are stock prices continuous or discrete

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The direct question "are stock prices continuous or discrete" is answered in two parts: in practice observed prices and trade times are discrete (they move in ticks, trades occur at discrete timestamps and in integral quantities), but financial theory commonly treats prices as continuous in time and value because continuous models are mathematically tractable and often good approximations. This article explains both meanings of the question "are stock prices continuous or discrete", compares US equities and many cryptocurrencies, surveys the modelling implications, and gives practical guidance for traders, quants, and developers.

As of 2023-06-01, according to the AEA conference submission by Sida Li & Mao Ye (AEA 2023), research on tick and lot constraints highlights how minimum price increments and discrete quantities shape trading outcomes and nominal price choices.

Definitions and conceptual distinction

Continuous vs. discrete variables

A variable is discrete when it can take only separate, distinct values (examples: integer share counts, price ticks). A variable is continuous when it can take any value in an interval of real numbers (example in models: a continuous price that can vary smoothly over a range). When people ask "are stock prices continuous or discrete" they may mean two different things: (1) Is the underlying state-space of prices continuous or discrete? and (2) Are observed market prices and timestamps recorded on a discrete grid?

• Discrete value example: A stock priced in cents can only take values that are integer multiples of the cent tick (historically larger ticks applied). Quantity discreteness means you trade whole shares (or allowed fractional shares).
• Continuous value example: In a continuous-time model, a price process S(t) takes values on the real line and evolves continuously in time according to a stochastic differential equation such as geometric Brownian motion.

Understanding the distinction between the mathematical state-space (model) and the recorded data (market observations) is central to answering "are stock prices continuous or discrete" in applied settings.

State space vs. observation granularity

The state space is the theoretical set of possible price values used in a model. Observation granularity is how exchange systems and data feeds record prices, quantities, and times. Even when a model assumes a continuous state space, market microstructure imposes discrete ticks, limited decimal precision, and timestamp granularity. Thus a continuous model can generate price realizations that are then rounded, truncated, or binned by the market infrastructure.

When resolving "are stock prices continuous or discrete" keep both layers in mind: continuous models can be excellent approximations to the underlying economics, while the recorded market data you use for calibration and execution are discrete.

Practical discreteness in real markets

Price ticks and minimum increments

On real exchanges prices move on a grid defined by the minimum price increment or tick size. In US equity markets, regulations and exchange rules have standardized small minimum increments (cent-level ticks for many securities), but those minimum increments are still discrete steps. Tick size determines which nominal prices are attainable without splitting orders across price levels.

Tick size matters for liquidity and execution. If ticks are large relative to the asset’s spread and volatility, traders face coarser pricing and wider effective spreads; if ticks are small, the market looks finer-grained and continuous approximations are more accurate. Empirical studies and market rules (including the regulatory environment that followed decimalization) show that changes in tick size change quoting behaviour and liquidity provision.

Lot sizes, fractional shares, and quantity discreteness

Share quantities have historically been discrete: round lots (often 100 shares) were the standard trading unit and odd-lot trading had different settlement and execution patterns. Over recent years, retail platforms and broker-dealers introduced more flexible fractional-share trading, reducing quantity discreteness for retail-sized trades. Institutional trading still operates on integral share counts but frequently uses block executions and algorithms to achieve effective fractional exposure via execution strategies.

Fractional-share services reduce the practical impact of quantity discreteness for small retail trades but do not eliminate tick-related discreteness in price.

Time discretization — trade timestamps and market hours

Trades and quotes are timestamped discretely (microseconds, nanoseconds, etc., depending on the feed). Exchange matching engines operate on discrete event processing: orders arrive, are matched, or queued at discrete timestamps. Equities also have defined trading hours (regular session, pre- and post-market sessions), so you cannot trade outside those windows on standard exchanges.

In contrast, many cryptocurrency venues offer near-continuous access (24/7 trading), but even there order book events are discrete and recorded with finite timestamp resolution. Thus when answering "are stock prices continuous or discrete" for time, the practical answer is discrete-event driven markets with finite timestamp granularity.

Exchange and ledger precision (crypto)

Cryptocurrency exchanges and on-chain ledgers often record values at high decimal precision (many digits after the decimal point). However, each exchange or smart contract can impose a price step (tick) and a minimum tradeable token amount. On-chain liquidity pools and order books therefore still implement discrete steps in practice, even if finer precision reduces visible discreteness.

For cryptocurrencies the combination of continuous 24/7 access and fine decimal precision makes continuous approximations more plausible in many contexts, but order-matching and settlement remain discrete processes.

Theoretical continuous models and why they are used

Continuous-time stochastic processes (Brownian motion, geometric Brownian motion)

Classical finance often models asset prices as continuous-time stochastic processes. The most famous example is geometric Brownian motion: dS = mu S dt + sigma S dW, where dW is Brownian motion. These models assume continuously evolving prices and continuous trading, which underpin results such as Black–Scholes option pricing and delta hedging formulas.

Continuous-time models are appealing because they provide closed-form solutions or tractable partial differential equations, facilitate comparative statics, and connect with powerful mathematical tools (Ito calculus, martingale theory). They also serve as limiting approximations of discrete-time trading models when trade frequency becomes very high.

Jump and mixed models

Recognizing that real prices can jump (overnight news, discrete events, large trades), researchers extend continuous diffusions to include jumps (jump-diffusion models) or completely discontinuous Lévy processes. These mixed models relax pure-continuity assumptions and better capture fat tails and sudden price moves. Jump models are still formulated in continuous time but allow discontinuities in price paths.

The existence of jumps provides a partial reconciliation: models can be continuous with stochastic components but still admit discrete discontinuities that match observed market jumps.

Rationale: mathematical tractability and limiting approximations

Continuous models are often justified as limits of high-frequency discrete-time models. When the tick size is small relative to price and trades are frequent, the discrete jump sizes and event times can be approximated by continuous diffusions in the limit. This limiting argument supports the practical use of continuous models even when the true data is discrete.

In short, continuous models are analytical approximations that work well when discretization is fine relative to the variation of interest. The modeling choice depends on the problem scale: for microstructure issues, discrete models are necessary; for long-horizon derivatives pricing, continuous models are usually sufficient.

Reconciling models with discrete reality

Discrete observations, model calibration and discretization

Practitioners fit continuous-time models to discrete data. Calibration requires careful handling of observation intervals, rounding, and microstructure noise. Common steps include:

• Sampling at lower frequencies (e.g., daily or 5-minute returns) to mitigate microstructure noise.
• Adjusting likelihoods or estimation methods to account for discrete ticks and rounding.
• Using discrete-time analogues (GARCH, ARIMA) where continuous assumptions fail at high frequency.

When implementing continuous hedging or replication, realize that continuous rebalancing is impossible; discrete-time implementations produce hedging errors that must be measured and controlled.

Replication errors and temporal granularity

Important literature (for example, work by Bertsimas, Kogan, and Lo) studies replication errors when continuous-time hedging strategies are implemented in discrete time. The finer the trading grid (higher frequency), the lower the replication error in stable markets, but costs, latency, and tick constraints limit how finely practitioners can rebalance.

These replication errors are one concrete way the practical discreteness affects the theoretically continuous strategies often used in textbooks.

Microstructure implications (bid-ask spreads, tick effects, stock splits)

Microstructure effects arise from ticks, order matching, and liquidity providers. Consequences include:

• Bid-ask spreads driven by discrete quoting and inventory costs.
• Price clustering at round numbers because of tick and human preferences.
• Stock splits as a way firms influence nominal stock price and therefore liquidity and tick-relative costs—research shows firms consider tick and lot constraints when setting nominal prices to make trading more attractive.

When answering "are stock prices continuous or discrete" from a microstructure point of view, discreteness is central for execution and short-term trading.

Empirical perspective

When discreteness matters (small-sample, low-price, low-volume assets)

Discreteness matters most when ticks, minimum quantities, or timestamp resolution are large relative to price changes or when volume is low. Examples include:

• Low-priced stocks where tick size is a significant fraction of the price (for example, when a one-cent tick is a material step for a $0.15 security).
• Thinly traded securities where trades are infrequent and the order book is sparse.
• Short intraday horizons where microstructure noise dominates statistical signals.

In such cases, continuous-time approximations can be misleading and discrete-event or agent-based models are more appropriate.

When continuous approximations are adequate (large samples, liquid high-price assets)

For liquid, high-priced assets with small tick sizes relative to price movements and high trading frequency, continuous approximations often produce accurate results for risk measures, derivative pricing, and portfolio evaluation. Over longer horizons (days or weeks), the cumulative effect of many small discrete events is well represented by continuous diffusions.

Thus answering "are stock prices continuous or discrete" depends on the time scale, liquidity, and choice of application.

Differences between equities and cryptocurrencies

Trading hours and market structure

US equity markets have regulated trading hours (regular session, plus limited pre-/post-market). Cryptocurrencies trade round-the-clock on many venues. This continuous access changes intraday dynamics and the applicability of certain models. However, continuous trading does not remove event discreteness: order arrivals, block trades, and on-chain transactions remain discrete.

Tick size, decimal precision, and on-chain considerations

Cryptocurrency markets often allow finer tick steps and many decimals, which reduces coarse discreteness in price. On-chain transactions add settlement delays and discrete ledger events that matter for custody and for strategies interacting directly with smart contracts. Some decentralized venues use continuous-time like pricing formulas (AMMs) but still settle in discrete on-chain transactions.

When answering "are stock prices continuous or discrete" for crypto, emphasize that the trading window and decimal precision make continuous approximations more plausible, but the matching, settlement, and ledger events remain discrete.

Regulation and market rules

Equities are subject to formal rules (e.g., Reg NMS in the US) that shaped tick structures and execution priorities; research like Li & Ye (AEA 2023) shows these institutional rules affect optimal nominal prices and trading outcomes. Crypto markets have heterogeneous regulatory coverage across jurisdictions and venue types; the lack of uniformly applied rules can increase variability in discreteness across venues.

Practical implications for practitioners

Modeling and risk management

• Model selection: use continuous diffusions for long-horizon pricing and risk when markets are liquid; use discrete-time or microstructure-aware models for intraday execution, market-making, and small-cap assets.
• Calibration: always calibrate with the same frequency and tick-adjusted data you will use in implementation. Account for rounding and microstructure noise explicitly.
• Risk measurement: include execution risk, rounding error, and discrete hedging costs when estimating real-world P&L.

Algorithmic trading and execution

Execution algorithms must respect ticks, lot sizes, and exchange rules. Latency matters: a discrete order book changes state at event times; algorithms that assume continuous price movement can be gamed by high-frequency participants. When building execution strategies on Bitget or other venues, design algorithms that submit orders at allowable ticks and avoid impossible price levels.

Bitget’s advanced order types, execution tools, and institutional-grade APIs (recommended for execution-intensive strategies) help practitioners manage tick-related constraints and latencies while prioritizing best execution.

Backtesting and data considerations

• Use tick or message-level data when studying microstructure effects and short-horizon strategies.
• Avoid interpolating prices across large gaps without accounting for spreads and order-book depth—straight-line interpolation can mislead.
• In backtests, simulate rounding to tick size and minimum quantity rules exactly to capture discrete effects.

When your backtest asks "are stock prices continuous or discrete" implicitly, enforce the discrete rules of the venue you plan to trade on.

Historical and regulatory context

Evolution of tick/lot rules and market microstructure

Market microstructure has evolved from large fixed ticks and round-lot norms to decimalization, smaller tick sizes, and more flexible lot sizes. These changes reduced the visible discreteness of prices for many securities and influenced liquidity provision and quoting behaviour.

Historical changes show how policy and technology alter the practical answer to "are stock prices continuous or discrete" by changing the granularity of market data and execution rules.

Research findings on optimal nominal prices

Research, including the AEA 2023 work by Sida Li & Mao Ye, models how tick and lot constraints influence firm and market choices. Firms may choose nominal share prices (via splits or dividends) to reach preferred liquidity characteristics because discrete ticks and nominal prices interact to affect spreads and execution costs.

These results demonstrate that discreteness is not just a technicality—it influences real economic decisions.

Summary and recommended reading

Answering the question "are stock prices continuous or discrete" requires clarity about whether you mean theoretical models or observed market data. Observed prices, quantities, and timestamps are discrete: exchanges enforce tick sizes, lot conventions, and timestamp resolutions. Financial theory often assumes continuity—either pure diffusions or jump-extended continuous-time processes—because continuous models are tractable and often good approximations when ticks are small and trading is frequent. When ticks, sparsity, or short horizons dominate, modelers and traders must explicitly account for discreteness.

Recommended further reading (selected foundational and applied work):

• Hull, excerpt: "A Model of the Behavior of Stock Prices" — overview of continuous-time modeling concepts.
• Bertsimas, Kogan & Lo — "When is time continuous?" — analysis of replication error and temporal granularity.
• Sida Li & Mao Ye (AEA 2023) — research on tick and lot constraints and their market effects.
• Surveys on Brownian models and jump processes (technical references and encyclopedia entries).

Further explore execution and custody solutions tailored to market microstructure needs: for trading and custody, consider Bitget’s exchange services and Bitget Wallet for secure custody and order execution tools.

See also

• Market microstructure
• Continuous-time finance
• Tick size
• Reg NMS (regulatory context for US equities)
• Brownian motion and jump processes

References

• Sida Li & Mao Ye (2023). "Discrete price, discrete quantity, and the optimal nominal price of a stock." AEA conference submission (2023).
• Hull, J. (excerpt). "A Model of the Behavior of Stock Prices." (textbook excerpt on continuous-time models).
• Bertsimas, D., Kogan, L., & Lo, A. "When is time continuous?" (paper on replication errors and temporal granularity).
• Wikipedia. "Brownian model of financial markets." (overview of Brownian-motion based market models).
• Quant.StackExchange and Economics.StackExchange threads discussing continuous vs discrete modeling and practical considerations.
• Investopedia. "Probability distributions: discrete vs continuous" (background on distributions and variables).

Actionable next steps

If you build trading algorithms or risk models, begin by asking where on the time/price/liquidity scale your strategy lives. If microstructure matters, adopt tick-aware backtests and consider Bitget’s execution APIs and Bitget Wallet for integrated custody and order management. To dive deeper into research, start with the listed references and examine how tick size and market structure affect the instruments you trade.

Want tailored guidance for your strategy implementation or data setup on Bitget? Explore Bitget’s developer resources and institutional services to align your models with live market constraints.