Guide · Derivatives mechanics · Updated 2026-09-19

Short convexity is a position whose hedge is always wrong in the direction that hurts

Every rupee figure here was computed from one stated position under one stated model, with the inputs printed below so the arithmetic can be rebuilt rather than believed. All figures are illustrative of the shape of the exposure, not a measurement of or forecast about any position.

Gamma is the rate at which a hedge goes stale

Delta answers one question: how many units of the underlying does this position behave like. A short call with a delta of 0.529 per unit behaves, for a small move, like being short 39.7 units of the index. Buy that many and the position is, for an instant, indifferent to direction.

Gamma answers how fast that number is changing. It is the derivative of delta with respect to the underlying, and for a short option it is negative. When the underlying rises the call gains delta and the short position loses it: you are shorter than you were, and restoring the hedge means buying above the price you last traded. When it falls the call sheds delta and the short position gains it: you are longer, and restoring the hedge means selling below it.

There is no path along this curve on which the re-hedge is favourable. That is short convexity, stated without metaphor. The position is not wrong because the market went the wrong way. It is wrong in both directions, and the only question is by how much.

The position, and every input behind it

The gamma line is the one everything below depends on. For every point the index rises this position loses 0.0641 units of delta, so across a two hundred and forty point move, one per cent, it acquires 15.4 units of index exposure on the wrong side without an order being placed.

The curvature, computed rather than asserted

Revalue the same option across a range of instantaneous moves and compare each result against what the starting delta alone predicted. The difference is the curvature, in rupees, and it is a cost to the seller at every move in both directions.

Every figure in the final column is a loss, on both sides of zero. That column is the curvature, and it is not proportional to the move: two per cent costs Rs 6,603 against Rs 1,779 for one per cent, near four times rather than twice. The straight line describes the position only in a neighbourhood, and the neighbourhood is small.

Run the same computation over one calendar day rather than instantaneously and the shape that matters appears, because it carries the credit for the day that passed.

The peak is Rs 1,197, what the day is actually worth if nothing happens. That is slightly more than the Rs 1,159 in the table above, because theta is the rate at the start and the rate accelerates into expiry, so a whole day realises a little more than the opening rate implies. The curve crosses zero at -0.77 and 0.80 per cent, and outside that band the day ends behind at an accelerating rate.

Theta is not the reward for gamma. It is gamma, priced

The most persistent error in this subject is the idea that a short option collects theta and suffers gamma, as though those were two accounts that could be managed separately. They are one quantity written two ways. Under this model the volatility part of theta is exactly minus one half of volatility squared, times the level squared, times gamma. Not approximately. Below is the value from the full pricing formula, the value from that expression, and the difference.

The two agree to a fraction of a rupee. What remains is the interest term, a financing effect with nothing to do with curvature. The credit is not payment for taking on curvature as a side effect; the credit is the curvature, priced at the volatility the model was given. Raise that input and both numbers rise together. No arrangement of strikes or expiries keeps one and discards the other.

One consequence is worth stating flatly. The breakeven move above is not an empirical finding about this position. Set one half of gamma times the move squared equal to one day of the volatility part of theta, cancel gamma from both sides, and what is left is the level times volatility times the square root of one divided by three hundred and sixty five. On these inputs that is 176 points, or 0.73 per cent, which is where the curve crosses zero. The breakeven is the one day implied move, by construction.

A gap is the same distance in its most expensive form

Option pricing rests on an argument about continuous hedging: that the seller can adjust at every price the underlying passes through, and that doing so reproduces the payoff. A gap is the event in which that fails. The price does not pass through the intervening levels, it arrives at the far end of them. The cost is arithmetic: curvature depends on the square of each unhedged step, so the same distance gets cheaper the more pieces it arrives in.

Twenty five small steps cost about 4 per cent of what one jump of the same size costs. That is the warning: a gap is not a larger ordinary move but a differently shaped event, and the difference is a multiplier rather than an increment.

The final column is the one to sit with. A gap of 2 per cent consumes about 6 days of credit on a contract with seven days to live; a gap of 5 per cent consumes about 28. Credit accrues in a straight line with time, cost with the square of distance. Those rates do not stay in proportion, and a gap is where they come apart.

The fifth column warns about rules of thumb. The square-law estimate holds gamma at its starting value, true only for small moves; at larger ones gamma collapses as the option leaves the strike, so the estimate overstates the cost. It is the one approximation here that errs in the seller's favour.

The hedging argument assumes an open market

An overnight gap is not exotic in India. It is the ordinary way the market moves from one session to the next, because there is no session in between. The normal equity session runs six and a quarter hours, leaving a great deal of clock in which a price forms somewhere and no hedge can move. The structure of the Indian trading day is what turns a modelling assumption into a real exposure.

For 81.4 per cent of every week, continuous hedging describes something that cannot be done. A weekend alone is 65.75 hours, the equivalent of 10.5 sessions of clock, and a long weekend is longer. Decay is measured in calendar time, so the model pays across that span at the rate of a trading day. The credit arrives with no ability to hedge through it.

That is the clean version of the trade. Over a weekend the credit is three days of theta, but the move that consumes it is not three times the daily breakeven. It is the square root of three times it, about 1.27 per cent on these inputs, because the credit grows with time while the tolerance grows only with the square root of time.

The loss is not the mirror of the gain

Two asymmetries sit inside this position, and they are different from each other. The first is a ceiling. The most that can ever be received is the premium, Rs 15,063, fixed at the moment of sale. The curvature cost has no such ceiling. Here the instantaneous move at which curvature alone consumes the whole premium is about 3.22 per cent up or 3.06 per cent down. Double that move and the loss quadruples while the gain stays where it was. That widening is why the tail of the distribution does a disproportionate share of the work, and why behaviour inside the ordinary range says little about behaviour outside it.

The second asymmetry is directional, and it runs opposite to most intuitions. At 3 per cent the gap table shows Rs 13,431 up against Rs 14,520 down: the downside is worse here, because the hedge bought against the short call is long index and in a sharp fall it loses faster than the option can gain. The lesson is not that down is always worse. It is that the sides are unequal, that which one is worse depends on strike, hedge and horizon, and that a position called symmetric because it is delta neutral has been described only to first order.

Inside a ban, your allowance is a delta fixed at two o'clock

All of that assumes you can trade when you want to. In Indian single stock derivatives there is a defined state in which you cannot: the ban period, which begins when market wide open interest crosses ninety five per cent of the market wide position limit and lifts only when it falls back to eighty. Inside it, fresh positions and rollovers are barred and you may act only to reduce.

The rule changed in a way that matters specifically to short gamma. Under the framework in force from October 2025, open interest is measured at portfolio level as a delta adjusted figure, the future equivalent open interest, rather than as a count of contracts. The quantity you must reduce is therefore a delta, and a short option's delta changes on its own.

The mechanics are unusually specific, and they are where competing explanations of gamma go quiet. The position held when the stock entered the ban is stored as a base. On the following day both that base and your closing position are converted into delta terms using the same contract delta, taken from the clearing corporation's risk parameter file at two in the afternoon. If your closing delta is at or below the base with no change of sign, there is no violation. Two clauses follow and both are about gamma. A passive increase caused purely by the underlying moving is expressly not a breach, which is the framework conceding that your delta moves without you trading. And a change of sign is not accepted as a reduction, so a position cannot be hedged straight through zero in one step.

So between two o'clock and the close your real delta keeps moving and the yardstick does not. The clearing corporation's own guidance to members is to monitor that window against the two o'clock parameter file and place corrective trades before the close. A passive increase from near month contracts expiring on expiry day is treated as a passive breach rather than a violation.

Note what that does to the hedging argument. Continuous hedging assumes you may trade at every price; here the trades available are defined by a delta computed at a fixed moment in the afternoon, and the one you want, a full re-hedge through neutral, may be the one the rule declines. The mechanics of the ban period and what it actually restricts are worth reading alongside this, because the constraint binds hardest when the position most needs adjusting.

Margin moves on the side that is already hurting

A third element has nothing to do with the option's value. A short option carries a margin requirement set from the risk it currently represents. When the underlying moves against it the position becomes more sensitive and sits closer to the money, and the requirement rises. The cash call arrives on the same side as the loss, in the same session.

One component can be computed exactly, because it is a stated percentage rather than a scenario calculation. On the day an index option expires, an additional extreme loss margin of two per cent applies to short index option positions, both those open at the start of the day and those taken during it for that expiry. Here, on index exposure of Rs 18,00,000, that component alone is Rs 36,000 against a premium of Rs 15,063: about 2.4 times the entire amount the position was sold for.

The rest comes from a scenario based calculation that cannot be reduced to a percentage and is not computed here. The direction is the point: the requirement follows the risk you now carry, and short gamma means that risk grows whichever way the market has just moved. On how margin is computed and what maintenance means, separately: a position can be solvent and still be closed, because what binds is cash on the day rather than value at expiry.

What the premium is actually payment for

None of this argues against selling options. Short convexity is a real and legitimate position, taken deliberately by people who have worked out what they are accepting and sized it against capital they can keep committed. It is one of the few positions anywhere where what is sold can be written down precisely.

Written down precisely, it is this. You receive a credit accruing in a straight line with calendar time. You accept a cost growing with the square of the distance the underlying travels between the moments you can act. Those moments cover 18.6 per cent of the week. In certain restricted states the available trades narrow, measured against a delta fixed at two in the afternoon. And the cash demanded rises on the side that is already losing.

Whether that exchange is worth making depends on the distribution of moves that actually occurs, and nothing here claims to know it. What the computation establishes is the shape: linear in, quadratic out. Every honest account begins by conceding that shape and most published accounts do not. What follows is not a formula but a decision about how much quadratic exposure to carry, given a credit you can calculate exactly and a distribution you cannot. That decision is the skill, and it is unreachable for anyone who has not first done this arithmetic on their own position.