An index options book has delta of plus 800 index units, gamma of minus 1.3 units per point and vega of minus Rs 3 lakh per vol point, with the index at 22,000. Build the P&L grid for index moves of minus 2%, 0 and plus 2% and volatility moves of minus 2, 0 and plus 2 points.

The worst cell is the index down 2% with volatility up 2 points, a loss of about Rs 10.78 lakh; the best is up 2% with volatility down 2, a gain of about Rs 8.26 lakh. A 2% move is 440 points: delta is worth Rs 3.52 lakh, gamma costs Rs 1.26 lakh either way, and each 2 points of volatility is Rs 6 lakh. The worst cell is also the likely one, because index sell-offs usually lift implied volatility.

A household budget hit by a price rise, a pay cut and a rent increase adds the three effects to find the month's shortfall. A book's profit for small moves is the same kind of sum: delta times the index move, plus half of gamma times the move squared, plus vega times the change in volatility. A 2% move on 22,000 is 440 points. Delta of 800 units makes Rs 3.52 lakh on a rise and loses it on a fall. Gamma of minus 1.3 costs half of 1.3 times 440 squared, Rs 1.26 lakh, on a move in either direction. Vega of minus Rs 3 lakh makes Rs 6 lakh if volatility falls 2 points and loses Rs 6 lakh if it rises 2.

The grid treats the nine cells as equals, but markets do not. When an equity index falls sharply, implied volatility usually rises, so the down-2%, volatility-up cell is the ordinary shape of a bad day, not a coincidence of two separate shocks. The up-2%, volatility-down cell, the book's best, is the ordinary shape of a calm rally. A manager who reads the grid as nine equally likely outcomes would underweight the bottom-left corner, which is exactly where this book is hurt most: Rs 10.78 lakh against Rs 4.78 lakh if volatility had held still.

Long delta tilts the curve up to the right; short gamma bends it down at both ends. The two balance at about 615 points, a 2.8% rise, where the book makes its most, about Rs 2.5 lakh. On the downside the two Greeks point the same way, so there is no fall the book profits from, while on the upside it only gains until the bend overtakes the tilt at about a 5.6% rise. This is the profile of a desk that has sold options to clients and is carrying some net long delta, perhaps by design, perhaps by drift.

Separate what can be hedged cheaply from what cannot. Selling 800 units of index futures removes delta at almost no cost; the worst cell then becomes Rs 7.26 lakh, gamma plus vega, and the up-2% cells lose that protection too. The rest of the worst cell is short vega and short gamma, which only buying options can reduce, and that costs premium and time decay. Then say the limits plainly. The grid is a second-order Taylor estimate: it ignores how vega itself changes when the index moves, the cross effect between price and volatility, and a day's theta, which for a short-gamma book is a gain. For 2% moves it is a fair first read; for a 5% gap the full revaluation would be needed, because the gamma figure itself will have changed by then.