Vega

Motivation: why this matters in quant finance

Vega (

\mathcal{V}

\nu

) is the option price's sensitivity to volatility:

\mathcal{V} := \frac{\partial V}{\partial\sigma}.

(Despite the Greek-letter name, vega is not a Greek letter — volatility trading is sufficiently important to have earned its own symbol.)

Why vega matters:

Volatility is the primary traded quantity in options. Market makers quote IV, not price, for most OTC transactions. Vega tells you how much $ you make/lose when implied vol moves one vol-point.
Vol trading. Pure vol bets are made by constructing vega-neutral or vega-directional positions — straddles, strangles, calendars, and risk-reversals are all vega-structured.
Risk decomposition. A trading book's risk is typically aggregated by delta, gamma, vega (and theta + rho) into a risk dashboard. Vega is usually the second-largest exposure after delta.
Model calibration. The volatility surface calibration matches market prices; its sensitivity (the Jacobian) is fundamentally about vega across strikes and maturities.
Volatility risk premium. Long vega = long volatility insurance. The historical excess of implied over realised vol (see implied volatility) is the compensation to short-vega traders for bearing tail risk.

The informal idea

Vega measures the option's $ value per 1-point change in volatility (i.e. 1% change in annualised vol). If vega is $0.35 per vol-point and implied vol rises from 20% to 21%, the option's value rises by $0.35.

Long calls/puts are long vega: higher vol $\Rightarrow$ higher option premium (more time-value, more optionality). Short calls/puts are short vega.

Vega is always positive for long vanilla options, regardless of moneyness. But its magnitude depends strongly on $S$ and $T$ :

Maximal ATM, decays away from strike. Deep ITM/OTM options are nearly linear in $S$ — vol matters less.
Grows with $\sqrt{T - t}$ . Long-dated options are far more vol-sensitive than near-expiry ones.

Black-Scholes vega

\mathcal{V} = S e^{-q(T-t)}\phi(d_1)\sqrt{T - t},

where

\phi

is the standard normal PDF and

d_1

is as in the delta lesson.

Call and put vega are equal (put-call parity: they differ by linear-in-

S

terms that have zero vol sensitivity).

Properties

Always positive for long vanilla options.
Peak ATM ( $d_1 \approx 0$ ).
Decays to zero for deep ITM/OTM.
Grows with $\sqrt{T - t}$ — long-dated ATM options are very vega-sensitive.
Vega vanishes at expiry. At $T - t = 0$ , vega = 0 — there's no vol left for the option price to depend on.

Gamma-vega identity

A beautiful Black-Scholes identity:

\mathcal{V} = \sigma S^2 (T - t)\Gamma.

So vega and gamma are proportional (via $\sigma S^2 (T-t)$ ). Both peak ATM, both decay for ITM/OTM. This is not a coincidence: both reflect the same underlying "optionality" of being near the strike with significant time to move around.

Proof sketch

$C = Se^{-qT}\Phi(d_1) - Ke^{-rT}\Phi(d_2)$ (set $T \equiv T - t$ for brevity):

\frac{\partial C}{\partial\sigma} = Se^{-qT}\phi(d_1)\frac{\partial d_1}{\partial\sigma} - Ke^{-rT}\phi(d_2)\frac{\partial d_2}{\partial\sigma}.

Using the identity $S e^{-qT}\phi(d_1) = Ke^{-rT}\phi(d_2)$ , the two terms combine using $\partial d_1/\partial\sigma - \partial d_2/\partial\sigma = \sqrt T$ :

\mathcal{V} = Se^{-qT}\phi(d_1)(\partial d_1/\partial\sigma - \partial d_2/\partial\sigma) = Se^{-qT}\phi(d_1)\sqrt T.

Numerical example

$S = 100, K = 100, T = 0.5, r = 0.05, q = 0, \sigma = 0.2$ :

$d_1 \approx 0.247$ , $\phi(0.247) \approx 0.3866$ .

\mathcal{V} = 100\cdot 0.3866\cdot \sqrt{0.5} \approx 100\cdot 0.3866\cdot 0.7071 \approx 27.34.

This is vega per unit change in $\sigma$ (i.e. vol going from 0.20 to 1.20). Industry convention is to quote vega per 1% vol point:

\mathcal{V}/100 = 0.2734

. So 1 vol-point rise (20% → 21%) raises this call by

\approx

$0.27.

Common pitfalls

"Vega is a Greek letter." Nope — it's a trader-invented name. The Greek letters used in practice are

\Delta, \Gamma, \Theta, \nu

(nu is used for vega in some textbooks),

\rho

. "Vega" comes from the finance-world convention, not math.

"Quotation conventions differ." Always check: is vega in absolute terms (per unit of

\sigma

) or percentage points (per 1% vol change)? Divide or multiply by 100 as needed.

"Black-Scholes vega is the market's vega." Only approximately. Real options markets exhibit a volatility surface with skew and term-structure. When vol shifts, different strikes and maturities move differently — the single-number "vega" is a first-order approximation to a far richer object.

"Vega-neutral means vol-neutral." Only locally. A calendar spread (long long-dated, short short-dated) can be locally vega-neutral but exposed to term-structure shifts of the vol surface. Practitioners decompose vega by maturity bucket and often by skew component.

"Delta-hedged options have zero vol exposure." Incorrect. Delta hedging removes spot risk; you're still long vega. Long-vega positions profit from vol going up, regardless of what the stock does.

Volatility surface and vega

In real markets, implied vol varies across strikes and maturities (the "volatility surface"). When a trader speaks of vega, they often mean:

Surface vega — flat-shift vega: PV change if the whole vol surface moves up by 1%.
Skew vega — PV change per unit of skew (slope of vol vs. strike).
Term vega — PV change per unit of term-structure slope.

For a single vanilla option, the traditional Black-Scholes vega is a reasonable first-order proxy; for a book of options, all three exposures matter separately.

Where this goes next

Implied Volatility: Vega is the building block for inverting price $\to$ IV (via Newton's method on vega).
Delta and Gamma: Vega is the third pillar of the first-order Greeks triad.
Stochastic Volatility Models: Heston/SABR explicitly model vol as a second state variable — their vegas are model-dependent.
The Derivation of the Black-Scholes Formula: Vega comes from differentiating the BS formula in $\sigma$ .