Vega is the first derivative of option price with respect to implied volatility, so it is only a local slope, valid at the volatility level where it was measured. Because option value is convex in volatility, that slope steepens as implied volatility rises, which is the quantity called volga. In an illustrative revaluation of a 16 delta 30 day put, vega per contract rises from 45.99 at 17.3 per cent implied volatility to 62.74 at 47.3 per cent, an increase of 36 per cent, while delta moves from minus 0.19 to minus 0.35 with no change in the underlying price at all. For a short option position both effects point the same way: exposure to volatility grows precisely as volatility rises.
Every options platform reports a number called vega. It is usually the second thing a short premium operator looks at after delta, and it is treated as the measure of volatility exposure. It is a useful number, but it is a slope, not a quantity. It tells you the rate at which the position value changes for the next small move in implied volatility, measured at the implied volatility that happens to be in effect right now. Move implied volatility somewhere else and the slope is different.
That is not a subtlety. For a short option book it is the difference between a loss estimate that scales the way you expected and one that runs away from you while the report on your screen still shows the number you sized against.
What the illustration shows
The table below revalues a single 16 delta 30 day put at successively higher levels of implied volatility, holding everything else fixed. The figures are illustrative, computed from a pinned snapshot on XSP with a multiplier of 100. Read the columns across: what happens to vega, to delta and to gamma as the volatility input alone is raised.
Table 23.4: Vega is not constant in volatility. A 16 delta 30 day put, revalued at successively higher implied volatility.
| Implied volatility | Vega per contract | Delta | Gamma |
|---|---|---|---|
| 17.3% | 45.99 | -0.19 | 0.0093 |
| 22.3% | 53.08 | -0.24 | 0.0083 |
| 27.3% | 57.04 | -0.28 | 0.0073 |
| 37.3% | 60.98 | -0.33 | 0.0057 |
| 47.3% | 62.74 | -0.35 | 0.0046 |
Take the middle row. At 27.3 per cent implied volatility the same contract shows vega of 57.04, delta of minus 0.28 and gamma of 0.0073. Nothing about the option changed between that row and the first row at 17.3 per cent, where vega was 45.99 and delta was minus 0.19. The strike is the same, the expiry is the same, the underlying price is the same. Only the volatility input moved, and every risk number on the line moved with it.
Volga: the slope steepens as conditions worsen
Vega rises by 36 per cent as implied volatility goes from 17 to 47. For a long option position that is pleasant. For a short position it is the wrong direction, because it means that as volatility rises, the exposure to volatility grows.
The practical consequence is in the cost of each additional volatility point. The first point costs $46 and the twentieth costs $61. A linear estimate built at the starting level, which is what a single reported vega figure invites you to build, understates the total by a margin that widens the further volatility travels. The estimate is not slightly wrong at the point where it matters most; it is wrong by more and more as the move extends.
This curvature is the quantity called volga, the sensitivity of vega to volatility itself. It does not need to be computed to be understood. It says the position becomes more sensitive as conditions get worse, which is a sentence that describes most of the ways short premium positions actually fail.
Vanna: delta moves without the spot moving
The delta column carries the companion effect. Delta moves from minus 0.19 to minus 0.35 across the table with no change in the underlying price at all, purely from the volatility repricing. That is vanna, the sensitivity of delta to volatility.
The plain statement is this: in a falling market your short options move toward the money, so your volatility exposure grows, at exactly the moment volatility itself is rising. An option that was comfortably out of the money on the delta report at the start of the session can be carrying nearly twice the directional exposure by the middle of it without the index having done anything the volatility surface had not already priced.
Note the gamma column as well, because it moves the other way. Gamma falls from 0.0093 at 17.3 per cent to 0.0046 at 47.3 per cent. Higher implied volatility spreads the distribution out and flattens the curvature of the delta profile. Operators sometimes read that decline as reassurance. It is not: gamma is falling while vega and delta are both growing, so the position is becoming less twitchy per point of spot and more exposed per point of volatility at the same time.
The two effects multiply rather than add
In a real dislocation these do not arrive one at a time. Spot falls, so vanna increases vega. Volatility rises, so volga increases vega. Both are happening in the same hour, in the same direction, on a position that was sized against a vega number computed before either started.
That multiplication is the mechanism that converts a manageable loss into a terminal one. It is not an exotic tail scenario and it does not require a market event of historic size. It is what an ordinary dislocation does to an ordinary short premium book, and it is invisible on every risk report that shows a single vega figure, because a single figure has no way to express that the figure is about to change.
Where this way of looking fails
Naming volga and vanna does not solve the problem, and it is worth being explicit about how a second-order view breaks down, because the failure modes are real.
- They are model outputs, not observations. Volga and vanna come from the same pricing model that produced the vega you already distrust. If the model's assumptions about the distribution of returns are wrong, the second derivatives are wrong in the same direction as the first, and often by more.
- The illustration holds spot constant, which never happens. The table isolates the volatility effect on purpose. In a live move the underlying is travelling at the same time, so the numbers a book actually experiences are a path, not a revaluation grid.
- A parallel shift in implied volatility is a convenient fiction. Surfaces do not move up in one piece. Skew steepens, term structure inverts, and the volatility attached to one strike can move several times as far as the volatility attached to another. A single implied volatility column cannot represent that.
- More greeks can produce false precision. A report with five sensitivities on it feels more rigorous than a report with two. It is still a local expansion around a current state, and every additional term extends the range over which the expansion is tolerable without ever making it exact.
- Offsetting the convexity is not free. The only instruments that carry positive volga and vanna are options, so reducing the convexity of a short book means paying premium for it, which changes the return profile of the whole position and introduces its own basis and liquidity problems. There is no version of this where the exposure simply disappears.
What the second-order view does give you is the right shape of the problem. A single vega figure implies that a ten point rise in implied volatility costs ten times what a one point rise costs. The table says otherwise, and it says it in a direction that is unfavourable to anyone who is short. Knowing that the estimate degrades as the move extends is more useful than any particular replacement number, because it tells you which way the error runs when the error matters.
The next question is what this costs in currency terms on a book rather than on a single contract, which is the subject of the chapter that follows this one.
Key points
- Vega is a local slope measured at the current implied volatility, not a fixed quantity that survives a repricing.
- In an illustrative revaluation of a 16 delta 30 day put, vega per contract rises 36 per cent as implied volatility goes from 17 to 47.
- The first volatility point costs $46 and the twentieth costs $61, so a linear estimate built at the starting level understates the total by a widening margin.
- Delta moves from minus 0.19 to minus 0.35 across the same revaluation with no change in the underlying price, which is vanna.
- Volga and vanna push in the same direction during a selloff, so their effects multiply rather than add on a short premium book.
- Second-order greeks describe the shape of the error but remain model outputs, and offsetting the convexity requires buying options rather than trading linear instruments.
Related questions
what is volga in options
Volga is the sensitivity of vega to implied volatility, that is, the rate at which an option's volatility exposure itself changes as volatility moves. It exists because option value is convex in volatility rather than linear in it. For a short option position volga works against the holder, because it means each additional point of rising implied volatility costs more than the point before it.
what is vanna and why does it matter to a short option book
Vanna is the sensitivity of delta to implied volatility, equivalently the sensitivity of vega to the underlying price. It matters because it means the directional exposure of an option can change when implied volatility reprices, with no move in the underlying at all. In a falling market, short out of the money puts drift toward the money through vanna at the same time as volatility is rising, so the two effects compound rather than offset.
is vega accurate for large moves in implied volatility
No. Vega is a first derivative, so it is only accurate for small moves around the volatility level at which it was measured. For large moves the true change in option value diverges from the linear estimate, and for a short position the divergence runs against the seller, understating the loss by a widening margin as the move extends.
why does gamma fall when implied volatility rises
Higher implied volatility widens the distribution of possible outcomes at expiry, which flattens the curvature of the option's delta profile around any given strike. In the illustrative revaluation of a 16 delta 30 day put, gamma falls from 0.0093 at 17.3 per cent implied volatility to 0.0046 at 47.3 per cent. Falling gamma should not be read as falling risk, because vega and delta are both growing over the same range.
can you hedge volga and vanna
Only with other options, because linear instruments such as futures and the underlying itself have no volatility convexity to offset. That means reducing volga and vanna exposure involves paying premium, which changes the return profile of the whole position and introduces its own basis, liquidity and model risk. There is no combination of linear hedges that removes the convexity of a short option book.
why do risk reports show only one vega number
Because a single figure is easy to aggregate across strikes, expiries and underlyings, and because it is the standard output of the pricing models most platforms use. The cost is that a single number cannot express that the number itself changes as market conditions change. That limitation is invisible in calm conditions and most visible during a dislocation, which is when the report is being relied on most.