Vega is quoted per volatility point, peaks slightly above the spot and grows with the square root of time; a finite difference checks it in one line.
Vega is the change in an option's price for a change in volatility: in Black-Scholes, vega = S × φ(d1) × √T, where φ is the standard normal density. For a one-year at-the-money call on a stock at 100, with a 4 per cent rate and 24 per cent volatility, φ(d1) is 0.382882, so vega is 38.2882 per unit of volatility, or 0.3829 per volatility point. On 10,000 options one point is worth 3,829.
Worked in full in Quantitative Finance by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
Vega is quoted per point because volatility is quoted in points. Going from 24 to 25 per cent is a change of 0.01 in the formula's units, so the derivative is divided by 100 before anyone reads it. Most of the confusion around vega comes from forgetting that step, or from treating a figure that is accurate for one point as accurate for eight.
| Input | Value |
|---|---|
| Spot, S | 100 |
| Strike, K | 100 |
| Rate, r | 4% |
| Volatility, σ | 24% |
| Time to expiry, T | 1 year |
| Call price | 11.4541 |
| Position | 10,000 options |
d1 = [ln(S/K) + (r + σ²/2)T] ÷ (σ√T)
Vega = S × φ(d1) × √T, and per point = vega ÷ 100
In Excel, with d1 in B1, spot in B2 and years in B3: =B2*NORM.S.DIST(B1,FALSE)*SQRT(B3)/100. The FALSE returns the density, not the cumulative probability.
Step 1, d1. ln(100/100) is 0.0000 and (0.04 + 0.24²/2) × 1 is 0.0688, divided by 0.2400 gives d1 = 0.2867, the same d1 as in the Black-Scholes price.
Step 2, the density. φ(0.2867) = e−d1²/2 ÷ √(2π) = 0.382882. The peak of the normal density, at d1 = 0, is 0.3989.
Step 3, vega. 100 × 0.382882 × 1 = 38.2882, or 0.3829 per point. That is 3.34 per cent of the premium for every point of volatility.
Step 4, check it numerically. Reprice at 25 and 23 per cent: 11.8370 and 11.0713. The central difference, (11.8370 − 11.0713) ÷ 2, is 0.382872, within 0.00001 of the formula. This second route is how to check a vega reported by a system you did not build.
| Measure | Per option | 10,000 options |
|---|---|---|
| Premium | 11.4541 | 114,541 |
| Vega, one point | 0.3829 | 3,829 |
| Two points | 0.7658 | 7,658 |
A two-point rise in implied volatility, a move that is far from unusual, shifts the position by 7,658 without the stock moving at all. Vega is how a risk report says that.
The put on the same strike and expiry has exactly the same vega. Put-call parity makes the call minus the put independent of volatility, so a change in volatility moves both by the same amount. A long call and a long put, a straddle, therefore carries twice the vega of either: 0.7658 a point per pair. That is why a straddle, not a single option, is the usual way to buy volatility: it doubles the vega while carrying much less delta than the call alone.
Vega is not a constant of the option. It is largest near the money and grows with the square root of time.
| Strike | 3 months | 6 months | 1 year | 2 years |
|---|---|---|---|---|
| 80 | 0.0268 | 0.0892 | 0.1904 | 0.3207 |
| 90 | 0.1184 | 0.2010 | 0.3066 | 0.4367 |
| 100 | 0.1974 | 0.2764 | 0.3829 | 0.5197 |
| 107.12 | 0.1819 | 0.2764 | 0.3989 | 0.5527 |
| 110 | 0.1614 | 0.2645 | 0.3965 | 0.5598 |
| 120 | 0.0774 | 0.1929 | 0.3567 | 0.5593 |
Three things to read off it. At one year vega peaks not at the spot but at a strike of 107.12, where d1 is zero: K = S × e(r + σ²/2)T. There it equals S × 0.3989 × √T, the source of the trader's rule of thumb that at-the-money vega is about 0.4 × S × √T per hundred. The at-the-money two-year option has 1.36 times the vega of the one-year, a little under √2 = 1.41 because the peak drifts to higher strikes as maturity lengthens. And the three-month option has 0.52 times the one-year vega: short-dated books carry gamma, long-dated books carry vega.
Vega is also the denominator of Newton's method for implied volatility, which is why the iteration struggles, overshooting or stalling, on far out-of-the-money options with almost no vega. That iteration is worked in how to calculate implied volatility with Newton's method.
The five Greeks with a finite-difference check beside each, and vega across strikes and maturities, are live in the free workbooks for this book.
Yes, in Black-Scholes. Put-call parity makes the call minus the put independent of volatility, so both move by the same amount when volatility changes. For the one-year at-the-money option on a stock at 100, with 4 per cent rates and 24 per cent volatility, both the call and the put have a vega of 0.3829 per point.
Because vega is proportional to the normal density at d1, which peaks when d1 is zero. That happens at a strike of S times e^((r + vol squared / 2)T), 107.12 for the one-year example, slightly above the spot. There vega is 0.3989 per point, against 0.1904 at a strike of 80.
It grows roughly with the square root of time. In the example the at-the-money vega is 0.1974 per point at three months, 0.3829 at one year and 0.5197 at two years. Long-dated options carry most of a book's volatility exposure, short-dated ones most of its gamma.
This article is one calculation from Quantitative Finance. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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