Understand volatility skew and smile — Hinglish notes

5.3.7 · Stock-Market › The Greeks

WHAT hai yeh?

WHY exist karta hai? (First principles)

Black-Scholes assume karta hai ki asset returns lognormal hote hain → constant $σ$ , koi jumps nahi, symmetric bell curve. Agar yeh literally sach hota, toh IV saare strikes ke across ek flat horizontal line hoti.

Lekin real markets teen tarikon se is baat ko violate karte hain:

Fat tails — extreme moves normal distribution ke prediction se zyada baar hote hain. Options jo sirf tails mein payoff dete hain (deep OTM) isliye BS ke comparison mein zyada valuable hote hain → us higher price ko match karne ke liye zyada $σ$ chahiye → IV wings mein badhti hai → smile.
Stock returns ka negative skew — stocks rally karne se zyada tezi se crash karte hain. Toh left tail right se moti hoti hai. Traders downside protection ke liye (OTM puts) zyada pay karte hain → un low strikes ka IV inflated ho jaata hai → skew/smirk.
Crash fear / leverage effect — 1987 ke baad, dealers permanently downside par disaster insurance price karte hain. Girti hui prices leverage badhati hain → volatility badhti hai, ek built-in correlation create karta hai.

HOW padhein — moneyness

Hum moneyness ke against plot karte hain, raw strike ke against nahi, taaki saare assets fairly compare ho sakein.

Figure — Understand volatility skew and smile

Derive karna kyun higher IV ⇒ higher price (taaki hum invert kar sakein)

Vega ka derivation sketch: Shuru karo $c = S N (d_{1}) - K e^{- r T} N (d_{2}), d_{1} = \frac{l n ( S / K ) + ( r + \frac{1}{2} σ ^{2} ) T}{σ T}, d_{2} = d_{1} - σ T .$ $σ$ ke w.r.t. differentiate karo. Tricky cross-terms cancel ho jaate hain kyunki identity ki wajah se $S ϕ (d_{1}) = K e^{- r T} ϕ (d_{2}),$ sirf clean term $S ϕ (d_{1}) T$ bacha rehta hai. (Yeh identity hi Greeks ko itna neat banati hai.)

Worked Examples

Common Mistakes (Steel-manned)

Recall Feynman: 12-saal ke bache ko explain karo

Socho ek weather app jo guess karti hai ki kal kitna wild hoga. Black-Scholes ek lazy app hai jo kehti hai "har tarah ka din equally wild hai." Lekin log jaante hain ki storms (crashes) sunny days se zyada takleef dete hain, aur woh ek storm se bachne ke liye extra pay karenge. Toh ek bure crash-day ki insurance lazy app ki expectation se zyada cost karti hai. Jab hum plot karte hain ki har "din" kitna cost karta hai, toh storm side upar stick karti hai — woh skew hai. Agar dono super-hot aur super-cold din logon ko dartein hain, toh dono sides upar stick karti hain jaise ek smile. Bumpy curve sirf logon ka humein batana hai ki unhe kaunsi disasters se sabse zyada darr lagta hai.

Flashcards

Implied volatility kya hai?

Woh volatility

σ

jo Black-Scholes price ko option ki observed market price ke barabar kar de.

IV ko strike/moneyness ke against plot karne par BS ki tarah flat kyun nahi hoti?

Kyunki real returns mein fat tails aur skew hote hain, toh alag strikes par options lognormal prediction se zyada/kam valuable hote hain, prices match karne ke liye alag

σ

chahiye.

Volatility skew aur smile mein kya fark hai?

Skew = asymmetric downward slant (low strikes higher IV); Smile = symmetric U-shape (dono tails higher). Equities skew karti hain, FX smiles karta hai.

Equity skew kyun slant karti hai jisse OTM puts ka IV zyada hota hai?

Crash fear / leverage effect / downside insurance ki demand left tail ko mota banati hai, toh puts bid up ho jaati hain → higher IV.

Har option price ke liye unique IV ki guarantee kya karta hai?

Positive vega (

\partial c / \partial σ = S ϕ (d_{1}) T > 0

), toh price

σ

mein strictly increasing hai.

Log-moneyness ka formula?

m = ln (K / F)

, jahaan

m = 0

ATM,

m < 0

OTM puts,

m > 0

OTM calls.

Kya ek high-IV OTM put automatically overpriced hai?

Nahi. High IV market ka probability/crash view reflect karta hai; richness apne realized vol ke forecast ke against judge ki jaati hai, ATM ke against nahi.

Smile ki shape physically kya represent karti hai?

Underlying ki market ki implied probability distribution — bumps dikhate hain ki market kaunse price outcomes ko lognormal se zyada likely/valuable samajhti hai.

Smile hone ke bawajood Black-Scholes kyun ab bhi use hota hai?

Yeh ek quoting/translation language hai (IV↔price per strike), true distribution ka claim nahi.

Woh vega identity kaun si hai jo uski derivation simplify karti hai?

S ϕ (d_{1}) = K e^{- r T} ϕ (d_{2})

, jisse cross-terms cancel ho jaate hain aur

ν = S ϕ (d_{1}) T

bach jaata hai.

Connections

Implied Volatility
Black-Scholes Model
Vega
The Greeks
Lognormal Distribution and Fat Tails
Leverage Effect and Crash Risk
Volatility Surface and Term Structure
Put-Call Parity