5.3.6
Intuition Ek-sentence mein baatHistorical volatility batata hai ki stock past mein actually kitna move kiya , jabki implied volatility batata hai ki options market ko lagta hai future mein kitna move hoga . Ek rear-view mirror mein dekhta hai; doosra market ka forecast hai.
Volatility ek option ki price ka sabse important input hai (Black–Scholes ke zariye). Lekin "volatility" ambiguous hai:
JO actually stock ke saath hua — measurable, backward-looking → Historical Volatility (HV) .MARKET kya expect karti hai — live option prices se infer kiya gaya, forward-looking → Implied Volatility (IV) .
Intuition Do numbers kyun matter karte hainEk option ek future movement par bet hai. Uski price expected future volatility reflect karni chahiye, past volatility nahi. HV "sahi" data hai; IV "priced-in" expectation hai. Dono ke beech ka gap wahan hai jahan traders paisa banate hain.
HOW hum ise banate hain, step by step:
Step 1 — Daily log returns. Logs kyun? Kyunki returns multiplicatively compound hote hain, aur logs multiplication ko addition mein badal dete hain (isliye ye time ke saath additive hain aur roughly normal hain).
r t = ln ( S t − 1 S t )
Step 2 — Un returns ki standard deviation. Volatility = returns ka spread, isliye hum measure karte hain ki returns apne mean se kitna door scatter karte hain.
σ daily = N − 1 1 ∑ t = 1 N ( r t − r ˉ ) 2 N − 1 r ˉ
Step 3 — Annualize karo. 252 time ke saath add hota hai, isliye variance T T σ annual = σ daily × 252
Definition Implied volatilityσ Black–Scholes formula mein plug karni padti hai taaki uski theoretical price option ki actual market price ke barabar ho jaye.
HOW: Black–Scholes price ko function C = f ( S , K , T , r , σ ) sivaay σ invert karte ho: wo σ f ( … , σ ) = market price
Market Price = f ( S , K , T , r , σ I V ) ⟹ solve for σ I V
Formula ko seedha rearrange kyun nahi kar sakte? Kyunki σ N ( d 1 ) , N ( d 2 ) koi closed-form inverse nahi hai. Hum numerically solve karte hain (Newton–Raphson, vega ∂ C / ∂ σ
Intuition IV ek fear/demand gauge haiZyada option demand → traders option prices bid up karte hain → zyada price justify karne ke liye, plug-in σ IV badhti hai . Isliye IV earnings, elections, Fed meetings se pehle spike karta hai — chahe stock abhi tak move na hua ho. Isliye IV ko market ka "fear gauge" kaha jata hai (VIX exactly yahi hai S&P 500 ke liye).
Historical (HV)
Implied (IV)
Direction
Backward-looking
Forward-looking
Source
Actual past prices
Current option prices
Method
Log returns ki Std dev
Black–Scholes invert karo
Meaning
Kya hua tha
Market kya expect karti hai
Uses
Baseline / reality check
Pricing, trades timing karna
Intuition Trade logic (80/20 core)
IV ≫ HV: options expensive lagte hain relative to stock actually kitna move karta hai → premium selling ki taraf jhukao (straddles/spreads becho).IV ≪ HV: options saste lagte hain → options buying ki taraf jhukao.Sabse valuable ek habit: har option trade se pehle IV ko HV se compare karo.
Worked example Example 1 — HV compute karoEk stock ke 5 din ke daily log returns: 0.01 , − 0.02 , 0.015 , − 0.005 , 0.02 Step 1: mean r ˉ = ( 0.01 − 0.02 + 0.015 − 0.005 + 0.02 ) /5 = 0.004 Kyun? Spread measure karne ke liye centre chahiye.
Step 2: mean se deviations hain 0.006 , − 0.024 , 0.011 , − 0.009 , 0.016 ( 0.006 ) 2 + ( − 0.024 ) 2 + ( 0.011 ) 2 + ( − 0.009 ) 2 + ( 0.016 ) 2 = 0.000036 + 0.000576 + 0.000121 + 0.000081 + 0.000256 = 0.00107 Step 3: σ daily = 0.00107/4 = 0.0002675 = 0.01636 /4 N − 1 = 4 Step 4: annualize = 0.01636 × 252 = 0.01636 × 15.87 = 0.2597 ≈ 26.0% 252
Worked example Example 2 — IV vs HV signal padhoStock XYZ: HV = 20%, lekin ek at-the-money option ka IV = 35%.
Reasoning: Market zyada future movement price kar rahi hai jitna stock historically dikhata hai (agle hafte earnings?). Options relatively expensive hain.
Action bias: premium-selling strategies favour karo — agar tumhara maanna hai ki actual move 20% ke qareeb hoga. Ye step kyun? Tum inflated premium collect karoge aur fayda uthaoge agar realized volatility priced-in 35% se neeche aaye.
Worked example Example 3 — Time scalingDaily σ = 1.5% σ 4 = 1.5% × 4 = 1.5% × 2 = 3.0% × 4 T T
Common mistake "Daily vol ko annualize karne ke liye, 252 se multiply karo."Kyun sahi lagta hai: 252 trading days hote hain, aur count se scale karna natural lagta hai.
Fix: Volatility 252 Variance time ke saath scale karta hai; std dev time
Common mistake "High IV matlab stock definitely upar jayega (ya crash karega)."Kyun sahi lagta hai: IV scary times mein spike karta hai, isliye ye directional lagta hai.
Fix: IV expected movement ka magnitude measure karta hai, direction nahi . High IV ek bada move expect karta hai — upar ya neeche.
Common mistake "Agar IV > HV, toh hamesha options becho."Kyun sahi lagta hai: option 'overpriced' lagta hai reality ke comparison mein.
Fix: IV > HV sahi bhi ho sakta hai — ek real catalyst ise justify kar sakta hai. Tum tabhi jeetoge jab realized future volatility IV se neeche aaye. Ye probabilistic edge hai, guarantee nahi.
Recall Feynman: 12-saal ke bachche ko samjhaoEk bouncy ball imagine karo. Historical volatility hai ki wo kitna bounce karta raha jab tumne usse dekha — tumne ise measure kiya. Implied volatility hai ki har koi bet karta hai ki ye aage kitna bounce karega, based on kitna wo bouncing game khelne ke liye pay kar rahe hain. Agar log bahut pay kar rahe hain (high IV) lekin ball gently bounce karti rahi hai (low HV), toh game overpriced hai — shayad tum tickets bechne wale banana chahte ho khareedne wale ki bajaye.
Mnemonic Yaad rakho kaun sa kaun sa hai"HISTORY is HAppened, IMPLIED is IMagined."
Aur scaling ke liye: "vol takes the square-ROOT of the ROUTE (time)."
Historical volatility kya measure karta hai? Past log returns ki annualized standard deviation — stock actually kitna move kiya.
Implied volatility kya measure karta hai? Market ki expected future volatility, option ki market price se Black–Scholes ko invert karke infer ki gayi.
√252 se annualize kyun karte hain, 252 se kyun nahi? Variance time ke saath scale karta hai, isliye standard deviation time ke square root ke saath scale karta hai; ~252 trading days/year.
IV algebra se kyun nahi nikaali ja sakti? σ Black–Scholes ke N(d₁), N(d₂) normal terms ke andar baitha hai; koi closed-form inverse nahi, isliye numerically solve karte hain.
Is framework mein options 'expensive' kab hote hain? Jab IV ≫ HV — market stock historically jitna dikhata hai usse zyada movement price karti hai.
Kya high IV direction predict karta hai? Nahi — ye movement ka magnitude predict karta hai (upar ya neeche), direction nahi.
HV daily std dev ka formula? σ_daily = √( Σ(rₜ − r̄)² / (N−1) ), rₜ = ln(Sₜ/Sₜ₋₁).
Agar daily σ = 1.5% ho toh expected 4-day move? 1.5% × √4 = 3.0%.
VIX conceptually kya hai? S&P 500 ki market-implied 30-day volatility — ek broad IV/'fear' gauge.
Black-Scholes Model — wo engine jisse IV invert ki jaati haiVega — wo Greek jo volatility ke sensitivity ko measure karta hai (IV solve karne mein slope use hoti hai)The Greeks — parent chapterVIX Index — market-wide implied volatilityLog Returns and Random Walks — kyun volatility √t ke saath scale karta haiStraddles and Volatility Trading — IV vs HV gap kaise trade hota hai
Volatility - key BS input
