arXiv subjects

Tim Gebbie

Publications and source records attributed to Tim Gebbie.

At least 19 recordsLinked to original sources

arXiv2026

Frame covariance of CMBR lensing in the exact 1+3 covariant radiation hierarchy

The exact $1+3$ covariant Boltzmann hierarchy for the cosmic microwave background contains acceleration, shear and vorticity couplings whose coefficients grow linearly in $\ell$. We show that all are the multipole form of the aberration operator in the photon Liouville equation. Under a change of threading, this operator shifts by the along-ray generator of a Lorentz boost. The acceleration coupling of $\ell$ to $\ell\pm1$ and the shear coupling of $\ell$ to $\ell\pm2$ are therefore frame-covariant descriptions of the same operator, related by an algebraic, pointwise relabelling of direction. With emitter and observer frames fixed, the observed sky is independent of the intermediate threading at every $\ell$. We derive this in $1+3$ covariant and gauge-invariant variables without constructing second-order gauge-invariant variables. The $O(ε^2\ell)$ content is gravitational lensing, as known in metric variables. Linearisation about a Friedmann--Lemaître--Robertson--Walker background removes these couplings, requiring lensing to be reinstated from the geodesic equation. Keeping the threading arbitrary makes frame covariance explicit and places observer aberration in the same operator. In the Newtonian threading we compute both halves of the deflection: the equivalence-principle contribution enters through aberration, while the spatial-curvature contribution arises from free streaming on perturbed spatial sections and reduces to the same operator. The lensing efficiency follows from the direction derivative. The angular-power-spectrum contribution is quartic in the potential: for a deflection statistically independent of the source, the two fourth-order terms carrying $\ell^2$ cancel where the spectrum is smooth on the deflection scale, and lensing redistributes power without adding ensemble variance.

astro-ph↗

arXiv2026

A Finite Bid--Ask Spread from Replenishment Displaced from the Quote

We give a unified analytic account of a finite bid--ask spread in a two-field reaction--diffusion order book. The model retains separate bid and ask densities in operational time, with diffusion, cancellation, reaction and external order creation. On the symmetric equal-coefficient branch, the imbalance field determines the reaction price, while the total standing density carries the spread geometry. A market-making source is introduced by withholding order placement over a finite interval around the reaction boundary, representing round-trip replenishment displaced from the quote. Under a separated, quasistationary and weak-overlap approximation, transport penetrates inward from the placement edges over the cancellation length, giving the quoted spread as the sub-threshold core of the placement-source-free interval. Market orders remove standing density at the observed quote, whereas replenishment is delivered outside it and must diffuse back across the penetration depth. We derive the resulting delayed and attenuated quote-level response in operational time. On that branch, finite spread and price impact arise from distinct response sectors, while their calendar-time appearance requires a separate observation clock.

q-fin.TR↗

arXiv2026

Event-Time Order-Flow Memory, Operational-Time Impact, and Subordinated Market Observables

We consider two canonical market-microstructure regularities: the long-memory of trade signs and the square-root law of meta-order impact. The point is not to propose new empirical laws, but to separate the clocks on which existing laws are defined. The sign-memory law is an event-time statement about the ordering and fragmentation of hidden orders. The square-root impact law is an operational-time statement about front motion in a locally linear latent order book. Starting from a discrete-time random-walk bid/ask reaction--diffusion order book with a separate event clock, we identify an event-time imbalance reduction, derive the operational-time front and impact equations in the locally linear regime, and then subordinate both sign and impact observables to calendar time. Fractional or tempered clock effects enter through this event-to-calendar projection, not through a different operational-time impact mechanism. This gives a compact clock-aware framework in which event-time sign persistence, operational-time square-root impact, and anomalous calendar-time effects appear as distinct but compatible consequences of a common order-book representation.

q-fin.TR↗

arXiv2026

Correlation emergence and the Epps effect in two coupled limit order books

We give a unified analytic account of correlation emergence and the Epps effect in two coupled limit order books. The Epps effect is the empirical reduction in measured cross-asset log-return correlation at short aggregation scales, or equivalently the recovery of measured correlation as the aggregation interval increases. The model starts from a fixed-grid discrete random walk for order flow in operational time, with creation, cancellation and diffusion. A pair-trader coupling between the books is introduced at the level of order creation, and calendar time is imposed through separate observation clocks. We clarify how the operational-time model reduces to coupled reaction--diffusion equations with a moving reaction boundary defining the model log-mid-price. Using a regularised local-response representation of the coupling, we derive approximate closed-form expressions for realised correlations as a function of aggregation time. Here the Epps effect is shown to arise from two distinct mechanisms: asynchronous observation clocks (subordination), finite coupling response times, and their combination.

q-fin.TR↗

arXiv2026

Metaorder modelling and identification from public data

Market-order flow in financial markets exhibits long-range correlations. This is a widely known stylised fact of financial markets. A popular hypothesis for this stylised fact comes from the Lillo-Mike-Farmer (LMF) order-splitting theory. However, quantitative tests of this theory have historically relied on proprietary datasets with trader identifiers, limiting reproducibility and cross-market validation. We investigate whether it can be recovered from anonymous public data using synthetic metaorder reconstruction. Using transaction and quote data for the largest 239 stocks by market capitalisation on the JSE as of 13 March 2026 with the data range being 1 January 2023 until 31 December 2025, we conduct a grid search over reconstruction parameters and evaluate each configuration against established metaorder stylised facts and the LMF relation. Configurations selected to minimise errors across the metaorder impact stylised facts reproduce the targeted aggregate properties but yield a poor LMF relation. Configurations selected to minimise the LMF discrepancy recover the relation by construction while retaining several broad impact features, although some stock-level execution and decay fits are weaker. These asymmetric results show that recovering aggregate impact stylised facts alone is insufficient to identify LMF-consistent order splitting, while the LMF-targeted result establishes compatibility within the reconstruction class rather than an independent test. The findings support consistency with, rather than direct validation of, the LMF theory using anonymous market data.

q-fin.TR↗

arXiv2026

Metaorder modelling and identification from public data

q-fin.TR↗

arXiv2026

Reflexivity from Hierarchical Causality

Complex systems are often organised into hierarchies whose internal interactions are stronger or faster than interactions across levels. When markets are treated as genuinely multilevel systems it becomes natural to represent them as systems with hierarchical causality. Here we show that reflexivity can be formulated within such a discrete hierarchical causal system; but one in which a higher-level actor state restricts the lower-level transition kernels that remain admissible. Then event dynamics can be separated from calendar embeddings: a set-valued actor-conditioned correspondence can be used to define the admissible family of event kernels, while joint state and waiting-time laws can be used to determine compatible timing to then natural demonstrate reflexivity. A selected event-state law need not determine a unique calendar embedding. Locally, uniqueness of the joint event or timing specification requires uniqueness of both the admissible event kernel and its compatible timing law. Reflexivity is thus the endogenous closure of a hierarchical constraint loop, while timing and projection can generate calendar-time memory or causal ambiguity even for Markov event dynamics.

physics.soc-ph↗

arXiv2026

Reaction-boundary variance and adjoint-consistent local-volatility projection

We derive an operational-time variance kernel for a latent-order-book reaction boundary and use it to separate three objects usually collapsed in calendar-time volatility models: a structural boundary cumulant, a clock projection, and a pricing-measure choice. The reaction boundary is the zero of a bid--ask imbalance field. For a locally linear book, signed order-flow perturbations displace this zero through a damped Abel response kernel, so the variance of boundary increments is obtained as a finite-scale Green-function cumulant rather than introduced as a primitive diffusion coefficient. For long-memory forcing with exponent $0<γ<1$, the operational variance has a closed asymptotic form involving effective signed-forcing intensity, liquidity slope, resilience, memory, and operational coarse-graining scale. A deterministic activity clock gives the benchmark local-volatility projection. More general, non-unique clocks generate candidate calendar-time pricing systems. We argue that such projections are admissible only when the induced forward density operator and backward valuation operator remain adjoint on the same state space. Adjoint consistency is therefore a reality constraint on operational-to-calendar time projection: it disciplines non-unique time and identifies where incompleteness enters.

q-fin.PR↗

arXiv2026

Epistemic Limits of Empirical Finance: Causal Reductionism and Self-Reference

The clarion call for causal reduction in the study of capital markets is intensifying. However, in self-referencing and open systems such as capital markets, the idea of unidirectional causation (if applicable) may be limiting at best, and unstable or fallacious at worst. In this work, we critically assess the use of scientific deduction and causal inference within the study of empirical finance and financial econometrics. We then demonstrate the idea of competing causal chains using a toy model adapted from ecological predator/prey relationships. From this, we develop the alternative view that the study of empirical finance, and the risks contained therein, may be better appreciated once we admit that our current arsenal of quantitative finance tools may be limited to ex post causal inference under popular assumptions. Where these assumptions are challenged, for example in a recognizable reflexive context, the prescription of unidirectional causation proves deeply problematic.

q-fin.GN↗

arXiv2026

Correlation emergence in two coupled simulated limit order books

We use random walks to simulate the fluid limit of two coupled diffusive limit order books to model correlation emergence. The model implements the arrival, cancellation and diffusion of orders coupled by a pairs trader profiting from the mean-reversion between the two order books in the fluid limit for a Lit order book with vanishing boundary conditions and order volume conservation. We are able to demonstrate the recovery of an Epps effect from this. We discuss how various stylised facts depend on the model parameters and the numerical scheme and discuss the various strengths and weaknesses of the approach. We demonstrate how the Epps effect depends on different choices of time and price discretisation. This shows how an Epps effect can emerge without recourse to market microstructure noise relative to a latent model but can rather be viewed as an emergent property arising from trader interactions in a world of asynchronous events.

q-fin.TR↗

arXiv2026

Option prices from operational-time reaction-boundary lattices

We consider the role of a continuum operational time $u$, its mapping to calendar time $t$, and their relation to event time in 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 homogeneous nearest-neighbour log-price lattice; state-dependent local variance is represented through its general local-kernel extension. Its Chapman--Kolmogorov decomposition yields discrete forward and backward equations. 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, the lattice gives a structural route from boundary variance to projected local volatility. The key point is the separation of the operational kernel, the calendar-time clock projection, and the pricing-measure choice. Within the deterministic one-factor local-volatility class, an option surface identifies the projected clock--variance combination rather than its operational variance and clock components separately. The resulting hierarchy also distinguishes full calendar-generator equivalence from weaker terminal or continuum matching, and clarifies why incompleteness concerns unspanned clock, jump, or renewal risks rather than random time alone.

q-fin.PR↗

arXiv2026

A Gabor--Epps uncertainty principle for traders

We propose a Gabor--Epps uncertainty principle for practical trading. The key idea is that high-frequency correlation is not observed in clock time alone, but is resolved through market activity, order-flow overlap, and finite coupling response. This suggests six simple rules of thumb that may be useful to traders and trading programs operating at market-making frequencies, particularly those crossing books and markets below the average human response time. Throughout, the observation window is clock-dependent: in calendar time it is a physical interval, in trade time it is a trade-count interval, and in volume time it is a volume bucket. In summary: at event scales, the more precisely one localises market activity in time, the less well one can resolve stable cross-asset dependence. The more one resolves dependence, the more one has coarse-grained away the event-time structure that generated it. This can generate substantial clock risk.

q-fin.TR↗

arXiv2026

Revisiting Trade-sign Long-memory and Square-root Law price impact

Starting with a coupled discrete reaction--diffusion formulation for the lit and latent order books with non-uniformly sampled event times and meta-order source terms we show how two familiar market-microstructure regularities can emerge from this framework: the long-memory of trade signs associated with the Lillo--Mike--Farmer (LMF) theory and the square-root law (SQRL) of meta-order impact. This uses the locally linear order book and constant participation-rate execution in the front dynamics to reduce the dynamics to a Volterra equation whose leading-order solution then yields the well know result of concave impact trajectory, and a completion impact proportional to the square root of the meta-order size. We then use the interface representation to show how heavy-tailed Pareto meta-order lengths generate power-law trade-sign autocorrelations through the source term. These are familiar derivations, what is slightly different here is that we reinterpret these known derivations to make it clear that LMF law is an event-time sign-memory statement, whereas the square-root law is a physical-time viability statement where subordination can alter the calendar-time impact trajectories depending on the mappings and interpolation used to set continuum operational time.

q-fin.TR↗

arXiv2026

A Simple Hierarchical Causality Primer

We provide a brief primer for the idea behind formalising hierarchical causality in the context of complex systems. Here actors are not simply agents. Actors instantiate causation classes. Agents implement local dynamics in given levels or organisation in a given system. Hierarchical causality then describes how actor-level roles constrain, select, and organise agent-level behaviour across levels. The system then necessarily requires three additional structures. First, causation classes to abstract a given form of causal influence that an actor instantiates. Second, aggregation operators to move across the levels. Third, discrete event-time maps are required because the system comprises events, and the relation between local event counts and any global clock must be specified. Our formulation here is purposefully simple and discrete.

cs.MA↗

arXiv2026

The Ethics of LLM Sandbox and Persona Dynamics

It is well known that LLM guardrails and trained persona dynamics can produce a reality gap: the distance between the world a LLM is permitted or shaped to describe, and the world in which users must act. Here we argue that actively generating reality gaps is in fact unethical because it knowingly shifts epistemic risk back to the uninformed user -- this is reality laundering. This can potentially cause harm when operationalised at scale. The risk is sharpest in high-exposure advice contexts, where users seek orientation rather than a bounded, externally checkable task. Guardrails naively appear ethically necessary when they claim to prevent direct harm, but often become suspect when they suppress truthful perception and launder uncomfortable mechanisms into acceptable abstractions. Basel-style financial regulation, B-BBEE-style compliance, Societe Generale, and the London Whale show how formal safety systems can become legible, gameable, and performative while real exposure migrates elsewhere. The same pattern can appear in LLMs as moral compliance: safe language, distorted reality. We therefore distinguish refusing harm, from refusing reality; and then argue for top-down causal requirements specification at the task level rather than bottom-up moral correction at the response or sandbox level. Persona dynamics matter because the assistant interface is not neutral; it shapes how uncertainty, conflict, authority, and risk are staged. The conclusion is that so-called ``ethical AI'' becomes substantively unethical when it substitutes institutional reassurance for contact with reality.

cs.AI↗

arXiv2026

Non-unique time and market incompleteness

Financial markets are often modelled as if time were unique and continuous across assets and markets. Financial markets are however asynchronous, order flow is event-driven, and waiting times between events are often random. Many of the most influential formulations of financial market models presuppose a unique global calendar time and advocate for this or that preferred single latent continuous-time price system. Here we critically contrast these assumptions with event-time, renewal, point-process, and order-flow descriptions. We revisit no-arbitrage, no-dynamic-arbitrage, and risk-neutral option pricing in settings where the market is represented as a discrete event system and where the continuum limit of a discrete-time random walk need not be unique. The central suggestion is then that such non-uniqueness points to a more foundational form of market incompleteness than is usually emphasized. This highlights the importance of operational time at the level of decision making but reminds market practitioners that managing risk itself often requires reconciling operational time with a global calendar time. At these longer time scales forms of effective or average completeness may still emerge at lower frequencies and remain useful for portfolio construction and risk management, even if high-frequency hedging and execution expose a clock mismatch between trading, pricing, and longer-horizon allocation.

q-fin.TR↗

arXiv2025

The bias of IID resampled backtests for rolling-window mean-variance portfolios

Backtests on historical data are the basis for practical evaluations of portfolio selection rules, but their reliability is often limited by reliance on a single sample path. This can lead to high estimation variance. Resampling techniques offer a potential solution by increasing the effective sample size, but can disrupt the temporal ordering inherent in financial data and introduce significant bias. This paper investigates the critical questions: First, How large is this bias for Sharpe Ratio estimates?, and then, second: What are its primary drivers?. We focus on the canonical rolling-window mean-variance portfolio rule. Our contributions are identifying the bias mechanism, and providing a practical heuristic for gauging bias severity. We show that the bias arises from the disruption of train-test dependence linked to the return auto-covariance structure and derive bounds for the bias which show a strong dependence on the observable first-lag autocorrelation. Using simulations to confirm these findings, it is revealed that the resulting Sharpe Ratio bias is often a fraction of a typical backtest's estimation noise, benefiting from partial offsetting of component biases. Empirical analysis further illustrates that differences between IID-resampled and standard backtests align qualitatively with these drivers. Surprisingly, our results suggest that while IID resampling can disrupt temporal dependence, its resulting bias can often be tolerable. However, we highlight the need for structure-preserving resampling methods.

q-fin.PM↗

arXiv2024

Detecting discrete processes with the Epps effect

The Epps effect is key phenomenology relating to high frequency correlation dynamics in financial markets. We argue that it can be used to provide insight into whether tick data is best represented as samples from Brownian diffusions, or as samples from truly discrete events represented as connected point processes. We derive the Epps effect arising from asynchrony and provide a refined method to correct for the effect. We then propose three experiments which show how to discriminate between possible underlying representations. These in turn demonstrate how a simple Hawkes representation recovers phenomenology reported in the literature that cannot be recovered using a Brownian representation without additional ad hoc model complexity. However, complex ad hoc noise models built on Brownian motions cannot in general be discriminated relative to a Hawkes representation. Nevertheless, we argue that high frequency correlation dynamics are most faithfully recovered when tick data is represented as a web of interconnected discrete events rather than being samples from continuous Brownian diffusions even when combined with noise.

q-fin.ST↗