Price Formation, the Bid–Ask Spread, and Liquidity
Where prices come from when there is no single ‘price’, and why the spread is the market maker’s compensation for real risks.
Leads to: 16.3 turns the spread’s adverse-selection component into a model of impact.
Learning Objectives
Click a status chip to cycle: Not started → In progress → Studied → Practiced → Needs review → Mastered.
- Explain price formation as the aggregation of order flow and information into the mid-price.
- Decompose the bid–ask spread into order-processing, inventory, and adverse-selection components.
- State Glosten–Milgrom’s insight that informed traders force a spread even with zero costs.
- Define liquidity operationally via spread, depth, and resiliency.
- Estimate an effective spread from trade-and-quote data (Roll’s model).
Key Vocabulary
- Price formation
- The process by which order flow and information are incorporated into the quoted mid-price over time.
- Bid–ask spread
- The gap \(P_a-P_b\); the round-trip cost of immediacy and the market maker’s gross compensation.
- Order-processing cost
- The fixed operational cost (fees, technology) of providing quotes, a component of the spread.
- Inventory cost
- Compensation a market maker demands for holding an unwanted position away from its target.
- Adverse selection
- The risk of trading against a better-informed counterparty; a component of the spread even absent costs.
- Liquidity
- The ease of trading size quickly at low cost; measured by spread (tightness), depth, and resiliency.
- Effective spread
- \(2|P_{\text{trade}}-\text{mid}|\); the realized round-trip cost, often below the quoted spread.
Intuition & Motivation
Price formation
The mid-price evolves as a running estimate of value updated by order flow. Signed trades (buys positive, sells negative) predict short-horizon returns; a stylized model writes the efficient price as a random walk plus flow information,
where \(x_t\) is the signed trade size and \(\lambda\) (Kyle’s lambda) is the price impact per unit of net order flow. The permanent part of impact is exactly the market learning from informed flow; the transient part is inventory pressure that later mean-reverts.
Decomposing the spread
The quoted half-spread compensates the liquidity provider for three distinct things:
| Component | What it pays for | Behavior |
|---|---|---|
| Order-processing | Fixed costs of quoting (tech, fees) | Roughly constant per trade |
| Inventory | Risk of holding an unwanted position | Grows with volatility & position |
| Adverse selection | Trading vs the informed | Grows with information asymmetry |
Liquidity: three dimensions
Liquidity is not one number. It has:
- Tightness - the spread; how much you pay to trade a small amount immediately.
- Depth - the size available near the touch; how much you can trade before moving the price.
- Resiliency - how fast the book refills after a trade consumes it.
A market can be tight but shallow (small spread, little size) or deep but wide. Execution algorithms must read all three.
Estimating the effective spread (Roll’s model)
Even without quote data, Roll (1984) showed that bid–ask bounce induces negative serial covariance in trade-price changes. If the efficient price is a random walk and trades bounce \(\pm s/2\) around it, then
Interactive: decompose an effective spread
- Thinking the spread is an exchange fee: it is the liquidity provider’s compensation for costs, inventory, and adverse selection.
- Ignoring adverse selection: even a costless, risk-neutral market maker must quote a spread (Glosten–Milgrom).
- Reducing liquidity to the spread alone: depth and resiliency matter as much for anything larger than a tiny order.
- Applying Roll’s formula when autocovariance is positive: informational trends violate its pure-bounce assumption.
- Report the effective spread (relative to the mid at trade time), not just the quoted spread - it captures price improvement.
- Watch how the spread widens with volatility and around news: the adverse-selection and inventory components spike.
- Estimate \(\lambda\) (Kyle’s lambda) from signed flow to size your orders relative to the market’s information sensitivity.
- Assess liquidity in all three dimensions before choosing between aggressive and passive execution.
Knowledge Check
Practical Exercise
(a) Explain, using Glosten–Milgrom, why a market maker widens quotes when it suspects more informed flow (e.g. around an earnings release). (b) Trades on a name give \(\text{Cov}(\Delta p_t,\Delta p_{t-1})=-0.16\) cents\(^2\); estimate the implied spread. (c) Two venues both quote a \(1\)-cent spread, but one shows \(50\) shares at the touch and the other \(5{,}000\). Which is more liquid for a \(2{,}000\)-share order and why?
(a) Around a news event a larger fraction of incoming orders are informed, so conditioning on a buy (or sell) shifts \(\E[V\mid\text{trade}]\) more strongly. To avoid systematically buying from those who know \(V\) is high (and selling to those who know it is low), the maker widens the spread - raising the adverse-selection component - until quoting is again break-even.
(b) Roll: \(s=2\sqrt{0.16}=2\times0.4=0.8\) cent implied spread.
(c) For a \(2{,}000\)-share order the second venue is far more liquid despite the identical quoted spread: with only \(50\) shares at the touch the first venue forces the order to walk deep into the book (large slippage), whereas \(5{,}000\) shares of depth lets the full order fill near the touch. Liquidity is spread and depth (and resiliency), not spread alone.
Lesson Summary
Formula Sheet Additions
- Did I treat the spread as compensation, not a fee?
- Did I include adverse selection, not just costs and inventory?
- Did I check Roll’s covariance is negative before applying it?
- Did I assess depth and resiliency, not just the quoted spread?
Retrieval Practice
Close the lesson and answer from memory before checking. This is deliberate, effortful recall - the single highest-yield study action.
A: Order-processing cost, inventory cost, and adverse-selection cost.
A: Because a buy signals the asset is more likely undervalued, \(\E[V\mid buy]\gt \E[V\mid sell]\); competitive quotes must straddle the expectation, giving a positive spread (Glosten–Milgrom).
A: \(s=2\sqrt{-\text{Cov}(\Delta p_t,\Delta p_{t-1})}\); it assumes price changes come from bid–ask bounce around a random-walk efficient price (needs negative autocovariance).
Flashcards
Click to flip. These feed the site-wide spaced-repetition queue.
Completion Checklist
- I can explain the core ideas in my own words
- I worked the derivations/examples by hand
- I completed the interactive workbench(es)
- I passed the knowledge check
Source References
This lesson synthesizes and paraphrases concepts from the sources below. No copyrighted text, problem sets, or solutions are reproduced. Return to the originals for full depth.
- Algorithmic and High-Frequency Trading (Cartea, Jaimungal & Penalva, 2015) current - Ch. 2–4 - Ch. 2–4: price formation, spread decomposition, Glosten–Milgrom and Kyle models of informed trading.
- Time Series Analysis (James Hamilton, 1994) foundational - Ch. 3–4 - Ch. 3–4: autocovariance structure used in Roll’s effective-spread estimator.
- Stochastic Calculus for Finance II (Steven Shreve, 2004) foundational - Ch. 2–3 - Ch. 2–3: martingale/random-walk models of the efficient price.