Nonlinear Intraday Return Surface: Trades and Volume Interaction

Introduction

Ané and Geman (2000) showed that clocking returns by cumulative trade count can substantially improve the approximation to normality relative to calendar time. Their result suggests a close relationship between return variance and trade count. The surface below challenges this one-dimensional picture: by treating both trade count and traded volume as jointly ranked activity measures, a markedly nonlinear landscape emerges — one that no single-axis conditioning can reveal.

Figure 1. Nonlinear intraday return surface: trades and volume interaction.

Note: The floor axes show log-normalised within-session activity for each bar of length \(\Delta \in \{30\text{ s},\, 60\text{ s}\}\): the horizontal axis is \(\tilde{x} = (\ln N - \ln N_{\min})/(\ln N_{\max} - \ln N_{\min}) \in [0,1]\), where \(N\) is the number of trades in the bar; the depth axis is \(\tilde{y} = (\ln V - \ln V_{\min})/(\ln V_{\max} - \ln V_{\min}) \in [0,1]\), where \(V\) is the total number of shares traded. The vertical axis shows the mean standardised bar return \((r - \mu_r)/\sigma_r\) within each cell of a \(5\times 5\) uniform grid over \([0,1]^2\), where \(r = \ln(P_t/P_{t-1})\) is the bar-to-bar log return computed from consecutive last transaction prices, and \(\mu_r\), \(\sigma_r\) are the global mean and standard deviation across all bars.

Source: Author’s own elaboration using LOBSTER message file data (AAPL, 21 June 2012, one full regular trading session, 09:30–16:00 ET; 23,658 ticks of executions of a visible limit order).

Trades–Volume Interaction: 30-Second Bars

The dominant feature of the surface is a rise from high volume to low volume: bars with large traded volume are associated with lower average standardised returns within the displayed sample (dark region of the surface), while bars with small volume sit at above-average returns (light region). Along the trades axis the picture is more nuanced. At low volume, the contour lines are approximately parallel to the trades axis, indicating that the number of trades appears to have little additional impact on returns once volume is small. At moderate-to-high volume the contours curve, revealing a nonlinear interaction: for the same volume level, a higher trade count is associated with a different return than a lower trade count. The surface therefore suggests that a purely additive specification in trades and volume may be inadequate.

Trades–Volume Interaction: 60-Second Bars

At the one-minute horizon the interaction between trades and volume appears to become more pronounced. Returns are highest in the low-volume, high-trades corner (light region) and lowest in the opposite corner of high volume and few trades (dark region). The surface therefore runs diagonally across the floor rather than along a single axis, and neither trades nor volume alone appears sufficient to explain the return surface. This joint dependence is supported by the strongly curved contour lines projected onto the floor: they fan out from the high-volume/low-trades region and converge toward the high-trades/low-volume corner, a pattern suggestive of nonlinear interaction between the two activity measures.

References

Ané, T., & Geman, H. (2000). Order flow, transaction clock, and normality of asset returns. Journal of Finance, 55(5), 2259–2284.