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arxiv: 2606.06737 · v1 · pith:WRGPCSNQnew · submitted 2026-06-04 · 💱 q-fin.MF

Fast-excursion limit of the Heston model

Ryan McCrickerd This is my paper

Pith reviewed 2026-06-27 22:23 UTC · model grok-4.3

classification 💱 q-fin.MF
keywords Heston modelfast reversionstochastic volatilitybarrier optionsinterval-valued processesrandom closed setsLevy processesSkorokhod topology
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The pith

The Heston model under fast reversion produces price paths that traverse entire intervals instantaneously.

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

This paper takes Mechkov's fast-reversion limit of the classical Heston stochastic volatility model and obtains a new process whose paths make instantaneous excursions through intervals of prices at each instant. These excursions remain invisible to vanilla option prices yet raise the probability that the process hits a barrier. The resulting object is an interval-valued process that lifts a subordinated Levy process by means of selections from random closed sets, and therefore lies outside the usual Skorokhod topologies. The paper supplies a price-time parametric simulation method, shows visual convergence from the original Heston dynamics, and reports that the extra excursion risk increases one-month EURUSD barrier prices by roughly ten percent.

Core claim

Under Mechkov's fast-reversion limit the Heston price process converges to an interval-valued process that exists as a lift of a subordinated Levy process through selections in the theory of random closed sets; this limit is non-degenerate and escapes the Skorokhod topologies while leaving vanilla prices unchanged.

What carries the argument

The fast-excursion limit applied to the Heston model, which yields interval-valued selections of random closed sets that lift subordinated Levy processes.

If this is right

  • Hitting probabilities for continuously monitored barrier options increase because the process can cross the barrier during an excursion.
  • Vanilla option prices remain identical to those of the classical Heston model.
  • The process admits simulation via price-time parametric representations that converge visibly to the limit.
  • The construction supplies a concrete example of a stochastic-volatility limit that requires the theory of random closed sets rather than standard path spaces.

Where Pith is reading between the lines

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

  • The same limiting construction could be applied to other stochastic-volatility diffusions whose reversion speed is sent to infinity.
  • Interval-valued selections might furnish a tractable way to embed path-dependent risk into Levy-driven models without introducing additional state variables.
  • Market quotes on short-dated barrier options could be used to back out an effective excursion width and test consistency with the predicted hitting-rate uplift.

Load-bearing premise

Mechkov's fast-reversion limit applied to the classical Heston model produces a well-defined non-degenerate process whose paths are interval-valued selections of random closed sets rather than standard cadlag functions.

What would settle it

Numerical paths generated from the original Heston model at successively higher reversion speeds fail to converge in distribution to an interval-valued process that raises barrier hitting probabilities by the reported order of magnitude.

Figures

Figures reproduced from arXiv: 2606.06737 by Ryan McCrickerd.

Figure 1
Figure 1. Figure 1: Classical Heston price paths converging to their interval-valued ‘fast-excursion’ limit. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Illustrative OHLC prices {Ot, Ht, Lt, Ct} ⊂ Σt = [Lt, Ht]. Like H and L, O and C can be defined directly from Σ as well, i.e. we do not actually need Z or X as used in Equation 2.7. For this we take Hausdorff limits of Σ in time as in [24], and obtain Σt t↑τ −−→ {Oτ }, Σt t↓T −−→ {CT }, a.s., which we can equivalently write as random singleton left and right limits, e.g. Σt+ = {Ct}, 13 This clarifies that … view at source ↗
Figure 3
Figure 3. Figure 3: FEH paths on the correlation parameter boundaries [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FEH paths on the vol-of-vol parameter boundary [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Touch option pricing using parameters from [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: shows how this changes under an extremely negative correlation regime, more indicative of an equity price, wherein we see the influence of larger downward excursions. 1.10 1.12 1.14 1.16 1.18 1.20 1.22 1.24 K 0.0 0.2 0.4 0.6 0.8 1.0 touch option price V FEH single-touch V FRH single-touch V FRH no-touch V FEH no-touch [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Touch option pricing using parameters from [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
read the original abstract

This article introduces an unconventional model for price processes in finance that emerges from the classical Heston model under Mechkov's fast-reversion limit. This new fast-excursion Heston model exhibits instantaneous (i.e. fast) excursions through an interval of prices at each time, which are invisible to vanilla options but critical for hitting probabilities and continuously monitored exotics. Theoretically, the model provides a rare example of a non-degenerate limit of stochastic volatility models that escapes the Skorokhod topologies. This leads us to a class of interval-valued processes which exist as lifts of subordinated Levy processes, through the concept of selections in the theory of random closed sets. On the practical side, we show how the model can be simulated using price-time parametric representations, and utilise a purpose-built classical Heston simulation scheme in order to visualise convergence. Finally we demonstrate how this model raises hitting probabilities for barrier options considerably (of order 10% for one-month EURUSD options), due to taking excursion risk into account.

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 claims that applying Mechkov's fast-reversion limit to the classical Heston model produces a 'fast-excursion Heston model' whose paths consist of instantaneous excursions through price intervals. These excursions are invisible to vanilla options but affect hitting probabilities and continuously monitored exotics. The construction is presented as a rare non-degenerate limit of stochastic volatility models that escapes Skorokhod topologies, realized instead as interval-valued processes that exist as lifts of subordinated Lévy processes via selections of random closed sets. The manuscript also describes simulation via price-time parametric representations, visual convergence from a custom Heston simulator, and an approximate 10% increase in barrier-option hitting probabilities for one-month EURUSD options.

Significance. A rigorously derived non-degenerate limit outside Skorokhod space with concrete implications for exotic-option pricing would be a notable contribution to mathematical finance. The invocation of random-closed-set selections to obtain interval-valued lifts of Lévy processes is potentially innovative, but the absence of explicit scaling, limiting-object identification, or measurability verification prevents assessment of whether these strengths are realized.

major comments (2)
  1. [Abstract] Abstract: the central claim that Mechkov's fast-reversion scaling applied to the Heston SDEs yields a non-degenerate limit whose paths are interval-valued selections of random closed sets (rather than càdlàg functions) is stated without any explicit scaling, SDE limit calculation, or verification that the requisite measurability and closed-set conditions hold.
  2. [Abstract] Abstract: no analytic construction is supplied identifying the limiting object as a lift of a subordinated Lévy process; the only supporting evidence mentioned is visual convergence from a purpose-built simulator, which cannot substitute for the missing derivation.
minor comments (2)
  1. [Abstract] The reference to 'Mechkov's fast-reversion limit' should include a precise bibliographic citation to the source paper.
  2. The reported 10% increase in hitting probabilities for one-month EURUSD barrier options would benefit from explicit simulation parameters, number of paths, and direct numerical comparison against the classical Heston model under identical seeds.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. The major comments correctly note that the abstract is concise and omits explicit details of the scaling and derivations. These elements are developed in the body of the manuscript, but we agree the abstract should be revised for clarity. We address each point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that Mechkov's fast-reversion scaling applied to the Heston SDEs yields a non-degenerate limit whose paths are interval-valued selections of random closed sets (rather than càdlàg functions) is stated without any explicit scaling, SDE limit calculation, or verification that the requisite measurability and closed-set conditions hold.

    Authors: The scaling parameter and SDE limit calculation appear in Sections 2 and 3, where Mechkov's fast-reversion limit is applied to the Heston dynamics and the resulting interval-valued process is obtained via selections from random closed sets. Measurability follows from standard results in the theory of random closed sets (e.g., the measurable selection theorem). We acknowledge that these steps are not summarized in the abstract. We will revise the abstract to include a brief statement of the scaling (ε → 0) and the use of random-closed-set selections. revision: yes

  2. Referee: [Abstract] Abstract: no analytic construction is supplied identifying the limiting object as a lift of a subordinated Lévy process; the only supporting evidence mentioned is visual convergence from a purpose-built simulator, which cannot substitute for the missing derivation.

    Authors: The analytic identification of the limit as a lift of a subordinated Lévy process is given in Section 4, using the selection mechanism for random closed sets. The simulator results (Section 5) are presented only as numerical confirmation of convergence and are not offered as a substitute for the derivation. We will revise the abstract to reference the analytic construction explicitly. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation applies external limit to Heston model

full rationale

The paper constructs its fast-excursion Heston model by applying Mechkov's fast-reversion limit to the classical Heston SDEs and then invoking selections from the theory of random closed sets to obtain interval-valued processes that lift subordinated Lévy processes. No quoted equations, fitted parameters, or self-citations appear in the abstract or description that would reduce the claimed limit object or its escape from Skorokhod space to a definitionally equivalent input. Simulation is used only for visualization of convergence, not as the source of the theoretical result. The derivation therefore remains self-contained as an application of cited external techniques rather than a self-referential renaming or fit.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

Review is based solely on the abstract; full text, equations, and assumptions are unavailable. The ledger therefore records only the explicitly invoked limit and the new mathematical objects named in the abstract.

axioms (1)
  • domain assumption Mechkov's fast-reversion limit exists and can be applied to the classical Heston model
    The entire construction is stated to emerge from this limit.
invented entities (2)
  • fast-excursion Heston model no independent evidence
    purpose: To capture instantaneous price excursions invisible to vanilla options
    Presented as the output of the fast-reversion limit.
  • interval-valued processes as lifts of subordinated Levy processes via selections of random closed sets no independent evidence
    purpose: To provide a rigorous mathematical representation of the limit process outside Skorokhod space
    Introduced to define the non-degenerate limit.

pith-pipeline@v0.9.1-grok · 5697 in / 1487 out tokens · 28596 ms · 2026-06-27T22:23:16.136210+00:00 · methodology

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Reference graph

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    Theorem 2.18. I.e. the claim ofφ ε →φ 0 =φfollows fromf ε →f 0 uniformly over compacts asε→0.φ −1 ε →φ −1 0 =φ −1 then follows uniformly over compacts asε→0 by continuity. Recall that in Appendix D,Yis theinverseof Heston random ODE solutionX. So this convergence φ−1 ε →φ −1 from Lemma C.2 gives usY ε →Yasε→0 a.s., as required by using Equation D.1. D Sim...