Guide · Derivatives mechanics · Updated 2026-09-19

Theta is a rate that changes, and almost all of the change arrives in the last three sessions

Every figure below is computed from the Black-Scholes formula on stated inputs: an index at 25,000 points, a volatility input of 14 per cent a year, a financing rate of 6.5 per cent, no dividend, and a clock of 252 trading sessions in a year. Those are a worked example, not a quotation from any market. The point amounts belong to them; the percentages, as a later section computes, barely move when they change.

A rent is a constant. Time value is a square root.

The rent metaphor is not merely loose, it inverts the behaviour. Rent is a fixed charge per period, and the period at the start costs the same as the period at the end. Option time value does the opposite, for a structural reason rather than a conventional one.

An option is paid for the distribution of index levels still reachable before settlement. Under the model's assumptions the standard deviation of the index over a horizon grows with the square root of that horizon, so four times the time buys twice the spread of outcomes, not four times. At the money the option is very nearly proportional to that spread, and therefore to the square root of the time left.

A quantity proportional to the square root of time has a peculiar property: the fraction destroyed by one more unit of time depends on how much time is left. Going from 30 sessions to 29 removes 1.98 per cent of what is standing. Going from two sessions to one removes 30.33 per cent. Going from one to none removes all of it. The rate is not a number. It is a function of where you are.

The computed figures sit slightly above the pure square root throughout, and the gap is not noise. Option value also carries a financing term on the discounted strike, which adds a near constant amount per session however much time is left. A later section separates the two.

The curve, computed

Read the last column downward. A flat rate of 19.41 points a session fairly describes exactly one session in the contract's life, the 1th from the end, where the multiple crosses one. Before that the contract decays more slowly than a rent model says. After it, faster, and then very much faster.

The fifth column drops the financing term and keeps only the square root. It carries almost the whole of the behaviour, and the residual is that constant charge, which weighs more the more time is left.

With half the contract elapsed a flat model has written off half the premium and the computed value has lost 33.0 per cent. That gap of 17.0 percentage points is a misreading of the position's state, and it runs the same way for anyone marking by rule of thumb rather than by the model.

The last three sessions do the work of 8

Concentration is the practical consequence of curvature. The final five sessions of this contract remove 213.08 points, which is 36.6 per cent of the opening premium, against the 16.7 per cent a flat rate predicts. The final three remove 162.15 points, or 27.8 per cent, which is the same amount a flat rate would charge over 8.4 sessions.

A contract that has run four fifths of its life still holds most of the decision in front of it. Someone who has held a position for weeks without much happening has not been sitting through the expensive part of the contract. They have been sitting through the cheap part, and the expensive part has not started.

Tripling the volatility input moves the second last session's share from 30.73 per cent to 29.77 per cent. Changing the index level moves it not at all, because the level cancels out of a ratio. The percentages here are properties of the model's shape, not artefacts of the numbers chosen to demonstrate it. The point amounts are, and should never be lifted off this page.

The money and the wings are not the same instrument

The claim that at the money decays fastest is true and misleading at the same time, because it is true of one measurement and false of another, and the two measurements lead to opposite conclusions about risk.

In absolute points the at the money strike has the largest theta on the table, 22.98 points a session. In proportional terms it has close to the smallest, 10.8 per cent of its own time value, and it holds by far the largest share of that value on the final morning: 42.8 per cent, against 2.2 per cent one standard deviation out of the money and 0.0 per cent two out.

The wings are not slow decayers, they are small ones. A strike two standard deviations out of the money is worth 5.22 points with five sessions left and essentially nothing with one. Calling that slow because the point amount is small confuses the size of a quantity with the rate at which it is going.

The last column warns about the in the money side. For the strike 1,000 points in the money, only 24 per cent of quoted theta is the decay of optionality at all. The rest is the financing charge on the discounted strike, which is not time value going anywhere. A screen showing a large theta on a deep in the money option is mostly reporting interest.

The weekend is neither free nor two days

The weekend argument is usually conducted between two positions that are both wrong. One says no decay happens because the market is shut. The other says two full days disappear because the calendar says so. Both mistake a modelling choice for a fact.

The market prices the variance it expects to be realised between now and settlement. No trading variance is realised while the exchange is closed, so a weekend is worth less than two sessions. But information does not stop: policy statements, overseas sessions, results and events land across a weekend and arrive in the Monday opening auction, so a weekend is worth more than nothing. Call its contribution w, in ordinary trading sessions. Then w equals zero is the pure trading clock, and for a contract expiring the following Tuesday, w equals two reproduces the pure calendar clock exactly. The table takes the Friday close as the observable, fits the volatility input to it under each assumption about w, and reports Monday's close.

The Monday close ranges from 91.20 points to 62.08 points, a spread of 31.9 per cent, from one assumption that never appears on a screen. Nothing in the market moved in any row: the index, the strike and the Friday price are identical throughout by construction.

The last two columns explain something practitioners see constantly and misattribute. Read on a calendar clock, the same option shows 11.67 per cent on Friday and 15.06 per cent on Monday at a weekend weight of half a session, a jump of +3.39 volatility points. That is not the market repricing risk. It is the calendar clock having removed three days when the market removed the equivalent of one and a half sessions, with the volatility number absorbing the difference. The direction reverses going into the weekend, which is where the impression of a Friday afternoon volatility collapse comes from.

So a Monday quote cannot be projected from a Friday quote without taking a position on w, and that position is a view about how much can happen while the market is shut. A weekend carrying a scheduled policy decision and a quiet one are not the same object, and no clock convention knows the difference.

In 2025 the weekend moved, and it moved toward the cliff

By a circular dated 26 May 2025 the market regulator required every recognised exchange to settle all of its equity derivative contracts on one of two weekdays, Tuesday or Thursday, and to seek prior approval before changing that day. Each exchange keeps a single weekly benchmark index options contract expiring on its chosen day, and every other contract, index futures and single stock derivatives included, carries a minimum tenor of one month and settles in the last week of the month on the same weekday.

From 1 September 2025 the two main exchanges swapped. One moved its weekly and monthly settlements from Thursday to Tuesday and the other took Thursday, with contracts expiring on or before 31 August 2025 keeping their original days and live contracts re-dated after the close on 28 August 2025. Almost every page on option decay written before that date, and a good number written since, assumes a Thursday expiry. That assumption is now wrong for one of the two exchanges, and wrong about the most consequential detail of all: where the weekend falls relative to expiry.

Under a Thursday expiry the final five sessions run Friday, Monday, Tuesday, Wednesday, Thursday, so the weekend sits between the first and second with four sessions still standing behind it, on the shallow part of the curve. Under a Tuesday expiry they run Wednesday, Thursday, Friday, Monday, Tuesday, so the weekend sits between the fourth and fifth with two sessions behind it, on the steep part.

At a weekend weight of half a session the same weekend removes 11.16 per cent of a Tuesday-expiring contract's Friday premium and 6.36 per cent of a Thursday-expiring one's, a ratio of 1.76. The weekend has not changed. The exchange calendar moved it to a steeper part of the curve, and the arithmetic did the rest.

The third column is the case worth marking. Where a designated day is an exchange holiday and settlement moves to the preceding trading day, a Tuesday cycle lands on the Monday. The weekend then stands immediately in front of expiry with one session behind it and removes 18.91 per cent of the Friday premium. The holiday calendar, which most people treat as administrative detail, is a first order input into the final week. Confirm the settlement date for the specific series against the exchange's own calendar rather than counting weeks forward, and read the Sources note below on what that rule rests on here.

Collecting theta is the curvature term with the sign reversed

Under the model, theta is not an independent quantity that happens to be related to gamma. It is gamma, restated. For a European option the relationship is exact:

theta = one half times the squared volatility times the squared index level times gamma, plus the financing term on the discounted strike.

With 30 sessions left the curvature term is 69.0 per cent of theta. With one session left it is 93.0 per cent. The instrument whose decay is fastest is, necessarily and by the same arithmetic, the instrument whose gamma is largest. They are one measurement written two ways.

This is what makes collecting theta an accounting description rather than a source of return. Theta is the deterministic part of a change in value whose other part is driven by the same curvature, and the model sets the two to offset when the index moves in each session by precisely the amount the volatility input implies. What changes hands when an option is traded is a claim on realised variance against a price for variance, plus the liquidity to strike the trade at all. The identity says nothing about which term turns out larger over any particular week, and nothing here should be read as suggesting it does.

It also disposes of the idea that fast decay and large risk are opposite conditions to be traded off. They are the same condition on two axes, and the point at which one peaks is the point at which the other does.

The move that cancels a session

Both sides of that identity can be put in one unit. For each session count, solve for the index move that leaves the option's value exactly unchanged from one close to the next, so the curvature gain and the time loss offset.

With 20 sessions left the option is unchanged if the index moves 23.7 points, which is 0.095 per cent. On the final session it needs 91.2 points, or 0.365 per cent. Both the decay and the movement required to cancel it grow together, which is the same statement as the previous section made about theta and gamma, expressed in index points instead of greeks.

A quoted theta is a slope, not a forecast

The failure mode generic pages omit is the one that bites hardest in the window they spend the most words on. Theta as quoted is a first derivative, the rate of change of value with respect to time at one instant. Extrapolating it across a whole session assumes that rate holds for the session. It does not, and the curve bends in the direction that makes the error worst near expiry.

Twenty sessions out, the slope is a fine approximation and the error is 0.9 per cent. On the final session it predicts 47.24 points against an actual 91.21, a ratio of 1.93. For a pure square root the limiting ratio is exactly two, and the computed figure approaches it because that is what the mathematics requires: the slope at the start of the last interval is half the average slope across it.

Two corollaries follow. The quoted number is least reliable exactly when a position's fate is being decided, so projecting a day ahead from it in the last two sessions uses it outside the range where it means anything. And platforms differ: some print the analytic derivative, which carries the error above, while others print a full revaluation one day forward, which does not but instead assumes the index and the volatility input are unchanged tomorrow. Which one a screen shows changes how its number reads, and the difference is largest in the final week.

What this computation does not contain

Everything above holds the volatility input fixed. Real markets do not. Observed decay in a price is the sum of the time effect computed here and a volatility effect that can be larger than it, particularly around a scheduled event, and the two are routinely confused. A premium that fell further than this page's curve says is not evidence that decay accelerated; more often it is evidence that the volatility input came down.

One volatility for every strike is also a modelling convenience. Across a real surface the wings carry different inputs from the money, which changes their point amounts, though not the result that the wings surrender a far larger share of a far smaller number.

Three further limits. The model assumes no dividend, which near a heavy distribution period shifts the forward and with it the strike that is genuinely at the money. Settlement mechanics are ignored, and Indian single stock derivatives settle physically, which makes the last session of an in the money contract a delivery question as well as a pricing one. And the clock of 252 sessions is a convention: a different session count rescales the point amounts and leaves every percentage here where it is.

Within those limits the arithmetic supports something narrow and useful. It says where on the curve a position sits, what the next session costs in proportional terms, how much of that is optionality rather than financing, and how much of the week's decay is parked on one side of a weekend whose calendar position changed in 2025. It does not say what any of that is worth.