Understand bid-ask spread components — Stock-Market notes

WHAT is the bid-ask spread?

The market maker (MM) quotes both sides simultaneously. He earns the spread by buying low (at bid) and selling high (at ask) many times a day. WHY does a positive spread survive competition? Because the MM faces genuine costs; if the spread were zero he'd go bankrupt.

WHY does the spread exist? The three components

The spread is the sum of three compensations the MM demands. This is the Stoll / Glosten–Harris decomposition:

$S = \underbrace{2\,C_o}_{\text{order processing}} + \underbrace{2\,C_i}_{\text{inventory holding}} + \underbrace{2\,C_a}_{\text{adverse selection}}$

We write each as a half-spread around the mid, so the full spread is twice each half.

1. Order-processing cost ( $C_o$ )

Fixed, mechanical costs: exchange fees, clearing, technology, the MM's labor/profit margin. HOW it behaves: roughly constant per trade, so it makes spreads wider for cheap/illiquid stocks (fixed cost spread over small price).

2. Inventory-holding cost ( $C_i$ )

When you sell to the MM, he now holds a long position he didn't want. To hold it he bears price risk and ties up capital. HOW it behaves: he skews his quotes to unload inventory — if he's too long, he lowers both bid and ask to attract buyers. Grows with volatility $\sigma$ and holding time.

3. Adverse-selection cost ( $C_a$ )

This is the subtle one. Some traders are informed (they know news the MM doesn't). If an informed trader buys, price is about to rise — the MM loses on that fill. He can't tell informed from noise traders in advance, so he widens the spread to break even on average. HOW it behaves: grows with the probability of informed trading and with information asymmetry.

Figure — Understand bid-ask spread components

DERIVATION: the adverse-selection spread from first principles

Let's build the spread instead of memorizing it. This is a stripped Glosten–Milgrom model.

Deriving the ask. The MM must set $P_a$ = expected value of $V$ given that a buy order arrived. WHY? Because a rational competitive MM earns zero expected profit; the ask must equal the conditional expectation, else competitors undercut or he loses money.

$P_a = E[V \mid \text{buy}] = v_H\,P(v_H\mid\text{buy}) + v_L\,P(v_L\mid\text{buy}).$

Compute $P(\text{buy}\mid v_H)$ . If value is High, informed traders ( $\alpha$ ) all buy; noise ( $1-\alpha$ ) buy with prob $1/2$ : $P(\text{buy}\mid v_H) = \alpha + (1-\alpha)\tfrac12 = \tfrac{1+\alpha}{2}.$ If value is Low, informed all sell, noise still buy half the time: $P(\text{buy}\mid v_L) = (1-\alpha)\tfrac12 = \tfrac{1-\alpha}{2}.$

Total prob of a buy $= \tfrac12\cdot\tfrac{1+\alpha}{2} + \tfrac12\cdot\tfrac{1-\alpha}{2} = \tfrac12$ (symmetric, good sanity check).

Bayes: $P(v_H\mid\text{buy}) = \frac{\tfrac12\cdot\frac{1+\alpha}{2}}{\tfrac12} = \frac{1+\alpha}{2}.$

So $\boxed{P_a = v_H\frac{1+\alpha}{2} + v_L\frac{1-\alpha}{2} = m + \frac{\alpha}{2}(v_H-v_L).}$

By symmetry (a sell order): $\boxed{P_b = m - \frac{\alpha}{2}(v_H-v_L).}$

The spread: $S = P_a - P_b = \alpha\,(v_H - v_L).$

Worked examples

Forecast-then-Verify

Common mistakes (Steel-manned)

Flashcards

What are the three components of the bid-ask spread?

Order-processing cost, inventory-holding cost, and adverse-selection cost.

Define the quoted bid-ask spread.

S = P_a - P_b

, the ask minus the bid.

What is the mid-price and what does it represent?

m=(P_a+P_b)/2

; the market's prior best estimate of fair value.

In the Glosten–Milgrom model, what does the spread equal?

S=\alpha(v_H-v_L)

: fraction of informed traders times the value uncertainty.

Why does a competitive market maker set the ask to

E[V\mid \text{buy}]

Zero-profit condition — competition forces the quote to equal the conditional expected value; otherwise he's undercut or loses money.

Which component makes spreads widen right before an earnings release?

Adverse selection — higher chance/impact of informed trading.

Why multiply each half-spread component by 2 to get the spread?

Each cost is charged on both sides of the mid (bid and ask), so the full spread is twice each half-spread.

Effective spread formula?

2\times|P_{\text{trade}}-m|

; the actual round-trip cost, can be smaller than quoted if you get price improvement.

Adverse-selection cost rises with which two quantities?

Probability of informed trading

\alpha

and information asymmetry / value uncertainty

(v_H-v_L)

After a buy order arrives, which direction does expected true value move?

Upward — toward the ask; the order itself is informative (price impact).

Recall Feynman: explain to a 12-year-old

Imagine you run a lemonade stand where you both buy used cups and sell fresh lemonade. You sell for 5 rupees, buy back for 4 rupees — that 1-rupee gap is your spread. Why keep a gap? (1) Cups, sugar, and your time cost money (order processing). (2) If you buy a cup you don't need, you might get stuck with it and it could go bad (inventory). (3) Sometimes a kid buys all your lemonade because he knows it's about to get super hot and everyone will want it — you sold too cheap and lost out (adverse selection). To not go broke against those clever kids, you keep the gap a little wider. That gap is the bid-ask spread!

Connections

Market Microstructure — parent chapter.
Market Makers and Liquidity Provision — who quotes the spread.
Glosten-Milgrom Model — full information-based derivation.
Order Flow and Price Impact — how trades move the mid.
Adverse Selection and Information Asymmetry — the deepest spread component.
Effective vs Quoted vs Realized Spread — measurement variants.
Volatility and Liquidity — inventory cost driver.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, jab tum kisi stock ko turant buy karke turant sell karo, toh thoda paisa loss ho jata hai — kyunki tum ask par khareedte ho aur bid par bechte ho. Yeh gap hi bid-ask spread hai. Yeh koi random number nahi hai; market maker (MM) yeh spread teen real costs cover karne ke liye rakhta hai. Isko yaad rakho OIA: Order-processing (exchange fees, technology, MM ka margin), Inventory (jo position MM ko majboori mein hold karni padti hai uska risk), aur Adverse-selection (jab saamne wala trader ko news pehle se pata ho aur MM ko na pata ho).

Sabse interesting part adverse selection hai. Socho true value ya toh High ( $v_H$ ) hai ya Low ( $v_L$ ), 50-50 chance. Kuch traders informed hain (fraction $\alpha$ ) — unhe pata hai; baaki noise traders random buy/sell karte hain. Ab MM zero-profit ke liye ask ko $E[V\mid\text{buy}]$ ke barabar set karta hai. Bayes lagao toh nikalta hai $P_a = m + \frac{\alpha}{2}(v_H-v_L)$ aur $P_b = m - \frac{\alpha}{2}(v_H-v_L)$ , matlab spread $=\alpha(v_H-v_L)$ . Simple baat: jitne zyada informed log, aur jitni zyada value uncertainty, utna wide spread. Isiliye earnings ya bade news se pehle spread achanak fat jaata hai — MM darta hai ki koi "smart" banda usse loot na le.

Yeh matter kyun karta hai? Kyunki spread tumhara hidden trading cost hai — commission se alag. Illiquid, chhote stocks mein order-processing cost fixed hone ki wajah se spread bada dikhta hai. Aur jab tum "effective spread" ( $2\times|P_{trade}-m|$ ) dekhte ho, kabhi kabhi tumhe price improvement mil jata hai, matlab actual cost quoted se kam. Toh trade karne se pehle spread samajhna = apna real cost samajhna. Yeh 80/20 rule ka core insight hai: spread ka bada hissa often adverse selection hota hai, sirf ek chhota fee.

Test yourself — Market Microstructure