Pith. sign in

REVIEW 4 major objections 6 minor 52 references

Correct implied volatility shapes and reliable pricing in the rough Heston model

T0 review · 4 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Popular Fourier pricing can manufacture spurious smiles in rough Heston; a benchmark calibration is a ghost fit.

desk verdict A serious numerical-methods paper whose 'ghost calibration' claim about El Euch–Rosenbaum is plausible but not yet fully verified; it deserves peer review, with code and an independent short-maturity check required. read the letter →

arxiv 2412.16067 v1 pith:PT3XU7QT submitted 2024-12-20 q-fin.MF q-fin.CP

Svetlana Boyarchenko , Sergei Levendorskiǐ This is my paper · ORCID
classification q-fin.MFq-fin.CP MSC 60-0860E1060G1060G2265C2065D3065G2091G20
keywords roughHestonmodelfractionalAdamsmethodsinh-accelerationghostcalibrationimpliedvolatilitysurfaceFourierinversionConformalBootstrapprincipleRiccatiequation
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

The paper argues that a widely cited calibration of the rough Heston model, the one in reference [23], is a numerical artifact rather than a genuine fit. Using its own corrected pricing machinery, a modified fractional Adams method with sinh-accelerated Fourier inversion, it constructs the implied volatility surface for those parameters and finds it markedly flatter and a poor fit to the original data. The mechanism is ghost calibration: model error and numerical error almost exactly cancel. The paper also proposes the Conformal Bootstrap principle, a practical agreement-based error check for models whose characteristic function lacks proven analyticity or decay. If correct, published empirical claims about the rough Heston smile and any calibrations built on standard Fourier methods need to be re-examined.

What carries the argument

Three components carry the argument. First, modifications of the fractional Adams method for the fractional Riccati equation replace the naive first predictor with the small-time asymptotics $-0.5(\xi^2+i\xi)t^\alpha/\Gamma(\alpha+1)$, so that large spectral parameter $|\xi|$ does not corrupt short-maturity prices. Second, $\sinh$-acceleration for Fourier inversion deforms the integration contour into a hyperbolic curve $i\omega_1+b\sinh(i\omega+y)$, making the oscillatory integrand decay exponentially and allowing a short trapezoid sum to achieve high accuracy; an ad-hoc bound, equation (3.20), is used to choose the truncation $N$. Third, the Conformal Bootstrap principle accepts a computed price only when two well-separated contour deformations agree to about $10^{-m}$, because agreement on distant grids makes a common large error essentially impossible. The paper also uses the identity $\Phi=\exp[\int_0^\tau(\gamma\theta h+vF)\,ds]$, which avoids a separate fractional integration when assembling the log-characteristic function.

What would settle it

Take the paper's calibrated parameter set (1.1), compute option prices with an independent high-accuracy method such as a very fine Monte Carlo simulation of the rough Heston process or a completely different quadrature with rigorous error bounds, for maturities from one week to one month and strikes within 20% of spot, and compare the implied volatilities to both the published surface from [23] and the paper's SINH surface. If the independent surface matches the published one rather than the flatter SINH one, the ghost-calibration claim is refuted; a cheaper check is to increase the Adams grid size $M$ and the Fourier truncation $N$ simultaneously and see whether the paper's prices move by more than the claimed $10^{-8}$.

Watch

Extended reading notes

Core claim

The central claim is that the benchmark rough Heston calibration in reference [23] is an example of ghost calibration. For the same parameter set, the paper's numerical procedures produce an ATM skew several times lower than the published one, decaying quickly with maturity, and an implied volatility surface that is flatter and fits the market data poorly. It diagnoses the sources of the discrepancy: the standard fractional Adams method mishandles large Fourier frequencies in the predictor step for short maturities, and fixed-parameter Carr-Madan, COS, and Lewis inversions introduce systematic errors that can reshape a straight volatility slope into a smile. The paper asserts that its method evaluates thousands of vanilla prices in milliseconds with relative errors around $10^{-3}$ or better, and that the disagreement is large enough to invalidate calibration conclusions drawn from the standard methods.

Load-bearing premise

The load-bearing premise is that the rough Heston characteristic function is analytic in a cone around the real axis and decays there fast enough; the paper's only support for this is an explicitly ad-hoc bound, so if that analyticity or decay fails, the quoted error floors and the ghost-calibration claim could change.

Editorial extensions

If this is right

  • For the benchmark rough Heston parameters in [23], the published volatility smile and ATM skew are replaced by a flatter, poorly fitting surface, so any calibration conclusions drawn from that surface need to be rechecked.
  • Short-maturity out-of-the-money options, where rough-volatility effects are claimed to be strongest, are exactly where standard CM, COS, and Lewis schemes are least reliable, so empirical tests of rough-volatility skew behavior based on those schemes are suspect.
  • The modified Adams plus sinh-acceleration pipeline evaluates thousands of option prices in milliseconds with relative errors around $10^{-3}$ or better on a standard laptop, making reliable calibration computationally feasible.
  • Using a fixed set of Fourier-grid parameters across strikes and maturities, as in the common CM and COS practice, can manufacture smiles and skews that are not present in the true model prices; the Conformal Bootstrap principle provides a practical check against such artifacts.
  • Option prices outside the no-arbitrage bounds, which the paper reproduces for FFT-based and COS schemes, offer a simple diagnostic that a calibration is being driven by numerical error rather than model content.

Reading between the lines

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

  • The ghost-calibration mechanism likely extends beyond the rough Heston model to any pipeline coupling a numerically solved Riccati equation with fixed-parameter Fourier inversion, including affine models and deep-learning pricing surrogates trained on such prices.
  • A testable consequence of the paper's claim is that recalibrating the same dataset with a certified-accurate pricer should move the fitted parameters to a different region of parameter space and worsen the apparent fit, directly confirming error cancellation.
  • The Conformal Bootstrap principle could be automated as an online error certificate in production systems: reject any batch of prices unless two distinct contour deformations agree, then adaptively expand grids until they do.
  • A natural extension is to test the paper's short-maturity accuracy claims in higher-precision or interval arithmetic, especially where the explicitly ad-hoc decay bound (3.20) is the only support for the chosen truncation.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The paper develops fast and accurate pricing procedures for vanilla options in the rough Heston model, combining modified fractional Adams methods for the fractional Riccati equation with sinh-accelerated Fourier inversion. It presents numerical evidence that several popular Fourier inversion schemes (Carr-Madan, Lewis, COS, flat iFT) produce sizable pricing errors, and it uses the new method to compute implied volatility surfaces for the Euch-Rosenbaum calibrated parameters. The central empirical claim is that the correct implied volatility surface for those parameters is much flatter than reported in the original calibration, with an ATM skew several times lower, making the original calibration an example of 'ghost calibration'. The paper also proposes a Conformal Bootstrap principle for error assessment and outlines calibration procedures based on it.

Significance. If correct, the paper's central claim is significant: it challenges a widely cited rough Heston calibration and attributes the apparent fit to numerical error cancellation rather than model performance. The proposed pricing methodology is also practically valuable: it reports millisecond-scale pricing for moderate maturities, uses a mathematically motivated improvement of the standard Adams method, and includes detailed comparisons with the hybrid method of [16]. The paper is honest about the conditional nature of its error control, but this honesty cuts both ways: the headline claim rests on numerical error assessment that is explicitly heuristic. The absence of machine-checked proofs or independently certified short-maturity benchmarks means the current evidence is suggestive rather than conclusive.

major comments (4)
  1. [Section 3.8, Eq. (3.20)] The truncation parameter N in the pricing algorithm is selected from the ad-hoc bound (3.20), and the paper itself states that this bound is 'far from perfect' and that 'any numerical procedure is a conditional one' (Section 6). Because no rigorous proof of (3.18)-(3.20) is provided, the reported error levels, especially at T=1/52 and T=1/252 where decay is slowest, are not certified. This is load-bearing: the headline claim that the true ATM skew is several times lower than in [23] depends on accurate short-maturity prices produced with this unproven truncation control.
  2. [Section 2.2 and Section 3.9] Modifications I-III of the fractional Adams method all build on the same asymptotic ansatz (2.13), and the product-integration coefficients used in the corrections are shared. The Conformal Bootstrap principle compares prices evaluated on different contours and with different modifications, but if the ansatz (2.13) is biased for large |ξ| and small t, each modification inherits that bias and agreement between them does not detect it. The paper provides no independent test of (2.13) in exactly the regime where the ghost-calibration conclusion is decided.
  3. [Tables 16-17] The external comparisons with the hybrid method of [16] agree well at moderate maturities (Tables 12-15), but at T=1/52 and T=1/252 the hybrid prices diverge strongly from the paper's benchmarks. For example, in Table 16 at K=1.05 the hybrid price is 4.113E-04 versus the benchmark 3.752E-04, and in Table 17 at K=1.05 the hybrid price is 6.39E-07 versus the benchmark 3.31E-08. These comparisons therefore do not validate the paper's benchmarks in the maturity range where the 'ghost calibration' claim is decided; an independent short-maturity benchmark is missing.
  4. [Section 4.3 and Tables 12-17] The benchmark prices are described as calculated using 'much finer and longer grids', but no error certificate for these benchmarks is provided beyond agreement between two sinh-deformation parameter sets. Because both parameter sets are generated by the same family of Adams modifications and the same unproved decay bound, the agreement is an internal consistency check rather than an accuracy certificate. Please provide reproducible code, exact or high-precision independent benchmarks, or a rigorous error bound for at least the short-maturity ATM skew that is central to the paper's main conclusion.
minor comments (6)
  1. [Section 3.6] The section title contains a typo: 'inifinite trapezoid rule' should read 'infinite trapezoid rule'.
  2. [Section 3.3] The sentence 'CM method caused problems in the financial industry, and has net been used by practitioners ever since' should read 'has not been used'.
  3. [Appendix B, Table 17] The header of Table 17 appears to contain a duplicated 'VH Err VH' column and lacks the 'Vfast' column that is described in the table notes; please correct the table formatting.
  4. [Figure 3] The paper states that the ATM skew is 'several times lower' than in [23], but Figure 3(a) does not provide the numerical values of the skew at T=1/52 and T=1/252. Reporting these values explicitly would make the central comparison quantitative and easier to verify.
  5. [Figure 5 caption] The caption of Figure 5 is grammatically garbled in the sentence beginning 'the true difference between the empirical implied volatilities...' and should be rewritten for clarity.
  6. [Abstract] There is a typographical error in the spelling of Levendorskiĭ in the abstract; the same issue appears in several other places in the text.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the ghost-calibration claim is a new numerical consequence of the Euch-Rosenbaum parameters, though short-maturity accuracy rests on a self-consistent bootstrap rather than an independent benchmark.

full rationale

The paper's derivation chain is: take the calibrated parameters (1.1) from [23], solve the fractional Riccati equation (2.7) with modified Adams methods whose predictor uses the known leading small-time asymptotic (2.13), form the characteristic function via (2.8), invert with sinh-accelerated Fourier transforms (3.15), and then read off the implied volatility surface. No parameter is fitted to the IV surface being predicted; the flatness and poor fit of the surface are consequences computed from fixed inputs, so the central claim is not equivalent to an input by construction. The self-citations to the authors' sinh-acceleration and ghost-calibration papers are descriptive and not load-bearing: the method is specified in the paper, and moderate-maturity prices are checked against the independent hybrid method of [16] and against much finer grids. The main caveat is that the shortest maturities (T = 1/52 and T = 1/252) that drive the ATM-skew comparison are certified by the Conformal Bootstrap principle, i.e., agreement among the authors' own modifications sharing the same h_as ansatz, rather than by an external benchmark, and the paper itself concedes that 'any numerical procedure is a conditional one' (Section 6). That is a validation limitation, not a reduction of the output to the input; the numerical claim could in principle be wrong even though it is not circular. Accordingly no circular step is exhibited.

Assumptions & free parameters 4 free parameters · 3 assumptions · 0 invented entities

The central numerical results depend on several hand-chosen method parameters and unproven analytical assumptions about the rough Heston characteristic function. These are not fitted to market data but affect the computed prices and therefore the corrected IV surface.

free parameters (4)
  • C (constant in truncation bound) = 10
    Set ad hoc in Section 3.8 to compute the truncation parameter Lambda_0; affects the number of terms in the Fourier series.
  • kd (safety factor for contour distance) = 0.9 or 0.95
    Chosen by hand in Sections 3.7 and 3.1 to set the step size zeta; affects discretization error.
  • omega and gamma_plus/minus (sinh-deformation angle) = 0.1 or 0.2
    Chosen by hand in Section 5, Pre-calibration Step II; controls the contour deformation and required grid size.
  • M (number of time steps in modified Adams method) = varies (e.g., 9, 100, 317, 20000)
    Chosen by hand for each numerical example; controls the accuracy of the solution of the fractional Riccati equation.
assumptions (3)
  • domain assumption The fractional Riccati equation (2.7) has a unique solution and the characteristic function representation (2.8) is valid.
    Standard for the rough Heston model, but the paper does not prove existence and uniqueness here; it relies on the model construction in [23].
  • domain assumption The characteristic function admits analytic continuation to a strip and a cone, with exponential decay at infinity.
    The paper states this is unknown for the rough Heston model (Section 6), yet the sinh-acceleration error control relies on it. Section 3.8 makes a heuristic assumption about this.
  • ad hoc to paper The Conformal Bootstrap principle (Section 3.9) reliably indicates accuracy when two independent numerical schemes agree.
    This is a heuristic principle proposed in the paper, not a theorem. It is used to justify the claimed error bounds without rigorous proof.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Correct implied volatility shapes and reliable pricing in the rough Heston model." pith.science (2026). https://pith.science/paper/PT3XU7QT

@misc{pith2026241216067,
  author       = {Pith},
  title        = {Pith review of: Correct implied volatility shapes and reliable pricing in the rough Heston model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PT3XU7QT}},
  note         = {Machine review of arXiv:2412.16067}
}
read the original abstract

We use modifications of the Adams method and very fast and accurate sinh-acceleration method of the Fourier inversion (iFT) (S.Boyarchenko and Levendorski\u{i}, IJTAF 2019, v.22) to evaluate prices of vanilla options; for options of moderate and long maturities and strikes not very far from the spot, thousands of prices can be calculated in several msec. with relative errors of the order of 0.5\% and smaller running Matlab on a Mac with moderate characteristics. We demonstrate that for the calibrated set of parameters in Euch and Rosenbaum, Math. Finance 2019, v. 29, the correct implied volatility surface is significantly flatter and fits the data very poorly, hence, the calibration results in op.cit. is an example of the {\em ghost calibration} (M.Boyarchenko and Levendorki\u{i}, Quantitative Finance 2015, v. 15): the errors of the model and numerical method almost cancel one another. We explain how calibration errors of this sort are generated by each of popular versions of numerical realizations of iFT (Carr-Madan, Lipton-Lewis and COS methods) with prefixed parameters of a numerical method, resulting in spurious volatility smiles and skews. We suggest a general {\em Conformal Bootstrap principle} which allows one to avoid ghost calibration errors. We outline schemes of application of Conformal Bootstrap principle and the method of the paper to the design of accurate and fast calibration procedures.

Figures

Figures reproduced from arXiv: 2412.16067 by the authors.

Figure 1
Figure 1. Parameters of the model α = 0.62, γ = 0.1, ρ = −0.681, θ = 0.3156, ν = 0.331, v = 0.0392. (A): Re ϕ calculated using modification II of the Adams method; (B): the difference between Re ϕ produced by the method [23] and Re ϕ on Panel (A). In both cases, T = 1/52 and M = 1000. The nodes ξ are on the line {Im ξ = −1.5}. The differences shown on Panel (A) translate into the relative errors of evaluation of the terms in … view at source ↗
Figure 2
Figure 2. Panel (A). Dots: h = κ∞ = p 1 − ρ 2/(γν), other lines: h = h(ξj , t)/|ξj |, for ξ15 = 1.1299−0.5i, ξ30 = 4.3712−0.5i, ξ45 = 15.5301−0.5i, ξ56 = 39.1661 − 0.5i. Panel (B). Dots: t 7→ − max{G1(t) cos ωy, G2(t) cos(2ω)y 2} (the curve defined by the RHS of (3.20) is higher and gives a more accurate bound), other lines: t 7→ Re ϕ(ξj )/|ξj |. Parameters from [23, Example 5.1]. 2) find the positive solution Λ02 of the equa… view at source ↗
Figure 3
Figure 3. ATM skew; the parameters are in (1.1) [PITH_FULL_IMAGE:figures/full_fig_p029_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Implied volatility curves; the parameters are in (1.1) [PITH_FULL_IMAGE:figures/full_fig_p030_4.png]
Figure 5
Figure 5. Figure 5: Implied volatility curves. The parameters α = 0.512, γ = 0.88, ρ = −0.7, ν = 0.96, θ = 0.016, v = 0.148, are the result of calibration to the real data in [18, p.27]. The implied volatilities calculated using the Lewis and Adams methods and shown on [PITH_FULL_IMAGE:f…
Figure 6
Figure 6. Figure 6: Implied volatility curves in the rough Heston model (Example in [21, Sect. 6.2]); parameters α = 0.6, γ = 2, ρ = −0.6, θ = 0.025, ν = 0.2, v0 = 0.025; S0 = 1 [PITH_FULL_IMAGE:figures/full_fig_p032_6.png]
Figure 7
Figure 7. Figure 7: Implied volatility surfaces in the rough Heston model [23, Example 5.1]. for time to maturity in the range (1 day, 1 week); spot S0 = 1. If the price is outside the no-arbitrage bounds, σIMP is set to 0. Panel (A): surface is calculated using the SINH-acceleration and …
Figure 8
Figure 8. Figure 8: Volatility surface in the rough Heston model with parameters α = 0.6, γ = 2, θ = 0.0225, ν = 0.2, ρ = −0.6, v = 0.0225, for time to maturity in the range (1 day, 1 week); spot S0 = 1. If the price is outside the no-arbitrage bounds, σIMP is set to 0. Panel (A): calcula…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

52 extracted references · 42 canonical work pages

  1. [16]

    Callegaro, M

    G. Callegaro, M. Grasselli, and G. Pag` es. Fast Hybrid Schemes for Fractional Riccati Equations (Rough is not so Tough). Mathematics of Operations Research. Articles in advance , pages 1–34, - 2020. 26 SVETLANA BOYARCHENKO AND SERGEI LEVENDORSKI ˘I

    work page 2020

  2. [23]

    Euch and M

    O.E. Euch and M. Rozenbaum. The characteristic function of rough Heston models. Mathematical Finance, 29:3–38, 2019

    work page 2019

  3. [1]

    Barndorff-Nielsen

    O.E. Barndorff-Nielsen. Processes of Normal Inverse Gaussian Type. Finance and Stochastics , 2:41–68, 1998

    work page 1998

  4. [2]

    Barndorff-Nielsen and S.Z

    O.E. Barndorff-Nielsen and S.Z. Levendorski ˇi. Feller Processes of Normal Inverse Gaussian type. Quantita- tive Finance, 1:318–331, 2001

    work page 2001

  5. [3]

    Bayer, P

    C. Bayer, P. Fritz, and J. Gatheral. Pricing under rough volatility. Quantitative Finance, 15(1):1–18, 2015

    work page 2015

  6. [4]

    Berner, P

    J. Berner, P. Grohs, G. Kutiniok, and P. Petersen. The modern mathematics of deep learning. In P. Grohs and G. Kutiniok, editors, Mathematics aspects of deep learning , 1–111. Cambridge University Press, Cam- bridge, UK, 2022

    work page 2022

  7. [5]

    Boyarchenko and S

    M. Boyarchenko and S. Levendorski ˘i. Ghost Calibration and Pricing Barrier Options and Credit Default Swaps in spectrally one-sided L´ evy models: The Parabolic Laplace Inversion Method.Quantitative Finance, 15(3):421–441, 2015. Available at SSRN: http://ssrn.com/abstract=2445318

    2015

  8. [6]

    Boyarchenko and S

    S. Boyarchenko and S. Levendorski ˘i. On rational pricing of derivative securities for a family of non- Gaussian processes. Preprint 98/7, Institut f¨ ur Mathematik, Universit¨ at Potsdam, January 1998. Available at http://opus.kobv.de/ubp/volltexte/2008/2519

    work page 1998

Show all 52 references
  1. [7]

    Boyarchenko and S

    S. Boyarchenko and S. Levendorski ˘i. Generalizations of the Black-Scholes equation for truncated L´ evy processes. Working Paper, University of Pennsylvania, April 1999

    1999

  2. [8]

    Boyarchenko and S

    S. Boyarchenko and S. Levendorski ˘i. Option pricing for truncated L´ evy processes.International Journal of Theoretical and Applied Finance, 3(3):549–552, July 2000

    2000

  3. [9]

    Boyarchenko and S

    S. Boyarchenko and S. Levendorski ˘i. New efficient versions of Fourier transform method in applications to option pricing. Working paper, 2011. Available at SSRN: http://ssrn.com/abstract=1846633

    2011

  4. [10]

    Boyarchenko and S

    S. Boyarchenko and S. Levendorski˘i. Efficient variations of Fourier transform in applications to option pric- ing. Journal of Computational Finance , 18(2):57–90, 2014. Available at http://ssrn.com/abstract=1673034

    2014

  5. [11]

    Boyarchenko and S

    S. Boyarchenko and S. Levendorski ˘i. Sinh-acceleration: Efficient evaluation of probability distri- butions, option pricing, and Monte-Carlo simulations. International Journal of Theoretical and Applied Finance , 22(3):1950–011, 2019. DOI: 10.1142/S0219024919500110. Available...

    1950 doi

  6. [12]

    Boyarchenko and S

    S. Boyarchenko and S. Levendorski ˘i. Conformal accelerations method and efficient evaluation of stable distributions. Acta Applicandae Mathematicae , 169:711–765, 2020. Available at SSRN: https://ssrn.com/abstract=3206696 or http://dx.doi.org/10.2139/ssrn.3206696

    2020 doi

  7. [13]

    Boyarchenko and S

    S. Boyarchenko and S. Levendorski ˘i. Static and semi-static hedging as contrarian or conformist bets. Mathematical Finance , 3(30):921–960, 2020. Available at SSRN: https://ssrn.com/abstract=3329694 or http://arXiv.org/abs/1902.02854

    2020 arXiv

  8. [14]

    Boyarchenko and S

    S. Boyarchenko and S. Levendorski ˘i. Efficient evaluation of double barrier options. Interna- tional Journal of Theoretical and Applied Finance , 27(2):2450007, 2024. Available at SSRN: http://ssrn.com/abstract=4262396 or http://arxiv.org/abs/2211.07765

    2024 arXiv

  9. [15]

    Boyarchenko and S

    S. Boyarchenko and S. Levendorski ˘i. Efficient evaluation of expectations of functions of a L´ evy process and its extremum. Finance and Stochastics , 2024. To appear. Working paper version is available at SSRN: https://ssrn.com/abstract=4140462 or http://arXiv.org/abs/2209.02793

    2024 arXiv

  10. [17]

    Carr and D.B

    P. Carr and D.B. Madan. Option valuation using the Fast Fourier Transform. Journal of Computational Finance, 2(4):61–73, 1999

    1999

  11. [18]

    Coprechot

    P.-A. Coprechot. Study of the joint S&P 500/VIX smile calibration problem within rough volatility models. MSc Thesis, Department of Mathematics, Imperial College, London, 2020

    2020

  12. [19]

    de Innocentis and S

    M. de Innocentis and S. Levendorski ˘i. Calibration Heston Model for Credit Risk. Risk, pages 90–95, Sep- tember 2017. Available at SSRN: http://ssrn.com/abstract=2757008

    2017

  13. [20]

    Duffie, D

    D. Duffie, D. Filipovi´ c, and K. Singleton. Affine processes and applications in finance. Annals of Applied Probability, 13:984–1053, 2002

    2002

  14. [21]

    European option pricing under the rough Heston model us- ing the COS method

    Kamuran Emre Erkan. European option pricing under the rough Heston model us- ing the COS method. Master thesis. Delft University of Technology, 2020. Available at https://repository.tudelft.nl/record/uuid:721510e7-6393-4b5e-af07-75063ced210c

    2020

  15. [22]

    Euch and M

    O.E. Euch and M. Rozenbaum. Perfect hedging in rough Heston models. Working paper, 2017. Available online at: https://arxiv.org/abs/1703.05049

    2017 arXiv

  16. [24]

    Fang and C.W

    F. Fang and C.W. Oosterlee. A novel pricing method for European options based on Fourier-Cosine series expansions. SIAM Journal on Scientific Computing , 31(2):826–848, 2008

    2008

  17. [25]

    Feng and V

    L. Feng and V. Linetsky. Pricing discretely monitored barrier options and defaultable bonds in L´ evy process models: a fast Hilbert transform approach. Mathematical Finance, 18(3):337–384, July 2008

    2008

  18. [26]

    Forde, B

    M. Forde, B. Smith, and L. Viitasaari. Rough volatility and CGMY jumps with a finite history and the rough Heston model - Small time asymptotics in the k = √ t regime. Quantitative Finance, 21(4):541–563, 2021

    2021

  19. [27]

    Forde and H

    M. Forde and H. Zhang. Asymptotics for rough stochastic volatility models.SIAM J. Financial Mathematics, 8(1):114–145, 2017

    2017

  20. [28]

    P.K. Friz, P. Gassiat, and P. Pigato. Precise asymptotics: Robust stochastic volatility models. Ann. Appl. Probability, 31(2):896–940, 2021

    2021

  21. [29]

    P.K. Friz, P. Gassiat, and P. Pigato. Short-dated smile under rough volatility: Asymptotics and numerics. Quantitative Finance, 22(3):463–480, 2022

    2022

  22. [30]

    Gatheral, T

    J. Gatheral, T. Jaisson, and M. Rosenbaum M. Volatility is rough. Quantitative Finance , 18(6):933–949, 2018

    2018

  23. [31]

    Gatheral and R

    J. Gatheral and R. Radoiˇ ci´ c. Rational approximation of the rough Heston solution.Intern. Journal Theo- retical and Applied Finance , 22(3):13–27, 2019

    2019

  24. [32]

    Gerhold, C

    S. Gerhold, C. Gerstenecker, and A. Pinter. Moment explosions in the rough Heston model. Decisions in Economics and Finance , 42(2):575–608, 2019

    2019

  25. [33]

    S. L. Heston. A closed-form solution for options with stochastic volatility with applications to bond and currency options. The Review of Financial Studies , 6(2):327–343, 1993

    1993

  26. [34]

    Horvath, A

    B. Horvath, A. Muguruza, M. Roome, and F. Shi. Deep learning volatility. Quantitative Finance, 21(1):–,

  27. [35]

    Abi Jaber, M

    E. Abi Jaber, M. Larsson, and S. Pulido. Affine Volterra processes. Ann. Appl. Probab. , 29(5):3155–3200, 2019

    2019

  28. [36]

    Jaber and S

    E.A. Jaber and S. Li. Volatility models in practice: Rough, Path-dependent or Markovian. Working paper, January 2024. Available online at: https://arxiv.org/abs/2401.03345

    2024 arXiv

  29. [37]

    Jacquier, C

    A. Jacquier, C. Martini, and A. Muguruza. Pricing under rough volatility. Quantitative Finance, 18(1):45– 61, 2018

    2018

  30. [38]

    R. Lee. Option pricing by Transform methods: Extensions, Unification, and Error control. Journal of Computational Finance, 7(3):51–86, 2004

    2004

  31. [39]

    Levendorski ˘i

    S. Levendorski ˘i. Efficient pricing and reliable calibration in the Heston model. International Journal of Theoretical and Applied Finance, 15(7), 2012. 125050 (44 pages)

    2012

  32. [40]

    Levendorski ˘i

    S. Levendorski ˘i. Pitfalls of the Fourier Transform method in Affine Models, and remedies. Applied Math- ematical Finance , 23(2):81–134, 2016. Available at http://dx.doi.org/10.1080/1350486X.2016.1159918 or http://ssrn.com/abstract=2367547

    2016

  33. [41]

    A. Lewis. A simple option formula for general jump-diffusion and other exponential L´ evy processes, Sep- tember 2001. Available at: http://papers.ssrn.com/abstract=585451. RELIABLE PRICING IN AND CALIBRATION OF THE ROUGH HESTON MODEL 27

    2001

  34. [42]

    Alan L. Lewis. Option valuation under stochastic volatility: with Mathematica code. Finance Press, Newport, 2000

    2000

  35. [43]

    A. Lipton. Mathematical Methods For Foreign Exchange: A Financial Engineer’s Approach . World Scien- tific, Singapore, 2001

    2001

  36. [44]

    V. Lucic. On singularities in the Heston model. Working paper, 2007. Available at http://ssrn.com/abstract=1031222

    2007

  37. [45]

    G. Pag` es. Multi-step Richardson-Romberg extrapolation: Remarks on variance control and complexity. Monte Carlo Methods Appl. , 13(1):37–70, 2007. https://arxiv.org/abs/1901.09647

    2007 arXiv

  38. [46]

    S.I. Romer. Empirical analysis of rough and classical stochastic volatility models to the SPX and VIX markets. Quantitative Finance, 22(10):1805–1838, 2022

    2022

  39. [47]

    F. Stenger. Numerical Methods based on Sinc and Analytic functions . Springer-Verlag, New York, 1993. Appendix A. A.1. Grids depending on ξ. The accuracy of calculations can be increased using grids de- pending on ξ. To understand what a proper dependence of the grid on a (lar...

    1993

  40. [49]

    with small T , when Φ( ξ, T) decays very slowly, and the truncation error is large

  41. [50]

    far from the tails, where the OTM option prices are small, and the integrand highly oscillates

  42. [51]

    close to maturity, where even marginally accurate calculations are possible only in a very small vicinity of the spot

  43. [52]

    (This is an effect typical for the Heston model; one expect that the same effect can be observed for the rough Heston model)

    for T from a moderately long intervals, if the strip of analyticity shrinks as T increases. (This is an effect typical for the Heston model; one expect that the same effect can be observed for the rough Heston model)

  44. [53]

    Call” prices (rounded) in the rough Heston model; parameters are α = 0.6, γ= 0.1, θ= 0.3156, ν= 0.331, ρ= −0.681, v= 0.0392, r = 0.3, S0 = 100, T = 1. Errors shown are for OTM “put

    as the fixed ω1 is getting closer to the boundaries of the admissible interval ( µ−(T ), µ+(T )), the discretization error explodes. Appendix B. Figures and tables RELIABLE PRICING IN AND CALIBRATION OF THE ROUGH HESTON MODEL 29 0 0.05 0.1 0.15 0.2 0.25 0.05 0.1 0.15 0.2 0.25 ...

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.