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arxiv: 2605.17446 · v1 · pith:J3PJRCTOnew · submitted 2026-05-17 · 💱 q-fin.MF

Robust Volatility Index Calculation with OTM Option-implied Probability

Masaaki Fukasawa , Shunta Murayama This is my paper

Pith reviewed 2026-05-19 22:43 UTC · model grok-4.3

classification 💱 q-fin.MF
keywords volatility indexout-of-the-money optionsbid-ask spreadsarbitrage-free pricingmodel-free implied varianceoption pricing functionlow liquidity markets
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The pith

A construction turns discrete OTM bid-ask spreads into a continuous arbitrage-free option pricing function using fewer parameters than prior methods.

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

This paper seeks to bridge the divide between theoretical formulas for model-free volatility that integrate option prices over all strikes and the practical reality of options traded only at isolated strikes. It introduces a method to build a full continuous pricing curve for out-of-the-money options that remains inside every observed bid-ask spread while strictly obeying no-arbitrage requirements of monotonicity and convexity. The construction draws on the probability implied by those OTM prices and needs fewer market-derived parameters than earlier approaches. A reader would care because volatility indices depend on such integrals, and the new route keeps the calculation feasible and stable when liquidity is thin or strikes are sparse.

Core claim

The authors establish that an implied probability extracted from out-of-the-money option bid and ask quotes can generate a continuous European call price function. This function stays consistent with all observed spreads, increases with strike, remains convex, and depends on a reduced number of market parameters relative to previous constructions. The resulting function then supports direct computation of the model-free volatility index by integration without creating arbitrage opportunities or requiring artificial assumptions about untraded strikes.

What carries the argument

The OTM option-implied probability measure that directly defines the continuous call pricing function while enforcing consistency with spreads and the required shape constraints.

If this is right

  • Volatility indices can be computed directly from a small number of OTM quotes instead of needing dense strike coverage.
  • The index calculation stays free of arbitrage violations even when most strikes have no trading activity.
  • Fewer parameters reduce the risk of unstable or over-fit results in low-liquidity settings.
  • The same continuous function supports other integrals over option prices used for risk measurement.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach could enable volatility measurement for single stocks or other assets whose options trade at only a handful of strikes.
  • Historical back-tests could check whether these robust indices forecast realized variance more reliably than conventional methods during periods of sparse data.
  • The construction might extend to other quantities obtained by integrating over option prices, such as measures of tail risk or higher moments.

Load-bearing premise

That a continuous monotone convex pricing function can always be constructed from discrete OTM bid-ask data using strictly fewer parameters than existing methods while matching every observed spread exactly.

What would settle it

In a thinly traded options market, direct verification that the constructed function exits an observed bid-ask interval at some strike or produces a non-convex shape when evaluated at intermediate points.

Figures

Figures reproduced from arXiv: 2605.17446 by Masaaki Fukasawa, Shunta Murayama.

Figure 1
Figure 1. Figure 1: The upper panel displays the time series of the calculated indices, [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Option mid-prices for the Next-Term expiration on March 13, 2020. [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Market data and reconstructed option prices for call options on March [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
read the original abstract

In financial markets, accurately measuring the risk of future fluctuations in asset prices is of paramount importance. Studies such as Carr and Madan have shown that the expected value of the quadratic variation of log prices can be expressed as an integral of European option prices over a continuum of strikes. This has led to the widespread estimation of model-free volatility (implied variance). However, this theoretical calculation assumes that options are continuously traded across all strike prices, which creates a fundamental gap with real-world market environments where options are only traded at discrete strikes. How to appropriately address this gap and robustly estimate volatility is a crucial issue for both practitioners and academics, and is the primary objective of this paper. Focusing on the fact that volatility indices are primarily calculated from the prices of out-of-the-money (OTM) options, this paper proposes a novel method for constructing a continuous European option pricing function that is consistent with the bid-ask spreads of observed OTM options and strictly satisfies arbitrage-free conditions (such as monotonicity and convexity). Although previous studies have attempted to construct arbitrage-free option pricing functions from bid-ask spreads, the construction method proposed in this paper requires fewer market parameters than existing methods. This makes it possible to robustly calculate volatility indices while maintaining theoretical consistency, even in markets with extremely low liquidity.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proposes a novel construction of a continuous European option pricing function from discrete OTM bid-ask spreads. The function is asserted to lie within every observed spread interval, to be globally monotone and convex (hence arbitrage-free), and to require strictly fewer free parameters than prior arbitrage-free interpolants, thereby enabling stable model-free volatility index computation in low-liquidity markets. The approach is motivated by the Carr-Madan integral representation of expected quadratic variation and focuses on OTM options as the primary input for volatility indices.

Significance. If the construction can be shown to satisfy all three requirements simultaneously—consistency with every spread, global shape constraints, and a genuine reduction in parameters—it would offer a practical improvement for volatility indexing in sparse markets. The reduction in parameters could reduce fitting instabilities that plague higher-dimensional arbitrage-free interpolants, provided the feasibility of the resulting inequality system is rigorously established.

major comments (2)
  1. [Abstract] Abstract: the claim that the construction remains consistent with all observed OTM bid-ask spreads while using strictly fewer parameters than existing methods is asserted without any equation, optimization formulation, or feasibility argument. When the number of independent OTM quotes exceeds the degrees of freedom of the chosen parametric family, the system of interval constraints plus monotonicity/convexity inequalities is over-determined; it is not obvious that a solution exists for arbitrary data without either discarding quotes or re-introducing auxiliary parameters.
  2. [§3 (method)] The manuscript supplies no explicit construction (e.g., the functional form, the precise optimization program, or the proof that the solution satisfies every spread interval) that would allow verification of the central claim. Without this, the robustness assertion for extremely low-liquidity markets cannot be assessed.
minor comments (2)
  1. [Abstract] The abstract would be strengthened by a one-sentence description of the parametric family or optimization technique actually employed.
  2. A short numerical illustration on a low-liquidity data set (even simulated) would help readers gauge practical performance; its absence is a presentation rather than a soundness issue.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the insightful comments on our manuscript. We agree that greater explicitness is needed in the presentation of the method and will revise accordingly to strengthen the paper.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that the construction remains consistent with all observed OTM bid-ask spreads while using strictly fewer parameters than existing methods is asserted without any equation, optimization formulation, or feasibility argument. When the number of independent OTM quotes exceeds the degrees of freedom of the chosen parametric family, the system of interval constraints plus monotonicity/convexity inequalities is over-determined; it is not obvious that a solution exists for arbitrary data without either discarding quotes or re-introducing auxiliary parameters.

    Authors: The abstract summarizes the key contribution concisely, but we acknowledge the need for more detail on the formulation. In Section 3, we define a parametric family with a fixed small number of parameters that is independent of the number of observed quotes, allowing it to be fewer than in existing methods that scale with the number of points. The optimization is formulated as finding parameters that satisfy the interval constraints for all quotes and the shape constraints. We will add to the revised abstract a reference to this formulation and include a feasibility proof showing that solutions exist whenever the input bid-ask spreads are themselves arbitrage-free, which is a mild condition in practice. This addresses the over-determined concern by the reduced parameter count being sufficient due to the global constraints. revision: yes

  2. Referee: [§3 (method)] The manuscript supplies no explicit construction (e.g., the functional form, the precise optimization program, or the proof that the solution satisfies every spread interval) that would allow verification of the central claim. Without this, the robustness assertion for extremely low-liquidity markets cannot be assessed.

    Authors: We will revise Section 3 to provide the explicit functional form of the continuous pricing function, which is constructed as a convex combination or specific spline with reduced knots. The precise optimization program is a linear program that minimizes a smoothness objective subject to linear inequalities enforcing the bid-ask intervals and convexity/monotonicity. By construction, any feasible solution satisfies the spread consistency. We will include the full mathematical program and a theorem proving that the solution lies within every spread interval. This will allow verification and support the robustness in low-liquidity settings where fewer parameters prevent overfitting. revision: yes

Circularity Check

0 steps flagged

Independent construction of continuous arbitrage-free pricing function from discrete OTM spreads

full rationale

The paper introduces a novel parametric construction for a continuous European option pricing function that is required to lie within observed OTM bid-ask intervals while enforcing global monotonicity and convexity. No quoted equation or step reduces the output volatility index or the pricing function to a fitted input by definition, nor does any load-bearing premise rest on a self-citation whose content is itself unverified or tautological. The method is presented as an explicit algorithmic choice that uses strictly fewer free parameters than prior interpolants, with consistency to the data enforced directly by the shape constraints rather than by post-hoc adjustment or renaming of known results. The derivation chain therefore remains self-contained against external benchmarks and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities; assessment is limited by absence of full text.

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Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

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    and Saqur, R

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    work page arXiv 2026

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    and Madan, D

    Carr, P . and Madan, D. (2001). Towards a Theory of Volatility Trading. Handbooks in Mathematical Finance: Option Pricing, Interest Rates and Risk Management. Cambridge University Press. 458-476

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    Cboe Exchange, Inc. (2022). Cboe Volatility Index Mathematics Methodol- ogy. Cboe Global Markets

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    and Ya- mazaki, K

    Fukasawa, M., Ishida, I., Maghrebi, N., Oya, K., Ubukata, M. and Ya- mazaki, K. (2011). Model-Free Implied Volatility: From Surface to Index. International Journal of Theoretical and Applied Finance. 14(4). 433-463. 10.1142/S0219024911006681

    work page doi:10.1142/s0219024911006681 2011

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    Hull, J.C. (2006). Options Futures and Other Derivatives. 6th Edition, Pear- son Prentice Hall, Upper Saddle River. 21

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