Understand Delta and directional exposure — Hinglish notes

5.3.1 · Stock-Market › The Greeks

Delta kya hai?

Sign kyun? Calls tab value gain karti hain jab stock rise kare → positive slope. Puts tab value gain karti hain jab stock fall kare → negative slope.

Delta ko First Principles se derive karna

Chalte hain intuition build karte hain option ki value se.

Step 1: Option value stock price par depend karti hai

Ek option ki value $V (S, t, σ, r, K)$ kai variables par depend karti hai, lekin stock price $S$ sabse dominant driver hai.

Agar stock thodi si amount $Δ S$ move kare, to option value approximately itni change hoti hai: $Δ V \approx \frac{\partial V}{\partial S} \cdot Δ S$

Woh partial derivative $\frac{\partial V}{\partial S}$ hi delta hai.

Step 2: Partial derivative kyun?

Dusre variables (time $t$ , volatility $σ$ , interest rate $r$ , strike $K$ ) bhi $V$ ko affect karte hain. Hum yeh pooch rahe hain: baaki sab constant rakhe, jab sirf $S$ move kare to $V$ kaise change hoti hai?

Yahi exactly ek partial derivative measure karta hai.

Step 3: Delta as a hedge ratio

Black-Scholes se hum derive kar sakte hain: $Δ_{call} = N (d_{1})$ jahan $N (d_{1})$ cumulative normal distribution function hai jo evaluate ki gayi hai: $d_{1} = \frac{l n ( S / K ) + ( r + σ ^{2} /2 ) T}{σ T}$

Yeh formula kyun? Black-Scholes assume karta hai ki tum ek option ko $Δ$ shares of stock hold karke replicate kar sakte ho. Replication portfolio ki stock moves ke liye same sensitivity honi chahiye → woh sensitivity $N (d_{1})$ hai.

Puts ke liye: $Δ_{put} = N (d_{1}) - 1$ 1 kyun subtract karte hain? Put-call parity: $C - P = S - K e^{- r T}$ . Derivatives lete hain: $Δ_{C} - Δ_{P} = 1$ . Toh $Δ_{P} = Δ_{C} - 1$ .

Delta tumhe directional exposure ke baare mein kya batata hai

Yeh kyun matter karta hai?

Risk management: Tumhe apni exposure "stock equivalents" mein pata hoti hai
Hedging: Delta-neutral banne ke liye tumhe -300 shares se offset karna hoga
Expected P&L: Agar stock $1 m o v e k a r e, t u m e x p ec t k a r t e h o$ 300 banana/khona (first-order approximation)

Moneyness ke across Delta ka behavior

Moneyness	Call Delta	Put Delta	Intuition
Deep ITM	~1.0	~-1.0	Almost exactly stock jaisa behave karta hai
ATM	~0.50	~-0.50	ITM finish karne ka 50/50 coin flip
Deep OTM	~0.0	~0.0	Stock move barely worthless option ko affect karta hai

Yeh pattern kyun?

Deep ITM: Exercise hone ka almost certain hai → stock own/short karne jaisa behave karta hai
ATM: Maximum uncertainty → balanced sensitivity
Deep OTM: ITM finish karne ki tiny probability → small moves matter nahi karte

Worked Examples

Common Mistakes

Delta in portfolio construction

Multi-leg strategies

Kisi bhi options portfolio ke liye: $Δ_{portfolio} = \sum_{i = 1}^{n} Δ_{i} \times q_{i} \times m_{i}$

jahan:

$Δ_{i}$ = position $i$ ka delta
$q_{i}$ = quantity (long ke liye positive, short ke liye negative)
$m_{i}$ = multiplier (standard contracts ke liye 100)

Example: Iron Condor

Long 1 put, K=$95, Δ=-0.15
Short 1 put, K=$100, Δ=-0.30
Short 1 call, K=$110, Δ=0.30
Long 1 call, K=$115, Δ=0.15

Portfolio delta: $Δ = [(- 0.15) - (- 0.30) - (0.30) + (0.15)] \times 100 = 0$

Zero kyun? Iron condors delta-neutral banne ke liye design kiye jaate hain, theta decay aur low volatility se profit karte hain.

Delta decay aur time

Jaise expiration paas aata hai, delta zyada "digital" ho jaata hai:

ITM options: Delta → ±1.0
OTM options: Delta → 0
ATM options: Delta ±0.50 ke paas rehta hai lekin zyada sensitive ho jaata hai (higher gamma)

Kyun? Kam time matlab kam uncertainty. ITM options almost certainly exercise honge. OTM options almost certainly worthless expire honge.

Recall Ek 12-saal ke bacche ko explain karo

Socho tumhare paas ek magic ticket hai jo tumhe ek toy $10 mein kharidne deta hai, lekin sirf tab jab tum chahte ho. Toy ki price roz change hoti rehti hai.

Delta yeh poochhne jaisa hai: "Agar aaj toy ki price $1 badh jaaye, toh mera magic ticket kitna zyada valuable ho jaata hai?"

Agar tumhare ticket ka delta 0.50 hai, toh jab toy $10 se$ 11 ho jaaye, tumhara ticket 50 cents aur valuable ho jaata hai. Agar delta 0.80 hai, tumhara ticket 80 cents zyada valuable ho jaata hai.

Yeh change kyun hota hai? Jab toy $5 k ihaia u r t u mha r a t i c k e tt u mh e u sse$ 10 mein kharidne deta hai, tumhara ticket almost worthless hai (delta near 0). Jab toy $20 k ihai, t u mha r a t i c k e t s u p er v a l u ab l e haik y u nk i t u m$ 20 wali toy sirf $10 mein khareed sakte ho (delta near 1.0).

Delta tumhe batata hai tumhara magic ticket abhi actually toy own karne jaisa kitna behave kar raha hai.

Connections

Black-Scholes Model — Delta B-S mein replication argument se aata hai
Gamma — Delta ki khud ki rate of change
Delta-neutral trading — Market-neutral positions construct karne ke liye delta ka use
Portfolio Greeks — Multiple positions mein delta aggregate karna
Hedging strategies — Dusre risk factors isolate karne ke liye delta hedging
Option moneyness — ITM/OTM delta ko kaise affect karta hai
Implied volatility — $d_{1}$ mein $σ$ term ke through delta ko affect karta hai

#flashcards/stock-market

Options mein delta kya measure karta hai? :: Option price ki rate of change stock price ke respect mein. Yeh batata hai ki underlying stock mein $1 move hone par tumhara option value kitna change hoga.

Call options ke liye delta range kya hai?

0 se +1.0 tak (ya percentage ke roop mein express karte waqt 0 se 100). Deep OTM calls ka delta 0 ke paas hota hai, deep ITM calls ka delta 1.0 ke paas.

Put options ke liye delta range kya hai?

0 se -1.0 tak (ya -100 se 0 tak). Negative sign indicate karta hai ki puts tab value gain karti hain jab stock price gire.

Position delta kaise calculate karte hain?

Position Delta = Option Delta × Number of Contracts × Contract Size (standard equity options ke liye 100).

Agar tumhare paas delta 0.65 wale 4 call contracts hain, toh tumhara position delta kya hai?

0.65 × 4 × 100 = 260. Iska matlab hai tumhare paas stock ke 260 shares own karne ke equivalent directional exposure hai.

Delta-neutral hone ka kya matlab hai?

Net position delta zero hona, matlab small stock price moves tumhari position ki value ko affect nahi karte. Tum directional risk se isolated ho jaate ho.

-300 delta wali short position ke liye delta-neutral hedge kaise create karte hain?

Underlying stock ke 300 shares kharido (kyunki har share ka delta = +1.0 hota hai), jo puts ke -300 delta ko offset karne ke liye +300 delta deta hai.

ATM option ka delta approximately kya hota hai?

Approximately ±0.50 (calls ke liye 0.50, puts ke liye -0.50), jo in-the-money finish karne ki 50% probability represent karta hai.

Stock price move hone par delta kyun change hota hai?

Kyunki option ki ITM finish karne ki probability change hoti hai. Jaise yeh zyada ITM hota hai, delta ±1.0 ke paas jaata hai. Jaise yeh zyada OTM hota hai, delta 0 ke paas jaata hai. Delta ka yeh change gamma se measure hota hai.

Delta ka sign tumhe kya batata hai?

Tumhari position ki directionality. Positive delta = bullish exposure (stock rise hone se profit). Negative delta = bearish exposure (stock fall hone se profit).

Black-Scholes se call delta ka formula kya hai?

Δ_call = N(d₁), jahan N cumulative normal distribution function hai aur d₁ = [ln(S/K) + (r + σ²/2)T] / (σ√T).

Call delta aur put delta ka kya relationship hai?

Δ_put = Δ_call - 1. Yeh put-call parity se aata hai. Same strike par, put delta hamesha call delta se 1.0 kam hota hai.

Expiration paas aane par delta ka kya hota hai?

Yeh zyada "digital" ho jaata hai — ITM options ka delta ±1.0 ke paas jaata hai, OTM options ka delta 0 ke paas. ATM ke paas transition sharper ho jaata hai (higher gamma).

Delta ko "hedge ratio" kyun kehte hain?

Kyunki yeh batata hai ki option ke price movement ko replicate ya hedge karne ke liye tumhe kitne shares of stock hold karne hain. Yeh risk-free portfolio mein options ke against shares ka ratio hai.

Deep ITM call options ka delta kya hota hai?

Approximately 1.0 (ya 100), matlab yeh stock ke saath almost dollar-for-dollar move karte hain, almost stock ownership jaisi behave karte hain.