Learn about adverse selection — Hinglish notes

6.3.4 · Stock-Market › Market Microstructure

Adverse selection KYUN hoti hai?

WHY yeh unfair lagta hai (aur hai bhi): Trade tabhi hoti hai jab informed trader choose karta hai aapke quote ko hit karne ke liye. Toh jab bhi aapka quote "galat" hota hai, ek informed trader wahin khada hota hai usse exploit karne ke liye. Aap systematically select against ho rahe ho — isliye adverse selection.

MM iske baare mein KYLA karti hai: Woh sharks ko identify nahi kar sakti, toh woh har order ko thoda informed maankar chalti hai. Kisi ka kharidna khud mein buri khabar hai (shayad value zyada hai); kisi ka bechna achchi khabar hai (shayad value kam hai). Woh apna belief update karti hai aur accordingly price set karti hai.

First principles se spread KAISE derive karein (Glosten–Milgrom)

Maano asset ki sahi value ek random variable $V$ hai jiske sirf do outcomes hain:

$V = V_{H} (high) with prob \frac{1}{2}, V = V_{L} (low) with prob \frac{1}{2}$

Prior expected value: $μ = \frac{1}{2} V_{H} + \frac{1}{2} V_{L}$ .

Do tarah ke traders aate hain, ek-ek karke:

Probability $α$ ke saath trader informed hai — $V$ exactly jaanta hai.
Probability $1 - α$ ke saath trader noise hai — 50/50, randomly khareedta ya bechta hai.

Informed behaviour (KYUN): Agar woh jaante hain $V = V_{H}$ , asset mid se zyada worth hai — woh khareed-te hain. Agar $V = V_{L}$ , woh bechte hain. Woh kabhi "galat" taraf trade nahi karte.

Step 1 — BUY aane ki probability kya hai?

Yeh step KYUN? MM ka ask price uski expected value equal honi chahiye given ki buy hoti hai. Toh humein $P (buy)$ aur $P (V_{H} ∣ buy)$ chahiye.

$P (buy ∣ V_{H}) = α \cdot 1 + (1 - α) \cdot \frac{1}{2}$ Kyun: agar $V = V_{H}$ , informed hamesha khareedta hai; noise aadha time khareedta hai. $P (buy ∣ V_{L}) = α \cdot 0 + (1 - α) \cdot \frac{1}{2}$ Kyun: agar $V = V_{L}$ , informed kabhi nahi khareedta; sirf noise (aadha) khareedta hai.

$P (buy) = \frac{1}{2} P (buy ∣ V_{H}) + \frac{1}{2} P (buy ∣ V_{L}) = \frac{1}{2}$ Kyun: symmetry ki wajah se buy utni hi likely hai jitni ki overall sell.

Step 2 — Buy ke baad Bayes update

Yeh step KYUN? Buy evidence hai jo $V_{H}$ ki taraf jhukta hai. Ask woh fair value hai us evidence ko condition karke.

=\frac{\left(\tfrac{1+\alpha}{2}\right)\tfrac12}{\tfrac12}=\frac{1+\alpha}{2}$$ ### Step 3 — Ask price $$\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}$$ Perfect symmetry se (sell buri khabar hai): $$\boxed{\text{Bid}=\frac{1-\alpha}{2}V_H+\frac{1+\alpha}{2}V_L}$$ ### Step 4 — Spread > **Yeh step KYUN?** Subtract karo dekho ki adverse selection *kitni cost* karti hai. $$\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 ki trader ==informed== hai. > - $(V_H-V_L)$ = ==uncertainty== (possible values kitni door hain ek doosre se). > Spread exist karta hai **purely information asymmetry ki wajah se** — koi inventory cost nahi, koi fee nahi, koi processing cost nahi. Agar $\alpha=0$ (koi informed nahi), 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 $\mu=\tfrac12(102+98)=100$. *Kyun:* prior fair value. > - $\text{Ask}=\tfrac{1.2}{2}(102)+\tfrac{0.8}{2}(98)=61.2+39.2=\$100.40$. *Kyun:* buy belief ko $V_H$ ki taraf dhakelta hai. > - $\text{Bid}=\tfrac{0.8}{2}(102)+\tfrac{1.2}{2}(98)=40.8+58.8=\$99.60$. > - Spread $=0.2\times4=\$0.80$. ✔ $\alpha(V_H-V_L)$ se match karta hai. > [!example] Example 2 — zyada informed traders > Same values lekin $\alpha=0.5$. > - Spread $=0.5\times4=\$2.00$. *Kyun:* aadha flow toxic hai, toh MM ko khud ko do baar zyada protect karna padta hai. **Zyada sharks ⇒ wider spread.** > [!example] Example 3 — time ke saath price discovery > Ek buy observe karne ke baad, *new mid* Ask ban jaata hai? Nahi — new **expected value** hai $E[V\mid\text{buy}]=\text{Ask}=\$100.40$. *Yeh kyun important hai:* mid buys ke baad **upar drift** karta hai aur sells ke baad **neeche**. Order flow prices ko move karta hai — yeh drift market ka $V$ **seekhna** hai. Isliye lagatar buys price ko upar push karti hain chahe koi news headline na ho. --- ## Steel-manned mistakes > [!mistake] "Spread sirf MM ka profit margin hai." > **Kyun sahi lagta hai:** MMs noise traders se spread earn karti hain, toh yeh *margin jaisa dikhta* hai. > **Fix:** Adverse-selection component **profit nahi** hai — yeh informed traders ko expected losses ka *compensation* hai. Ek *informed* trade par MM ka net expected profit **negative hota hai aur exactly cancel karta hai** noise traders se hone wale gain ko. Pure Glosten–Milgrom mein MM break even karti hai (zero-profit / competitive quotes). > [!mistake] "Agar prices efficient hain, toh order flow price nahi hilana chahiye." > **Kyun sahi lagta hai:** efficient markets = price pehle se hi info reflect karti hai. > **Fix:** Efficiency *order flow ke zariye* *achieve* hoti hai. Har informed trade information **reveal** karti hai; MM ka Bayes update *hi* price ka efficient banana hai. Order flow mechanism hai, koi violation nahi. > [!mistake] "Adverse selection = inventory risk." > **Kyun sahi lagta hai:** dono spreads ko wide karte hain. > **Fix:** Yeh **alag** components hain. Inventory cost tab aati hai jab MM ek unwanted position hold karti hai; adverse selection trade ke *information content* se aati hai. Empirical spread $\approx$ order-processing + inventory + **adverse selection**. --- ## Feynman check > [!recall]- Ek 12-saal ke bachche ko samjhao (click to reveal) > Socho tum ek lemonade stall chalate ho aur tum cups bechte bhi ho aur wapas khareed bhi lo. Kuch bachche ek secret jaante hain — ki bahut garmi aane wali hai (toh lemonade ki value badh jayegi) — ya baarish aane wali hai (value ghategI). Agar ek "secret-jaannewala" bachcha tumse khareedta hai, toh shayad tumne price kam rakhi thi. Agar woh bechta hai, toh tumne shayad price zyada rakhi thi. Kyunki tum nahi pehchan sakte ki kaun secret jaanta hai, tum apna **buy price thoda neeche aur sell price thoda upar** set karte ho khud ko bachane ke liye. Yahi gap isliye hai kyunki kuch customers tumse zyada jaante hain. Yahi adverse selection hai! --- ## Active recall > [!recall] Quick self-test > 1. Kaunsi cheez ek trade ko market maker ke liye "adverse" banati hai? > 2. $P(\text{buy}\mid V_H)$ scratch se derive karo. > 3. Spread $\alpha(V_H-V_L)$ kyun hai aur sirf $\alpha$ kyun nahi? > 4. Buy ke baad mid price move karta hai? Kaunsi direction mein aur kyun? #flashcards/stock-market Market mein adverse selection kya hai? ::: Yeh risk ki aapke quote ke doosri taraf wala trader zyada informed hai, toh market maker systematically informed traders se harti hai aur spread widen karna padta hai. Glosten–Milgrom mein $\alpha$ kya represent karta hai? ::: Probability ki aane wala trader informed hai (sahi value $V$ jaanta hai). Adverse-selection spread formula kya hai? ::: $\text{Spread}=\alpha(V_H-V_L)$ — informed fraction times value uncertainty. Buy order market maker ki expected value kyun badhata hai? ::: Buy evidence hai ki value high hai (informed traders sirf tabhi khareed-te hain jab $V=V_H$), toh Bayes $E[V]$ ko upar update karta hai — mid upar drift karta hai. Agar $\alpha=0$ ho, toh spread kya hoga? ::: Zero — koi informed trader nahi toh koi adverse selection nahi. Kya adverse-selection spread market maker ka profit hai? ::: Nahi — yeh informed traders ko expected losses ka compensation hai; competitive zero-profit case mein MM break even karti hai. $P(\text{buy}\mid V_H)$ kiske barabar hai? ::: $\alpha + (1-\alpha)\tfrac12 = \tfrac{1+\alpha}{2}$: informed hamesha khareedta hai plus aadhe noise traders. Adverse selection aur inventory risk mein kya fark hai? ::: Adverse selection = trade ke information content ki cost; inventory risk = unwanted position hold karne ki cost. Dono spreads wide karte hain lekin alag hain. Jab $\alpha$ badhta hai toh spread ka kya hota hai? ::: Yeh linearly wide hota hai — zyada informed ("toxic") flow MM ko khud ko zyada protect karne par majboor karta hai. --- ## Connections - [[Bid-Ask Spread]] — adverse selection uske teen components mein se ek hai. - [[Glosten-Milgrom Model]] — yahan derive kiya gaya formal sequential-trade model. - [[Kyle Model]] — continuous version jisme ek strategic informed trader aur market depth $\lambda$ hai. - [[Bayes Theorem]] — quote-setting ke peeche ka update engine. - [[Price Discovery]] — order flow prices ko efficient kaise banata hai. - [[Information Asymmetry]] — general economic condition (Akerlof ka "lemons"). - [[Market Maker Inventory Risk]] — doosra main spread component. > [!mnemonic] Yaad rakho > **"SHARKS Set Spreads"** — **S**ome traders **H**old **A**dvance **R**eal **K**nowledge, **S**o the MM widens. Aur size = *kitne sharks* ($\alpha$) × *paani kitna gehra hai* ($V_H-V_L$). ## 🖼️ 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 ```