNO Gonon et al. (2024) leveraged graph neural operators (Li et al., 2020) to map observed data [(k), 11), (kg, 16,,2,,00 02)..).... } to a smoothed surface. In their original work, all contracts are treated equally, with the reference set encompassing all observed option contracts in a given interval. To adapt this method to our setting, we limit the reference set to those 9 selected contracts and modify the graph construction as follows: (i) reference set contracts form fully connected directed graph; (ii) each reference contract has directed edges pointing to all non-reference contracts; (iii) non-reference contracts neither point back to reference contracts nor connect to other non-reference contracts. This graph architecture guarantees that the pre-
4.3. Qualitative Results
We begin with an illustration of the generated implied volatil- ity surface using SPX end-of-day data on the first testing day JO3-J..- The grey surface in Fig. 1 (left) is con- structed using only the 9 reference contracts shown in Ta- ble 1. Despite these contracts covering only a small region of log moneyness and maturity, the surface demonstrates excellent fit to the unobserved data, as illustrated in Fig. I (right). Further validation is provided in Fig. 2, where we
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HyperIV: Real-time Implied Volatility Smoothing
Index
Interval
SSVI
VAE
GNO
HyperTV
Index
Interval
SSVI
VAE
GNO
HyperlV
SPX
I-min
0.0283
0.0222
0.0140
0.0167
SPX
1-min
3.2698
3.6925
2.2398
2,5115
NDX
1-min
0.0273
0.0258
0.0162
0.0156
NDX
1-min
15,2799
17.3612
10.2187
10.0043
SPX
I-day
0.0312
0.0162
0.0085
0.0075
SPX
1-day
6.6257
4.0799
1.7349
166766
NDX
I-day
0.0498
0.0493
0.0117
0.0113
NDX
1-day
25.8445
32.2850
7.7103
7.2159
RUT
1-day
0.0566
0,0414
0.0133
0.0107
RUT
1-day
2.9377
4,5851
1.3467
0.9554
VIX
1-day
0.2771
0.0826
0.1345
0.0336
VIX
1-day
0,5256
0.1734
0.2889
0,0771
MXWLD
I-day
0.0511
0.0651
0.0362
0,0166
MXWID
I-day
2.9746
8.7424
3.2839
1.6886
MXEF
1-day
0.0551
0.0697
0.1040
0.0172
MXEF
I-day
3.1641
5.7376
11.6957
1.75%
Table Testing MAE (Implied Volatility)
Table 5. Testing MAE (Price)
Total implied variance
Maturity 3 days
VAE
GNO
HyperlV
Matufit 13 days
0.10
Macufity2 23 days
SPX-NNDX 0.0257
0,0185
0.0223
Density
1.5
Macufiy 32 days
NDX SSPX
0.0315
0,1272
0.0349
Macutry 42 days
10
Maturit 52 days
0.05
Maturit 62 days
5
Macufity 11 days
Macufit 41 days
Table 6. Generalization (MAE) on one-minute interval data
Macutiy 91 days
0.00
-1.0
0.0
-1.0
-055
0.0
0.5
SPX
NDX
RUT
VIX
MXWLD
MXEF
Log moneyness
Log moneyness
To
SPX
0.0075
0.0154
0.0124
0.1810
0.01.59
0.0284
NDX
0.0186
0.0113
0.0170
0.1874
0.0202
0.0355
RUT
0.0157
0.0134
0.0107
0.1927
0.0164
0.0299
Figure 3. Left: the slices of total implicd variance p²t - the ab- sence of crossed lines suggests no calendar spread arbitrage. Right: the inferred density from implied volatility surface the absence of negative values suggests no butterfly arbitrage.
VIX
0.6082
0.5606
0.20228
0.0336
0.5377
0.2922
MXWLD
0.0242
0.0230
0.0232
0.1821
0.0166
0.0203
MXFF
0.0335
0.0267
0.0309
0.1643
0.0221
0.0172
Table 7. Generalization (MAE) on one-day interval data of Hy- perlV
plot several surface slices corresponding to maturities with reference contracts. The volatility smiles are accurately re- covered even without observations for tails, confirming that the hypernetwork has successfully learned the stylised facts.
To investigate the sources of errors, we partition each dataset into 160 two-dimensional bins (10 maturity intervals x 16 log moneyness intervals), ensuring approximately equal numbers of contracts per bin. Fig. 4 displays the mean im- plied volatilities and mean absolute errors from different models across these bins. Fig. 4 reveals a clear correla- tion between error magnitude and implied volatility value, with larger errors concentrated in regions of higher implied volatilities (e.g., short maturities and extremely small strikes in the top-left corner). VIX and MXEF exhibit notably dif- ferent implied volatility distributions compared to other assets, with significant values in the top-right corner (short maturities and extremely large strikes), explaining the poor performance of baseline models on these two assets. In contrast, HyperIV maintains strong performance across all regions. Additional analysis of aggregated errors over time, log moneyness, and maturity intervals is provided in Ap- pendix C, confirming these findings.
To verify the absence of static arbitrage, we examine two key aspects. First, Fig. 3 (left) displays the total implied volatilities (w- = o²t) across multiple maturities. According to Eq. 10, total implied volatility should increase monoton- ically with maturity to prevent calendar spread arbitrage, which is confirmed in Fig. 3. Second, we analyse the in- ferred density (Eq. 13) for the same set of maturities in Fig. 3 (right). The non-negative nature of these densities rules out butterfly arbitrage.
4.4. Quantitative Results
To evaluate the performance of HyperIV and other baselines, we compute the test mean absolute error (MAE) for both im- plied volatilities and option prices, with results summarised in Table 4 and Table 5. HyperIV achieves superior perfor- mance across all assets except for minute-level SPX data, where GNO performs better. Notably, HyperIV demon- strates remarkable stability across all datasets, while SSVI fails on VIX and GNO fails on both VIX and MXEF. It is worth highlighting that SSVI, despite having only four pa- rameters, outperforms VAE (300K parameters) on MXWLD and MXEF
4.5. Further Analysis
Generalization We investigate the cross-assed generaliza- tion by examining whether model trained on one asset can effectively estimate implied volatilities for another asset. Results are summarised in Table 6 and Table 7, with addi- tional baseline comparisons provided in Appendix D. For minute-level data, GNO exhibits strong SPX-NNDX gener-
7

---

HyperIV: Real-time Implied Volatility Smoothing
Implied Volatility
SSVI Abs. Error
VAE Abs. Error
GNO Abs. Error
HyperIV Abs. Error
(NDX-Min) (SPX-Min) Maturity
Maturity
0.3
0.2
0.3
0.3
0.08
0.2
0.05
0.2
0.03
(SPX- (PX)-Dy)) Maturity
0.20
0.15
0.0
0.10
0.05
(NDX- Day)
Maturity
0.40
0.4
Maturity
0.6
(RUT)
0.10
(VIX)
Maturity
1.5
0.00
0.20
(MXWLD) Maturity
0.7
0.40
0.5
0.20
0.2
1.0
0.60
(MXEF) Maturity
0.8
0.40
0.6
0.4
Log Moneyness
Log Moneyness
Log Moneyness
Log Moneyness
Log Moneyness
Figure Mean values grouped by log moneyness and maturity. The log moneyness increasing from to right and the maturity increasing from top to bottom. The assets are (top-down) SPX, NDX (1-min), SPX, NDX, RUT, VIX, MXWLD, MXEF (1-day).
alization but fails catastrophically in the reverse direction (NDX->PXX). Both VAE and HyperIV demonstrate robust bidirectional performance across directions. For end-of-day data, HyperlV exhibits robust generalization across most assets, with the notable exception of VIX an expected limitation given the sigmificant domain gap. Remarkably, excluding VIX, HyperlV's generalization performance in any-to-any transfer scenarios surpasses that of SSVI cali- brated directly the target datasets.
are strictly enforced, as demonstrated by these near-zero loss values. The primary residual error arises from the integrate-to-one constraint (Eq. 17), reflecting minor nu- merical inaccuracies inherent to trapezoidal integration. In practice, market realities like transaction costs render any residual arbitrage opportunities economically unexploitable, with additional safeguards from standard risk management protocols further mitigating operational risks. This approach of using auxiliary loss to promote arbitrage- free surfaces is common in contemporary deep learning models (Ackerer et al., 2020; Zheng et al., 2021; Gonon et al., 2024), unlike simpler parametric models such as SSVI, where arbitrage- free conditions are typically guaranteed by analytically con- straining the model's parameter space.
Auxiliary Loss While our approach does not provide hard guarantees for arbitrage-free conditions, the auxiliary losses remain notably small: approximately 10-8 during training and 1 during testing. The calendar spread arbitrage (Eq. 15) and positive density function (Eq. 16) constraints
8

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HyperIV: Real-time Implied Volatility Smoothing
4.6. Limitation
ing. While hypemetwork training incurs substantial upfront computational costs (still significantly lower than baseline methods), the climination of frequent recalibration reduces long-term operational overhead. This efficiency extends to environmental sustainability, as reduced computational de- mands translate into lower energy consumption for real-time trading systems.
Interpretability Unlike SSVI, our method lacks trans- pareney and interpretabilitty. It has significantly smaller number of parameters (337) in the implied volatility surface network, but these parameters do not carry meaningful in- terpretations or provide actionable insights for traders, as SSVT's parameters do. This limitation could be partially ad- dressed through certain architecture designs of the implied volatility model, such as mixture of SVI/SABR models instead of MLP.
References
Ackerer, D., Tagasovska, N., and Vatter, T. Deep smoothing of the implied volatility surface. In NeurIPS, 2020.
Computational Cost While SSVI only needs basic CPU, our method requires modern GPU or NPU. When using an M2 chip instead of an A 100 GPU, inference time increases from ms to ms for HyperIV, from 8 ms to 300 ms for GNO, and from 0.34 ms to 15 ms for VAE, Interestingly, SSVI performs better on the M2 chip (from 13 ms 10 ms) due to better single-core performance.
Baschetti, F., Bormetti, G., and Rossi, P. Deep calibration with random grids. Quantitative Finance, pp. 1263-1285, 2024.
Bayer, C.,, Horvath, B., Muguruza, A., Stemper, B., and Tomas, M. On deep calibration of (rough) stochas- tic volatility models. arXiv preprint arXiv:1908 08806, 2019.
Data Accessibility Due to licence restrictions, we cannot redistribute the data used for model training. Academic researchers may access the end-of-day data through their institution's subscription to WRDS (which includes Option- Metrics). Minute-level data considerably more expensive and presents greater preprocessing challenges. Theoreti- cally, our method can be verified using synthetic data, which can be generated at any desired frequency.
Benth, F. E.., Detering, N., and Lavagnini, S. Accuracy of deep learning in calibrating hjm forward curves. Digital Finance, 3(3-4):209-22 240, 2021.
Bergeron, M., Fung, N., Hull, J., and Poulos, Z. Variational autoencoders: A hands-off approach to volatility, arXiv preprint arXiv: 102.03945, 2021.
5. Conclusion
Blanka Horvath, A. M. and Tomas, M. Deep learning volatil- ity: deep neural network perspective pricing and calibration in (rough) volatility models. Quantitative Finance, 21(1):11-27, 2021.
We present a novel approach to implied volatility smoothing that effectively addresses the challenges of sparse data and real-time computational constraints. The proposed model, HyperIV, demonstrates how hypernetworks can accurately capture complex implied volatility patterns while simultane- ously enforcing arbitrage-free conditions and maintaining computational efficiency. Future research directions include explicitly incorporating historical data and modelling tem- poral dynamics in implied volatility surfaces.
Cao, Z., Zhao, S., and Huang, S. One-X property con- jecture, stochastic orders and implied volatility surface construction. Available at SSRN 4744237, 2024.
Carr, P., Geman, H., Madan, D. B., and Yor, M. The fine structure of asset returns: An empirical investigation. Journal of Business, 75(2):305-332, 2002.
The authors would like to thank Vin- cent Zhao at Winton for initially highlighting the problem setting of fitting the implied volatility surface using sparse data, and for insightful discussions throughout this research.
Acknowledgements
Carr, P., Geman, H., Madan, D. B., and Yor, M. Stochastic volatility for lévy processes. Mathematical Finance, 13 (3):345-382, 2003.
Impact Statement
Christian Bayer, P. F. and Gatheral, J. Pricing under rough volatility. Quantitative Finance, 16(6):887-904, 2016.
The democratisation of options trading through platforms like Robinhood, which has dramatically increased retail investor participation, underscores the critical need for effi cient volatility surface smoothing tools. HyperIV provides a streamlined solution where practitioners need only execute a single forward pass on available option quotes to gen- erate a complete arbitrage-free implied volatility surface eliminating traditional optimisation and hyperparameter tun-
Cohort, P., Corbetta, J., Martini, C., and Laachir, I. Robust calibration and arbitrage-free interpolation of ssvi slices. Decisions in Economics and Finance, 42:665-61 667666777775 2019,
El Euch, O. and Rosenbaum, M. Perfect hedging in rough heston models. The Annals of Applied Probability, 28(6), 2018.

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HyperTV: Real-time Implied Volatility Smoothing
Fukasawa, M. Normalization for implied volatility. arXiv preprint arXiv: /OX8.:0008.5000 5005, 2010.
Liu, S., Borovykh, A., Grzelak, L. A., and Oosterlee, C. W A neural network-based framework for financial model calibration. Journal of Mathematics in Industry, 9(1):9 2019.
Gatheral, J. A parsimonious arbitrage- free implied volatil- ity parameterization with application the valuation of volatility derivatives. Presentation Global Derivatives & Risk Management, Madrid, 2004.
Madan, D. B., Carr, P. P., and Chang, E. C. The variance gamma process and option pricing. Review of Finance, 2 (1):79-105, 1998.
Gatheral, and Jacquier, A. Arbitrage-free svi volatility surfaces. Quantitative Finance, 14(1):59-71, 2014.
Martini, C. and Mingone, A. No arbitrage svi. SIAM Journal on Financial Mathematics, 13(1):227-261, 2022.
Gatheral, J., Jaisson, T., and Rosenbaum, M. Volatility is rough. Quantitative Finance, 18(6):933-949) 2018.
Mingone, A. No arbitrage global parametrization for the essvi volatility surface. Quantitative Finance, 22(12): 2205-2217, 2022.
Glasserman, P. and Pirjol, D. W-shaped implied volatility curves and the gaussian mixture model. Quantitative Finance, 2344)5557-577, 2023.
Mundinger, K., Zimmer, M.. and Pokutta, S. Neural param- eter regression for explicit representations of pde solution operators. arXiv preprint arXiv:2403.1276. 2024.
Gonon, L., Jacquier, A., and Wiedemann, R. Operator deep smoothing for implied volatility. arXiv preprint UrXiv:2406.11522 2024.
Neri, C. and Schneider, L. Maximum entropy distributions inferred from option portfolios on an asset. Finance and Stochastics, 16(2):293-318, 2012.
Guo, G., Jacquier. A., Martini, C., and Neufcourt, L. Gener- alised arbitrage- free svi volatility surfaces. SIAM Journal on Financial Mathematics, 7(1):619-641, 2016.
Ning, B., Jaimungal, S., Zhang, X., and Bergeron, M. Arbitrage-free implied volatility surface generation with variational autoencoders. SIAM Journal on Financial Mahematics, 14(4): 1004:1004 2023.
Ha, D., Dai, A., and Le, Q. V. Hypernetworks. arXiv preprint arXiv: 009.09906, 2016.
Hagan, P. S., Kumar, D., Lesniewski, A. S., and Woodward, D. E. Managing smile risk. Wilmott Magazine, pp. 84- 108, September 2002.
Sitzmann, V., Martel, J., Bergman, A., Lindell, D., and Wet- zstein, G. Implicit neural representations with periodic activation functions. In NeurIPS, 2020.
Hendriks, L. and Martini, C. An extended ssvi volatility surface model. Journal of Computational Finance, 22(5); 25-39, 2019.
Vaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones, I.,, Gomez, A. N., Kaiser, L. u., and Polosukhin, I. Atten- tion is all you need. In NeurIPS, 2017.
Hernandez, A. Model calibration with neural networks. Available at SSRN 2812140, 2016.
Yang, Y. and Hospedales, T. M. On calibration of mathemat- ical finance models by hypernetorrs.. In ECML PKDD, 2023.
Heston, S. L. A closed-form solution for options with stochastic volatility with applications to bond and cur- rency options. The Review of Financial Studies, 6(2): 327-343, 1993.
Zaheer, M., Kottur, S., Ravanbakhsh, S., Poczos, B., Salakhutdinov, R. R., and Smola, A. J. Deep sets. In NeurIPS, 2017.
Homescu, C. Implied volatility surface: Construction methodologies and characteristics. arXiv preprint arXiv: / /07. 1834, 2011.
Zaugg. N. F., Grzelak, L. A., and Perotti, L. Volatility parametrizations with random coefficients: Analytic flex- ibility for implied volatility surfaces. arXiv preprint arXhv:2411.0404. 2024.
Kou, S. G. A jump-diffusion model for option pricing. The Review of Financial Studies, 15(1):223-22 224,), 2002.
Zetocha, V. The longitude: managing implied volatility of illiquid assets. Available at SSRN 4208856, 2022.
Lee, J., Lee, Y., Kim, J., Kosiorek, A.. Choi, S., and Teh, Y. W. Set transformer: A framework for attention- permutation-invarii neural networks. In ICML, 2019.
Zetocha, V. Volatility transformers: an optimal transport-
inspired approach to arbitrage- free shaping of implied volatility surfaces. Available at SSRN 4623940, 2023.
Li, Z., Kovachki, N., Azizzadenesheli, K., Liu, B., Bhat- tacharya, K., Stuart, A., and Anandkumar, A. Neural operator: Graph kernel network for partial differential equations. arXiv preprint arXiv:2003 2003.03488. 2020.
Zheng, Y., Yang. Y., and Chen, B. Incorporating prior finan- cial domain knowledge into neural networks for implied volatility surface prediction. In KDD, 2021.
10

---

HyperIV: Real-time Implied Volatility Smoothing
A. Architecture
A.I. Neural Network for Implied Volatility Surface
Our model for the implied volatility surface maps coordinates (k, t) to volatility or through three fully connected layers. The first two layers employ hyperbolic tangent activation functions, while the final layer uses softplus activation function to ensure positive outputs. This architecture contains 337 parameters in total. The PyTorch implementation is shown below:
iv_network torch. nn. Sequentiia
torch. nn. Linnea (2, 16), forch.n... Tanh(),
Norch.nn.Lines. (16, 16), torch.nn. Tanh (),
torch.nn. (16, 1), norch.nn.Softpae ()
A.2. Hypernetwork
The hypernetookk functions as a set embedding network that transforms data from dimension [B, M, to [B, PJ. Here, represents the batch size, M denotes the number of elements in each set (set in our implementation). D indicates the number of features per instance (3 features: and for each option contract in the reference set), and P corresponds to the number of parameters in our implied volatility surface network (337 specified above).
The network architecture consists of several sequential transformations. Initially, a fully-connected layer maps each option's 's features to И hidden neurons (B, M) DJ - [B, M, H). These representations then pass through multiple transformer encoder layers, enabling crosscoommunicttion between set elements while maintaining dimensionality ([B, M, H] + [B, M, H)). The transformer's output then averaged across set dimension (B, M, H - [B, H), and final fully- connected layer produces the desired parameter vector ([B, H]] [B, P)). Notably, the absence of positional embeddings in the transformer encoder ensures the network's output remains invariant the ordering of set elements. The PyTorch implementation follows below.
class Set.Bmbeddingketworrkk (nn.Medul))
def weiiiit__ii (self, input_dim, output_dim, num hum_heads=2, num lam_layers=2, hidden_dim=122)
super superr/@tttnbecddingeetttt.. self) init_ ()
self.fel s Ln (inputtt (input_dim, hidden_dimi
self self.attention_layerr an Mo..oullelist
hwm..Traanfformmrrrcooddeeraaa
d_model-hidden_ddnn, nhead=num_headds,
din_.fesfforrrrdhhiddded..ddd
batch_first=True,,
dropout=0,
activation="zeli")
for in range (num_layers)))
self.for fc2 = mn. Linear (hidden_dim, output_dim)
def (srwa x):
x self.fcl (x)
for layer in self self.
layer (x)
x.mean (dim-1)
x = self. fc2 (x)
return

---

HyperIV: Real-time Implied Volatility Smoothing
F
K
V
k:
A
3827.0768
0.0192
0.0399
3755
17.0000
-0.0190
0.2100
-0.2519
2
3827.0768
0.0192
0.0399
3825
42.4000
-0.0005
0.2056
-0.4867
3
3827.0768
0.0192
0.0399
3900
15.0000
0.0189
0.1955
0.2471
4
3836.3443
0.0822
0.0411
3675
39.4500
-0.0430
0.2335
-0.2498
5
3836.3443
0.0822
0.0411
3840
92.1500
0.0010
0.2148
0.5061
6
3836.3443
0.0822
0.0411
4000
28.3000
0.0418
0.1936
0.2343
3853.7277
0.2384
0.0439
3575
74.4000
-0.0751
0.2512
-0.2503
3853.7277
0.2384
0.0439
3855
163.4000
0.0003
0.2209
0.5203
3853.7277
0.2384
0.0439
4125
50.4000
0.0680
0.1917
0.2483
Table 8. A snapshot of the reference set 2 for SSVI, GNO, and HyperlV
F
K
V
k
A
3827.0768
0.0192
0.0399
3746.5496
18.7273
-0.0213
0.2333
-0.2500
2
3827.0768
0.0192
0.0399
3827.0768
43.7309
0.0000
0.2070
0.5057
3
3827.0768
0.0192
0.0399
3896.947 9478
14.8118
0.0181
0.1900
0.2500
4
3836.3443
0.0822
0.0411
3669.8340
40.9677
-0.0444
0.2419
-0.2500
5
3836.3443
0.0822
0,0411
3836.3443
90.5037
0.0000
0,2070
0.5118
6
3836.3443
0.0822
0.0411
3983.0089
29.7121
0.0375
0.1866
0.2500
7
3854.9641
0,2466
0.0440
3573.3798
75.0444
-0.0758
0.2494
-0.2500
8
3854.9641
0.2466
0.0440
3854.9641
156.2998
0.0000
0.2070
0.5205
9
3854.9641
0.2466
0.0440
4117.5128
49.7415
0.0659
0.1842
0,2500
Table 9, A snapshot of virtual options for VAE
B. Virtual Options for VAE
SSVI, GNO, and HyperIV can handle continuous data inputs, only requiring the selection of the closest available option contracts from the market. In our implementation, while we target 90-day maturity for options indexed (7,8,9), the actual contracts have 87-day maturity. Similarly, ATM options have log-moneyness (k) values close to, but not exactly, zero. For the 25 de-delta options, select contracts with delta values nearest to -0.25 for puts and 0.25 for calls. All options presented in the table represent actual market contracts. The reference set Z for the specific example in Table 8 is:
Z {t-((0(090))000 0.0192, 0.2100), (-0.0005, 0.0192, 0.2056). (0.0189, 0.0192, 0.1955),
(- 0.04430,00 0.0822, 0.2335), (0.0010, 0.0822, 0.2148), (0.0418, 0.0822, 0. 0.133)),
(- 0-0.70751 0.2384, 0.2512), (0.0003, 0.2384, 0.2209), (0.0680, 0.2384, 0.1917)
While the ordering of triplets (k, f, within Z is flexible, the component order within each triplet must be maintained.
In contrast to GNO and HyperIV, VAE requires fixed-grid data inputs. To accommodate this requirement, we employ arbitrage-free interpolation (Cohort et al., 2019) to generate virtual options with precise k: = 0 for ATM options at desired maturities. The interpolation process determines both implied volatilities and corresponding prices, as shown for indices 5, and 8 in Table 9. Subsequently, optimise i values achieve exact delta targets: -0.25 for put options (indices 1, 4, 7) and 0.25 for call options (indices 3, 6, 9). For this particular example, the VAE input takes the form of an ordered 9-element vector:
z [0. 0022333,00 0.2070, 0.1900, 0.2419, 0.2070, 0.1866, 0.2494, 0.2070, 0.1842]
(18)
Unlike the previous case, the order of elements in this vector must strictly follow specific (A, t) grid pattern: starting with 7-day options (25 A Put, ATM, 25 A Call), followed by 30-day options in the same delta sequence, and concluding with 90-day options in the same pattern.
12

---

HyperIV: Real-time Implied Volatility Smoothing
MáS Log Monsyrnes for 5PS58in
0.05
100
0.20
0.04
0.30
HAE
0.75
MAXE
3.04
0.50
0.02
0.36
0.25
160
150
1..22
10
00.5
Timestamo Index
Log Moneynesa
Matuniy
by Timefor NDX Min
202
8.15
0.5
MAE by Motufit for NOX Nin
10.20
0.00
0.20
3.10
Timplies Vitve
0.06
0.26
3.04
2.05
0.02
0.250
100
150
1.00
0.25
0.50
0.20
Timestamp
Moneyness
MAE for
MAE by Log Monoyness for SPX
NAL by Maturrty for 5PM
0.100
0.28
0.4
100
imot volaliity
0.
0.10
MIKE
HI
0.50
9.05
0.025
0.000
100
150
0.00
0.0
1.5
15
2.5
0.22
Timestarp Index
Moncyness
Maturiy
0.08
0.52
1.00
0.75-
WAE by Monsyness for NDX
Werlg
1.5
Implees Vo willity
0:15
0.40
0.50
0.300
035
0.04
0.02
0.26
0.25
0.05
0.00
A22
0.00
0.25
100
150
0.0
0.
0.
2/
Timestamo Index
Log Monuymess
Maturty
HAF Time
108
M&E Haturty for NUT
Imstied Waiilitry
0.100
0.30
0.333
0.3
0.015
0.30
0.30
0.05
0.275
0.1
0.055
0.00
5D
150
0250
1.5
1.0
3.5
0.0
0.5
0.2
0.00
5.5
1.5
2.0
"Emesdamp Indes
Log Monmyneex
Haturty
HAT by Time for
Log Moneyness for VIX
HAE by Mat VoX
1.00
blent now gon
1.1
led Wite
1.5
Impllett Moattiy
0.4
Ingilee Momaaay
MAE
0.50
0.2
0.25
0.3
0.00
0.01
100
150
1.0
0'0
0.5
3.0
2.5
0.6
H4B by Time for HOWLD
0.30
Imppted Viteriit
0.8
MAE Log Moneyness for WAWLD
8
Implant Whirriyy
0.3-
MAE by Maturit for HXWLD
00.
0.13
0.5
Impled MO0
MAE
MAE
0.05
0.0
0.0
200
130
Timestamo
Moneyness
HAE by Time for WXEF
M&E by Log Moncyness for NXEF
MAE Haturiy for MXEF
500 wC LN0
Inpare 1R0led VOMRY
2.0
0.5
2.30
Implied Valaniiiyy
0.3
MAE
0.5
Impi
150
1.20
6.9
-0.4
D.0
0.4
0.5
0.6
Timestamp Ihdex
Figure 5. Absolutc errors aggregated by Time Log Moneyness/ / Maturity. The Y-axis on the right is for implied volatility.
C. MAE by Time, Log Moneyness, Maturity
To analyse the sources of error, we aggregate the absolute errors across different intervals of time, log moneyness, and maturity in Fig. 5. The first column provides insight into model stability over time. HyperlV demonstrates the most
13

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Hyper Real-time Implied Volatility Smoothing
consistent performance across all assets. GNO also performs well generally, with exceptions in VIX and single outlier in MXEF. While SSVI and VAE show stable performance, their accuracy falls short of both GNO and HyperlV.
The second column illustrates model performance across different ranges of log moneyness (strike prices). The results reveal a clear positive correlation between errors and implied volatility levels. All models struggle with extreme values of log moneyness, with SSVI and VAE showing particular sensitivity to these cases.
The third column presents model performance across different maturity ranges. While less pronounced than the moneyness effect, there remains a notable relationship between implied volatility and errors. All models face challenges in fitting very short maturities, consistent with our findings in Fig. 4 of the main paper. HyperIV maintains consistent performance across all maturities except very short ones, though minute-level data exhibits inherently higher noise levels.
14

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Hyper Real-time Implied Volatility Smoothing
From
SPX
NDX
RUT
VIX
MXWLD
MXEF
To
SPX
0.0162
0.0444
0.0454
0.2600
0.0508
0.0636
NDX
0.0392
0.0493
0.0526
0.2461
0.0584
0.0650
RUT
0,0308
0.0422
0.0414
0.2435
0,0481
0.0551
VIX
0.6901
0.5960
0.6433
0,0826
0.6170
0.5640
MXWLD
0.0355
0.0573
0.0631
0.2702
0.0651
0.0783
MXEF
0.0502
0.0570
0.0607
0.2242
0.0661
0.0697
Table 10. Generalization (MAE) one-day interval data of VAE
From
SPX
NDX
RUT
VIX
MXWLD
MXEF
To
SPX
0,0085
0,0133
0,0146
0.0471
0.0291
0.0797
NDX
0.0136
0.0117
0.0175
0.0568
0.0468
0.0812
RUT
0.0142
0.0127
0.0133
0.0587
0.0440
0.0765
VIX
65.0678
322.2718
305.4467
0.1345
885.5610
200.7394
MXWLD
0.0196
0.0196
0.0192
0.0508
0.0362
0.0621
MXEF
0.0285
0,0279
0,0319
0.0529
0.1592
0.1040
Table 11. Generalization (MAE) on one-day interval data of GNO
D. Further Results on Generalization
We extend our analysis by examining the generalization capabilities of VAE and GNO across different assets. The results reveal that models trained on standard assets fail to generalize to VIX, and conversely, models trained VIX perform poorly on other assets. This limitation is expected given the distinct implied volatility distribution of VIX compared to other assets.
Interestingly, we observe that models trained on larger datasets sometimes outperform those trained specifically on smaller datasets (such as MXWLD and MXEF). This phenomenon suggests that VAE and GNO benefit significantly from increased training data. In contrast, HyperIV does not exhibit such data dependency, maintaining consistent performance regardless of training dataset size. This distinction highlights key advantage of HyperIV: its effectiveness in less liquid markets where extensive training data may not be available.
15


Reviews:
[{"review_id": "z9d7sCyKon", "reviewer": "Reviewer_yHpq", "summary": "This paper presents a framework based on hypernetwork to perform the implied volatility smoothing with very few reference samples and small computational cost. The robustness and reliability of the proposed approach is evaluated under a special circumstance, where the smoothing needs to be completed within milliseconds with only a limited number of reference samples.\n\n## update after rebuttal\n\nThe authors have resolved most of my questions. However, the original contributions to the ML community are not very strong, and the use cases of this method are limited to a special condition, e.g., small data size with limited computation time allowed. I have increased my score from 2 to 3.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your insightful comments.\n\n* Justification of real use cases.\n\nThe implied volatility surface is a starting point for option trading and hedging. HyperIV's ability to generate an arbitrage-free surface in ~2 ms from sparse data (9 contracts) is useful for intra-day option traders. Specifically, it enables:\n\n1. Updating the market view based on the latest transitions (e.g., the past minute).\n\n2. Providing timely option quotes.\n\n3. Calculating real-time Option Greeks for dynamic hedging (e.g., delta hedging).\n\n4. Using the latest surface for anomaly detection in subsequent quotes, potentially identifying trading opportunities.\n\n* Connection to Yang & Hospedales (2023) [ECML PKDD]\n\nBoth papers use HyperNetworks, but their work focuses on accelerating calibration for models like rough Bergomi, still requiring iterative optimization (~5 seconds). Our method is calibration-free at inference time, constructing a surface in ~2 ms via a single forward pass. This fundamental difference in approach and speed is why their method wasn't selected as a direct baseline for our specific calibration-free, real-time, sparse-data setting.\n\n* Performance with more data points and computation power.\n\nHyperIV is specifically designed for the challenging scenario of sparse data and high-frequency updates. For general cases like fitting end-of-day surfaces with thousands of options, where time sensitivity is low (once per day), directly training a standard network or calibrating a model might suffice, possibly without needing a hypernetwork. However, the sparse-data setting is a realistic reflection of high-frequency trading conditions where only a few contracts have reliable quotes at any instant, making HyperIV's speed and data efficiency valuable.", "claims_and_evidence": "The claims that the proposed HyperIV is \"particularly valuable for real-time trading applications.\" is not clearly justified in the current manuscript. The authors should provide more specific examples or discussions about how to make use of this fast estimation of the implied volatility surface in financial applications or quantitative tradings.", "methods_and_evaluation_criteria": "The evaluations are restricted to a special setup, where only a small number of reference samples are provided with limited computational resources. The authors need to clarify how common and important is this setup in real-world applications.", "theoretical_claims": "No theory proof involved.", "experimental_designs_or_analyses": "The condition of the experimental setup is restricted to one special setup. See my comments in \"Methods And Evaluation Criteria\".", "relation_to_broader_scientific_literature": "The key contributions of this work is limited to one specific finance problem, e.g., the implied volatility smoothing.", "essential_references_not_discussed": "The method proposed in this work is very similar to that in a previous paper [1], which also utilized hypernetwork to build a financial model. This work is only briefly mentioned in the literature review. More detailed discussions regarding the key differences between these two works should be added.\n\n[1] Yang, Y. and Hospedales, T. M. On calibration of mathematical finance models by hypernetworks. In ECML PKDD, 2023.", "other_strengths_and_weaknesses": "Strength:\n- the problem and method is clearly explained\n- the structure of the paper is well organized\n- experimental results seem promising\n\nWeakness:\n- The value of this work in real-world finance applications/tradings is not clearly justified.\n- The advantage of the proposed framework is not clear in other more general cases, e.g., with enough data points and computation power\n- Difference from previous similar works is not clearly discussed", "other_comments_or_suggestions": "N/A", "questions_for_authors": "1. How is the performance of the proposed method compared to other SOTA methods if we have more data points and allow for more computational time? Does the performance advantage of HyperIV still hold?\n2. How can this technique be used to create values in real-world financial applications/tradings? More discussions and examples should be given.\n3. What is the key difference between HyperIV and HyperCalibration [1], which is not sufficiently discussed in the current manuscript?\n\n[1] Yang, Y. and Hospedales, T. M. On calibration of mathematical finance models by hypernetworks. In ECML PKDD, 2023.", "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": "WBvJTdqb1E", "reviewer": "Reviewer_tu8c", "summary": "This paper introduces a new method called HyperIV, designed to quickly construct accurate and arbitrage-free implied volatility surfaces using minimal market data.\nMain findings and results include:\n1. HyperIV generates high-quality implied volatility surfaces in real-time—approximately within just 2 milliseconds—using only 9 observed market option prices, making it highly suitable for fast-paced trading environments.\n2. It outperforms other well-known approaches such as the SSVI model, Variational Autoencoders (VAE), and Graph Neural Operators (GNO), both in terms of computational speed and predictive accuracy.\n3. It uses one neural network (a hypernetwork) to instantly generate parameters for a smaller, compact neural network that builds the implied volatility surface. And the model incorporates built-in mechanisms that prevent common arbitrage issues, like calendar spread and butterfly arbitrage, by applying specialized auxiliary loss functions during training.\n\nOne notable feature of HyperIV is its ability to generalize well across different markets, requiring very few data points (only nine contracts) at high-frequency intervals (every minute). This capability addresses practical challenges encountered in real-world trading, where only limited data is reliably available at high frequencies.", "questions": "", "limitations": "", "rating": 5, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your insightful comments.\n\n* Dividend modelling\n\nThe method itself does not rely on specific dividend assumptions like proportional dividends. It uses log forward moneyness ($k = \\log(K/F)$), where the forward price $F$ (taken from data vendors in our study) already incorporates the impact of rates and dividends. We will clarify this in the revised manuscript's Preliminaries section.\n\n* Literature on illiquid options\n\nThank you for recommending these papers. We agree they are relevant and will add them to the literature review in the revised version.\n\n* More experiments on 2024 data\n\nThe original work used data available up to late 2023, as the 2024 data snapshot was not yet released by the vendor at the time of experimentation. We have run preliminary experiments on available 2024 futures options data. The results support our original findings:\n\nAverage MAE on 2024 Data (%)\n\n| Asset   | HyperIV | SSVI  |\n|---------|---------|-------|\n| TNOT10Y | 0.52%   | 0.74% |\n| BONDS   | 0.78%   | 1.18% |\n| CRUDE   | 0.68%   | 1.18% |\n| JYEN    | 0.41%   | 0.57% |\n| EURO    | 0.34%   | 0.40% |\n| GOLD    | 0.49%   | 0.74% |\n| SPX     | 0.47%   | 0.97% |\n\n90th Percentile MAE on 2024 Data (%) (representing the tail/worst cases)\n\n| Asset   | HyperIV | SSVI  |\n|---------|---------|-------|\n| TNOT10Y | 1.16%   | 1.90% |\n| BONDS   | 1.69%   | 2.52% |\n| CRUDE   | 1.56%   | 3.19% |\n| JYEN    | 0.79%   | 1.21% |\n| EURO    | 0.85%   | 0.94% |\n| GOLD    | 1.59%   | 2.04% |\n| SPX     | 1.09%   | 1.87% |\n\nWe will add these results to the revised paper.", "claims_and_evidence": "The major claims in this paper include real-time performance and computational speed, accuracy of implied volatility surfaces, non-arbitrage, and generalization across markets. And they are clearly supported based on the claimed testing results from the authors.", "methods_and_evaluation_criteria": "The proposed method and evaluation criteria are standard in evaluating the goodness of the implied vol surface. Another criteria that can be added is the ratio of the fitted implied vols within the best bid/ask, this is also very important.\n\nOne of the baseline is the SSVI model which is known to suffer several issues, and it not enough for today's financial market; it is OK to use it as a representative of the parametric fitting methods, however, if possible, the authors can also compare their methods with more advanced parametric method using more parameters and constrains such as the vola dynamic's products.\n\nThe authors may also want to study the stability of the vol surfaces across the day, i.e. the vol surface should not change too much, in particular, on the wings, if there is no significant market news happening. This can be done by calculating the variance swap price using the fitting implied vol surfaces.", "theoretical_claims": "There is no essentially any new theoretical results; the whole paper is more on the fine application of hypernetwork in learning the shape of implied vol smiles; the trick on adding penalty functions to reduce the arbitrage possibility (formula 14-17) is standard.\n\nOne of the claim which is not essential here is that the author only assume proportional divided on Page 3; this may be OK in this paper as the examples are all index; however, it seems that the assets discussed in the paper are not constrained to index, the author may also want to mention the general affine dividend modeling in the literature.", "experimental_designs_or_analyses": "The proposed experiments look good in general; as mentioned above, the authors could check the values of the variance swap from their fitted implied vol surfaces.", "relation_to_broader_scientific_literature": "This paper discusses how to apply hypernetwork in implied vol surface fitting; this helps to enrich the literature on the applications on deep learning in the implied vol surface fitting. Moreover, the authors achieved a high-speed which is rarely discussed in the previous literature while keeping a good fitting quality, making it more possible to apply such deep learning based method in real-life trading This is a very interesting key contribution.", "essential_references_not_discussed": "The authors discussed how to fit a vol surface given only few data and the non-arbitrage conditions. The following papers also discussed how to fit implied vol surfaces for illiquid names and conditions to guarantee non-existence of calendar arbitrage which should be included in the literature review and discussions:\n\nThe Longitude: Managing Implied Volatility of Illiquid Assets (it discuss how to fit illiquid names)\n\nVolatility Transformers: an optimal transport-inspired approach to arbitrage-free shaping of implied volatility surfaces (it discuss how to transfer implied vols/densities from one maturity to another) \n\nOne-X Property Conjecture, Stochastic Orders and Implied Volatility Surface Construction (it discuss sufficient and necessary conditions to eliminate calendar arbitrage for implied densities over different expiries, and provide a deep discuss on the theoretical side on conditions to eliminate arbitrage.)", "other_strengths_and_weaknesses": "The main strength of this paper, based on its claimed testing results, is the introduction of a fast, robust, deep-learning based vol fitting method, making it more possible to apply it in real-life trading.\n\nHowever, they are also some potential weakness:\n1. the tested underlying are all index which is known to be liquid and easy to fit in general; it is better to test some other real illiquid names such EEM.\n\n2. The author mentioned W-shape in the introduction part, however, did not really dig into it. This is an important topic and appears in single stocks a lot; the authors may want to study the performance of their method on single stocks around earning dates (notice that the options are American options).\n\n3. the testing period is relatively short, only covering half a year of 2023; if possible, the authors should consider their method for 2024 year's data, as there are many macro events making the market volatile.", "other_comments_or_suggestions": "There seems no essential typo or gramma issues in this paper; in general, it is written nice and clearly.", "questions_for_authors": "Besides, the comments and suggestions above, some of the main questions are:\n\n1. if possible, could you also test against the data from 2024, especially, the data in the 2nd half of 2024? Also, add the calculation of variance swap, if possible.\n\n2. if possible, could you try the algorithm for single stocks such as AAPL, around earning dates, and also for very illiquid ETF names? (minor)\n\n3. Enrich the literature review as mentioned above.", "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": "dK2FtcCJkt", "reviewer": "Reviewer_drJn", "summary": "The paper studies the problem of fitting the implied volatility surface. They consider a challenging setting when the time interval is reduced to one minute. This would mean the sample size is much smaller, therefore making the problem challenging. Toward this goal, they use hyper-network, the practice of using one neural network to generate the weights of another network, separating training data from test input.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your insightful comments.\n\n* Data allocation between 1-day and 1-minute intervals\n\nTo clarify the data usage: while the 1-day dataset covers more *assets* (8 vs 2), the 1-minute data constitutes the majority (about 87%) of surfaces analyzed (~130,000 out of ~150,000 total, see Table 2). We included the diverse 1-day assets partly because end-of-day data is more accessible, aiding reproducibility, and also to provide a testbed for the cross-asset generalization study.\n\n* Performance comparison on 1-minute SPX data\n\nIt is true that GNO achieved a slightly lower MAE on 1-minute SPX data (0.0140 vs 0.0167, Table 4). However, this should be considered alongside computational costs (Table 3): HyperIV uses about 1/4 the memory and is nearly 4x faster at inference. Furthermore, HyperIV demonstrates superior *robustness*, performing consistently well across all assets, whereas GNO's performance degrades significantly on VIX and MXEF. For context, if we scaled up HyperIV to use half of GNO’s resources (memory/runtime), its MAE could be reduced to 0.0128, easily surpassing GNO. We presented the lightweight version as this marginal accuracy difference did not outweigh the significant efficiency advantage.\n\n* Using more market features\n\nYes, it is possible to extend HyperIV (and potentially the baselines) to include additional features beyond moneyness and maturity. This could be an interesting direction for future work.\n\n* Density function in Equation (11) and the second derivative\n\nThe term $p(k,t)$ in Eq. (11) represents the **risk-neutral probability density function** of the terminal log-moneyness, $\\log(S_t/F)$, for maturity $t$.\n\nThe second derivative of the *undiscounted* call price $C(K)$ with respect to the strike price $K$ yields the risk-neutral probability density function $Q(K)$ of the terminal asset price $S_T$, evaluated at $K$:\n\nThe undiscounted call price is:\n \n$$\nC(K) = \\mathbb{E}^Q[\\max(S_T - K, 0)] = \\int_{K}^{\\infty} (S_T - K) Q(S_T)  dS_T\n$$\n\nTaking the first derivative w.r.t. $K$:\n\n$$\n\\frac{\\partial C(K)}{\\partial K} = - \\int_{K}^{\\infty} Q(S_T)  dS_T\n$$\n\nTaking the second derivative w.r.t. $K$:\n\n$$\n\\frac{\\partial^2 C(K)}{\\partial K^2} = Q(K)\n$$\n\nTherefore, the second derivative yields the density of the underlying asset price under the risk-neutral measure $Q$.\n\nThis derivation is model-agnostic. Our $p(k,t)$ is directly related to this density $Q(K)$ through a change of variable from strike price $K$ to log-moneyness $k=\\log(K/F)$.", "claims_and_evidence": "The key claim is that in the setting the authors care about, the proposed method gives the smallest MAE loss on the test split. This claim itself is supported by the Table 4 and Table 5.", "methods_and_evaluation_criteria": "I have very little experience in option pricing literature and am unable to confidently judge if MAE is enough.", "theoretical_claims": "There are no theoretical claims. There are some mathematical results that cite existing literatures.", "experimental_designs_or_analyses": "The experiment uses data from 6 index funds and perform the experiment. They use 2 funds for 1-minitue interval and 6 for 1-day interval. The experiment design is a bit strange, as the author claims to study the setting where the interval is 1-minitue. This is the main setting they would like to focus on, but more data is devoted to the setting when the interval is 1-day. In the 1-minitue setting, the proposed methods only beat the other baselines in one of the example.", "relation_to_broader_scientific_literature": "The discussion of related literature is extensive.", "essential_references_not_discussed": "No.", "other_strengths_and_weaknesses": "The experiment results are a bit weak for reasons discussed above.", "other_comments_or_suggestions": "The author should discuss extensions of their algorithm where we can use more features from the market to better predict the IV surface.", "questions_for_authors": "What is the density from equation (11)? Why the second derivative w.r.t. k makes a density function?", "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:

Meta-review: Congratulations for having your paper accepted!

Two of the reviewers engaged in a healthy discussion with the authors and provided valuable comments that, arguably have improved the quality, presentation and the support of the claims of the paper.

I would advise the authors take the time till the camera ready version to incorporate the reviewers feedback and the extra comparisons in the final version of the manuscript.

Once again, congratulations
Decision: Accept (poster)