Arbitrage-Free Multi-Maturity Risk-Neutral Marginals

Charlie Che; Hao Qin; Liming Feng; Ruozhong Yang

REVIEW 3 major objections 8 minor 50 references

Arbitrage-Free Multi-Maturity Risk-Neutral Marginals

T0 review · 3 major / 8 minor · reviewed 2026-07-08 · glm-5.2

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

Pith's one-line read Arbitrage-free option prices yield ready-to-use probability distributions

desk verdict Solid construction with one real gap in the calendar-freedom proof read the letter →

arxiv 2607.06204 v1 pith:E65OME4L submitted 2026-07-07 q-fin.CP q-fin.MFq-fin.RM

Hao Qin , Ruozhong Yang , Charlie Che , Liming Feng This is my paper · ORCID

classification q-fin.CPq-fin.MFq-fin.RM PACS 02.50.-r89.65.Gh

keywords marginalsoptionpricesrisk-neutralarbitrage-freeconstructiondistributionefficient

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 proves that if you start with a set of option prices that are free of internal contradictions (no butterfly arbitrage) and are properly ordered across maturities (no calendar arbitrage), you can construct a complete probability distribution for the underlying asset at each maturity that simultaneously satisfies five demanding properties: it is a valid probability law with unit total mass, it reprices every input option exactly, it is free of both butterfly and calendar arbitrage at every strike (not just at the observed quotes), it is guaranteed to be constructible without numerical failure, and it admits closed-form densities, distribution functions, quantiles, and Monte Carlo sampling. The construction works by shifting attention from the call-price curve itself to its slope and curvature. The slope of the call-price curve encodes discounted tail probabilities, and the curvature encodes the probability density. By estimating slopes at observed strikes and then choosing non-negative piecewise-constant curvature on each interval between strikes, the method assigns probability mass interval by interval, matching the input prices exactly. Power-law tails complete the distribution outside the observed strike range, matching boundary prices and slopes while allocating residual mass and keeping moments finite. Across maturities, the construction proceeds sequentially: each earlier maturity's call curve serves as a lower bound for the next, and a three-piece curvature construction is deployed only on intervals where the simpler two-piece form would violate that bound. The key theoretical result is that, under a mild chord-based input condition on calendar ordering, the per-interval feasibility problem always has a solution, so the construction never gets stuck.

What carries the argument

Two-piece and three-piece piecewise-constant curvature construction; power-law tail extrapolation with boundary-matching; chord-based calendar reduction to scalar curvature floor

What would settle it

Feed the construction market data where the upstream de-arbitrage step produces prices satisfying pointwise calendar conditions but not the stricter chord-based version, and check whether the construction fails to find a feasible solution or produces calendar violations.

Watch

Extended reading notes

Core claim

The central mechanism is the reduction of marginal construction to non-negative curvature assignment on call-price slopes. Because the Breeden-Litzenberger identity ties the second derivative of the call-price curve to the risk-neutral density, and because the endpoint slopes of that curve fix the total probability mass, the entire problem of building a valid distribution reduces to choosing non-negative curvature on each strike interval whose integral matches the prescribed slope increment. For a single maturity, a two-piece constant curvature per interval suffices to match both endpoint prices and slopes in closed form. For multiple maturities, the inherited calendar lower bound from the先前

Load-bearing premise

The construction requires that input prices across consecutive maturities satisfy a chord-based calendar inequality at every observed strike whose forward-aligned strike falls within the previous maturity's observed range. This condition is stronger than simple pointwise calendar freedom because the chord between observed prices can absorb available slack on coarse strike grids, so an upstream de-arbitrage step that produces pointwise calendar-free prices might not satisfy it

Editorial extensions

If this is right

Downstream methods that consume risk-neutral marginals—such as martingale optimal transport, Bass local-volatility calibration, and tail-risk measurement—can receive distribution-level inputs directly from this construction, eliminating the ad hoc numerical differentiation and normalization steps that currently bridge arbitrage-free price surfaces to probability laws.
The closed-form density, distribution function, and quantile availability mean that Monte Carlo sampling from the constructed marginals is straightforward, enabling scenario analysis and stress testing directly from de-arbitraged market quotes.
The O(n) per-maturity computational cost and the decoupled interval structure make the method suitable for real-time or near-real-time marginal construction in production pricing and risk systems.
The modular separation between upstream de-arbitrage and downstream marginal construction means that improvements in either layer—better surface smoothers or richer tail models—can be adopted independently without breaking the interface between them.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Bottom line: this paper does what it says — builds arbitrage-free risk-neutral marginals from discrete call prices with exact repricing, closed-form density/CDF/quantile, and guaranteed feasibility. The single-maturity construction is clean and correct. The multi-maturity calendar argument has a genuine gap in its proof, though the construction itself may well be fine in practice. It deserves a serious referee who can close or fix that gap. Here is my reasoning. The paper fills a real gap between upstream surface methods and downstream marginal-consuming methods (MOT, Bass LV, tail-risk). The single-maturity construction is the strongest part: anchoring at call-price slopes, using two-piece piecewise-constant curvature with a closed-form solution (Theorem 1), and attaching power-law tails with boundary matching (Propositions 1–2). The proofs are correct — I checked the feasibility bracket, the tail integration, and the slope-mass identity. The reduction to boundary conditions on C' is elegant and the O(n) closed-form per-maturity is a real computational advantage. The numerical experiments on SSVI and S&P 500 data are reasonable and show the method works in practice. Now the soft spot. The stress-test concern about Proposition 3 and Theorem 3 lands. The calendar-freedom argument works as follows: Proposition 3(b) constructs a specific curve C* whose gap below the chord IC is exactly ρ²A²/(2h₁). Part (c) shows C* is calendar-free when h₁ ≥ h_cal₁. Lemma 3 then proves feasibility by exhibiting this same C*. So far so good. But Theorem 3's proof says the per-interval optimization (23) has a non-empty feasible set, so by Proposition 3(c) the solution is calendar-free. The problem: Proposition 3(c) certifies calendar freedom for the specific C* with h_mid = 0. A generic feasible point of (23) — one with h_mid > 0 — distributes curvature differently, and its gap below IC could exceed δ_min. The constraint h₁ ≥ h_cal₁ was derived from C*'s specific gap formula. The paper says 'the choice of F does not affect correctness,' but the correctness argument only covers C*, not arbitrary feasible points. This is the gap between 'a feasible point exists that is calendar-free' and 'every feasible point is calendar-free.' The fix is likely straightforward: either prove the gap bound holds for all feasible points (not just C*), or require the optimizer to use C* specifically when calendar constraints bind. The paper may already intend this — the construction is local and the explicit solution is available — but the proof as written does not close the loop. Assumption 1 (the chord-based input calendar condition) is a minor concern by comparison. The paper is transparent about it being stronger than pointwise calendar freedom on coarse grids, and the refinement argument is reasonable. Recommendation: send to peer review. The paper is a solid contribution with one fixable proof gap. A good referee should focus on whether the calendar-freedom guarantee extends from C* to all feasible points of (23), or whether the construction should be restricted to C* on calendar-sensitive intervals.

Referee Report

3 major / 8 minor

Summary. The paper proposes an explicit construction of risk-neutral marginal laws from discrete arbitrage-free call prices. At a single maturity, the method estimates call-price slopes at observed strikes, assigns non-negative piecewise-constant curvature on each strike interval to match endpoint prices and slopes, and completes the distribution with closed-form power-law tails. Across maturities, a sequential construction enforces calendar no-arbitrage: the earlier curve provides a lower bound, and a three-piece curvature construction is invoked locally only on intervals where the two-piece fit would violate the inherited calendar floor. The authors prove feasibility (Theorems 1–3), verify butterfly and calendar arbitrage-freeness, and demonstrate the method on synthetic SSVI data and S&P 500 market quotes with an upstream smoother (SANOS). The resulting marginals admit closed-form densities, distribution functions, quantiles, and Monte Carlo sampling.

Significance. The paper addresses a genuine gap between upstream arbitrage-free surface construction and downstream marginal-consuming applications (MOT, Bass local volatility, tail-risk measurement). The construction is explicit and O(n) per maturity, with closed-form per-interval solutions (Theorem 1) and a scalar curvature floor reducing the pointwise calendar constraint (Proposition 3). The power-law tails are parameter-free given boundary data (Propositions 1–2), and the density-anchored PL3 extension (Proposition 4) is a useful optional refinement. The numerical experiments include both synthetic benchmarks and real market data with an independent upstream smoother, and the calendar-slack diagnostics provide falsifiable checks. The stress-test concern regarding whether the calendar-freedom guarantee extends from the explicitly constructed feasible curve C* to a generic minimizer of (23) is a substantive issue that warrants attention; my assessment of it is given in the major comments.

major comments (3)

§5.1, Proposition 3 and the proof of Theorem 3 (§8.11): The calendar-freedom guarantee is established for the specific explicitly constructed curve C* (Eq. 46, with h_mid=0), whose maximum gap below the chord IC is exactly ρ²A_l²/(2h₁). The per-interval optimization (23) allows a generic minimizer with h_mid > 0 and a different curvature profile. The paper states 'the choice of F does not affect this correctness' (§4.3 and Remark 1), and the proof of Theorem 3 (§8.11) says the feasible set is non-empty and 'by Proposition 3(c) it is pointwise calendar-arbitrage-free.' However, Proposition 3(c) certifies calendar freedom for C* specifically, not for an arbitrary feasible point of (23). A generic feasible point satisfying h₁ ≥ h_cal_1, h_mid ≥ 0, endpoint matching, and width bounds could have a gap below IC exceeding δ_min, because the gap formula ρ²A_l²/(2h₁) was derived under the特定 h_mid
Assumption 1 (§5, Eq. 14): The chord-based calendar condition is stronger than pointwise calendar freedom on coarse grids, as the paper acknowledges. However, the practical impact is not quantified. For the S&P 500 experiment (§6.3), it would strengthen the paper to report whether the SANOS-de-arbitraged grid satisfies Assumption 1 directly or whether grid refinement was needed. If refinement was needed, how many additional strikes were required? This is load-bearing because the entire calendar-freedom guarantee rests on Assumption 1 holding.
§5.4, Eqs. (24)–(25): The global slope selection programs enforce monotonicity of tail indices (α_j non-increasing for right tail, β_j non-increasing for left tail). The feasible sets are shown to be non-empty by construction (constant α or β). However, the selected slopes feed back into the body construction as boundary conditions for the adjacent intervals. If the global QP selects slopes far from the finite-difference estimates, the body intervals adjacent to k₁ and k_n could require large curvature, potentially affecting the calendar floor h_cal_1 on those intervals. The paper does not discuss whether this feedback can create infeasibility of the per-interval problems (23) at the boundary intervals. A brief remark addressing this would close the loop.

minor comments (8)

Table 1: The comparison of method families is useful but the criteria are somewhat coarse. For instance, 'Arbitrage-free price/surface interpolation' is marked △ for 'Valid marginal' but ✓ for 'Exact fit' — some interpolation methods (e.g., Kahalé 2004) do produce valid densities by construction. A footnote clarifying the specific methods considered under each family would help.
§4.1, Algorithm 1: The bootstrap step (line 5) uses the power-law tail of Proposition 2 as a temporary construction to estimate the slope at k₁, but Proposition 2 itself requires the slope at k₁ as input. The text explains that the bootstrap estimate Ĉ′(k₁)^(0) from Eq. (1) is used in place of the final slope, but this circularity could be stated more explicitly.
§6.2: The multi-maturity SSVI experiment reports 16.282 seconds end-to-end for 11 maturities × 100 strikes. It would be useful to report the breakdown between the global slope QP, the per-interval constructions, and the tail assembly to confirm that the per-strike cost is dominated by the local constructions as claimed.
§6.3: The half-tick time-value filter discards strikes whose discounted time value is below half a tick. The sensitivity of the reconstructed marginals to this threshold is not reported. A brief robustness check or remark would be welcome.
Figure 5: The calendar-slack diagnostics are shown for two representative maturity pairs. It would be informative to also report the minimum calendar slack across all maturity pairs and all strikes, as a single summary statistic confirming non-negativity everywhere.
References: The paper cites Buehler et al. (2026) and Deschâtres (2026) with 2026 dates. If these are forthcoming or preprint works, the citation format should indicate the access date or preprint number.
§3, paragraph on normalization: The tilde notation is introduced and then dropped. A brief note that all subsequent equations are in normalized units would help readers who skip ahead.
§8.9 (proof of Lemma 3): The explicit feasible solution (46) sets h_mid = 0. The text notes that a smoothness objective 'yields a better-behaved minimizer that typically lies in the interior of the feasible set.' This is an informal claim; if it is meant to address the concern about generic minimizers, it should be stated more precisely or removed.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for a careful and substantive report. The referee correctly identifies the core tension in the calendar-freedom argument (Major Comment 1), and we agree the manuscript must be revised to close the gap between the existence proof for the specific curve C* and the guarantee for a generic minimizer of (23). We also agree that the S&P 500 experiment should report whether Assumption 1 holds directly (Major Comment 2) and that the tail-slope-to-body feedback at boundary intervals deserves explicit discussion (Major Comment 3). All three comments lead to concrete revisions.

read point-by-point responses

Referee: §5.1, Proposition 3 and the proof of Theorem 3 (§8.11): The calendar-freedom guarantee is established for the specific explicitly constructed curve C* (Eq. 46, with h_mid=0), whose maximum gap below the chord IC is exactly ρ²A_l²/(2h₁). The per-interval optimization (23) allows a generic minimizer with h_mid > 0 and a different curvature profile. The paper states 'the choice of F does not affect this correctness' (§4.3 and Remark 1), and the proof of Theorem 3 (§8.11) says the feasible set is non-empty and 'by Proposition 3(c) it is pointwise calendar-arbitrage-free.' However, Proposition 3(c) certifies calendar freedom for C* specifically, not for an arbitrary feasible point of (23). A generic feasible point satisfying h₁ ≥ h_cal_1, h_mid ≥ 0, endpoint matching, and width bounds could have a gap below IC exceeding δ_min, because the gap formula ρ²A_l²/(2h₁) was derived under the特定 h_mid

Authors: The referee is correct. Proposition 3(c) certifies calendar freedom for the specific curve C* (with h_mid = 0), not for an arbitrary feasible point of (23). The statement 'the choice of F does not affect this correctness' is accurate only in the sense that the feasible set is non-empty (Lemma 3 constructs C* explicitly), but it is misleading if read as claiming that every feasible point is calendar-free. We will revise the manuscript to state this distinction explicitly. revision: yes
Referee: Assumption 1 (§5, Eq. 14): The chord-based calendar condition is stronger than pointwise calendar freedom on coarse grids, as the paper acknowledges. However, the practical impact is not quantified. For the S&P 500 experiment (§6.3), it would strengthen the paper to report whether the SANOS-de-arbitraged grid satisfies Assumption 1 directly or whether grid refinement was needed. If refinement was needed, how many additional strikes were required? This is load-bearing because the entire calendar-freedom guarantee rests on Assumption 1 holding.

Authors: We agree this is a load-bearing point and should be documented. In the S&P 500 experiment, the SANOS-de-arbitraged grid was checked against Assumption 1 at every observed strike of each later maturity whose forward-aligned strike falls inside the observed range of the previous maturity. The check passed directly without grid refinement. We will add a sentence to §6.3 stating this explicitly, including the number of forward-aligned strikes checked and the minimum observed chord-slack. revision: yes
Referee: §5.4, Eqs. (24)–(25): The global slope selection programs enforce monotonicity of tail indices (α_j non-increasing for right tail, β_j non-increasing for left tail). The feasible sets are shown to be non-empty by construction (constant α or β). However, the selected slopes feed back into the body construction as boundary conditions for the adjacent intervals. If the global QP selects slopes far from the finite-difference estimates, the body intervals adjacent to k₁ and k_n could require large curvature, potentially affecting the calendar floor h_cal_1 on those intervals. The paper does not discuss whether this feedback can create infeasibility of the per-interval problems (23) at the boundary intervals. A brief remark addressing this would close the loop.

Authors: The referee raises a valid point about the feedback loop. The boundary intervals [k₁, k₂] and [k_{n-1}, k_n] use the globally selected tail slopes as endpoint conditions. In principle, if the QP shifts a boundary slope far from the finite-difference estimate, the slope increment A_l on the adjacent interval changes, which in turn affects h_cal_1. However, infeasibility cannot arise: the two-piece body construction (Theorem 1) is feasible for any butterfly-free endpoint slopes, and the three-piece construction (Lemma 3) is feasible for any h₁ ≥ h_cal_1. The only effect of a shifted boundary slope is that h_cal_1 may increase, which the three-piece construction absorbs by adjusting h₁. We will add a remark to §5.4 or §5.6 clarifying this. revision: yes

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the construction is transparently a fitting procedure whose guarantees are structural, not circular.

full rationale

The paper proposes an explicit construction of risk-neutral marginals from discrete arbitrage-free call prices. The exact repricing property is acknowledged as a construction feature (fitting endpoint prices and slopes), not a prediction. The no-arbitrage guarantees (butterfly freedom from non-negative curvature, calendar freedom from the three-piece construction with curvature floor h_cal_1) are structural properties of the construction class, not circular restatements of inputs. The key derivation chain is: (1) butterfly-free input prices imply non-negative slope increments A_i (Theorem 1); (2) the two-piece curvature construction with non-negative pieces guarantees C'' >= 0 (butterfly freedom by construction); (3) Assumption 1 provides input calendar slack at observed strikes, Lemma 2 converts this to strict positive slack against the constructed prior curve, Proposition 3 reduces the pointwise calendar constraint to a scalar curvature floor h_cal_1, and Lemma 3 exhibits an explicit feasible solution. The skeptic's concern about whether a generic minimizer of (23) (rather than the explicit C*) satisfies calendar freedom is a correctness/completeness concern about the proof, not a circularity issue. The paper does not fit a parameter to data and then call the fit a prediction. The one self-citation (Qin et al. 2025) appears in the literature review (Section 2.2) as an example of Bass local-volatility calibration and is not load-bearing for any theorem or construction in this paper. The proofs are self-contained, relying on standard results (Breeden-Litzenberger identity, Lee's moment formula) cited to their original authors. No step in the derivation chain reduces to its inputs by definition or by self-citation in a way that would constitute circularity.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

No new entities are introduced. The power-law tail forms are parametric choices, not postulated objects.

free parameters (6)

β_i (interval split point) = determined by price-matching constraint (6) and curvature-jump minimization
Per-interval parameter solving the two-piece construction; closed-form solution in Theorem 1(b)
h_i (first-piece curvature) = determined by price-matching constraint (6)
Per-interval curvature; non-negativity guarantees butterfly freedom
α (right tail index) = α = 2 + λk_n e^{-rT} / C_{k_n} (Eq. 9)
Determined by matching boundary call price and slope at last observed strike
β_L, a_L (left tail parameters) = determined by boundary residuals γ_L, δ_L via Eq. (11)
Determined by matching boundary conditions at origin and first observed strike
C'(k_i) (slope estimates at observed strikes) = Lagrange finite-difference estimates projected onto no-arbitrage brackets
Not observed directly; estimated from discrete call prices via Algorithm 1 and finite-difference formulas
Boundary slopes C'(k_1^Tj), C'(k_n^Tj) across maturities = solved via QP (24)-(25) to enforce tail calendar conditions
Global optimization over maturity-level variables to ensure non-increasing α_j and β_j

assumptions (5)

standard math Breeden-Litzenberger identity: d²C/dK² = e^{-rT} f(K)
Section 3; standard result linking call-price curvature to risk-neutral density
domain assumption Input call prices are butterfly-free (convex call-price curve)
Section 3; required for non-negative curvature construction
domain assumption Assumption 1: Input calendar condition (strict chord-based inequality at observed strikes)
Section 5; required for Theorem 3. Stronger than pointwise calendar freedom on coarse grids
domain assumption Forward moneyness alignment of strike grids across maturities
Section 5; required for Lemmas 4-5 on tail calendar conditions
domain assumption Constant risk-free rate r and dividend yield q
Section 3; simplifies boundary conditions and forward-moneyness alignment

how reviews work

0 comments

Cite this review

Pith. "Pith review of Arbitrage-Free Multi-Maturity Risk-Neutral Marginals." pith.science (2026). https://pith.science/paper/E65OME4L

@misc{pith2026260706204,
  author       = {Pith},
  title        = {Pith review of: Arbitrage-Free Multi-Maturity Risk-Neutral Marginals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E65OME4L}},
  note         = {Machine review of arXiv:2607.06204}
}

read the original abstract

Many quantitative finance methods and applications are formulated in terms of option-implied risk-neutral marginals rather than directly in terms of option prices. Representative examples include martingale optimal transport, Bass local-volatility calibration, scenario analysis, and option-implied tail-risk measurement. The desired risk-neutral marginals should define a genuine probability law on the entire support, reproduce the input arbitrage-free option prices exactly, be free of butterfly and calendar arbitrage, and admit efficient evaluation of the density, distribution function, and quantiles, as well as Monte Carlo sampling. Existing methods typically optimize only a subset of these properties, depending on their intended purpose. This leaves a gap between upstream arbitrage-free option prices and the readily usable risk-neutral marginals required by downstream applications. We propose an explicit construction of risk-neutral marginals from discrete arbitrage-free option prices. On the observed strike range, probability mass is assigned interval by interval to exactly reproduce the input option prices. Outside the observed range, closed-form power-law tails complete the distribution by satisfying price and slope boundary conditions and allocating the remaining probability mass. Butterfly- and calendar-arbitrage-freeness are guaranteed by construction. The construction is feasible by design and computationally efficient. The resulting marginal laws admit closed-form densities, distribution functions, quantiles, and efficient Monte Carlo sampling. Numerical experiments on synthetic SSVI data and S\&P~500 market data demonstrate that the proposed construction efficiently and robustly produces marginals satisfying all of these properties in practice.

Figures

Figures reproduced from arXiv: 2607.06204 by the authors.

**Figure 2.** Single-maturity SSVI benchmark comparison. The top row shows the reconstructed marginal density, and [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗

**Figure 3.** Performance of the density-anchored right tail in the single-maturity SSVI experiment. The left panel shows [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Multi-maturity SSVI benchmark comparison. The left panel shows the reconstructed implied-volatility [PITH_FULL_IMAGE:figures/full_fig_p022_4.png]

**Figure 5.** Figure 5: Calendar-slack diagnostics for the multi-maturity SSVI experiment. The left panel shows the calendar slack [PITH_FULL_IMAGE:figures/full_fig_p023_5.png]

**Figure 6.** Figure 6: S&P 500 market experiment. The left panel compares our reconstructed marginal densities with the SANOS [PITH_FULL_IMAGE:figures/full_fig_p024_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

50 extracted references · 50 canonical work pages

[1]

Building arbitrage-free implied volatility: Sinkhorn's algorithm and variants

Building arbitrage-free implied volatility: Sinkhorn's algorithm and variants , author=. arXiv:1902.04456 , year=

work page Pith review arXiv 1902
[2]

Robust and

Qin, Hao and Che, Charlie and Yang, Ruozhong and Feng, Liming , journal=. Robust and

work page
[3]

Risk , volume=

Bounds of probability , author=. Risk , volume=

work page
[4]

Arbitrage-free

Gatheral, Jim and Jacquier, Antoine , journal=. Arbitrage-free. 2014 , publisher=

work page 2014
[5]

Journal of Econometrics , volume=

Nonparametric option pricing under shape restrictions , author=. Journal of Econometrics , volume=. 2003 , publisher=

work page 2003
[6]

Journal of Banking & Finance , volume=

Testing the stability of implied probability density functions , author=. Journal of Banking & Finance , volume=. 2002 , publisher=

work page 2002
[7]

A simple and reliable way to compute option-based risk-neutral distributions , author=

work page
[8]

The Journal of Finance , volume=

Nonparametric estimation of state-price densities implicit in financial asset prices , author=. The Journal of Finance , volume=. 1998 , publisher=

work page 1998

Show all 50 references

[9]

The Journal of Business , volume=

Prices of state-contingent claims implicit in option prices , author=. The Journal of Business , volume=. 1978 , publisher=

1978
[10]

The Journal of Computational Finance , volume=

Arbitrage-free estimation of the risk-neutral density from the implied volatility smile , author=. The Journal of Computational Finance , volume=. 2003 , doi=

2003
[11]

The Journal of Derivatives , volume=

How useful are implied distributions? Evidence from stock index options , author=. The Journal of Derivatives , volume=. 2000 , publisher=

2000
[12]

Journal of Financial Economics , volume=

Approximate option valuation for arbitrary stochastic processes , author=. Journal of Financial Economics , volume=. 1982 , publisher=

1982
[13]

Mathematical and Computer Modelling , volume=

Retrieving risk neutral densities based on risk neutral moments through a Gram--Charlier series expansion , author=. Mathematical and Computer Modelling , volume=. 2007 , publisher=

2007
[14]

Hermite polynomial based expansion of

Xiu, Dacheng , journal=. Hermite polynomial based expansion of. 2014 , publisher=

2014
[15]

Journal of Econometrics , volume=

Sieve estimation of option-implied state price density , author=. Journal of Econometrics , volume=. 2021 , publisher=

2021
[16]

Annual Review of Financial Economics , volume=

Risk-neutral densities: A review , author=. Annual Review of Financial Economics , volume=. 2018 , publisher=

2018
[17]

Computational Statistics , volume=

On extracting information implied in options , author=. Computational Statistics , volume=. 2007 , publisher=

2007
[18]

Quantitative Finance , volume=

Arbitrage-free smoothing of the implied volatility surface , author=. Quantitative Finance , volume=. 2009 , publisher=

2009
[19]

and Tzavalis, Elias , title=

Rompolis, Leonidas S. and Tzavalis, Elias , title=. Journal of Financial and Quantitative Analysis , volume=. 2008 , publisher=

2008
[20]

Model-Independent Bounds for Option Prices---A Mass Transport Approach , journal=

Beiglb\". Model-Independent Bounds for Option Prices---A Mass Transport Approach , journal=. 2013 , publisher=

2013
[21]

Model-Free Hedging: A Martingale Optimal Transport Viewpoint , publisher=

Henry-Labord. Model-Free Hedging: A Martingale Optimal Transport Viewpoint , publisher=. 2017 , doi=

2017
[22]

and Thomas, Charles P

Melick, William R. and Thomas, Charles P. , title=. Journal of Financial and Quantitative Analysis , volume=. 1997 , publisher=

1997
[23]

Implied Exchange Rate Distributions: Evidence from OTC Option Markets , journal=

Campa, Jos. Implied Exchange Rate Distributions: Evidence from OTC Option Markets , journal=. 1998 , publisher=

1998
[24]

Recovering Risk-Neutral Probability Density Functions from Options Prices Using Cubic Splines and Ensuring Nonnegativity , journal=

Monteiro, Ana Margarida and T\". Recovering Risk-Neutral Probability Density Functions from Options Prices Using Cubic Splines and Ensuring Nonnegativity , journal=. 2008 , publisher=

2008
[25]

Local Volatility Calibration by Optimal Transport , booktitle=

Guo, Ivan and Loeper, Gr. Local Volatility Calibration by Optimal Transport , booktitle=. 2019 , doi=

2019
[26]

Applied Mathematics & Optimization , volume=

Eckstein, Stephan and Kupper, Michael , title=. Applied Mathematics & Optimization , volume=. 2021 , publisher=

2021
[27]

, title=

Hobson, David G. , title=. Paris-Princeton Lectures on Mathematical Finance 2010 , series=. 2011 , publisher=

2010
[28]

, title=

Malz, Allan M. , title=. The Journal of Derivatives , volume=. 1997 , publisher=

1997
[29]

arXiv preprint arXiv:2601.11209 , year=

Buehler, Hans and Horvath, Blanka and Kratsios, Anastasis and Limmer, Yannick and Saqur, Raeid , title=. arXiv preprint arXiv:2601.11209 , year=

arXiv
[30]

Convex Volatility Interpolation , journal =

Desch. Convex Volatility Interpolation , journal =. 2026 , month = feb, day =. doi:10.2139/ssrn.4831218 , url =

2026 doi
[31]

, title=

Lee, Roger W. , title=. Mathematical Finance , volume=. 2004 , publisher=

2004
[32]

Bass Construction with Multi-Marginals: Lightspeed Computation in a New Local Volatility Model , author=
[33]

Calibration of the

Acciaio, Beatrice and Marini, Antonio and Pammer, Gudmund , journal=. Calibration of the. 2025 , publisher=

2025
[34]

arXiv preprint arXiv:2402.05669 , year=

q -Bass Martingales , author=. arXiv preprint arXiv:2402.05669 , year=

arXiv
[35]

The Martingale

Hasenbichler, Manuel and Joseph, Benjamin and Loeper, Gr. The Martingale. arXiv preprint arXiv:2310.13797 , year=
[36]

Martingale

Backhoff-Veraguas, Julio and Beiglb. Martingale. The Annals of Probability , volume=. 2020 , publisher=

2020
[37]

The Annals of Applied Probability , volume=

A Stochastic Control Approach to No-Arbitrage Bounds Given Marginals, with an Application to Lookback Options , author=. The Annals of Applied Probability , volume=. 2014 , publisher=

2014
[38]

The Annals of Probability , volume=

On a Problem of Optimal Transport under Marginal Martingale Constraints , author=. The Annals of Probability , volume=. 2016 , publisher=

2016
[39]

Risk , volume=

An Arbitrage-Free Interpolation of Volatilities , author=. Risk , volume=
[40]

Risk , volume=

Volatility interpolation , author=. Risk , volume=. 2011 , publisher=

2011
[41]

Le Floc'h, Fabien , journal=. The. 2025 , publisher=

2025
[42]

The Thirteenth International Conference on Learning Representations (ICLR) , year=

Operator Deep Smoothing for Implied Volatility , author=. The Thirteenth International Conference on Learning Representations (ICLR) , year=
[43]

Mathematical Finance , volume=

Option Pricing with Orthogonal Polynomial Expansions , author=. Mathematical Finance , volume=. 2020 , publisher=

2020
[44]

Parametric Risk-Neutral Density Estimation via Finite Lognormal-

Li, Yifan and Nolte, Ingmar and Pham, Manh Cuong , journal=. Parametric Risk-Neutral Density Estimation via Finite Lognormal-. 2024 , publisher=

2024
[45]

Mathematical Finance , volume=

Regular Variation and Smile Asymptotics , author=. Mathematical Finance , volume=. 2009 , publisher=

2009
[46]

Mathematical Finance , volume=

The Log-Moment Formula for Implied Volatility , author=. Mathematical Finance , volume=. 2023 , publisher=

2023
[47]

The Journal of Finance , volume=

Tails, Fears, and Risk Premia , author=. The Journal of Finance , volume=. 2011 , publisher=

2011
[48]

The Annals of Probability , volume=

Complete Duality for Martingale Optimal Transport on the Line , author=. The Annals of Probability , volume=. 2017 , publisher=

2017
[49]

Bernoulli , volume=

Dual Attainment for the Martingale Transport Problem , author=. Bernoulli , volume=. 2019 , publisher=

2019
[50]

The Journal of Finance , volume=

Short-term market risks implied by weekly options , author=. The Journal of Finance , volume=. 2017 , publisher=

2017

Pith tools

Reviewed July 8, 2026 · model on record in the stance chip above.

[1] [1]

Building arbitrage-free implied volatility: Sinkhorn's algorithm and variants

Building arbitrage-free implied volatility: Sinkhorn's algorithm and variants , author=. arXiv:1902.04456 , year=

work page Pith review arXiv 1902

[2] [2]

Robust and

Qin, Hao and Che, Charlie and Yang, Ruozhong and Feng, Liming , journal=. Robust and

work page

[3] [3]

Risk , volume=

Bounds of probability , author=. Risk , volume=

work page

[4] [4]

Arbitrage-free

Gatheral, Jim and Jacquier, Antoine , journal=. Arbitrage-free. 2014 , publisher=

work page 2014

[5] [5]

Journal of Econometrics , volume=

Nonparametric option pricing under shape restrictions , author=. Journal of Econometrics , volume=. 2003 , publisher=

work page 2003

[6] [6]

Journal of Banking & Finance , volume=

Testing the stability of implied probability density functions , author=. Journal of Banking & Finance , volume=. 2002 , publisher=

work page 2002

[7] [7]

A simple and reliable way to compute option-based risk-neutral distributions , author=

work page

[8] [8]

The Journal of Finance , volume=

Nonparametric estimation of state-price densities implicit in financial asset prices , author=. The Journal of Finance , volume=. 1998 , publisher=

work page 1998