Arbitrage-Free Implied Volatility Surface Reconstruction via Additive Martingales

Haochen Luo; Sifan Lin; Xiaolong Luo; An Binh Minh; Yuan Zhang; Chen Liu

q-fin.CPOct 19, 2026

Arbitrage-Free Implied Volatility Surface Reconstruction via Additive Martingales

Authors: Haochen Luo, Sifan Lin, Xiaolong Luo, An Binh Minh, Yuan Zhang, Chen Liu

Organizations: City University of Hong Kong · Shanghai University of Finance and Economics

Abstract

This work focuses on reconstructing a continuous implied volatility (IV) surface from a small number of option quotes, a fundamental task for option pricing and risk management. We propose Additive Martingale (AddMart), a few-shot IV surface reconstruction framework that satisfies static no-arbitrage conditions by construction. Rather than predicting IVs directly, AddMart maps sparse quotes to non-negative increment densities of an exponential additive martingale, from which valid risk-neutral marginals and IVs are recovered. By incorporating the no-arbitrage structure directly into its architecture, rather than enforcing it through regularization in the learning objective, AddMart produces IV surfaces that satisfy static no-arbitrage conditions throughout the domain. Extensive experiments on ten real-option datasets demonstrate that AddMart achieves competitive reconstruction accuracy with no material static arbitrage violations, while directly enabling the estimation of VaR, CVaR, and crash probabilities.

Figures & tables

Figure 1: The ADDMART pipeline. Reference quotes are encoded into a latent representation for maturity-wise increment prediction, followed by parameter-free deterministic pricing and inversion.

Figure 2: Butterfly and calendar restrictions implied by static no-arbitrage.

		AddMart		HyperIV		GNO		VAE		SSVI ‡		Deep Sm. ‡
Scale	Asset	Reliable	Overall	Reliable	Overall	Reliable	Overall	Reliable	Overall	Reliable	Overall	Reliable	Overall
Daily	SPY	$\mathbf{0.56}{\scriptscriptstyle\,\pm 0.02}$	$\mathbf{0.89}{\scriptscriptstyle\,\pm 0.05}$	$0.57{\scriptscriptstyle\,\pm 0.02}$	$0.99{\scriptscriptstyle\,\pm 0.03}$	$0.62{\scriptscriptstyle\,\pm 0.05}$	$0.95{\scriptscriptstyle\,\pm 0.07}$	$1.33{\scriptscriptstyle\,\pm 0.15}$	$1.75{\scriptscriptstyle\,\pm 0.11}$	$1.29$	$2.24$	$1.07$	$2.23$
	QQQ	$\mathbf{0.68}{\scriptscriptstyle\,\pm 0.03}$	$\mathbf{1.25}{\scriptscriptstyle\,\pm 0.05}$	$0.74{\scriptscriptstyle\,\pm 0.03}$	$1.31{\scriptscriptstyle\,\pm 0.04}$	$0.90{\scriptscriptstyle\,\pm 0.10}$	$1.45{\scriptscriptstyle\,\pm 0.13}$	$1.30{\scriptscriptstyle\,\pm 0.15}$	$1.98{\scriptscriptstyle\,\pm 0.07}$	$1.20$	$2.19$	$1.11$	$2.57$
	IWM	$\mathbf{0.77}{\scriptscriptstyle\,\pm 0.03}$	$\mathbf{1.17}{\scriptscriptstyle\,\pm 0.03}$	$0.78{\scriptscriptstyle\,\pm 0.08}$	$1.27{\scriptscriptstyle\,\pm 0.09}$	$1.09{\scriptscriptstyle\,\pm 0.53}$	$1.77{\scriptscriptstyle\,\pm 0.94}$	$1.19{\scriptscriptstyle\,\pm 0.12}$	$1.93{\scriptscriptstyle\,\pm 0.14}$	$1.16$	$1.95$	$1.20$	$2.57$
	AAPL	$\mathbf{1.18}{\scriptscriptstyle\,\pm 0.07}$	$\mathbf{1.40}{\scriptscriptstyle\,\pm 0.05}$	$\mathbf{1.18}{\scriptscriptstyle\,\pm 0.07}$	$1.55{\scriptscriptstyle\,\pm 0.08}$	$1.52{\scriptscriptstyle\,\pm 0.25}$	$1.77{\scriptscriptstyle\,\pm 0.31}$	$1.76{\scriptscriptstyle\,\pm 0.04}$	$2.37{\scriptscriptstyle\,\pm 0.10}$	$2.08$	$3.14$	$2.90$	$5.11$
	GOOGL	$2.22{\scriptscriptstyle\,\pm 0.14}$	$\mathbf{2.37}{\scriptscriptstyle\,\pm 0.11}$	$\mathbf{2.07}{\scriptscriptstyle\,\pm 0.25}$	$2.43{\scriptscriptstyle\,\pm 0.24}$	$2.59{\scriptscriptstyle\,\pm 0.32}$	$2.76{\scriptscriptstyle\,\pm 0.24}$	$3.12{\scriptscriptstyle\,\pm 0.09}$	$3.42{\scriptscriptstyle\,\pm 0.10}$	$3.21$	$4.01$	$5.87$	$8.10$

Table 1: Reliable and overall IV-MAE (

\times 10^{-2}

, mean

\pm

std over 5 seeds). Lowest per metric within each asset row in bold. ‡ SSVI and Deep Smoothing are per-surface fits (one optimization per surface, not amortized over a training seed), reported as a single value.

		AddMart		HyperIV		GNO		VAE		SSVI ‡		Deep Sm. ‡
Scale	Asset	Calendar	Butterfly	Calendar	Butterfly	Calendar	Butterfly	Calendar	Butterfly	Calendar	Butterfly	Calendar	Butterfly
Daily	SPY	$\mathbf{0}$	$\mathbf{0}$	$0.7310$	$1.6295$	$15.6910$	$4.4026$	$0.0546$	$8.4843$	$\mathbf{0}$	$\mathbf{0}$	$\mathbf{0}$	$0.0113$
	QQQ	$\mathbf{0}$	$\mathbf{0}$	$1.0047$	$2.3451$	$22.2821$	$5.2844$	$0.2145$	$7.8386$	$\mathbf{0}$	$\mathbf{0}$	$\mathbf{0}$	$0.0222$
	IWM	$\mathbf{0}$	$\mathbf{0}$	$0.6008$	$4.8701$	$18.8903$	$7.6170$	$0.3043$	$6.8536$	$\mathbf{0}$	$\mathbf{0}$	$\mathbf{0}$	$0.0642$
	AAPL	$\mathbf{0}$	$\mathbf{0}$	$1.0690$	$6.6294$	$20.6356$	$5.0270$	$0.0785$	$5.0351$	$\mathbf{0}$	$\mathbf{0}$	$\mathbf{0}$	$0.0160$
	GOOGL	$\mathbf{0}$	$\mathbf{0}$	$0.9958$	$7.9404$	$14.6969$	$3.4538$	$0.2408$	$5.5603$	$\mathbf{0}$	$\mathbf{0}$	$0.0051$	$0.0195$

Table 2: Calendar and butterfly arbitrage-violation pair rates (%) on the common

801\times 30

full-support price grid, using a numerical tolerance of

10^{-10}

. Amortized methods are pooled over 5 seeds; ‡ SSVI and Deep Smoothing are per-surface fits and are reported as a single value. Lowest per metric within each asset row in bold.

Figure 3: Risk-neutral densities and CDFs. AddMart guarantees valid densities and monotone CDFs, while IV prediction-based baselines may violate these properties, particularly for one-year NVDA. Insets highlight the tails.

Reliable IV-MAE
Variant	SPY	NVDA
Linear exact	$0.56{\pm}0.04$	$1.84{\pm}0.06$
Linear crop	$0.56{\pm}0.02$	$1.82{\pm}0.06$
Circular	$\mathbf{0.55{\pm}0.04}$	$\mathbf{1.76{\pm}0.06}$
Calendar violation (%)
Linear exact	$\mathbf{0.000}$	$\mathbf{0.000}$

Table 3: Key ablations on SPY and NVDA daily (mean

\pm

std over five seeds). IV-MAE is reported in

\times 10^{-2}

. All mixture and training-loss variants use the exact linear decoder and have zero calendar and butterfly violations.

Figure 4: Effect of reference-set size on AddMart for SPY and NVDA on daily data. Lines and shaded regions report the mean and standard deviation of reliable-region IV-MAE over five seeds.

Appendix figures & tables18 assets

Supplementary material from the paper’s appendix.

Appendix

Symbol	Domain or type	Description
Option market and pricing
$B(k,s_{\mathrm{tot}})$	$[0,1]$	Forward-normalized Black call price.
$C(k,\tau)$	$[0,1]$	Call price in log-moneyness coordinates, $C(k,\tau)=c(e^{k},\tau)$ .
$c(z,\tau)$	$[0,1]$	Forward-normalized undiscounted call price, $c(z,\tau)=\mathbb{E}_{\mathbb{Q}}[(M_{\tau}-z)^{+}]$ .
$d_{+},d_{-}$	$\mathbb{R}$	Black variables, $d_{+}=-k/s_{\mathrm{tot}}+s_{\mathrm{tot}}/2$ and $d_{-}=-k/s_{\mathrm{tot}}-s_{\mathrm{tot}}/2$ .
$\mathrm{DF}(\tau)$	$(0,1]$	Discount factor for maturity $\tau$ , defined as $\mathrm{DF}(\tau)=\exp[-r(\tau)\tau]$ .

Appendix

Table 4: Notation used throughout the paper.

Component	Tensor mapping	Configuration
Quote embedding	$\mathbb{R}^{K\times 3}\rightarrow\mathbb{R}^{K\times 128}$	Linear projection
Set encoder	$\mathbb{R}^{K\times 128}\rightarrow\mathbb{R}^{K\times 128}$	2 layers, 2 heads
Surface pooling	$\mathbb{R}^{K\times 128}\rightarrow\mathbb{R}^{64}$	Mean pooling, linear map
Bucket conditioning	$\mathbb{R}^{64}\rightarrow\mathbb{R}^{66}$	Append $(\tau_{g},\sqrt{\tau_{g}})$
Density head	$\mathbb{R}^{66}\rightarrow\mathbb{R}^{3R}$	MLP $(66,128,128,3R)$
Increment law	$\mathbb{R}^{3R}\rightarrow\mathbb{R}^{X}$	$R=8$ , $X=2048$

Appendix

Table 5: AddMart architecture used in the main experiments. The number of reference quotes

K

may vary without changing the network.

Scale	Asset	Surfaces	Split (train/val/test)	Contracts	Avg. contracts	$\tau$ range	$k$ range
Daily	SPY	$992$	$601/67/324$	$920{,}977$	$928.4$	$[0.0027,1.9726]$	$[-2.482,\,0.696]$
	QQQ	$958$	$584/65/309$	$563{,}724$	$588.4$	$[0.0027,1.9726]$	$[-1.355,\,0.680]$
	IWM	$727$	$406/45/276$	$270{,}787$	$372.5$	$[0.0027,1.9726]$	$[-1.418,\,0.540]$
	AAPL	$465$	$198/22/245$	$85{,}619$	$184.1$	$[0.0027,1.9726]$	$[-2.454,\,0.798]$
	GOOGL	$573$	$321/36/216$	$134{,}580$	$234.9$	$[0.0027,1.9671]$	$[-1.587,\,0.903]$
	META	$591$	$282/31/278$	$190{,}279$	$322.0$	$[0.0027,1.9726]$	$[-2.715,\,1.488]$

Appendix

Table 6: Statistics of the final experimental datasets. “Split” reports the number of train/validation/test surfaces. A surface is one retained quote timestamp, and “Avg. contracts” is the mean number of retained contracts per surface, including the nine reference quotes. Maturity

\tau

is measured in ACT/365 years from the exact snapshot time to expiration at 16:00 ET, and

k=\log(K/F)

Method	Trainable parameters	Batch	Queries/surface	Epochs (daily/intraday)
HyperIV	$243{,}153$	16	1024	$1500/150$
GNO	$102{,}529$	20	1024	$300/60$
VAE	$332{,}015$	20	1024	$200/60$

Appendix

Table 7: Implementation settings for the amortized baselines. Epoch counts are maximum values for daily/intraday experiments.

	SPY		QQQ		NVDA
Method	180d	1y	180d	1y	180d	1y
AddMart	$\mathbf{1.87{\pm}.37}$	$\mathbf{2.27{\pm}.35}$	$3.05{\pm}.27$	$\mathbf{3.12{\pm}.19}$	$\mathbf{3.86{\pm}.20}$	$\mathbf{2.42{\pm}.21}$
HyperIV	$1.89{\pm}.18$	$4.76{\pm}.66$	$\mathbf{2.88{\pm}.24}$	$4.65{\pm}.39$	$4.72{\pm}.93$	$5.39{\pm}1.91$
GNO	$3.47{\pm}2.48$	$4.23{\pm}.97$	$19.74{\pm}9.79$	$6.73{\pm}2.57$	$5.23{\pm}3.74$	$3.44{\pm}.72$
VAE	$12.51{\pm}3.78$	$6.18{\pm}1.44$	$10.09{\pm}5.42$	$14.79{\pm}5.32$	$12.19{\pm}5.00$	$3.25{\pm}1.22$

Appendix

Table 8: Option-implied downside-CVaR error on daily test surfaces. Entries are CVaR-MAE (

\times 10^{-2}

), averaged over 50 test dates and

\alpha\in\{1\%,5\%,10\%\}

, with mean

\pm

std over five seeds. Lowest error in each column is bold.

Figure 5: Signed CVaR residuals over the test period. Columns denote reconstruction methods and rows denote assets. Within each panel, the heatmap rows combine the 180-day and one-year horizons with tail probabilities of $1\%$ , $5\%$ , and $10\%$ . Blue indicates that the model underestimates the full-information option-implied CVaR, while red indicates overestimation. Values are averaged over five seeds.

Figure 6: Daily reliable-region IV-MAE on the two SPY stress windows. Lines and shaded regions denote the five-seed mean and standard deviation. The April panel uses a logarithmic vertical scale because one GNO seed diverges during the most volatile days.

	Apr. 1–10, 2025		Mar. 23–31, 2026
Method	Reliable	Overall	Reliable	Overall
AddMart	$\mathbf{2.03{\pm}.23}$	$4.92{\pm}.84$	$\mathbf{0.72{\pm}.13}$	$\mathbf{0.98{\pm}.09}$
HyperIV	$2.23{\pm}.45$	$4.99{\pm}.86$	$0.80{\pm}.12$	$1.58{\pm}.19$
GNO	$16.74{\pm}28.25$	$23.85{\pm}37.66$	$0.82{\pm}.20$	$1.19{\pm}.16$
VAE	$2.68{\pm}.45$	$\mathbf{4.76{\pm}.53}$	$1.83{\pm}.24$	$2.06{\pm}.19$

Appendix

Table 9: SPY performance during two out-of-sample market-stress windows. IV-MAE is reported in

\times 10^{-2}

as mean

\pm

std over five seeds. Best results are shown in bold.

Scale	Method	B1 forward	B1 end-to-end	B32 forward	B32 surface/s	Peak MiB B1/B32	Weights MiB
Daily	AddMart	6.98	7.60	1.040	961	127.64 / 2939.44	0.918
	HyperIV	0.58	1.19	0.019	53,201	32.99 / 34.84	0.928
	GNO	1.99	2.60	1.874	533	36.69 / 36.85	0.402
	VAE	0.20	29.60	0.009	113,918	34.04 / 58.05	1.272
Minute	AddMart	6.80	7.43	1.042	960	127.64 / 2939.44	0.918
	HyperIV	0.57	1.21	0.019	52,994	32.99 / 34.84	0.928

Appendix

Table 10: Single-GPU inference performance. Forward and end-to-end latency are reported in milliseconds per surface. Throughput measures neural forward inference at batch size 32. Peak memory is the maximum allocated GPU memory for batch sizes 1 and 32, respectively.

Figure 7: Long-horizon behavior on LEAPS. Left: average ATM-skew term structure on the SPY test set, with the reliable-region cutoff at 14 days and the LEAPS region beginning at 365 days. Right: a representative AAPL LEAPS slice observed on December 10, 2025 (554 DTE; 18 market quotes). The shaded interval denotes the reliable moneyness region

|k|\leq 0.1

. AddMart increasingly underestimates the persistent long-dated skew, whereas HyperIV retains more of the observed skew but extrapolates more aggressively outside the reliable region.

Figure 8: Representative failure case on a VIX daily test surface observed on September 9, 2025. The surface is selected by the median per-surface MAE gap between AddMart and HyperIV rather than by an extreme error. AddMart substantially overestimates the short-maturity smile, while the discrepancy decreases with maturity.

Figure 9: Daily-data error profiles for SPY, QQQ, IWM, AAPL, GOOGL, META, and NVDA. Rows correspond to underlying assets. The columns report 20-surface rolling IV-MAE over the test period, IV-MAE by log-moneyness, and IV-MAE by maturity, respectively. IV-MAE is reported in

\times 10^{-2}

. For amortized methods, lines and shaded regions denote the mean and standard deviation over five seeds; SSVI is fitted independently to each surface.

Figure 10: Joint reconstruction-error profiles across the seven daily datasets. The first column shows market IV; the remaining columns show IV-MAE ( $\times 10^{-2}$ ) aggregated by log-moneyness and maturity. Error color scales are shared across methods within each asset row. The dashed box denotes the reliable region, and gray cells contain insufficient observations for reliable aggregation.

Figure 11: Static-arbitrage violation profiles across the seven daily datasets. Each cell reports the pooled calendar and butterfly pair violation rate (%) on a dense price grid, using a tolerance of $10^{-7}$ . Color scales are shared across methods within each asset row; the dashed box denotes the reliable region.

Figure 12: Joint reconstruction-error profiles on the SPY, QQQ, and IWM minute-scale datasets. Each heatmap reports IV-MAE ( $\times 10^{-2}$ ) by log-moneyness and maturity; notation and masking follow Figure 10 .

Figure 13: Static-arbitrage violation profiles on the SPY, QQQ, and IWM minute-scale datasets. Evaluation and notation follow Figure 11 .

Figure 14: SPY implied-volatility slices on 7 April, 2 May, and 10 June 2025. Each row corresponds to one trading day and each column to one reference maturity. Gray dots denote market quotes and gold stars denote the nine reference contracts used by AddMart, HyperIV, GNO, VAE, and SSVI.

Figure 15: NVDA implied-volatility slices on 7 April, 2 May, and 10 June 2025. Each row corresponds to one trading day and each column to one reference maturity. Gray dots denote market quotes and gold stars denote the nine reference contracts used by AddMart, HyperIV, GNO, VAE, and SSVI.

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