agg-dependence-problem
🔍 Derivatives and Temporal Mismatch​
Both state and flow variables can be differentiated (Δ or d/dt) to express their rate of change:
| Operation | On State Variable | On Flow Variable |
|---|---|---|
| Δ over time | Change in the snapshot (e.g., ΔOI) | Change in cumulative activity between windows (e.g., ΔVolume) |
| Units | “contracts per interval” | “change in contracts traded per interval” |
So the first derivative exists for both, but they describe different physical processes:
ΔOI→ how quickly positions are changing (creation / closure).- ΔVolume → how quickly activity levels are accelerating or fading.
⚠️ The Aggregation-Dependence Problem​
Because volume is defined over a chosen time interval, any ratio involving it inherits that time scale:
Even if open interest is unchanged, the ratio ( V / OI ) varies dramatically depending on the window.
That’s not an error — it’s a dimensional feature of mixing an instantaneous quantity with an aggregated one.
In other words, the ratio is scale-dependent:
- Over longer windows, volume accumulates while OI remains static.
- Over shorter windows, the same OI “supports” smaller chunks of traded volume.
This is why a “volume-to-OI” ratio isn’t a universal number — it’s a function of time aggregation.
You can absolutely compute it across multiple granularities (1d, 8h, 1h, 15m) to study how trading intensity evolves, but it’s crucial to keep the temporal dimension explicit.
🧠Summary​
- You can apply Δ to both states and flows, but their meanings differ.
- Ratios between them are time-scale sensitive, since one side accumulates with window length.
- Recognizing this mismatch is the key conceptual step before attempting any unified indicator such as volume-to-OI turnover or OI-flow efficiency.