Learn Vega and volatility sensitivity — Hinglish notes

5.3.4 · Stock-Market › The Greeks

Vega exist karta hi kyun hai?

Yahan volatility KYA hai? Past (realized) volatility nahi — option ki price mein implied volatility (IV) embed hoti hai, jo market ka forecast hai ki stock kitna swing karega (annualized, fraction ki tarah jaise $0.20 = 20%$ ).

Zyada volatility option value kyun badhati hai? Option ka asymmetric payoff hota hai:

Ek call sirf premium tak hi lose kar sakti hai (floor), lekin upside unbounded hai.
Toh future prices ki distribution ko spread karne se deep-in-the-money tail mein probability mass add hoti hai, bina matching downside pain ke.

Zyada spread ⇒ fatter profitable tail ⇒ higher expected payoff ⇒ higher premium. Yahi sensitivity Vega hai.

Note: "Vega" asli Greek letter nahi hai — isliye yeh sabse alag lagta hai.

Vega derive kaise karte hain (Black–Scholes call)

Black–Scholes call price se shuru karte hain: $C = S N (d_{1}) - K e^{- r T} N (d_{2})$ $d_{1} = \frac{l n ( S / K ) + ( r + \frac{1}{2} σ ^{2} ) T}{σ T}, d_{2} = d_{1} - σ T$

Humein $ν = \partial C / \partial σ$ chahiye. Differentiate karte hain: $\frac{\partial C}{\partial σ} = S N^{'} (d_{1}) \frac{\partial d _{1}}{\partial σ} - K e^{- r T} N^{'} (d_{2}) \frac{\partial d _{2}}{\partial σ}$

Key trick — do messy terms collapse ho jaate hain. Ek identity hai: $S N^{'} (d_{1}) = K e^{- r T} N^{'} (d_{2})$

Yeh step kyun? Gaussian $N^{'} (x) = \frac{1}{2 π} e^{- x^{2} /2}$ likhne par aur $d_{2} = d_{1} - σ T$ use karne par, exponents exactly $ln (S / K) + r T$ se differ karte hain, jo $S$ vs $K e^{- r T}$ factors se cancel ho jaata hai. (Niche verify kiya hai.)

Kyunki woh common factor equal hai, unhe group karte hain: $\frac{\partial C}{\partial σ} = S N^{'} (d_{1}) (\frac{\partial d _{1}}{\partial σ} - \frac{\partial d _{2}}{\partial σ})$

Kyunki $d_{2} = d_{1} - σ T$ : $\frac{\partial d _{1}}{\partial σ} - \frac{\partial d _{2}}{\partial σ} = \frac{\partial}{\partial σ} (σ T) = T$

Yeh step kyun? Saare ugly $\partial d_{1} / \partial σ$ terms cancel ho jaate hain; sirf explicit $σ T$ difference bachta hai. Ekdum beautiful.

Figure — Learn Vega and volatility sensitivity

Vega ko upar-neeche kya drive karta hai?

Factor badhta hai	Vega...	KYU
Time to expiry $T$	↑ (via $T$ )	zyada time = zyada cumulative swing possible
At-the-money-ness	↑ (ATM ke paas peak hota hai)	$N^{'} (d_{1})$ max hota hai jab $d_{1} \approx 0$ , yaani $S \approx K$
Deep ITM / OTM	↓→0	outcome lagbhag certain hai, vol usse mushkil se hilata hai
Spot price $S$	scale karta hai	Vega mein explicit $S$ factor hai

Worked Examples

Common Mistakes

Active Recall

Recall Q: Ek sentence mein, Vega kya hai?

Implied volatility mein 1% change per option price ka change; long options ke liye hamesha positive.

Recall Q: Same strike/expiry wale call aur put ka Vega same kyun hota hai?

Put–call parity ( $C - P$ ) $σ$ se independent hai, isliye unke $σ$ -derivatives equal hain.

Recall Q: Vega kahan (strike mein) aur kab (time mein) sabse bada hota hai?

At-the-money par sabse bada ( $d_{1} \approx 0$ , isliye $N^{'} (d_{1})$ peak karta hai) aur zyada time to expiry ke liye ( $T$ ki tarah badhta hai).

Recall Ek 12-saal ke bacche ko samjhao (hidden)

Socho tum apne dost se bet laga rahe ho ki ek bouncing ball ek line se aage jaayegi. Agar ball bahut wildly bounce kare, toh zyada chance hai ki woh line se kaafi aage jaaye — isliye us bet ka "ticket" zyada worth hoga. Vega yeh hai ki tumhare ticket ki price kitni badhti hai jab sab maan lete hain ki ball zyada wildly bounce kar rahi hai. Koi farq nahi padhta ball kis taraf jaaye, bas kitna jump karta hai woh matter karta hai.

Flashcards

Vega kya measure karta hai?

Implied volatility mein 1% change ke saath option price ki sensitivity,

ν = \partial V / \partial σ

Vega formula (Black–Scholes)?

ν = S N^{'} (d_{1}) T

, 100 se divide karke per 1% IV quote kiya jaata hai.

Kya Vega long option ke liye positive ya negative hota hai?

Positive — options khareedna long volatility hona hai.

Kya Vega calls aur puts ke liye same hota hai?

Haan; put–call parity mein koi

σ

term nahi hai.

Vega strikes mein kahan maximal hota hai?

At-the-money par, jahan

d_{1} \approx 0

aur

N^{'} (d_{1})

sabse bada hota hai.

Vega time to expiry ke saath kaise change karta hai?

T

ki tarah badhta hai; expiry par 0 ho jaata hai.

Straddle Vega 0.40, IV +15 pts jump kare — P&L?

\approx +\

6.00

p er s ha r e, s p o t se in d e p e n d e n t ." V e g a cr u s h " k y ahai ? ::: S c h e d u l e d e v e n t k e baa d I V co l l a p se h o na, j i sse l o n g - v o l p os i t i o n s v a l u e l ose k a r t ihain c hah es p o t u n c han g e d h o . I d e n t i t y

S N'(d_1)=Ke^{-rT}N'(d_2)

k y u n ? ::: G a u ss ian e x p o n e n t se x a c tl y

\ln(S/K)+rT

se d i f f er k a r t e hain, j o

v s

Ke^{-rT}$ factors se cancel ho jaata hai.

Mnemonic

Connections

The Greeks — Delta, Gamma, Theta, Rho ka overview
Black-Scholes Model — $N^{'} (d_{1})$ term ka source
Implied Volatility — woh input jiske against Vega differentiate karta hai
Theta and time decay — Theta aur Vega long-vol positions mein trade off karte hain
Gamma — dono ATM par peak karte hain; Gamma spot-curvature hai, Vega vol-sensitivity hai
Put-Call Parity — prove karta hai ki calls aur puts Vega share karte hain
Straddles and Strangles — pure Vega/volatility trades