Guide · Options and volatility · Updated 2026-09-19

Vega is not one number, and adding it across expiries is adding different units

Every figure here was computed from Black and Scholes on a stated grid rather than quoted: a broad index at an illustrative level of 24,000, a base volatility of 14 points flat across the curve, a continuously compounded financing rate of 6.5 per cent, and no distribution yield. All figures are illustrative.

A far dated option is not a longer bet on the same thing

The instinct that a three month option is a one month option with more time attached survives until the first time somebody hedges one with the other. It is wrong in a specific and measurable way, and the measurement is the whole subject.

Vega is the change in model value for a one point change in implied volatility. In the Black and Scholes model it is the index level, multiplied by the square root of the tenor, multiplied by the standard normal density at the usual moneyness term. The square root is there because volatility is quoted as an annual rate while what prices a contract is the standard deviation of the move over its life, which is that rate times the square root of the tenor. Uncertainty accumulates with the square root of time, so four times the tenor is twice the uncertainty. The density term is why growth is slightly slower than a pure square root: the moneyness term itself grows with the tenor and pushes the density down.

The square root of time, computed on a grid

Read the second and third columns together. At fourteen days the computed ratio is 1.410 against a pure square root of 1.414. At a year it is 6.278 against 7.221, a shortfall of about 13 per cent. The square root is the mechanism and the density term is the correction, so anybody using the square root as a mental shortcut at long tenors overstates the far leg's sensitivity by roughly that much.

More vega and less gamma is a substitution, not an increase

The last column of that table is exact rather than approximate. Divide vega by gamma in this model and every density term cancels, leaving the index level squared multiplied by volatility and by the tenor. That identity says the ratio of volatility exposure to path exposure grows linearly in time to expiry: move from a seven day contract to a ninety day one and the ratio rises by a factor of about 12.9, which is the tenor ratio exactly. The far dated contract has roughly 3.47 times the vega and about 3.70 times less gamma per unit, and those are two faces of the same square root.

So the common claim that far dated options are riskier is not wrong so much as unfinished. A trader who lengthens tenor to quiet the daily noise of gamma has bought a position that cares about something else entirely, and has usually not written down what that something else is. Neither contract is the safe one. Their risk arrives through different channels.

Vega across strikes collapses at a rate that depends on the tenor

Vega is not only a function of time. The density term falls away from the money, and how fast it falls depends on the same square root. At short tenors the distribution of outcomes is narrow, so a strike a few per cent away is nearly unreachable and barely sensitive to volatility. At long tenors it is comfortably inside.

The first row is the one to sit with. Six per cent out of the money with a week to run, the computed vega is 0.06 index points per volatility point, which is 0.5 per cent of the at the money figure. With a year to run the same relative strike carries 59.45, or 71.6 per cent of its own at the money figure.

Two things follow. A short dated position away from the money holds far less volatility exposure than its contract count suggests, which is why such a book can look diversified across strikes and behave as though it held almost nothing until the index moves and the strikes come alive. And a long dated wing carries most of an at the money contract's exposure, so vega on a near dated book sits at one strike while on a far dated book it is smeared across the surface.

The curve does not move in one piece

Everything so far treats volatility as a single number that could move. It is not a single number. It is a curve with one point per available expiry, and the points do not move together. This is the part that generic treatments skip, and it is the part that decides what a two leg position actually does.

The reason is structural rather than empirical. An implied volatility for a contract expiring in time T is, to a good approximation, the average of expected instantaneous variance over the interval from now to T. If variance reverts toward a longer run level at speed k, then expected instantaneous variance at a future time t is the long run level plus the current deviation decayed by the factor exp of minus k t. Averaging that over the interval gives the propagation factor directly: a shock of size v to instantaneous variance today changes the average variance for tenor T by v multiplied by one minus exp of minus k T, all divided by k T.

That factor is one at the very front and falls toward zero at the back. It is not an assumption about how markets behave. It is what averaging does to a quantity that decays, and it is why a parallel shift in a volatility curve is not a thing that can happen while variance reverts at all.

The fourth column is the uneven propagation, and it is severe. The front moves 6.00 points and the one year point moves 1.44, a ratio of about 4.2 to one. Anybody treating a volatility move as a level shift has already made an error of that size before doing any arithmetic.

The last column is the result almost nobody shows, and it runs the other way. Vega rises with the square root of the tenor while the move falls faster than that, so their product is not monotone. It rises from 79.3 at seven days to a maximum of about 177.1 near 96 days, then falls back to 119.8 at a year. The most exposed point on the curve, measured in money rather than in volatility points, is in the middle.

That maximum is not a constant of nature. It is a function of the mean reversion speed, the one input here that cannot be read off a screen. Slow reversion pushes the peak toward the back of the curve and fast reversion pulls it toward the front.

What a calendar spread is actually long

A calendar spread is long one expiry and short another at the same strike. Described that way it sounds like a position on time. In volatility terms it is something more specific: it is long the far leg's vega and short the near leg's vega, which makes it long the difference between two points on a curve.

Write the volatility term of its value change and the structure is visible immediately. It is the far leg's vega multiplied by the far leg's volatility move, minus the number of near legs multiplied by the near leg's vega and the near leg's volatility move. Two vegas and two moves, and the moves are not equal.

Now size the position so that the vegas add to zero, which is what a risk report that sums vega across expiries will tell you to do. That means the number of near legs is the far leg's vega divided by the near leg's vega. Substitute it back and the near leg's vega cancels completely. What is left is the far leg's vega multiplied by the far leg's move minus the near leg's move.

That expression is exact, and it settles the question. A calendar sized to be vega neutral by addition is not neutral to anything. It is short the spread between the two volatility moves, one for one, at a size equal to the far leg's vega. Since a front led shock always makes the near move larger, this position is structurally exposed to exactly the shock people buy calendars in anticipation of.

Read the second and third rows against each other. The position with a vega sum of exactly zero moves by -98.26 index points. The one that does not move at all has a vega sum of +16.38, which a report adding vega across expiries would flag as an open exposure. The measure and the risk point in opposite directions, and the measure is the one that is wrong.

The plain one for one calendar in the first row matters for a different reason. Its vega sum of +32.68 is positive, so it is called long volatility, and under this shock its volatility term is positive too. That agreement is a coincidence of this shock's shape. Make the front move sharply enough relative to the back and the same structure moves the other way while its vega sum still reads positive.

The hedge ratio is a view, and it should be written down as one

If adding vega is the wrong instruction, the obvious question is what the right ratio is. The honest answer is that there is no shock free ratio, because the ratio that leaves a position unmoved depends entirely on the shape of the move, and the shape depends on a mean reversion speed nobody can read off a screen.

The fourth column spans 0.964 to 2.880 near legs per far leg, a factor of about 3.0, and the only thing that changed across those rows is an assumption about how quickly a variance shock fades. The last column is the price of ignoring the question: the same addition neutral position that moves by -46.9 index points under slow reversion moves by -198.9 under fast reversion.

There is no computing your way out of this. What can be done is to state the assumed shape, size to it, and know the direction of the error if the shape is wrong. A position sized for slow reversion is short too little of the front if reversion turns out fast. That is a knowable sensitivity rather than a surprise, and the difference between the two is the whole of the practice.

Decay in the passage of time is a separate sensitivity with its own non linear behaviour, and it bears on this because the near leg is also the leg whose value erodes fastest. It is treated on its own terms elsewhere, since mixing the two is how a volatility result gets attributed to time.

Which points exist on the Indian curve, and what that permits

All of the above assumes the tenors are available. In India they are not, and the constraint tightened in 2025.

By a circular on the final settlement day for equity derivatives contracts, issued in May 2025, expiries of all equity derivatives contracts on an exchange are limited to a single weekday, either Tuesday or Thursday, chosen by that exchange. Each exchange may maintain one weekly benchmark index options contract on its chosen day. Every other equity derivatives contract, which is to say benchmark index futures, non benchmark index futures and options, and single stock futures and options, is to be offered with a minimum tenor of one month, with expiry in the last week of every month on that chosen day. Exchanges were required to submit proposals by 15 June 2025, and an exchange must now obtain prior approval before modifying the settlement day of its derivatives contracts.

The consequence is concrete. There is exactly one instrument per exchange with a point at the short end, so a near leg with days rather than weeks to run comes from that series or it does not exist. Every other leg starts a month out, and successive far legs sit a month apart because they all fall in the last week of their month. The tenor gap in a calendar is therefore chosen from a short list rather than tuned to a view.

That matters because the propagation table above shows the near leg is where the shock is largest and the vega is smallest. Removing short dated points removes the leg doing the work on one side of the structure. Any article written before May 2025 that describes a ladder of weekly expiries across several instruments is describing a curve that no longer exists. Lot sizes bite here too: the legs are separate contracts, so a computed ratio will rarely be a whole number of lots.

The failure mode, stated plainly

The commonest way a supposedly hedged volatility position turns out not to be hedged is that its vegas were added across expiries as though they were the same unit. They are not, and the arithmetic here gives the size of the error rather than just its direction. On the stated shock the front moves about 1.6 times as far as the ninety day point, and a calendar sized until its vegas sum to zero carries a value change of -98.26 index points per unit of far leg from the volatility term alone. A risk report adding those vegas reports zero. The difference is not a modelling nicety. It is the position.

Three checks follow, and none needs a model beyond the one used here. Never report a single net vega for a book spanning expiries without also reporting it bucketed by tenor, because the net figure destroys the only information that matters. State the shock shape whenever a sizing is quoted, because a ratio without a shape is not a hedge ratio. And read the product of vega and the expected move at each tenor rather than either alone, since the point carrying the most exposure in money is usually neither the one with the most vega nor the one with the largest move.

None of the figures here describe what any position earns. They are one term of a sensitivity calculation on a stated grid, with the index level, the passage of time, financing and every other input held fixed. A real position experiences all of those terms at once, and the volatility term is simply the one most often measured in a unit that does not add.