Quadratic Smile Re-fit in Option Pricing

• The paper shows how the quadratic smile re-fit separates option total variance into level, skew, and curvature components using a parsimonious quadratic function.
• It employs a per-tenor refit on a fixed tenor–delta grid with iterative delta inversion to ensure convergence and alignment with target construction.
• Results indicate that while the method is computationally efficient and aligns with model targets, it can face arbitrage issues and errors under sparse data conditions.

Searching arXiv for papers on quadratic smile re-fit and closely related smile/refit methodologies. I’ll look up the most relevant arXiv records to ground the article in the cited literature. Quadratic smile re-fit denotes a family of smile-refitting procedures in which the strike dependence of option total variance, or an effective variance correction inside a pricing formula, is represented by a quadratic function of a moneyness-like coordinate at fixed maturity. In the literature considered here, the term appears in two closely related but distinct senses: as a direct per-tenor fit of total variance in log-moneyness, $w_{T_i}(k)=a_i+b_i k+c_i k^2$, and as a maturity-dependent effective-variance correction that is quadratic in the caplet moneyness variable $d_{(2)}$. In both cases the quadratic structure separates level, skew, and curvature in a way that is analytically convenient and operationally useful, while leaving unresolved questions about extrapolation, dynamic consistency, and static no-arbitrage outside the fitted region (Singh et al., 15 Jun 2026, Turfus et al., 2023).

1. Formal structure of the quadratic re-fit

The canonical quadratic specification is written in total variance rather than implied volatility. For a fixed expiry $T_i$, the smile is modeled as

$$w_{T_i}(k)=a_i+b_i k+c_i k^2, \qquad w(k)=\sigma^2(k)\,T,$$

with $k=\log(K/F)$. In this parameterization, $a_i$ captures level, $b_i$ captures skew, and $c_i$ captures curvature. The formulation is explicitly described as “deliberately less expressive than SVI,” which makes its role clear: it is a parsimonious local refit rather than a universal surface model (Singh et al., 15 Jun 2026).

A second quadratic form arises in analytic caplet pricing. There, the asymptotic price is re-expressed through an effective variance

$V_{\mathrm{effective}}=V_C+v(K,T_1,T_2),$

with the correction

$$v(K,T_1,T_2)\sim 2V_C^{1/2}\left(C_1(T_1,T_2)-C_2(T_1,T_2)\,d_{(2)}+C_3(T_1,T_2)\left(d_{(2)}^2-1\right)\right).$$

In that representation, the constant term is a level correction, the linear term in $d_{(2)}$0 is the skew correction, and the quadratic term in $d_{(2)}$1 is the smile correction. The paper explicitly interprets the framework as “essentially equivalent to imposing on a Hull–White model an effective variance which is a quadratic function of the moneyness parameter for any given maturity” (Turfus et al., 2023).

These two uses share the same organizing principle: a quadratic dependence is not primarily a claim about true market dynamics, but a way to encode the first three practically important smile features—level, skew, and curvature—with minimal dimensionality.

2. Per-tenor quadratic refitting on a tenor–delta grid

In the crypto volatility-surface setting, the quadratic smile re-fit is implemented as a deterministic per-tenor procedure on a fixed $d_{(2)}$2 tenor–delta grid with

$d_{(2)}$3

for a total of 42 cells per surface. For each snapshot and each listed expiry $d_{(2)}$4 with at least five train-grade strikes, the coefficients $d_{(2)}$5 are fitted separately by ordinary least squares on observed $d_{(2)}$6 pairs. For a target tenor $d_{(2)}$7, the fitted coefficients are then linearly interpolated between the two bracketing expiries, with flat extrapolation at the boundary tenors; the paper notes that this is “equivalent to a linear-in-$d_{(2)}$8 interpolation of total variance at fixed $d_{(2)}$9” (Singh et al., 15 Jun 2026).

The final step is delta inversion. For each target call delta $T_i$0, $T_i$1 is obtained by fixed-point iteration using the Black-76 delta relation

$T_i$2

with update

$T_i$3

until convergence, with a cap of 50 iterations. Any cell that fails to reproduce $T_i$4 to within $T_i$5 is rejected (Singh et al., 15 Jun 2026).

A distinctive feature of this implementation is that the re-fit is not merely a benchmark smile parameterization. It is the inverse of the gridding procedure used to create the model targets. The gridded surfaces themselves are built by applying this exact quadratic/interpolation/delta-inversion pipeline to the full observed chain. That is why the paper describes the baseline as “deliberately strong”: when a tenor row is sufficiently populated, the refit is effectively aligned with the target-construction map itself (Singh et al., 15 Jun 2026).

3. Effective-variance reformulation in short-rate option pricing

In analytic pricing of backward-looking compounded-rate caplets, the quadratic smile re-fit appears not as a direct regression in strike space, but as a derived effective-variance representation inside a Hull–White-like pricing formula. The short-rate model is

$T_i$6

where $T_i$7 follows an Ornstein–Uhlenbeck process

$T_i$8

The parameters are interpreted as follows: $T_i$9 controls the overall level, $$w_{T_i}(k)=a_i+b_i k+c_i k^2, \qquad w(k)=\sigma^2(k)\,T,$$0 is the skew adjustment, and $$w_{T_i}(k)=a_i+b_i k+c_i k^2, \qquad w(k)=\sigma^2(k)\,T,$$1 is the smile factor. When $$w_{T_i}(k)=a_i+b_i k+c_i k^2, \qquad w(k)=\sigma^2(k)\,T,$$2, the model reduces to Hull–White (Turfus et al., 2023).

The caplet price is re-written in terms of an effective variance whose baseline component is

$$w_{T_i}(k)=a_i+b_i k+c_i k^2, \qquad w(k)=\sigma^2(k)\,T,$$3

and whose strike dependence is absorbed into the quadratic correction $$w_{T_i}(k)=a_i+b_i k+c_i k^2, \qquad w(k)=\sigma^2(k)\,T,$$4. The coefficients $$w_{T_i}(k)=a_i+b_i k+c_i k^2, \qquad w(k)=\sigma^2(k)\,T,$$5 are maturity-dependent integrals built from the model kernel and the functions $$w_{T_i}(k)=a_i+b_i k+c_i k^2, \qquad w(k)=\sigma^2(k)\,T,$$6 and $$w_{T_i}(k)=a_i+b_i k+c_i k^2, \qquad w(k)=\sigma^2(k)\,T,$$7. Economically, the paper identifies $$w_{T_i}(k)=a_i+b_i k+c_i k^2, \qquad w(k)=\sigma^2(k)\,T,$$8 as the level correction, $$w_{T_i}(k)=a_i+b_i k+c_i k^2, \qquad w(k)=\sigma^2(k)\,T,$$9 as the skew correction, and $k=\log(K/F)$0 as the smile correction (Turfus et al., 2023).

The same structure carries over to term-rate or LIBOR caplets, where

$k=\log(K/F)$1

and $k=\log(K/F)$2 has the same constant-minus-linear-plus-quadratic dependence on the relevant moneyness variable. The paper therefore treats the quadratic refit as a model-derived implied-volatility translation rather than an ad hoc parametric overlay. This is important because it directly addresses a common misconception: in this setting the quadratic smile form is not introduced for convenience alone, but arises from asymptotic pricing formulas and is then used to obtain a more robust implied-volatility representation (Turfus et al., 2023).

The practical motivation is stability away from the money. The paper states that direct asymptotic price formulas can become unreliable for extreme strikes, whereas using the effective variance inside a Hull–White-style formula helps avoid pathologies such as negative option prices. It also notes a caveat: the quadratic effective-variance expressions could, in extreme parameter regimes, become negative, but this is regarded as outside the intended asymptotic regime and not practically relevant (Turfus et al., 2023).

4. Quadratic refitting within skew–smile decompositions

A related line of work does not impose a quadratic smile directly, but decomposes smile fitting into a skew-first stage and a subsequent smile correction. In modeling SPX and DAX index options, the Shifted Log-Normal (SLN) model—also known as displaced diffusion—is used to fit the near-forward region and extract a stable skew parameter $k=\log(K/F)$3, after which smile effects are added either perturbatively or by refitting with SABR. The paper’s practical recipe is explicitly summarized as “skew first, smile next” (Kuklinski et al., 2014).

The SLN model is attractive because it gives a good fit of near-at-the-forward strikes, especially for maturities of 60 days or more, and because its skewness has a simple algebraic form: $k=\log(K/F)$4 This allows skew to be estimated accurately from prices themselves, without directly computing the third moment from the full probability density. The paper emphasizes that the far tails are difficult to estimate directly because of poor liquidity and wider bid-ask spreads, so the near-forward fit serves as a stable first stage (Kuklinski et al., 2014).

Conceptually, this decomposition is closely related to quadratic smile refitting. Both approaches separate skew from higher-order curvature and treat the residual smile as a secondary correction. In the SLN/SABR framework the second-stage object is not literally a quadratic polynomial, but the operational logic is parallel: determine the leading asymmetry from liquid near-forward data, then add curvature only where needed. The theoretical connection is reinforced by the result that SLN trajectories and prices are exact solutions of SABR for $k=\log(K/F)$5, so the first-stage skew fit is dynamically meaningful rather than purely descriptive (Kuklinski et al., 2014).

Empirically, the same paper reports that SPX is strongly negatively skewed and can often be fit very well by SLN over a relatively wide strike range, while DAX exhibits stronger smile dominance, especially at short expiries, making smile adjustment more essential. This distinction is a reminder that quadratic or near-quadratic smile corrections are most effective when the surface is primarily a skew deformation plus moderate curvature, and less effective when curvature dominates the cross-section (Kuklinski et al., 2014).

5. Identifiability, missing data, and static no-arbitrage

The most explicit empirical evaluation of a quadratic smile re-fit under partial observability appears in the crypto surface-completion study. There the hybrid predictor uses a deterministic per-tenor rule: $k=\log(K/F)$6 The threshold of three observed cells is not heuristic: a quadratic $k=\log(K/F)$7 has three parameters, so below three distinct $k=\log(K/F)$8 points the row-wise fit is rank-deficient or not well posed (Singh et al., 15 Jun 2026).

This identifiability boundary explains the performance profile. On the BTC test set under random masking, hidden-cell RMSE for the smile re-fit only is $k=\log(K/F)$9 at $a_i$0, $a_i$1 at $a_i$2, $a_i$3 at $a_i$4, $a_i$5 at $a_i$6, and $a_i$7 at $a_i$8. The corresponding hybrid errors are $a_i$9, $b_i$0, $b_i$1, $b_i$2, and $b_i$3, while ConvVAE only yields $b_i$4, $b_i$5, $b_i$6, $b_i$7, and $b_i$8. The paper highlights the $b_i$9 masking case as an eightfold reduction versus the smile baseline, achieved at no additional inference cost (Singh et al., 15 Jun 2026).

Structured missingness isolates the same phenomenon more sharply. When each tenor row still contains at least five of seven cells, as in column holes or wing holes, the quadratic fit remains rank-sufficient and is essentially perfect. When an entire tenor row is missing, the fit must extrapolate $c_i$0 across tenors, and the error becomes catastrophic: the structured-hole table reports smile re-fit RMSE $c_i$1 for row_random and $c_i$2 for long_tenor. The paper explicitly states that under such row-hole scenarios the learned model is the only viable predictor (Singh et al., 15 Jun 2026).

A second misconception addressed directly in this literature is that a quadratic smile refit is automatically arbitrage-free. The crypto paper states the opposite: the quadratic smile re-fit alone does not reliably satisfy static no-arbitrage, especially at high mask rates. At $c_i$3, the smile re-fit produces calendar violations on $c_i$4 of BTC reconstructions and $c_i$5 of ETH reconstructions, and butterfly violations on $c_i$6 of BTC reconstructions and $c_i$7 of ETH reconstructions. By contrast, the ConvVAE and the deployed hybrid have $c_i$8 butterfly violations on both BTC and ETH, and are made calendar-free at the listed strikes after an essentially free $c_i$9 isotonic regression projection whose RMSE impact is at most about $V_{\mathrm{effective}}=V_C+v(K,T_1,T_2),$0 vol points for the learned predictors (Singh et al., 15 Jun 2026).

These results delimit the natural domain of the quadratic re-fit. It is strongest when the data-generating map is itself quadratic and the local cross-section is sufficiently observed; it becomes fragile under rank deficiency, cross-tenor extrapolation, and arbitrage checks performed beyond the immediate regression problem.

6. Relation to richer smile parameterizations and structural models

Quadratic smile refits occupy a specific place within a broader hierarchy of smile modeling. One direction extends the local polynomial idea while replacing global quadratic wings by asymmetric nonlinear controls. The sigmoid-based functional description of the volatility smile defines total implied variance as

$V_{\mathrm{effective}}=V_C+v(K,T_1,T_2),$1

with separate left- and right-wing parameters $V_{\mathrm{effective}}=V_C+v(K,T_1,T_2),$2 and $V_{\mathrm{effective}}=V_C+v(K,T_1,T_2),$3, and a center shift $V_{\mathrm{effective}}=V_C+v(K,T_1,T_2),$4. The paper emphasizes that around the critical point $V_{\mathrm{effective}}=V_C+v(K,T_1,T_2),$5, the smile behaves like a local low-order polynomial, while the sigmoid structure controls the wings; it is explicitly designed to improve on quadratic and SVI-style refits when one wants a term-by-term calibration assembled into an arbitrage-free surface on a grid (Itkin, 2014).

A second direction embeds smile refitting into a dynamic pricing model rather than a static surface parameterization. In the Markov-Functional interest-rate framework, the smile enters through UVDD digital mapping, which replaces the Black-Scholes digital mapping by a mixture of displaced lognormals. In the two-component parametrization, the displacement $V_{\mathrm{effective}}=V_C+v(K,T_1,T_2),$6 controls skew, while $V_{\mathrm{effective}}=V_C+v(K,T_1,T_2),$7 and $V_{\mathrm{effective}}=V_C+v(K,T_1,T_2),$8 control smile curvature or convexity. The reported average absolute errors—Black ATM model $V_{\mathrm{effective}}=V_C+v(K,T_1,T_2),$9, lognormal $$v(K,T_1,T_2)\sim 2V_C^{1/2}\left(C_1(T_1,T_2)-C_2(T_1,T_2)\,d_{(2)}+C_3(T_1,T_2)\left(d_{(2)}^2-1\right)\right).$$0, displaced diffusion $$v(K,T_1,T_2)\sim 2V_C^{1/2}\left(C_1(T_1,T_2)-C_2(T_1,T_2)\,d_{(2)}+C_3(T_1,T_2)\left(d_{(2)}^2-1\right)\right).$$1, UVDD $$v(K,T_1,T_2)\sim 2V_C^{1/2}\left(C_1(T_1,T_2)-C_2(T_1,T_2)\,d_{(2)}+C_3(T_1,T_2)\left(d_{(2)}^2-1\right)\right).$$2, and UVDD with restricted $$v(K,T_1,T_2)\sim 2V_C^{1/2}\left(C_1(T_1,T_2)-C_2(T_1,T_2)\,d_{(2)}+C_3(T_1,T_2)\left(d_{(2)}^2-1\right)\right).$$3 $$v(K,T_1,T_2)\sim 2V_C^{1/2}\left(C_1(T_1,T_2)-C_2(T_1,T_2)\,d_{(2)}+C_3(T_1,T_2)\left(d_{(2)}^2-1\right)\right).$$4—show how a richer structural mapping can dramatically outperform simpler static smile fits while preserving the Markov-Functional framework (Wang, 2014).

A third direction addresses coupled surfaces rather than a single asset smile in isolation. The quadratic rough Heston model combines rough volatility with a quadratic price-feedback mechanism $$v(K,T_1,T_2)\sim 2V_C^{1/2}\left(C_1(T_1,T_2)-C_2(T_1,T_2)\,d_{(2)}+C_3(T_1,T_2)\left(d_{(2)}^2-1\right)\right).$$5 and is presented as a continuous-time, continuous-path framework that can fit SPX and VIX smiles jointly. The paper reports a calibration to May 19, 2017 market data over short expirations from 2 to 5 weeks, with parameters

$$v(K,T_1,T_2)\sim 2V_C^{1/2}\left(C_1(T_1,T_2)-C_2(T_1,T_2)\,d_{(2)}+C_3(T_1,T_2)\left(d_{(2)}^2-1\right)\right).$$6

and states that VIX smiles fall systematically within bid-ask spreads while the SPX volatility surface is reproduced very well, including the extreme left tail. This suggests that once the refitting problem is broadened from a single smile to a jointly consistent SPX/VIX system, quadratic corrections in the smile itself are insufficient unless accompanied by roughness and asymmetric feedback in the state dynamics (Gatheral et al., 2020).

Taken together, these developments place quadratic smile re-fit in a precise role. It is a low-dimensional mechanism for isolating level, skew, and curvature; it can be exact relative to a chosen gridding map; and it can even emerge as a derived effective-variance representation from an analytic model. But the surrounding literature also shows its limits. When wing asymmetry, cross-maturity consistency, dynamic smile evolution, joint SPX/VIX calibration, or strong no-arbitrage requirements become central, richer parameterizations or structural stochastic-volatility models are typically introduced to absorb the phenomena that a purely quadratic refit leaves unresolved.