Learn Gamma and delta sensitivity — Hinglish notes

5.3.2 · Stock-Market › The Greeks

Overview

Gamma measure karta hai ki delta kitni tezi se change hota hai jab underlying stock price move karti hai. Delta abhi aapka directional exposure batata hai, lekin gamma batata hai ki woh exposure kitna unstable hai. High gamma matlab aapka delta wildly swing karta hai—ek chhoti si price move mein aap hedged se massively exposed ho sakte hain.

YEH definition kyun? Delta directional risk quantify karta hai. Agar aapke paas ek call hai $Δ = 0.6$ ke saath, toh aap $0.60 g aink a r t e h o ha r$ 1 ke liye jitna stock rise karta hai per share. Kyunki ek standard option contract 100 shares control karta hai, woh call $60 g aink a r t ahaiha r$ 1 move par contract level par. Yeh aapki equivalent stock position hai: us call ka owner hona 0.6 shares ka owner hone jaisa hai (ya 60 shares per contract).

Second derivative kyun? Kyunki risk linear nahi hai. Aapka hedge ratio (delta) shift hota hai jab market move karta hai. Gamma us curvature ko capture karta hai. High gamma = aapka hedge quickly break down hota hai.

Derivation from First Principles

Black-Scholes call value se shuru karte hain: $C = S N (d_{1}) - K e^{- r T} N (d_{2})$

jahan $d_{1} = \frac{l n ( S / K ) + ( r + σ ^{2} /2 ) T}{σ T}$ aur $d_{2} = d_{1} - σ T$ .

Step 1: $S$ ke respect mein differentiate karke delta find karo.

$Δ_{call} = \frac{\partial C}{\partial S} = N (d_{1}) + S N^{'} (d_{1}) \frac{\partial d _{1}}{\partial S} - K e^{- r T} N^{'} (d_{2}) \frac{\partial d _{2}}{\partial S}$

KYU? $S N (d_{1})$ par product rule aur cumulative normals par chain rule.

Partials compute karo: $\frac{\partial d _{1}}{\partial S} = \frac{1}{S σ T}, \frac{\partial d _{2}}{\partial S} = \frac{1}{S σ T}$

Standard normal PDF hai $N^{'} (x) = \frac{1}{2 π} e^{- x^{2} /2}$ . Black-Scholes put-call relationship use karke, $N^{'} (d_{1})$ aur $N^{'} (d_{2})$ wale terms cancel ho jaate hain:

$S N^{'} (d_{1}) = K e^{- r T} N^{'} (d_{2})$

Yeh cancel kyun hote hain? Kyunki $d_{2} = d_{1} - σ T$ , aur discount factor ke saath normal PDF mein substitute karne par yeh identity milti hai. Yeh Black-Scholes symmetry hai.

Is tarah: $Δ_{call} = N (d_{1})$

Ek put ke liye (put-call parity se ya direct differentiation se): $Δ_{put} = N (d_{1}) - 1 = - N (- d_{1})$

Step 2: Delta differentiate karke gamma find karo.

$Γ = \frac{\partial Δ}{\partial S} = \frac{\partial N ( d _{1} )}{\partial S} = N^{'} (d_{1}) \frac{\partial d _{1}}{\partial S}$

Substitute karo: $Γ = \frac{N ^{'} ( d _{1} )}{S σ T} = \frac{1}{S σ T} \cdot \frac{1}{2 π} e^{- d_{1}^{2} /2}$

Gamma positive kyun hai? $N^{'} (d_{1}) > 0$ hamesha. Chahe aap call own karo ya put, gamma same hota hai: long options convexity gain karte hain, short options use lose karte hain.

Gamma (calls aur puts dono ke liye same): $Γ = \frac{N ^{'} ( d _{1} )}{S σ T}$

Gamma at-the-money peak karta hai kyunki $d_{1} \approx 0$ jab $S \approx K$ , jo $N^{'} (d_{1})$ ko maximize karta hai.

Delta Sensitivity: Position Delta Kaise Change Hoti Hai

$n$ options ka ek portfolio consider karo, har ek ka delta $Δ_{i}$ aur gamma $Γ_{i}$ hai. Agar stock $Δ S$ se move kare:

$Δ_{new} \approx Δ_{old} + Γ \cdot Δ S$

KYU? Taylor expansion: $Δ (S + Δ S) \approx Δ (S) + Γ (S) \cdot Δ S$ . Yeh ek linear approximation hai; higher-order greeks (speed, etc.) ise refine karte hain.

Scenario: Stock $101 par move karta hai.

$Δ_{new} \approx 0.50 + 0.08 \cdot 1 = 0.58$

Yeh kyun matter karta hai: Per contract (100 shares), aap 50 shares short se hedge kiye hue the. Ab aapko 58 shares short chahiye. Ek volatile market mein, har $1 move par rebalancing expensive hai (bid-ask spread, commissions). High gamma = delta hedgers ke liye high transaction costs.

Scenario: Stock $125 par move karta hai.

$Δ_{new} \approx 0.92 + 0.01 \cdot 5 = 0.97$

KYU? Delta already 1 ke paas hai. Gamma chhota hai kyunki option strike se dur hai—almost certain hai ki exercise hoga, toh yeh stock ke saath nearly 1:1 move karta hai.

Stock $2 ba d h t ahai . N a y a d e l t a p er s ha r e :$ 0.50 + 0.05 \cdot 2 = 0.60 $. P or t f o l i o d e l t a =$ 100 \cdot 0.60 = 60$.

Hedge break kyun hua? Aap ab 60 deltas short ho lekin sirf 50 shares own karte ho. Aap 10 deltas short ho—aur upar jaane par exposed. Aapko 10 aur shares kharidne honge.

**Agar stock $2 g i r e : * * D e l t a p er s ha r e =$ 0.50 - 0.05 \cdot 2 = 0.40$. Portfolio delta = 40. Aap ab long 10 deltas ho (aap 50 shares own karte ho lekin sirf 40 deltas short ho). Aapko 10 shares bechne honge.

Gamma aapko high par kharidne aur low par bechne par majboor karta hai—profitability ka dushman.

Why Gamma Matters: The Greeks in Action

Market Makers short gamma hote hain (woh liquidity provide karne ke liye options sell karte hain). Volatile markets mein, unhe constantly rebalance karna padta hai, feedback loops create karte hue: market badhne par kharidna, girne par bechna. Yeh moves amplify karta hai—"gamma squeeze."
Volatility Traders long gamma chahte hain. Agar aap ek straddle kharidein (ATM call + ATM put), aapke paas positive gamma hota hai. Kisi bhi direction mein bade moves aapka delta badhate hain, agar rebalance karo toh profit hota hai.
Expiration Risk: Gamma explode karta hai jab expiry paas aati hai, khaaskar ATM. Ek chhoti si move ek option ko worthless se ITM mein flip kar sakti hai. Isliye "gamma risk" expiration se pehle aakhiri dinon mein dominate karta hai.

Kyun galat hai: Delta har price move ke saath change hota hai (yahi toh gamma hai). Ek static hedge sirf tiny moves ke liye kaam karta hai. Jitna bada gamma, utni tezi se aapka hedge deteriorate hota hai.

Fix: Dynamic hedging. Periodically ya jab delta shifts aapki risk tolerance se zyada ho tab rebalance karo. Yeh accept karo ki high-gamma positions = high hedging costs.

Kyun galat hai: Options wala koi bhi portfolio (including convertibles, warrants, structured products) gamma rakhta hai. Chahe aap ek "stock investor" ho covered call ke saath, aapke paas negative gamma hai—aapka hedge (short call) bade moves mein aapke against kaam karta hai.

Fix: Apne portfolio ke greeks samjho. Agar aapne income ke liye calls sell ki hain, aap short gamma ho. Risk jaano.

Recall Ek 12-saal ke bacche ko explain karo

Socho tum cycle chala rahe ho. Delta teri speed hai—abhi kitni tezi se ja rahe ho. Gamma batata hai ki teri speed pedals ke liye kitni sensitive hai. Low gamma wali cycle steady hoti hai: thoda pedal maaro, speed thodi change hoti hai. High gamma wali cycle twitchy hoti hai: ek chhota sa pedal push aur tum aage shoot kar jaate ho. Expiry ke paas wale options us twitchy cycle ki tarah hote hain—teri "speed" (delta) kisi bhi price move ke saath super fast change hoti hai, toh tumhe apna balance (apna hedge) constantly adjust karna padta hai. Agar careful nahi raho, crash ho jaoge (rebalancing costs se paise jaenge).

Connections

Delta hedging strategies — Dynamically rebalance kaise karein
Vega aur gamma ka relationship — High vega often high gamma ATM ke saath aata hai
Gamma scalping — Long-gamma position ko rebalance karke profit kamaana
Pin risk at expiration — Extreme gamma risk jab options exactly ATM expire hon
Implied volatility smile — ATM options ka highest gamma hota hai, IV structure affect karta hai

#flashcards/stock-market

Delta kya measure karta hai? :: Stock price ke respect mein option value ke change ki rate; aapka directional exposure (equivalent stock position). Delta 0.6 = $0.60 g ainha r$ 1 move par per share (= $60 per standard 100-share contract).

Gamma kya measure karta hai?

Stock price ke respect mein delta ke change ki rate; aapka directional exposure kitni tezi se change hota hai.

Long options ke liye gamma hamesha positive kyun hota hai?

Yeh value ka second derivative hai; convexity hamesha positive hoti hai long positions ke liye (aap dono directions mein bade moves se benefit karte ho ek linear hedge ke relative mein).

ATM call ka delta kya hota hai?

Approximately 0.5 (Black-Scholes mein in-the-money finish karne ki 50% probability).

Deep ITM call ka delta kya hota hai?

1.0 ke paas (almost stock own karne jaisa behave karta hai).

Gamma sabse zyada kahan hota hai?

At-the-money, khaaskar expiration ke paas.

Gamma rebalancing risk kyun create karta hai?

Kyunki delta har price move ke saath shift hota hai, aapko hedge maintain karne ke liye high par kharidna aur low par bechna padta hai.

Agar aap short gamma ho, toh stock move karne par kya hota hai?

Aapka delta aapke against shift hota hai—aapko stock badhne par kharidna aur girne par bechna padta hai, losses lock in karte hue.

"Gamma squeeze" kya hota hai?

Jab market makers (short gamma) apna delta hedge karne ke liye stock badhne par kharidne ke liye majboor hote hain, rally amplify hoti hai.

Time to expiry gamma ko kaise affect karta hai?

Gamma badhta hai jab expiration paas aata hai, khaaskar ATM; far-dated options ka gamma lower hota hai.