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arxiv: 2606.09564 · v2 · pith:TQ34ICCPnew · submitted 2026-06-08 · 💱 q-fin.PR · q-fin.MF· q-fin.TR

Option prices from operational-time reaction-boundary lattices

Chris Angstmann , Tim Gebbie This is my paper

Pith reviewed 2026-06-27 14:07 UTC · model grok-4.3

classification 💱 q-fin.PR q-fin.MFq-fin.TR
keywords option pricingoperational timeMarkov latticeBlack-Scholes-Mertonreaction-boundary modellimit order bookincomplete marketsChapman-Kolmogorov
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The pith

An operational-time Markov lattice on log-prices yields a generalized European option pricing PDE that recovers Black-Scholes-Merton under risk-neutral drift and constant volatility.

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

This paper establishes option pricing equations starting from a Markov lattice defined in operational time rather than calendar time. A nearest-neighbour log-price lattice with state- and time-dependent transitions yields discrete forward and backward equations through Chapman-Kolmogorov decomposition. These converge to the standard continuum equations under local finite-variance scaling. The backward equation in price variables produces a generalized European pricing PDE, which reduces to the Black-Scholes-Merton equation when the risk-neutral drift condition and constant volatility hold. The construction treats the lattice as a reaction-boundary model for limit-order-book mid-prices and separates the operational kernel, calendar-time projection, and pricing measure to address incomplete markets from unspanned risks.

Core claim

We derive option-pricing equations from an operational-time Markov lattice rather than from a calendar-time diffusion. The primitive model is a nearest-neighbour log-price lattice with state- and time-dependent transition probabilities. Its Chapman-Kolmogorov decomposition yields discrete forward and backward equations, which converge under local finite-variance scaling to the usual continuum adjoint pair. In price variables, the backward equation gives a generalized European pricing PDE and reduces to Black-Scholes-Merton under the risk-neutral drift restriction and constant volatility. Interpreted as a reaction-boundary model for limit-order-book mid-prices, the construction identifies loc

What carries the argument

The operational-time Markov lattice with state- and time-dependent transition probabilities, whose Chapman-Kolmogorov decomposition produces discrete forward and backward equations that converge to the continuum adjoint pair.

Load-bearing premise

The discrete forward and backward equations obtained from the Chapman-Kolmogorov decomposition of the operational-time lattice converge to the usual continuum adjoint pair under local finite-variance scaling.

What would settle it

A numerical check in which the discrete lattice equations fail to approach the Black-Scholes-Merton PDE as the time step shrinks under finite variance would falsify the convergence and reduction claims.

Figures

Figures reproduced from arXiv: 2606.09564 by Chris Angstmann, Tim Gebbie.

Figure 1
Figure 1. Figure 1: Schematic route from a discrete Markov lattice to the BSM [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Hierarchy of continuum limits arising from a discrete-time random walk (DTRW) / discrete Markov lattice. The central spine [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
read the original abstract

We consider the role of a continuum operational time u and its mapping to calendar time t and how these relate to event time for option pricing problems. We derive option-pricing equations from an operational-time Markov lattice rather than from a calendar-time diffusion. The primitive model is a nearest-neighbour log-price lattice with state- and time-dependent transition probabilities. Its Chapman-Kolmogorov decomposition yields discrete forward and backward equations, which converge under local finite-variance scaling to the usual continuum adjoint pair. In price variables, the backward equation gives a generalized European pricing PDE and reduces to Black-Scholes-Merton under the risk-neutral drift restriction and constant volatility. Interpreted as a reaction-boundary model for limit-order-book mid-prices, the construction identifies local volatility with an activity-rescaled risk-neutral bid-ask reaction-boundary variance. The framework separates the operational kernel, calendar-time projection, and pricing-measure choice, to clarify how unspanned clock, jump, or renewal risks can lead to incomplete-market pricing.

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

1 major / 1 minor

Summary. The manuscript derives option-pricing equations from a nearest-neighbour log-price Markov lattice in operational time with state- and time-dependent transition probabilities. Chapman-Kolmogorov decomposition produces discrete forward and backward equations claimed to converge, under local finite-variance scaling, to the standard continuum adjoint pair. The backward equation in price variables yields a generalized European pricing PDE that reduces to Black-Scholes-Merton under the risk-neutral drift restriction and constant volatility. The construction is interpreted as a reaction-boundary model for limit-order-book mid-prices, with local volatility identified as an activity-rescaled risk-neutral bid-ask reaction-boundary variance, and separates the operational kernel, calendar-time projection, and pricing-measure choice to address incomplete markets arising from unspanned clock, jump, or renewal risks.

Significance. If the convergence holds rigorously, the separation of operational time, calendar projection, and measure choice supplies a lattice foundation for pricing under incomplete markets that may be useful for limit-order-book modeling. The explicit reaction-boundary interpretation links local volatility to bid-ask activity in a manner that could generate testable implications for mid-price dynamics. The reduction to BSM is by construction once the risk-neutral restriction and constant volatility are imposed, so the primary contribution would reside in the generalized PDE and the operational-time framework rather than novel closed-form prices.

major comments (1)
  1. [Abstract] Abstract: the central claim that the discrete forward and backward equations converge to the usual continuum adjoint pair under local finite-variance scaling is load-bearing for recovery of the generalized PDE. With explicitly state- and time-dependent transition probabilities, the manuscript supplies no explicit Taylor expansion or remainder estimate confirming that higher-order and cross terms cancel; without this verification the limiting generator may acquire extraneous drift or diffusion contributions that prevent clean recovery of the risk-neutral operator.
minor comments (1)
  1. The abstract and introduction could more explicitly flag that the BSM reduction is obtained by imposing the risk-neutral drift and constant-volatility restrictions, to avoid any appearance that the lattice produces an independent derivation of BSM.

Simulated Author's Rebuttal

1 responses · 0 unresolved

Thank you for the opportunity to respond to the referee's report. We appreciate the identification of the need for a more explicit verification of the continuum limit, which is central to the paper's claims.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the discrete forward and backward equations converge to the usual continuum adjoint pair under local finite-variance scaling is load-bearing for recovery of the generalized PDE. With explicitly state- and time-dependent transition probabilities, the manuscript supplies no explicit Taylor expansion or remainder estimate confirming that higher-order and cross terms cancel; without this verification the limiting generator may acquire extraneous drift or diffusion contributions that prevent clean recovery of the risk-neutral operator.

    Authors: We agree that an explicit Taylor expansion with remainder estimates would strengthen the argument and remove any ambiguity about extraneous terms. In the revised manuscript we will insert a new appendix (or expanded section) that performs the local finite-variance scaling on the Chapman-Kolmogorov equations, expands the state- and time-dependent transition probabilities to the required order, demonstrates cancellation of all higher-order and cross terms, and confirms that the limiting generator is precisely the risk-neutral adjoint operator. revision: yes

Circularity Check

0 steps flagged

No significant circularity; standard lattice-to-PDE limit with explicit reduction check

full rationale

The derivation begins with a nearest-neighbour operational-time lattice whose transition probabilities are state- and time-dependent, applies the Chapman-Kolmogorov equation to obtain discrete forward and backward equations, invokes a local finite-variance scaling to pass to the continuum adjoint pair, transforms the backward equation into a generalized European pricing PDE in price variables, and finally imposes the risk-neutral drift restriction plus constant volatility to recover the Black-Scholes-Merton operator. Each step is a conventional limiting argument or algebraic substitution; the final reduction to BSM is an explicit consistency check under stated restrictions rather than a fitted input renamed as a prediction. No self-citation load-bearing step, self-definitional closure, or smuggled ansatz appears in the abstract or the described chain. The convergence claim is the weakest modelling assumption but is not shown to be equivalent to the target PDE by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The derivation rests on the Chapman-Kolmogorov equation for the lattice and on the local finite-variance scaling limit that produces the continuum PDE; no free parameters, new axioms, or invented entities are introduced in the abstract.

axioms (2)
  • standard math Chapman-Kolmogorov decomposition of the operational-time Markov lattice yields discrete forward and backward equations.
    Invoked in the abstract as the step that produces the discrete equations before the scaling limit.
  • domain assumption Local finite-variance scaling produces convergence to the usual continuum adjoint pair.
    Stated as the condition under which the lattice equations become the continuum PDE pair.

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