FRM Exam Part I · Option Sensitivity Measures: The "Greeks"

Vega and Volatility Sensitivity of Options Explained

Updated 11 October 2026 · Fact-checked

Vega is the change in an option's value for a 1-unit change in volatility, holding other inputs fixed. In Black-Scholes-Merton, vega = S0 × N'(d1) × √T for a non-dividend stock. It is always positive for long calls and puts, and highest near at-the-money. Hedge it with other options, not the underlying.

Understand Vega and Volatility Sensitivity

Vega measures how much an option's price changes when the volatility of the underlying changes. Volatility is the market's view of how widely the price may move. A wider spread of outcomes helps the option holder, because the loss is capped at the premium while the gain is not. So higher volatility raises the value of both calls and puts. Vega is therefore positive for long options.

Vega is the same for a European call and a put with the same strike and maturity on the same underlying. This follows from put-call parity: the call minus the put equals S0 − K e^(−rT), which does not depend on volatility. If the two differ by a term with no volatility in it, their sensitivities to volatility must be equal.

Vega depends on moneyness and maturity. It is largest for at-the-money (or near-the-money) options. Deep in-the-money and deep out-of-the-money options are almost sure to finish in or out of the money, so a change in volatility barely changes their value. Vega rises with time to maturity, because more time lets volatility have more effect. The rise follows √T in the formula, so it is slower than proportional to T.

Vega is quoted in different ways. The formula gives the change per 1.00 (100 percentage points) change in volatility. Traders usually quote the change for a 1 percentage point move, which is the formula result divided by 100. Check which one the question uses.

Vega is not a Greek letter, but it is treated as one. It matters for hedging. A delta-neutral position can still gain or lose a lot when implied volatility moves. To make a portfolio vega-neutral, you add options with an offsetting vega. The underlying stock and forwards have zero vega, so they cannot do this. Because an option also has gamma, fixing vega with an option changes gamma and delta too. You then need to rebalance delta with the underlying, and often use two options to neutralise both gamma and vega.

Key formulas to remember

Vega of a European option (no dividends): Vega = S0 × N'(d1) × √T; Same for calls and puts. N'(x) = (1 ÷ √(2π)) × e^(−x²÷2). Result is per 1.00 change in volatility; divide by 100 for a 1 percentage point change.
Vega with continuous dividend yield q: Vega = S0 × e^(−qT) × N'(d1) × √T; Use for index and currency options (in the currency case, q is the foreign risk-free rate).
Definition: Vega = ∂V ÷ ∂σ; Change in option value per unit change in volatility.
d1: d1 = [ln(S0 ÷ K) + (r + σ²÷2) × T] ÷ (σ × √T); Needed to compute N'(d1) in vega.
Approximate P&L from a volatility move: ΔV ≈ Vega × Δσ; Keep Δσ in the same units as the vega quote.
Vega-neutral hedge position: Number of hedge options = − Vega of portfolio ÷ Vega of one hedge option; A negative number means sell hedge options. Then rebalance delta with the underlying.

How to solve Vega and Volatility Sensitivity questions

Use this method for any question on vega, from a calculation to a hedging question.

1Identify what is asked: vega value, price change from a volatility move, the sign or size comparison, or a hedge quantity.
2Check the position: long options have positive vega, short options negative vega. Stock and futures positions have zero vega. Multiply by the number of options and the sign.
3If you must compute vega, find d1 first, then N'(d1), then multiply by S0 (and e^(−qT) if there is a dividend yield) and √T.
4Check units. Decide whether volatility is quoted as a decimal or in percentage points, and whether vega is per 1.00 or per 1%.
5For a price change, use ΔV ≈ Vega × Δσ, with the units matched.
6For a comparison question, think moneyness and maturity: near-the-money has the highest vega, longer maturity has higher vega.
7For a hedge, set the hedge quantity to cancel portfolio vega: N = −Vega_portfolio ÷ Vega_option. Then recompute delta and adjust with the underlying.
8Sanity check: sign, size and direction should match intuition before you pick an answer.

Quickest way: Shortcut for vega questions

When to use it: Use when time is short and the question gives vega directly or asks for a hedge ratio or qualitative comparison.

Remember that vega is positive for long options and zero for the underlying.
For a hedge, compute N = −portfolio vega ÷ option vega and fix the sign.
For price change, multiply vega by the volatility move, using the same units.
For ranking, pick the near-the-money, longest-maturity option as highest vega.
If a question gives N'(d1), skip d1 and use S0 × N'(d1) × √T directly.

Common mistakes in Vega and Volatility Sensitivity

Saying a put has negative vega because its price falls when the stock rises.
Students confuse vega with delta.
Fix: A long put gains value when volatility rises. Long call and long put have the same positive vega for the same strike and maturity.
Believing vega is highest for deep in-the-money options.
Students think a bigger premium means bigger sensitivity.
Fix: Vega follows N'(d1), the normal density, which peaks near d1 = 0. That is close to at-the-money.
Using the underlying stock to hedge vega.
Stock is the standard delta hedge.
Fix: Stock has zero vega. Use another option, then rebalance delta with the stock.
Mixing units: applying a per 1.00 vega to a 1 percentage point move.
Textbook formula and trader quotes differ.
Fix: Divide the formula vega by 100 for a 1% volatility move, or enter the move as a decimal (0.01).
Forgetting the sign when computing a vega-neutral hedge.
Students compute the ratio and stop.
Fix: If the portfolio has positive vega, you need negative vega: sell options. Write N = −Vega_p ÷ Vega_option.
Assuming vega rises in proportion to maturity.
Students think more time means linearly more sensitivity.
Fix: Vega is proportional to √T in the formula, so quadrupling maturity only doubles it, other inputs equal.

Worked examples

Example 1

A non-dividend-paying stock trades at $50. A European call has T = 1 year and N'(d1) = 0.36. Find its vega per 1 percentage point change in volatility, and the approximate change in price if volatility rises from 20% to 22%.

Show the solution

Vega = S0 × N'(d1) × √T = 50 × 0.36 × √1 = 18.0 per 1.00 change in volatility.
Per 1 percentage point: 18.0 ÷ 100 = 0.18.
Volatility rises by 2 percentage points.
ΔV ≈ 0.18 × 2 = 0.36.

Answer: Vega is 0.18 per 1 percentage point, and the call price rises by about $0.36.

Example 2

A trader is delta-neutral and has a portfolio vega of −1,200 (per 1 percentage point). A traded option has vega 0.60 per option, delta 0.50. How many options should be bought or sold to be vega-neutral, and how many units of the underlying are then needed to restore delta-neutrality?

Show the solution

N = −Vega_portfolio ÷ Vega_option = −(−1,200) ÷ 0.60 = +2,000.
A positive N means buy 2,000 options.
Their delta is 2,000 × 0.50 = +1,000 units of the underlying equivalent.
The portfolio was delta-neutral, so the new delta is +1,000.
To offset, sell 1,000 units of the underlying.

Answer: Buy 2,000 options, then sell 1,000 units of the underlying to restore delta-neutrality.

Exam tips

Know the three qualitative facts cold: vega is positive for long options, highest near the money, and increases with maturity.
Expect to see S0 × N'(d1) × √T and be ready to compute it from given N'(d1) values without recalculating d1.
Read the unit statement carefully. Vega per 1% versus per 1.00 is a common trap.
In hedging questions, work out the vega hedge first, then the delta fix with the underlying.
Remember that call and put vega are equal by put-call parity for the same strike and maturity.

Practice questions from Option Sensitivity Measures: The "Greeks"

Vega and Volatility Sensitivity in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

CFA Level IISensitivity and Scenario Risk Measures

Vega and Volatility Sensitivity: frequently asked questions

What is the vega of an option in simple terms?

It is how much the option price changes when volatility changes by one unit, holding everything else constant. A vega of 0.18 per percentage point means the price rises 0.18 if volatility rises 1 percentage point.

Why is vega highest for at-the-money options?

Vega depends on N'(d1), the normal density, which peaks when d1 is near zero, around at-the-money. There the outcome is most uncertain, so a change in volatility shifts the chance of finishing in the money the most.

How do I make a portfolio vega-neutral?

Add options with vega that offsets the portfolio vega, using N = −portfolio vega ÷ option vega. The underlying has no vega so it cannot help. Afterwards, rebalance delta using the underlying.

Do calls and puts have the same vega?

Yes, for European options on the same underlying with the same strike and maturity. Put-call parity shows their price difference does not depend on volatility, so their vegas are equal.