Non-unique time and market incompleteness

Abstract: 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.

• The paper demonstrates that mapping discrete market events to continuous time is non-unique, questioning the traditional calendar-time approach in finance.
• It employs event-driven frameworks to reveal that observed dependencies and risk measures vary with the choice of time representation.
• The study shows that no-arbitrage and risk-neutral pricing are framework-relative, highlighting inherent model risk in high-frequency trading environments.

Non-Unique Time and the Foundations of Market Incompleteness

Theoretical Background: Competing Ontologies for Financial Time

The paper "Non-unique time and market incompleteness" (2604.23608) rigorously examines the ontological and epistemological underpinnings of time modeling in mathematical finance. Classical price modeling predominantly adopts a single, global calendar-time axis, representing asset prices as (possibly jump-)diffusive semi-martingales. This approach, deeply ingrained in the FTAP and underpinning the logic of risk-neutral pricing and dynamic hedging, is contrasted with event-driven frameworks in which trades, order flow, quotations, and their inter-arrival times are the primitive market events.

The authors meticulously distinguish between the representational (ontological) choice—calendar time vs. event time—and the inferential (epistemological) question of what structure empirical high-frequency data actually supports. This dichotomy is critical; the prevailing calendar-time approach imposes a unique continuum a priori, whereas event-first models underscore discreteness and asynchrony as fundamental, especially at high frequencies.

Event-Time Models: Empirical Motivation and Technical Ramifications

Early literature, exemplified by Clark’s subordination hypothesis, transactional mixture-of-distributions approaches, and subsequently by autoregressive conditional duration (ACD) and point-process modeling, advocates for stochastic operational clocks linked to trading activity, information arrival, or order flow, rather than calendar time. Empirical evidence—e.g., the inability of volume or transaction clocks to perfectly "Gaussianize" returns [Ane and Geman (2000)], and the instability demonstrated in higher-moment recovery [Murphy and Izzeldin (2006)]—confirms that the selection of clock is both consequential and fundamentally unidentifiable from market data, manifesting as a form of representation-level (ontological) incompleteness.

Moreover, microstructure phenomena like the Epps effect demonstrate that covariance and dependence measures are not scale-invariant; their numerical values and economic meaning depend crucially on the choice of sampling scheme and synchronization conventions. This undermines the identification of a unique latent covariance at the shortest horizons. Dependence emerges as a clock-relative observable rather than a universal latent property.

Stochastic Process Limits: CTRW, DTRW, and Non-Uniqueness

In continuous-time random walk (CTRW) and discrete-time random walk (DTRW) frameworks, the passage from discrete event time to continuum calendar time is non-canonical. Distinct discretizations, jump-waiting dependencies, and synchronization schemes can yield non-unique limiting diffusive equations, fractional PDEs, or subordinated processes, depending on the scaling and pathwise structure imposed [Angstmann et al., 2015]. The continuous calendar-time price process thus becomes an artificial reduction, with no guarantee of uniqueness in its derivation from the underlying market events. This theoretical observation significantly weakens the common identification of the calendar-time representation as ontologically privileged.

No-Arbitrage, Risk-Neutral Pricing, and Market Viability

The principle of no-arbitrage, pivotal to market viability, is not inherently tied to calendar time or continuous trading. The Dalang–Morton–Willinger theorem and related discrete-time no-arbitrage results extend naturally to event-time frameworks, where self-financing strategies and gain processes are indexed by the event filtration rather than calendar time. Similarly, risk-neutral pricing and equivalent martingale measure arguments hold in both discrete time and under stochastic clocks, but any attempt to attach pricing measures to the physical market without clarifying the underlying clock is fundamentally model-relative.

A key claim advanced by the authors is that market incompleteness emerges at the level of representation, prior to the classical spanning-based definition. If the choice of clock is non-unique and not fully identifiable from observed data, then there is no universal state description under which all hedging or pricing strategies can be canonically formulated.

Effective Completeness, Practical Consequences, and Market Segmentation

The arguments do not invalidate calendar-time finance in aggregate. At low frequencies, or for aggregated risk-management and asset allocation, effective completeness can "emerge" as a coarse-grained property: calendar-time models often fit empirical data well, and asset managers' operational time matches the lower-frequency state descriptions. However, for high-frequency trading, hedging, and microstructure-aware execution, the lack of a unifying clock exposes basis risk, slippage, and model dependence that cannot be eliminated by improved calendar-time modeling.

Implications for risk management and trading are direct: slippage, basis risk, and funding costs become state- and clock-dependent, and model risk arises from ontological non-uniqueness rather than simply incomplete spanning. For practitioners, reconciling operational event time with reporting calendar time becomes a central challenge, especially for high-frequency execution and cross-asset hedging.

Conclusion

The paper demonstrates that the mapping from discrete market events to continuous-time price processes is inherently non-unique. The foundational incompleteness so revealed is ontological: the market may not select a unique physically meaningful clock for all purposes. Classical no-arbitrage and martingale pricing survive, but only as framework-relative constructs. Effective completeness remains useful at coarse scales, but cannot bridge all time representations seamlessly. This perspective motivates both new technical approaches to microstructure modeling and a reassessment of the scope of classical asset pricing theorems in operational trading environments.