Microprice¶
Definition¶
The microprice is a refined "fair value" estimator for an asset, built on top of the mid-price by weighting it with queue imbalance. It attempts to extract a continuous latent price from discrete book state — a value that sits within the spread, biased toward whichever side of the book carries more resting volume.
Introduced formally by Stoikov (2018) as a Markov-chain correction to the mid; the underlying intuition goes back to Cao, Hansch & Wang (2009) and appears under various names in the practitioner literature.
Core formulas¶
Basic queue-weighted microprice¶
The simplest form weights the best bid and ask by the opposite queue size — when the ask queue is large, the short-term fair value sits closer to the bid, and vice versa:
Equivalently, in terms of queue imbalance \(I = (q^b - q^a)/(q^b + q^a)\):
where \(m = (p^a + p^b)/2\) is the mid-price and \(s = p^a - p^b\) is the spread.
Stoikov (2018) Markov-chain microprice¶
Stoikov's refinement iterates the expected mid-price conditional on \((I, s)\) under a Markov model of queue dynamics:
where \(g\) is computed as a fixed-point of the expected mid-price shift when the next price-changing event fires. The key insight: the basic formula above is biased in regimes where spread is elevated; the Markov correction accounts for the fact that a wide spread will typically close from one side with probability biased by \(I\).
Why it's useful¶
- Predictive of next mid-move: the microprice leads the mid by a few milliseconds — crossing the microprice through the mid is a short-horizon direction signal.
- Better fair value for market makers: quoting around the microprice rather than the mid reduces adverse selection.
- Bridges discrete book state to continuous latent price: in the limit of large relative tick size, the microprice is essentially a readable version of the continuous efficient price (strongly validated empirically — see the W/USDT experiment in explainable-crypto-microstructure where spot OBI correlates at \(c = 0.94\) with the perp's implied continuous price).
- Natural benchmark for queue-imbalance signals: any QI-based predictor should be compared against a microprice-marked baseline rather than a mid-marked one.
Empirical properties¶
- Microprice position within the spread is ~linear in \(I\) for moderate imbalance; saturates toward the opposite best at high \(|I|\). This is the queue-imbalance-as-predictor curve studied in lipton-quote-imbalance and gould-bonart-queue-imbalance.
- Tick-size dependence: the microprice–mid gap carries more information in large-tick books where depth at the best concentrates; converges toward the mid in small-tick books where depth is dispersed. This is the same tick-size thread documented in deep-lob-forecasting and mlofi-xu-gould-howison.
- Consistent with Stoikov's Markov framework: empirical microprice profiles match the semi-analytic forms Stoikov derived for common parameter choices.
When to use / when not to use¶
Use when: - Marking inventory or computing fair value at the sub-second horizon. - Deciding whether to post passively or cross the spread in an execution algorithm. - Generating a short-horizon direction signal cheaper than a full OFI computation. - Market-making (quoting around microprice reduces adverse selection).
Avoid when: - Modelling long-horizon price dynamics — microprice is a microstructure primitive, not a macro signal. - Small-tick assets with small \(I\) — the microprice-mid gap is vanishingly small and the mid is fine. - You need a signal robust to spoofing; large visible queues that later cancel will move the microprice spuriously.
Computational complexity¶
- Basic formula: \(O(1)\) per update — read best bid/ask queues, apply a ratio.
- Stoikov Markov version: \(O(1)\) per update once the fixed-point \(g(I, s)\) table is precomputed (typically a small lookup grid over discretised \((I, s)\)).
- Trivially real-time on any HFT stack.
Implementations¶
- Native primitive in virtually every HFT market-making codebase.
- Open-source reference: Sasha Stoikov's companion code to the 2018 paper (Cornell FE).
LOBFrame(deep-lob-forecasting) implicitly uses microprice-like features via its input representation.- CFMM LP pricing (cfmm-liquidity-provision-pricing) uses a continuous price analogous to the microprice as the natural coordinate for the bonding curve.
Relation to other signals¶
| Signal | Computation | What it captures |
|---|---|---|
| Mid-price | \((p^a + p^b)/2\) | Naïve midpoint; ignores book state |
| Microprice | queue-weighted mid | Continuous latent price within spread |
| Quote Imbalance | \((q^b - q^a)/(q^b + q^a)\) | Direction signal only (not a price) |
| VWAP | \(\sum v p / \sum v\) | Execution-weighted price over a window; backward-looking |
| OFI-adjusted mid | \(m + \beta \cdot \text{OFI}\) | Cumulative event-flow adjustment |
The microprice is the natural fair-value companion to queue imbalance — QI tells you direction, microprice tells you the best scalar-valued fair price given that direction.
Connections¶
- Direct users: lipton-quote-imbalance (theoretical framework implying microprice); gould-bonart-queue-imbalance (logistic regressions on QI that are functionally equivalent to a non-parametric microprice); reality-gap-lob-simulation (uses \((I, n)\) state projection — microprice is a natural summary statistic of this state); explainable-crypto-microstructure (validates microprice empirically via W/USDT spot-vs-perp experiment); cfmm-liquidity-provision-pricing (CFMM spot-price is a microprice analogue on the bonding curve).
- Conceptual anchors: order-flow-imbalance (QI is the atomic ingredient of microprice); limit-order-book (the substrate); price-impact (microprice is what a low-impact execution should aim for).
- Sister method: queue-reactive-model — the Markov chain used to define Stoikov's microprice is closely related to QR dynamics.
- Foundational but non-arXiv references:
- Stoikov, S. (2018). "The Micro-Price: A High-Frequency Estimator of Future Prices." Quantitative Finance 18(12), 1959–1966. Not on arXiv; would need Semantic Scholar / SSRN retrieval to ingest directly.
- Cao, Hansch & Wang (2009). "The Information Content of an Open Limit-Order Book." Journal of Futures Markets 29(1), 16–41. Not on arXiv.