Learn about adverse selection

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WHY does adverse selection exist?

WHY it feels unfair (and is): Trade only happens when the informed trader chooses to hit your quote. So every time your quote is "wrong," an informed trader is standing there to exploit it. You are systematically selected against — hence adverse selection.

WHAT the MM does about it: She can't identify the sharks, so she treats every order as partly informed. The very act of someone buying is bad news (maybe value is higher); someone selling is good news (maybe value is lower). She updates her belief and prices accordingly.

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HOW to derive the spread from first principles (Glosten–Milgrom)

Let the true value of the asset be a random variable $V$ with only two outcomes:

$V = V_H \text{ (high) with prob } \tfrac12, \qquad V = V_L \text{ (low) with prob } \tfrac12$

Prior expected value: $\mu = \tfrac12 V_H + \tfrac12 V_L$.

Two kinds of traders arrive, one at a time:

• With probability $\alpha$ the trader is informed — knows $V$ exactly.
• With probability $1-\alpha$ the trader is noise — buys or sells 50/50, randomly.

Informed behaviour (WHY): If they know $V=V_H$, the asset is worth more than the mid — they buy. If $V=V_L$, they sell. They never trade the "wrong" way.

Step 1 — What is the probability a BUY arrives?

Why this step? The MM's ask price must equal her expected value given a buy occurs. So we need $P(\text{buy})$ and $P(V_H \mid \text{buy})$.

$P(\text{buy}\mid V_H)=\alpha\cdot 1 + (1-\alpha)\cdot\tfrac12$ Why: if $V=V_H$, informed always buy; noise buys half the time. $P(\text{buy}\mid V_L)=\alpha\cdot 0 + (1-\alpha)\cdot\tfrac12$ Why: if $V=V_L$, informed never buy; only noise (half) buys.

$P(\text{buy}) = \tfrac12 P(\text{buy}\mid V_H)+\tfrac12 P(\text{buy}\mid V_L)= \tfrac12$ Why: by symmetry a buy is as likely as a sell overall.

Step 2 — Bayes update after a buy

Why this step? A buy is evidence leaning toward $V_H$. The ask is the fair value conditional on that evidence.

=\frac{\left(\tfrac{1+\alpha}{2}\right)\tfrac12}{\tfrac12}=\frac{1+\alpha}{2}$$ ### Step 3 — The ask price $$\text{Ask}=E[V\mid \text{buy}]=P(V_H\mid \text{buy})V_H+P(V_L\mid \text{buy})V_L$$ $$\boxed{\text{Ask}=\frac{1+\alpha}{2}V_H+\frac{1-\alpha}{2}V_L}$$ By perfect symmetry (a sell is bad news): $$\boxed{\text{Bid}=\frac{1-\alpha}{2}V_H+\frac{1+\alpha}{2}V_L}$$ ### Step 4 — The spread > **Why this step?** Subtract to see what adverse selection *costs*. $$\text{Spread}=\text{Ask}-\text{Bid}=\alpha\,(V_H-V_L)$$ > [!formula] Adverse-selection spread (Glosten–Milgrom) > $$\text{Spread}=\alpha\,(V_H-V_L)$$ > - $\alpha$ = probability the trader is ==informed==. > - $(V_H-V_L)$ = ==uncertainty== (how far apart the possible values are). > The spread exists **purely from information asymmetry** — no inventory cost, no fee, no processing cost. If $\alpha=0$ (nobody informed), spread $=0$. --- ![[6.3.04-Learn-about-adverse-selection.png]] --- ## Worked examples > [!example] Example 1 — basic spread > $V_H=\$102$, $V_L=\$98$, $\alpha=0.2$. > - Mid $\mu=\tfrac12(102+98)=100$. *Why:* prior fair value. > - $\text{Ask}=\tfrac{1.2}{2}(102)+\tfrac{0.8}{2}(98)=61.2+39.2=\$100.40$. *Why:* buy pushes belief toward $V_H$. > - $\text{Bid}=\tfrac{0.8}{2}(102)+\tfrac{1.2}{2}(98)=40.8+58.8=\$99.60$. > - Spread $=0.2\times4=\$0.80$. ✔ matches $\alpha(V_H-V_L)$. > [!example] Example 2 — more informed traders > Same values but $\alpha=0.5$. > - Spread $=0.5\times4=\$2.00$. *Why:* half the flow is toxic, so the MM must protect herself twice as much. **More sharks ⇒ wider spread.** > [!example] Example 3 — price discovery over time > After observing a buy, the *new mid* becomes the Ask? No — the new **expected value** is $E[V\mid\text{buy}]=\text{Ask}=\$100.40$. *Why this matters:* the mid **drifts up** after buys and **down** after sells. Order flow moves prices — this drift is the market **learning** $V$. This is why a string of buys pushes price up even with no news headline. --- ## Steel-manned mistakes > [!mistake] "The spread is just the MM's profit margin." > **Why it feels right:** MMs do earn the spread from noise traders, so it *looks* like margin. > **The fix:** The adverse-selection component is **not profit** — it's *compensation for expected losses* to informed traders. The MM's net expected profit on an *informed* trade is **negative and exactly cancels** the gain from noise traders. In pure Glosten–Milgrom the MM breaks even (zero-profit / competitive quotes). > [!mistake] "If prices are efficient, order flow shouldn't move price." > **Why it feels right:** efficient markets = price already reflects info. > **The fix:** Efficiency is *achieved through* order flow. Each informed trade **reveals** information; the MM's Bayes update *is* the price becoming efficient. Order flow is the mechanism, not a violation. > [!mistake] "Adverse selection = inventory risk." > **Why it feels right:** both widen spreads. > **The fix:** They are **distinct** components. Inventory cost comes from the MM holding an unwanted position; adverse selection comes from the *information content* of the trade. The empirical spread $\approx$ order-processing + inventory + **adverse selection**. --- ## Feynman check > [!recall]- Explain to a 12-year-old (click to reveal) > Imagine you run a lemonade stand and you sell cups AND buy cups back. Some kids know a secret — that it's about to get super hot (so lemonade will be worth more) — or that rain is coming (worth less). If a "secret-knowing" kid buys from you, it's probably because you priced it too cheap. If one sells to you, you probably priced too high. Since you can't tell who knows the secret, you set your **buy price a bit low and sell price a bit high** to protect yourself. That gap is because some customers know more than you. That's adverse selection! --- ## Active recall > [!recall] Quick self-test > 1. What makes a trade "adverse" for the market maker? > 2. Derive $P(\text{buy}\mid V_H)$ from scratch. > 3. Why is the spread $\alpha(V_H-V_L)$ and not $\alpha$ alone? > 4. Does the mid price move after a buy? Which direction and why? #flashcards/stock-market What is adverse selection in a market? ::: The risk that the trader on the other side of your quote is better informed, so the market maker systematically loses to informed traders and must widen the spread. In Glosten–Milgrom, what does $\alpha$ represent? ::: The probability that an arriving trader is informed (knows the true value $V$). What is the adverse-selection spread formula? ::: $\text{Spread}=\alpha(V_H-V_L)$ — informed fraction times value uncertainty. Why does a buy order raise the market maker's expected value? ::: A buy is evidence the value is high (informed traders only buy when $V=V_H$), so Bayes updates $E[V]$ upward — the mid drifts up. If $\alpha=0$, what is the spread? ::: Zero — with no informed traders there is no adverse selection. Is the adverse-selection spread the market maker's profit? ::: No — it is compensation for expected losses to informed traders; in the competitive zero-profit case the MM breaks even. $P(\text{buy}\mid V_H)$ equals what? ::: $\alpha + (1-\alpha)\tfrac12 = \tfrac{1+\alpha}{2}$: informed always buy plus half of noise traders. How does adverse selection differ from inventory risk? ::: Adverse selection = cost of the trade's information content; inventory risk = cost of holding an unwanted position. Both widen spreads but are distinct. What happens to the spread as $\alpha$ rises? ::: It widens linearly — more informed ("toxic") flow forces the MM to protect herself more. --- ## Connections - [[Bid-Ask Spread]] — adverse selection is one of its three components. - [[Glosten-Milgrom Model]] — the formal sequential-trade model derived here. - [[Kyle Model]] — continuous version with a strategic informed trader and market depth $\lambda$. - [[Bayes Theorem]] — the update engine behind quote-setting. - [[Price Discovery]] — how order flow makes prices efficient. - [[Information Asymmetry]] — the general economic condition (Akerlof's "lemons"). - [[Market Maker Inventory Risk]] — the other main spread component. > [!mnemonic] Remember it > **"SHARKS Set Spreads"** — **S**ome traders **H**old **A**dvance **R**eal **K**nowledge, **S**o the MM widens. And the size = *how many sharks* ($\alpha$) × *how deep the water* ($V_H-V_L$). ## 🖼️ Concept Map ```mermaid flowchart TD AS[Adverse selection] -->|caused by| INFO[Informed traders know more] INFO -->|only trade when| WRONG[MM quote is wrong] WRONG -->|MM systematically| LOSS[Loses to informed] LOSS -->|forces MM to| SPREAD[Widen bid-ask spread] SPREAD -->|so gains from| NOISE[Uninformed noise traders] NOISE -->|cover the| LOSS AS -->|modeled by| GM[Glosten-Milgrom model] GM -->|assumes value| V[V is V_H or V_L 50/50] GM -->|fraction alpha| INFO GM -->|treats a buy as| BADNEWS[Buy is bad news] BADNEWS -->|via| BAYES[Bayes update] BAYES -->|yields| ASK[Ask = E of V given buy] BAYES -->|yields| BID[Bid = E of V given sell] ASK -->|difference is| SPREAD BID -->|difference is| SPREAD ``` ## 🔊 Hinglish (regional understanding) > [!intuition]- Hinglish mein samjho > Dekho, adverse selection ka core idea simple hai. Market maker (MM) do prices lagata hai — buy price (bid) aur sell price (ask). Problem ye hai ki kuch traders ko asset ki asli value pehle se pata hoti hai (informed traders), aur baaki random log hote hain (noise traders). MM ko pata nahi kaun kaun hai. Informed trader tabhi buy karega jab price sasta hai, aur tabhi sell karega jab price mehnga hai — matlab woh hamesha MM ko loss mein daalta hai. Isko "adverse" (ulta, khilaf) selection kehte hain, kyunki jo bhi aapke saath trade karta hai woh aksar aapse zyada janta hai. > > Ab MM kya kare? Woh har order ko thoda "suspicious" maanta hai. Jab koi buy karta hai, MM sochta hai "shayad value high hai" — isliye woh ask price thoda upar rakhta hai. Jab koi sell karta hai, "shayad value low hai" — bid thoda neeche. Ye gap hi spread hai. Formula nikalta hai: $\text{Spread}=\alpha(V_H-V_L)$, jahan $\alpha$ = kitne informed log hain, aur $(V_H-V_L)$ = value mein kitni uncertainty hai. > > Important baat — ye spread MM ka profit nahi hai. Ye toh informed traders se hone waale expected loss ki bharpaai hai. Noise traders se jo kamaata hai woh sharks ko cover karne mein chala jaata hai. Aur jitne zyada informed ("toxic") traders, utna bada spread. Ye bhi samjho: har buy ke baad price thoda upar drift karta hai, har sell ke baad neeche — kyunki order flow se market naya information seekh raha hai. Yahi price discovery ka asli mechanism hai. Exam aur real trading dono mein ye concept bahut kaam aata hai. ![[audio/6.3.04-Learn-about-adverse-selection.mp3]]