Non-unique time and market incompleteness

TL;DR

This paper challenges the traditional assumption of a unique, continuous global time in financial markets, emphasizing the importance of event-driven and operational time frameworks for understanding market incompleteness.

Contribution

It introduces a critical perspective on market modeling by contrasting continuous-time assumptions with event-time and order-flow descriptions, highlighting non-uniqueness and deeper market incompleteness.

Findings

Market models assuming a unique global time may overlook fundamental incompleteness.

Discrete event systems can lead to non-unique continuum limits affecting pricing.

Operational time considerations are crucial for risk management and portfolio strategies.

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.