Learn Vega and volatility sensitivity — Stock-Market notes

WHY does Vega exist at all?

WHAT is volatility here? Not past (realized) volatility — the option price embeds implied volatility (IV), the market's forecast of how much the stock will swing (annualized, as a fraction like $0.20 = 20\%$ ).

WHY does higher volatility raise option value? An option has asymmetric payoff:

A call can only lose the premium (floor), but the upside is unbounded.
So spreading out the distribution of future prices adds probability mass in the deep-in-the-money tail without adding matching downside pain.

More spread ⇒ fatter profitable tail ⇒ higher expected payoff ⇒ higher premium. That sensitivity is Vega.

Note: "Vega" is not a real Greek letter — that's why it's the odd one out.

HOW to derive Vega (Black–Scholes call)

Start from the Black–Scholes call price: $C = S\,N(d_1) - K e^{-rT} N(d_2)$ $d_1 = \frac{\ln(S/K) + (r + \tfrac12\sigma^2)T}{\sigma\sqrt{T}}, \qquad d_2 = d_1 - \sigma\sqrt{T}$

We want $\nu = \partial C/\partial\sigma$ . Differentiate: $\frac{\partial C}{\partial \sigma} = S\,N'(d_1)\frac{\partial d_1}{\partial \sigma} - Ke^{-rT}N'(d_2)\frac{\partial d_2}{\partial \sigma}$

Key trick — the two messy terms collapse. There is an identity: $S\,N'(d_1) = K e^{-rT} N'(d_2)$

Why this step? Writing out the Gaussian $N'(x)=\frac{1}{\sqrt{2\pi}}e^{-x^2/2}$ and using $d_2 = d_1-\sigma\sqrt T$ , the exponents differ by exactly $\ln(S/K)+rT$ , which is cancelled by the factors $S$ vs $Ke^{-rT}$ . (Verified below.)

Because that common factor is equal, group them: $\frac{\partial C}{\partial \sigma} = S\,N'(d_1)\left(\frac{\partial d_1}{\partial\sigma} - \frac{\partial d_2}{\partial\sigma}\right)$

Since $d_2 = d_1 - \sigma\sqrt T$ : $\frac{\partial d_1}{\partial\sigma} - \frac{\partial d_2}{\partial\sigma} = \frac{\partial}{\partial\sigma}(\sigma\sqrt T) = \sqrt T$

Why this step? All the ugly $\partial d_1/\partial\sigma$ terms cancel; only the explicit $\sigma\sqrt T$ difference survives. Beautiful.

Figure — Learn Vega and volatility sensitivity

WHAT drives Vega up and down?

Factor increases	Vega...	WHY
Time to expiry $T$	↑ (via $\sqrt T$ )	more time = bigger cumulative swing possible
At-the-money-ness	↑ (peaks near ATM)	$N'(d_1)$ max when $d_1\approx0$ , i.e. $S\approx K$
Deep ITM / OTM	↓→0	outcome nearly certain, vol barely moves it
Spot price $S$	scales	Vega has an explicit $S$ factor

Worked Examples

Worked example Example 1 — Compute Vega

$S=100$ , $K=100$ , $r=0$ , $T=0.25$ (3 months), $\sigma=0.20$ . Step 1: $d_1 = \frac{\ln(1) + (0 + 0.5\cdot0.04)(0.25)}{0.20\sqrt{0.25}} = \frac{0.005}{0.10}=0.05$ . Why? Plug into $d_1$ ; ATM with $r=0$ gives a tiny positive $d_1$ . Step 2: $N'(0.05)=\frac{1}{\sqrt{2\pi}}e^{-0.00125}=0.3988$ . Step 3: $\nu = S\sqrt T\,N'(d_1) = 100\cdot0.5\cdot0.3988 = 19.94$ (per 1.0 vol). Step 4: Per 1% IV: $19.94/100 = \mathbf{0.199}$ . Meaning: IV rising 20%→21% raises the option ≈ $0.20.

Common Mistakes

Active Recall

Recall Q: In one sentence, what is Vega?

The change in option price per 1% change in implied volatility; always positive for long options.

Recall Q: Why is Vega the same for a call and a put with same strike/expiry?

Put–call parity ( $C-P$ ) is independent of $\sigma$ , so their $\sigma$ -derivatives are equal.

Recall Q: Where (in strike) and when (in time) is Vega largest?

Largest at-the-money ( $d_1\approx0$ , so $N'(d_1)$ peaks) and for longer time to expiry (grows like $\sqrt T$ ).

Recall Explain to a 12-year-old (hidden)

Imagine betting a friend that a bouncing ball will end past a line. If the ball bounces wildly, there's a bigger chance it ends way past the line — so your "ticket" to that bet is worth more. Vega is how much your ticket's price goes up when everyone agrees the ball is bouncing more wildly. It doesn't matter which way the ball goes, just how much it jumps around.

Flashcards

What does Vega measure?

Sensitivity of option price to a 1% change in implied volatility,

\nu=\partial V/\partial\sigma

Vega formula (Black–Scholes)?

\nu = S\,N'(d_1)\sqrt{T}

, quoted per 1% IV by dividing by 100.

Is Vega positive or negative for a long option?

Positive — buying options is being long volatility.

Is Vega the same for calls and puts?

Yes; put–call parity has no

\sigma

term.

Where is Vega maximal across strikes?

At-the-money, where

d_1\approx 0

and

N'(d_1)

is largest.

How does Vega change with time to expiry?

Grows like

\sqrt{T}

; goes to 0 at expiry.

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

\approx +\

6.00

per share, independent of spot. What is "vega crush"? ::: IV collapse after a scheduled event, causing long-vol positions to lose value even if spot is unchanged. Why the identity

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

? ::: The Gaussian exponents differ by exactly

\ln(S/K)+rT

, cancelled by the

vs

Ke^{-rT}$ factors.

Mnemonic

Connections

The Greeks — overview of Delta, Gamma, Theta, Rho
Black-Scholes Model — source of the $N'(d_1)$ term
Implied Volatility — the input Vega differentiates against
Theta and time decay — Theta and Vega trade off in long-vol positions
Gamma — both peak ATM; Gamma is spot-curvature, Vega is vol-sensitivity
Put-Call Parity — proves calls and puts share Vega
Straddles and Strangles — pure Vega/volatility trades

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, Vega ka matlab simple hai: agar market ka "future me kitna hilega" ka andaaza — yaani implied volatility (IV) — 1% badhta hai, to option ka price kitna badlega, wahi Vega batata hai. Jab log sochte hain ki stock zyada wildly move karega, tab call aur put dono mehnge ho jaate hain, kyunki bade move ka chance badh jaata hai. Yaad rakho — Vega direction ki baat nahi karta, sirf movement ki magnitude ki.

Formula bhi cheez hai: $\nu = S\,N'(d_1)\sqrt{T}$ . Isko humne Black–Scholes se derive kiya, aur magic ye hai ki saare complicated terms cancel ho jaate hain kyunki $S\,N'(d_1)=Ke^{-rT}N'(d_2)$ ek identity hai. Isi wajah se call aur put ka Vega bilkul same hota hai (put-call parity me $\sigma$ hai hi nahi).

Practical baat: Vega at-the-money pe sabse zyada hota hai, aur jitna zyada time to expiry (LEAPS jaisa), utna bada Vega ( $\sqrt{T}$ ke saath badhta hai). Isiliye earnings ya budget jaise events se pehle traders straddle kharidte hain — IV upar jaata hai to profit, chahe stock hile ya na hile. Lekin event ke baad vega crush hota hai: IV gir jaata hai aur long-vol position ghaate me aa jaati hai, bhale spot na hila ho. Yehi Vega ka asli khel hai.

Test yourself — The Greeks

Connections