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REVIEW 5 major objections 5 minor 31 references

Integrating the implied regularity into implied volatility models: A study on free arbitrage model

T0 review · 5 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read A closed-form implied volatility curve that pins the Hurst exponent to 1/2 at the money and beats SABR on fit.

desk verdict A genuinely new closed-form IV smile with moneyness-dependent H, wrapped around a no-arbitrage proof that fails at a false monotonicity claim and should be cut or fixed before this is publishable. read the letter →

arxiv 2502.07518 v1 pith:NLIVJP7J submitted 2025-02-11 q-fin.CP

Daniele Angelini , Fabrizio Di Sciorio This is my paper · ORCID
classification q-fin.CP MSC 91G2091G6060G22
keywords impliedvolatilityHurstexponentmoneynessarbitrage-freesurfaceSABRmodelfractionalBrownianmotionsmileOptunacalibration
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proposes the Angelini-di Sciorio (AdS) model, a four-parameter closed-form expression for implied volatility as a function of moneyness $S/K$, with the Hurst exponent $H$ appearing explicitly inside the formula. The model's defining claim is that implied regularity peaks at the at-the-money point, $H(S/K=1)=1/2$, and decays into the in-the-money and out-of-the-money regions, an inverse smile the authors tie to deviations from market efficiency. They argue that when this volatility curve is plugged into the Black-Scholes pricing formula, the resulting call prices satisfy the five conditions for an arbitrage-free volatility surface, and they report that Optuna-calibrated fits to 30-day option data across multiple indexes beat SABR and fSABR on accuracy and curvature error metrics. If correct, the paper supplies a parsimonious, parameterizable, arbitrage-free implied volatility surface that also encodes a moneyness-dependent regularity.

What carries the argument

The load-bearing object is the moneyness-dependent Hurst function $H(S/K)$, a normalized inverse-smile shape that forces $H=1/2$ exactly at $S/K=1$ and dips toward $0$ in both wings, governed by the shape parameter $\delta$. It enters the volatility curve through the exponential decay term $e^{-\beta H(S/K)(S/K-S/K_{\min})}$, so that memory strength modulates how quickly implied volatility falls as moneyness moves away from the minimum-volatility strike; the quadratic prefactor $(S/K-S/K_{\min})^2$ supplies the smile's concavity. The parameter constraints $\alpha>0$, $\epsilon>0$, $\beta\in[-1,1]$, $\delta\in(0,1)$ are derived from concavity and convexity requirements on $H$ and $\sigma$, and the arbitrage-free certification then rides on standard Black-Scholes derivatives plus the martingale premise.

What would settle it

Compute the call-price bounds $(s-K)^+ \le C(T,K) \le s$ on any market-observed surface: pick $s$ = current spot, take a date when the market opens below $s$ (i.e., any downward day), and check whether a deep out-of-the-money call struck above the previous close prices below $(s-K)^+$; a single such violation refutes property (iii). Alternatively, directly test whether the fitted surface passes the butterfly-arbitrage condition $C_{KK} \ge 0$ across all strikes on the 30-day slices used in Table 1.

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Extended reading notes

Core claim

The central claim is that implied volatility can be represented without a stochastic-volatility engine: a single closed formula $\sigma(S/K)=\alpha(S/K - S/K_{\min})^2 e^{-\beta H(S/K)(S/K - S/K_{\min})} + \epsilon$, with $H(S/K)=\frac{1}{2}(1+|1-S/K_{\min}|^\delta)/(1+|S/K-S/K_{\min}|^\delta)$, reproduces the smile while satisfying butterfly- and calendar-arbitrage conditions when $\alpha, \epsilon>0$, $\beta\in[-1,1]$, $\delta\in(0,1)$. The paper's own verification walks through Zaugg's five conditions in Annex 4.3, with the martingale property imported from a multifractional Brownian motion setup in which $H(t)\to 1/2$ at maturity. The empirical section claims that over the fitted 30-day slices, the model's MSE, MAE, curvature-error and absolute-curvature-error metrics are smaller on average than those of SABR and fSABR, with the advantage most pronounced in the ITM and OTM tails.

Load-bearing premise

The arbitrage-free certification rests on the premise that the underlying price process is a martingale under the pricing measure, imported from a multifractional Brownian motion framework in which the Hurst parameter tends to 1/2 at maturity, and on a pathwise monotonicity claim about the underlying never falling below its starting level.

Editorial extensions

If this is right

  • If the claim holds, one can generate an arbitrage-free volatility smile from four parameters, removing the need for numerical simulation or density recovery in smile construction.
  • The $H=1/2$-at-the-money constraint gives a model-free anchor: any calibration must pinch the curve to the efficient-market value at $S=K$, and deviations in the wings are explicitly interpreted as memory and market-inefficiency effects.
  • The parameter restrictions ($\beta\in[-1,1]$, $\delta\in(0,1)$) come out of convexity demands, so fitting within these bounds automatically keeps the surface free of static arbitrage.
  • Across the 42-ticker test set, the reported curvature-error metrics suggest the functional form captures the second derivative of the smile better than SABR and fSABR in extreme moneyness regions.
  • The closed form is cheap enough for repeated recalibration, which matters for desks that re-fit surfaces intraday.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: the same functional form could be applied to VIX futures or variance swaps, where an implied-Hurst curve is less standard, to see whether the inverse-smile regularity pattern is a feature of equity index options or a broader market property.
  • Editorial inference: the curvature error metrics introduce a second-derivative target; a natural next step is to test whether calibration to ACE and RMSCE changes the fitted parameters in a way that degrades raw price fit, i.e., whether the curvature advantage is bought at the cost of price accuracy.
  • Editorial inference: because the model is closed-form with only four parameters, it could be embedded directly into an SVI-type parametrization or used as a prior in a neural-network calibration, reusing the invertible moneyness-to-H map as a structural regularizer.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 5 minor

Summary. The paper proposes a closed-form implied volatility (IV) model, called the AdS model, in which IV is written as a function of moneyness S/K, with a moneyness-dependent Hurst exponent H(S/K). The functional form is designed so that H(1)=1/2 and H decreases away from the money. The authors calibrate the model with Optuna on 30-day option data for many U.S. indices and stocks, compare its in-sample fit against SABR and fSABR using MSE, MAE, and curvature errors, and claim that the resulting volatility surface satisfies the five no-arbitrage conditions of Zaugg et al. The paper also claims an empirical regularity that H peaks at the at-the-money point and that this is a sign of market efficiency there.

Significance. If the claims were established, the paper would offer a very simple, closed-form IV surface with an explicit moneyness-dependent Hurst exponent, calibratable by standard global optimization. The explicit formula and the broad empirical exercise across many tickers are useful ingredients, and the use of Optuna with a clearly stated objective is a reproducible calibration strategy. However, the central advertised contribution is the no-arbitrage certificate, and that certificate is not valid: the proof of condition (iii) in Annex 4.3 relies on a false pathwise monotonicity statement and drops the discount factor, the martingale premise for multifractional Brownian motion is asserted rather than established, and the convexity condition (i) is only checked computationally rather than proved. In addition, the headline 'H=1/2 at the money' is built into the definition of H, not discovered from data, and the empirical comparison is in-sample and not uniformly favorable to the proposed model. The significance is therefore conditional on a substantial reworking of the proof and of the empirical design.

major comments (5)
  1. [Annex 4.3, condition (iii), Eq. (21)] The proof of the lower bound (s-K)+ <= C(T,K;p) is invalid. The text states that '(ST-K)+ >= (s-K)+ because the underlying price ST can only stay the same or increase over time.' This is false for any non-degenerate risk-neutral martingale: a non-constant martingale takes values below its starting point with positive probability. Moreover, the proof first writes C(T,K;p)=e^{-r(T-t)}E^Q[(ST-K)+] and then, two lines later, uses C(T,K;p)=E^Q[(ST-K)+], silently dropping the discount factor. Even with a correct Jensen argument, the discount factor must be handled, and the constant s must be the initial spot price. As written, condition (iii) is not proved, and the no-arbitrage certificate collapses at this point.
  2. [Annex 4.3, after condition (iv)] The martingale premise for the underlying is asserted, not established. The paper states that lim_{t->T} H(t)=1/2 for a multifractional Brownian motion 'ensures that St is a martingale,' but convergence of the Hurst parameter only at the terminal time does not imply H(t)=1/2 on [t0,T), and fractional Brownian motion with H != 1/2 is not a semimartingale. The cited reference [10] is not a proof of the required property within the manuscript. Since the risk-neutral pricing formula in condition (iii) requires a genuine martingale under Q, this is a load-bearing gap.
  3. [Annex 4.3, condition (i)] The convexity of the call price in K is not proved. Section 4.2 derives conditions on the first derivative of sigma with respect to 1/K, and Eq. (17) only imposes monotonicity of sigma. The proof of condition (i) then states a second-derivative inequality and refers to 'the computational study conducted' in Figure 4. A numerical check over a finite grid cannot certify convexity over the whole parameter domain, and no analytic argument for C_KK>0 is provided. This matters because the calibration domain D in Eq. (7) is asserted to be compatible with the no-arbitrage conditions, but that compatibility is not demonstrated.
  4. [Eq. (5) and Section 2, Figure 1] The claim that H approaches 1/2 when moneyness equals 1 is a consequence of the chosen functional form, not an independent empirical discovery. In Eq. (5), H(S/K) is constructed with a normalization factor that forces H(1)=1/2, and the denominator makes H decrease away from S/K=1. The 'inverse smile effect' is therefore baked into the model specification. The empirical Figure 1 plots the fitted implied regularity, but the paper does not provide an independent estimate of H that would test the functional form against market data.
  5. [Section 2.2, Table 2 and Eq. (7)] The empirical comparison does not support the broad claim of outperformance. First, all reported errors are in-sample calibration errors; there is no out-of-sample or cross-validation analysis. Second, Table 2 shows that AdS is not uniformly better than SABR on the per-ticker MSE: for example, the MA, MCD, MSFT, ACN, and UNH rows have lower SABR MSE than AdS MSE. Third, the model contains the extra data-dependent parameter Kmin, which is absent from the search space D in Eq. (7), and no estimation protocol for Kmin is given. Since the location of the minimum of sigma strongly affects the shape of the fitted curve, the comparison with SABR/fSABR is not on equal footing.
minor comments (5)
  1. [Abstract] The abstract contains the typo 'pratcical' instead of 'practical.'
  2. [Section 4, Eq. (12)] The sentence 'Let the option price of a call C(t,K;sigma)' is missing a verb; it should read 'Let the option price of a call be C(t,K;sigma).'
  3. [Section 1.1, Eq. (6)] The symbol C is used both for the scale parameter in the log-linear regression log sigma(n)=log C - H log n and for the call price C(T,K;p) elsewhere; this notation is confusing and should be changed.
  4. [Section 4.2, Eq. (19)-(20)] The derivation of the admissible interval for beta and the argument for choosing 0<delta<1 'as conservatively as possible' are not rigorous; the asymptotic passage St|g| -> infinity needs a precise statement, and the positivity condition on the second member of Eq. (19) should be proved rather than asserted.
  5. [Figure 4] The panels in Figure 4 lack axis labels and a description of the computational grid; the caption should state the parameter ranges, the number of sampled points, and the rule used to declare a condition satisfied.

Circularity Check

2 steps flagged · score 6.0 of 10

The ATM result H(1)=1/2 is built into Eq. (5) rather than empirically derived, and the arbitrage-free certificate imports its martingale premise from a same-author citation; the SABR/fSABR fitting comparison itself is not circular.

  1. self definitional [Section 2, Eq. (5); Abstract]
    "H(S/K)= 1/2 (1+|1−S/Kmin|^δ)/(1+|S/K−S/Kmin|^δ) ... The term 1/2(1+|1−S/Kmin|^δ) normalizes H in the region (0,1) and forces it to 1/2 when S/K=1. ... Our study reveals that H approaches 1/2 when moneyness equals 1, marking a critical point in market efficiency expectations."

    Substituting S/K=1 into Eq. (5) makes the denominator equal to the numerator, so H(1)=1/2 identically for all admissible parameters. The advertised ATM result is therefore a normalization imposed by the ansatz rather than a model prediction. The inverse-smile behavior away from the money is similarly produced by the chosen denominator |S/K−S/Kmin|^δ after the numerator is fixed, so presenting the fitted functional form as an empirical 'reveal' restates the input.

  2. self citation load bearing [Annex 4.3, proof of condition iv, after Eq. (21)]
    "The underlying price process St is assumed to follow a multifractional Brownian motion (mBm), where the Hurst parameter H(t) satisfies: lim t→T H(t)=1/2. This condition ensures that St is a martingale, making it consistent with the no-arbitrage condition and risk-neutral pricing framework. ... For more information see [10]."

    The arbitrage-free certification (conditions i–v) is built on S_t being a Q-martingale: condition iii uses E^Q[(S_T−K)+] and condition iv sets C(t0,K)=(s−K)+. The paper's only support for the martingale property is the assumed terminal limit H(t)→1/2 plus reference [10], which is Di Sciorio's own prior article. No independent proof or external check is provided, and the terminal limit alone does not establish the martingale property on [t0,T); the no-arbitrage conclusion is thus load-bearing on a same-author citation.

full rationale

The empirical comparison against SABR and fSABR (Section 2.2, Tables 1–2) is a normal calibration benchmark: the AdS parameters are fitted with Optuna to the same market IV data and the error metrics are direct outputs of that fit, so that part is self-contained and not circular. The circularity is in the interpretive and theoretical layers: the ATM value H(1)=1/2 is hard-wired into Eq. (5), yet the Abstract and Section 2 present it as an empirical discovery; and the arbitrage-free certificate imports the martingale premise from the authors' own reference [10] without an independent proof. I also flag, as a correctness risk rather than a circularity, that Annex 4.3(iii) proves (s−K)+≤C via the assertion 'the underlying price ST can only stay the same or increase over time,' which is false for a non-degenerate Q-martingale, and the same step drops the discount factor present in Eq. (21). That gap undermines the load-bearing no-arbitrage claim but is not an equivalence-by-construction. Because the fitting comparison has independent content, the score is 6 rather than 8.

Assumptions & free parameters 6 free parameters · 5 assumptions · 1 invented entities

The model needs four fitted constants plus the minimum-volatility strike Kmin, which is taken from data but not included in the search space. H is not an independent parameter: its shape is imposed by delta and by the fixed 1/2 normalization. The no-arbitrage proof relies on a multifractional Brownian motion assumption and on a false monotonicity statement. No new physical entities are introduced; the parametric H function is a modeling construct with no external falsifiable prediction.

free parameters (6)
  • alpha = per ticker, fitted by Optuna
    Scales overall volatility in Eq. (4); constrained alpha > 0.
  • beta = per ticker, fitted in [-1,1] by Optuna
    Controls how strongly H affects volatility decay in Eq. (4); the annex derives a broader admissibility bound.
  • delta = per ticker, fitted in (0,1) by Optuna
    Controls the steepness and shape of H(x) in Eq. (5); the range is justified by an asymptotic argument in Section 4.2.
  • epsilon = per ticker, fitted by Optuna
    Volatility floor in Eq. (4); constrained epsilon > 0.
  • Kmin = per ticker, taken from data
    Strike where sigma is minimal; appears in Eqs. (4) and (5) but is absent from the search space D in Eq. (7), so it is an implicit fitted or empirically chosen parameter.
  • C = estimated per asset following Bianchi et al.
    Scale constant in Eq. (6) used to estimate the empirical H values in Figure 1; not part of the final AdS calibration but part of the evidence for the H-moneyness pattern.
assumptions (5)
  • domain assumption Black-Scholes call pricing formula (12) is used to define C(T,K;sigma) from the AdS sigma(K).
    The arbitrage-free conditions are verified on this pricing function; if the market does not price options by this formula, the verification is moot.
  • domain assumption Empirical H is estimated from the linear regression log sigma(n) = log C - H log n (Eq. 6), assuming fBm self-similarity.
    This is the evidence for the claimed H-moneyness relationship in Figure 1; it relies on a scaling law that may not hold at all moneynesses.
  • domain assumption The underlying price process is a multifractional Brownian motion whose Hurst parameter satisfies lim_{t to T} H(t) = 1/2, making S_t a Q-martingale.
    Stated in Annex 4.3 after condition iv and cited to Di Sciorio [10]; the no-arbitrage claim depends on it.
  • ad hoc to paper In the proof of condition iii, ST can only stay the same or increase over time.
    This is used to justify (s-K)+ <= C; it is false for a risky asset and makes the proof invalid.
  • standard math Zaugg et al. conditions i-v characterize arbitrage-free surfaces.
    The paper adopts these conditions as the definition of no arbitrage; this is a reasonable external benchmark.
invented entities (1)
  • Moneyness-dependent Hurst function H(S/K)
    purpose: Makes the implied regularity vary with moneyness and enter the volatility formula in Eq. (4).
    The parametric form in Eq. (5) is new and is normalized so that H(1)=1/2; the empirical H-moneyness pattern in Figure 1 is not independent of the data and model used to motivate it, and the paper provides no falsifiable prediction outside the fitted surface.

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Pith. "Pith review of Integrating the implied regularity into implied volatility models: A study on free arbitrage model." pith.science (2026). https://pith.science/paper/NLIVJP7J

@misc{pith2026250207518,
  author       = {Pith},
  title        = {Pith review of: Integrating the implied regularity into implied volatility models: A study on free arbitrage model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NLIVJP7J}},
  note         = {Machine review of arXiv:2502.07518}
}
read the original abstract

Implied volatility IV is a key metric in financial markets, reflecting market expectations of future price fluctuations. Research has explored IV's relationship with moneyness, focusing on its connection to the implied Hurst exponent H. Our study reveals that H approaches 1/2 when moneyness equals 1, marking a critical point in market efficiency expectations. We developed an IV model that integrates H to capture these dynamics more effectively. This model considers the interaction between H and the underlying-to-strike price ratio S/K, crucial for capturing IV variations based on moneyness. Using Optuna optimization across multiple indexes, the model outperformed SABR and fSABR in accuracy. This approach provides a more detailed representation of market expectations and IV-H dynamics, improving options pricing and volatility forecasting while enhancing theoretical and pratcical financial analysis.

Figures

Figures reproduced from arXiv: 2502.07518 by the authors.

Figure 1
Figure 1. Implied regularity, IV, Moneyness relationship. conditions on convexity, behavior at the limits, and the consistency of pricing func￾tions. These conditions help guarantee that the pricing surfaces for both call and put options are arbitrage-free, meaning that no riskless profit opportunities can be exploited from the prices derived from the volatility surface. The detailed verification of these conditions is provid… view at source ↗
Figure 2
Figure 2. Implied Volatility (IV) fitting using SABR, fSABR, and AdS models. by: Ci = σi+1 − 2σi + σi−1 (Mi+1 − Mi) 2 . (9) Once the curvature is computed for both market data Cobs and model predictions Cmod, we can define error metrics: ACE (absolute curvature error) and RMSCE (root mean square curvature error) to evaluate the model’s performance. This metric measures the mean absolute error between the modeled and observed … view at source ↗
Figure 3
Figure 3. Violin plots for MSE, MAE, RMSCE, and ACE error metrics. The SABR and fSABR models, as highlighted in the literature, exhibit higher errors for OTM options, as confirmed by error metrics and fitting graphs. ITM 9 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Parameters condition • ii) For all T ∈ ΠT lim K→+∞ C(T, K; p) = St √ 2π lim K→+∞ d1 Z −∞ e − x 2 2 dx−e −r(T −t) lim K→+∞ K √ 2π d1−σ √ Z T −t −∞ e − x 2 2 dx. To compute lim k→+∞ d1 we need lim k→+∞ σ(K) and lim k→+∞ H(K): H∞ = lim k→+∞ H(K) = 1 2 [PITH_FULL_IMAGE:fi…

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Works this paper leans on

31 extracted references · 30 canonical work pages

  1. [10]

    Di Sciorio

    F. Di Sciorio. Option pricing under multifractional Brownian motion in a risk neutral frame- work. Studies of Applied Economics , 38(3), 2020

    work page 2020

  2. [1]

    J. Abadie. Application of the GRG algorithm to optimal control problems . North-Holland Publishing Company Amsterdam, 1970

    work page 1970

  3. [2]

    Akahori, X

    J. Akahori, X. Song, and T. H. Wang. Probability density of lognormal fractional SABR model. Risks, 10, 2022

    work page 2022

  4. [3]

    Akiba, S

    T. Akiba, S. Sano, T. Yanase, T. Ohta, and M. Koyama. Optuna: A next-generation hyper- parameter optimization framework. In Proceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining , pages 2623–2631. ACM, 2019

    work page 2019

  5. [4]

    Angelini and S

    D. Angelini and S. Bianchi. Nonlinear biases in the roughness of a fractional stochastic regularity model. Chaos, Solitons & Fractals , 2023

    work page 2023

  6. [5]

    Bianchi, F

    S. Bianchi, F. Di Sciorio, and R. Mattera. Forecasting VIX with Hurst exponent. In Methods and Applications in Fluorescence , pages 90–95. Springer, 2022

    work page 2022

  7. [6]

    Bianchi, A

    S. Bianchi, A. Pantanella, and A. Pianese. Modeling stock prices by multifractional brownian motion: an improved estimation of the pointwise regularity. Quantitative finance, 13(8):1317– 1330, 2013

    work page 2013

  8. [7]

    Chalamandaris and A

    G. Chalamandaris and A. E. Tsekrekos. Predictable dynamics in implied volatility surfaces from OTC currency options. Journal of Banking & Finance , 34(6):1175–1188, 2010

    work page 2010

Show all 31 references
  1. [8]

    Corrado and T

    C.J. Corrado and T. Su. Implied volatility skews and stock return skewness and kurtosis implied by stock option prices. The European Journal of Finance , 3:73–85, 1997

    1997

  2. [9]

    Couillard and M

    M. Couillard and M. Davison. A comment on measuring the Hurst exponent of financial time series. Physica A: Statistical Mechanics and its Applications , 348:404–418, 2005

    2005

  3. [11]

    Feunou, J.-S

    B. Feunou, J.-S. Fontaine, and R. T´ edongap. Implied volatility and skewness surface.Review of Derivatives Research, 20:167–202, 2017

    2017

  4. [12]

    Flint and E

    E. Flint and E. Mar´ e. Fractional Black–Scholes option pricing, volatility calibration and implied Hurst exponents in South African context. South African Journal of Economic and Management Sciences, 20(1):1–11, 2017

    2017

  5. [13]

    Forde and H

    M. Forde and H. Zhang. Asymptotics for rough stochastic volatility models. SIAM Journal on Financial Mathematics , 8.1:114–145, 2017

    2017

  6. [14]

    Gloter and M

    A. Gloter and M. Hoffmann. Stochastic volatility and fractional Brownian motion. Stochastic Processes and their Applications , 113:143–172, 2004. 18

    2004

  7. [15]

    D. E. Goldberg. Genetic Algorithms in Search, Optimization & Machine Learning . Addison- Wesley, 1989

    1989

  8. [16]

    B. Guo, Q. Han, and B. Zhao. The Nelson–Siegel model of the term structure of option implied volatility and volatility components. Journal of Futures Markets , 34(8):788–806, 2014

    2014

  9. [17]

    Hagan, A

    P. Hagan, A. Lesniewski, and D. Woodward. Probability distribution in the SABR model of stochastic volatility. In Large Deviations and Asymptotic Methods in Finance , pages 1–35. Springer, 2015

    2015

  10. [18]

    Horvath, A

    B. Horvath, A. Muguruza, and M. Tomas. Deep learning volatility: a deep neural network perspective on pricing and calibration in (rough) volatility models. Quantitative Finance , 21(1):11–27, 2021

    2021

  11. [19]

    Kachhara, J.K.E

    D. Kachhara, J.K.E. Markin, and A. Singh. Option smile volatility and implied probabilities: Implications of concavity in iv curves. Technical report, Stevens Institute of Technology, 2022. Supervised by Dr. Dan Pirjol

    2022

  12. [20]

    S. W. Kim and J. H. Kim. Volatility and variance swaps and options in the fractional SABR model. The European Journal of Finance , 26(17):1725–1745, 2020

    2020

  13. [21]

    Kirkpatrick, C

    S. Kirkpatrick, C. D. Gelatt Jr., and M. P. Vecchi. Optimization by simulated annealing. science, 220(4598):671–680, 1983

    1983

  14. [22]

    Lesniewski

    A. Lesniewski. Option smile and the SABR model of stochastic volatility. Technical report, MIT, 2014

    2014

  15. [23]

    A. A. Lund. The SABR model. Master’s thesis, Department of Finance Copenhagen Business School, 2023

    2023

  16. [24]

    B. B. Mandelbrot and J. W. Van Ness. Fractional Brownian motions, fractional noises and applications. SIAM Review, 10(4):422–437, 1968

    1968

  17. [25]

    D. W. Marquardt. An algorithm for least-squares estimation of nonlinear parameters. Journal of the Society for Industrial and Applied Mathematics , 11(2):431–441, 1963

    1963

  18. [26]

    Neuman and M

    E. Neuman and M. Rosenbaum. Fractional Brownian motion with zero Hurst parameter: a rough volatility viewpoint. Electronic Communications in Probability, 61, 2018

    2018

  19. [27]

    E. E. Peters. Fractal market analysis: applying chaos theory to investment and economics . John Wiley & Sons, 1994

    1994

  20. [28]

    J. M. Poterba and L. H. Summers. The persistence of volatility and stock market fluctuations. The American Economic Review , 76(5):1142–51, 1986

    1986

  21. [29]

    J. Stein. Overreactions in the options market. The Journal of Finance , 44(4):1011–1023, 1989. 19

    1989

  22. [30]

    B. Yuan, D. Brigo, A. Jacquier, and N. Pede. Deep learning interpretability for rough volatility. arXiv preprint arXiv:2411.19317 , 2024. Submitted on 28 Nov 2024

    2024 arXiv

  23. [31]

    Zaugg, A

    N.F. Zaugg, A. G. Lech, and L. Perotti. Volatility parametrizations with random coefficients: analytic flexibility for implied volatility surfaces. arXiv preprint, 2411.04041, 2024. 20

    2024

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