Trend arbitrage, bid-ask spread and market dynamics

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Trend arbitrage, bid-ask spread and market dynamics

Nikolai Zaitsev
Published on 7/10/2006
Equities
Stocks
Arbitrage
High frequency trading
Market timing
Trend-following

This paper introduces a novel approach to studying market microstructure by analyzing linear trends in tick-level bid-ask data. The author derives a trend arbitrage inequality that relates the bid-ask spread to the absolute trend slope, market reaction time, stock-specific latency, and transaction costs. Using tick data from 33 stocks across five European exchanges, the paper identifies trends via a linear fitting algorithm with a chi-squared trigger. The key empirical finding is that the distribution of the response time (τ_resp = spread/|trend| - average update time) is log-normal, with a sharp front edge that defines the market reaction time τ_M. This τ_M is found to be exchange-specific, ranging from about 1.2 seconds in Frankfurt to 5.5 seconds in Madrid, and is interpreted as the latency of the exchange system. The analysis also shows that the product of trend and update time (μ·τ_u) is normally distributed with a standard deviation such that three sigma lies within the tick size, indicating that price updates occur frequently enough to prevent trend arbitrage. When tick size is subtracted from the spread, the measured τ_M values across exchanges converge to a narrow range of 0.6-1.2 seconds, suggesting that tick size absorbs much of the stock-specific latency. The paper further identifies three investor groups (fast, medium, slow) with different fee and latency characteristics. A simulation using a random walk model reproduces the normal distribution of μ·τ_u, suggesting a duality between trend-based and return-based descriptions of price dynamics. Overall, the study provides a microstructure framework linking trend, spread, latency, and tick size, with implications for understanding market efficiency and arbitrage opportunities.

Highlights

  • 1Introduces linear trend as a novel tool for analyzing market microstructure and bid-ask spread dynamics.
  • 2Derives and tests a trend arbitrage inequality linking trend, spread, market reaction time, and update frequency.
  • 3Measures market reaction time (latency) from tick data, showing it is exchange-specific and ranges from 0.6 to 5.5 seconds.
  • 4Demonstrates that tick size influences price update frequency and absorbs stock-specific latencies.
  • 5Shows equivalence between trend-based and random walk descriptions of price dynamics via simulation.

Methods

  • M
    Linear trend fitting to intraday mid-prices with a chi-squared trigger to detect trend changes.
  • M
    Trend arbitrage inequality derived from a simple investor-specialist game, tested on tick data.
  • M
    Statistical analysis of distributions of response time (τ_resp) and product of trend and update time (μ·τ_u).
  • M
    Simulation of random walk model to compare trend properties with empirical data.

Results

  • R
    Market reaction time τ_M is exchange-specific: Amsterdam 1.9s, Frankfurt 1.22s, Madrid 5.5s, Milan 5.1s, Paris 2.1s.
  • R
    Distribution of τ_resp = Spr/|μ| - τ_u is log-normal, with a sharp front edge corresponding to τ_M.
  • R
    Product μ·τ_u follows a normal distribution with 3σ within the tick size, indicating efficient price updating.
  • R
    After subtracting tick size, τ_M values across exchanges converge to 0.6-1.2 seconds, suggesting tick size absorbs latency.
  • R
    Investor groups (fast, medium, slow) are identified in the τ_resp vs 1/|μ| plot, with distinct fee and latency parameters.
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