Maths 05 Investment Maths Without Fear

Vega: Sensitivity to Volatility

Translate volatility changes into a local option-value estimate.

Learn Options and GreeksAdvanced4 min

Long vanilla option; other inputs fixed

Plain English

The idea

Vega measures how option value changes when implied volatility changes. It isolates the volatility input while holding the other model inputs still.

Under the non-dividend European model, higher volatility raises long call and put values through the convex payoff. This is a value relationship, not a universal increase in the probability of finishing in the money.

Vega is usually quoted per one volatility point by market convention, but the Black-Scholes derivative is naturally per 1.00 volatility. That unit difference matters.

Implied volatility is the volatility input that makes a selected pricing model match a given option price. It is not a direct observation of future volatility, and different strikes or expiries can imply different values. Vega asks how model value changes for a small change in that input while the other inputs stay fixed.

The raw derivative uses volatility as a decimal. Many platforms instead quote a value change per one percentage point of volatility, so the unit must be checked before any multiplication.

Worked Example

Reading the volatility unit

Suppose an option platform shows vega of 0.08 per volatility point. If implied volatility rises from 20% to 21%, the option value is expected to rise by about 0.08 before other effects are considered.

In the raw calculus convention, the same sensitivity is 8.00 per 1.00 volatility because one percentage point is 0.01.

This does not mean volatility will rise. It only states the local model sensitivity to that input.

In the hypothetical quotation, vega is £0.08 per option unit per volatility percentage point. Moving from 20% to 21% is one percentage point, or 0.01 in decimal-volatility units. The first-order value effect is therefore approximately +£0.08, other inputs fixed.

For a half-percentage-point move from 20% to 20.5%, the estimate is +£0.04. A 1% relative increase from 20% would instead be 20.2%, a 0.2-percentage-point move, producing about +£0.016. The different answers come from different shock sizes, not different models.

Platform vega	0.08 per volatility point
Implied volatility move	20% to 21%
First-order option move	About +0.08
Raw derivative unit	About 8.00 per 1.00 volatility

Reading the result

Uncertainty and convex payoffs

The long vanilla option has a convex payoff: adverse expiry outcomes do not create an exercise obligation for the holder, while favourable outcomes can add value. Under the stated Black–Scholes assumptions, increasing volatility raises long call and put values. It need not raise the probability of finishing in the money in every starting situation.

For the same strike, expiry and other inputs, the model call and put have the same vega. This is a property of the model prices, not a claim that all calls and puts in the market have equal volatility exposure. Different contract inputs can give very different sensitivities.

Limits and assumptions

A quotation needs its convention

The raw formula reports value change per 1.00 of decimal volatility. Dividing it by 100 gives the sensitivity per one percentage point. In this example a raw vega of 8 corresponds to 0.08 per point; 1.00 is a derivative unit, not a proposed 100-percentage-point shock to apply linearly.

The derivation assumes positive underlying price, strike, volatility and time, European exercise, no dividends and constant model rate and volatility. Time is in years. Real implied-volatility surfaces can move unevenly across strikes and expiries, so a single parallel volatility change is a simplification. Larger changes also alter vega itself.

Common Mistake

Confusing volatility direction with price direction

Vega does not say the underlying price will rise or fall. It says the option value is sensitive to the market price of expected movement.

A call can lose value when the underlying rises if implied volatility falls enough and other effects dominate. Greeks help separate those forces.

An option can lose value despite a favourable underlying move if other contributions, including a volatility decline, outweigh it. Vega helps explain that possibility without forecasting the size or direction of future volatility changes.

Formula Summary

Black-Scholes vega and market quoting

European exercise, no dividends, positive S, K, sigma and T; time T is in years, sigma is an annualised decimal and r is a continuously compounded annual rate. Rates and volatility are constant model inputs; jumps and trading frictions are excluded. Do not apply these expressions directly at expiry or zero volatility.

N is the cumulative standard normal distribution; n is its density. The formulas describe long-option model values; signed positions change the exposure.

Vega

Vega = S n (d_{1}) \sqrt{T}

$S$: Current underlying price.
$n (d_{1})$: Standard normal density at d1.
$T$: Time to expiry in years.
$Vega / 100$: Approximate value change per one percentage-point volatility move.

Some texts use the Greek letter nu for vega, but option screens usually label it Vega.

Optional derivation Explore the derivation

The compact derivation uses the relationship between d1 and d2 to avoid expanding every derivative of d1.

01

Differentiate the call price with respect to volatility

Only d1 and d2 depend on volatility in the non-dividend call price.

Volatility Derivative

$\frac{\partial C}{\partial σ} = S n (d_{1}) \frac{\partial d_{1}}{\partial σ} - K e^{- r T} n (d_{2}) \frac{\partial d_{2}}{\partial σ}$
02

Relate the two volatility derivatives

Because d2 equals d1 minus sigma times square root of T, the d2 volatility derivative is the d1 derivative minus square root of T.

d2 Relationship

$\frac{\partial d_{2}}{\partial σ} = \frac{\partial d_{1}}{\partial σ} - \sqrt{T}$
03

Use the same density identity

The same identity from delta lets the scaled density terms match, leaving only the square-root time term.

Cancellation Identity

$S n (d_{1}) = K e^{- r T} n (d_{2})$
04

Read the result and convert units

The raw derivative is per 1.00 volatility. Market screens often show the amount per one percentage-point move, which is one hundredth of the raw derivative.

Vega Result

$\begin{matrix} {Vega}_{raw} = S n (d_{1}) \sqrt{T} \\ {Vega}_{point} = {Vega}_{raw} / 100 \end{matrix}$

Self-check

Check your understanding

How much is a move from 20% to 21% in decimal volatility?

0.01, equal to one percentage point. It is a 5% relative increase, which is a different way of measuring the same change.

What is the value estimate for a move from 20% to 20.5% at £0.08 vega per point?

Approximately +£0.04 per option unit, because the move is half a percentage point. The estimate holds other inputs fixed.

Why is raw vega divided by 100 for a per-point quotation?

The raw derivative uses a 1.00 decimal-volatility unit. One percentage point is 0.01 of that unit, so its first-order value contribution is one hundredth as large.

General learning context

Where this helps a public investor

Vega helps readers understand why an option can move around earnings, policy events, or market stress even before the underlying price has moved much.

Disclaimer

Educational Use Only

This article is for informational and educational purposes only. Options involve risk and are not suitable for every investor. Nothing here is a recommendation to buy, sell, write, or trade an option.

Path

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Terms

Vega: value sensitivity to the volatility input.Implied volatility: a model input inferred from an option price.Volatility point: one percentage point of volatility.Convex payoff: a payoff whose slope does not decrease as the underlying rises.

Sources

The idea

Reading the volatility unit

Uncertainty and convex payoffs

A quotation needs its convention

Confusing volatility direction with price direction

Black-Scholes vega and market quoting

Differentiate the call price with respect to volatility

Relate the two volatility derivatives

Use the same density identity

Read the result and convert units

Check your understanding

Where this helps a public investor

Educational Use Only