On Wednesday 10 Jun 2026 the Nifty future stood at 23,240.10 and the 23,250 call on the 30 Jun expiry settled at Rs 321.65. Its delta was 0.50. A seller with no view on direction buys 0.50 of the future for every unit sold, and then delta no longer matters. Six days later, on Tuesday 16 Jun, the future had climbed 761 points. The call had gone to Rs 785.65, and the hedge earned most of that loss back: the net came to −Rs 82.92 a unit. A hedge does not remove the loss. It tells me which part of the loss is left, and that part has a formula.
3By the end you can
- Write the P&L of a delta hedged option as gamma times the gap between realised and implied variance.
- Rebuild a hedge ledger from dated settlement prices and read the option, hedge and net columns.
- Explain why hedging more often narrows the scatter, and what it costs in whole lots.
The idea, plainly
Module F21 gave delta, module I09 gave gamma, and module I12 added them across legs. A delta hedge sets the position delta to zero and stops there. Gamma, theta and vega stay. Of the three, gamma and theta are one thing seen twice: the price of the curve is the price of time.
The identity
Over a short step, a delta hedged option earns the following. Take half the gamma, multiply by the future squared, and multiply by the gap between the squared move that happened and the squared move the implied volatility had priced.
Hedged P&L = Gamma*F*F/2
* (r*r-s*s*dt)
The maths: why theta and gamma cancel
Value changes with the future, time and volatility. For a small step:
dV = Delta*dF + Gamma*dF*dF/2
+ Theta*dt + Vega*dvol
A hedge of minus Delta in the future removes the first term. The Black 76 equation ties the rest together: with the interest term set aside, theta equals minus half of s squared, times F squared, times gamma. Put that in and the two terms join:
dV = Gamma*F*F/2
* (r*r-s*s*dt)
Vega is left out because the identity holds at a fixed implied volatility. A change in implied volatility adds its own term, which is the subject of module A11.
Read 2 more paragraphs, 145 words
Here F is the future, r the fractional move in the step, s the implied volatility and dt the step in years. The buyer wins when the index moves more than priced, the seller when it moves less. The sign of the direction never enters. That is why a hedged option is a bet on volatility alone.
A small picture helps. Think of the hedge as a shop that keeps its shelf stocked to a target level. Each time the index moves, delta changes and the shelf is wrong by a little. Restocking after a rise means buying at the higher price, and after a fall means selling at the lower. A gamma buyer restocks in the direction that earns and a gamma seller in the direction that costs. The identity adds up those little restocking gains and losses, and the formula is their sum.
Three things decide the result
- Realised against implied. Over the life of the option the squared moves add up. If the index moved at the implied pace, the hedged result is near zero before costs.
- Where the index spends its time. Gamma is largest at the strike, so a move near the strike counts for more than the same move far from it. Two paths with the same realised volatility can pay differently.
- How often the hedge is reset. Between resets the delta drifts, and the drift is gamma doing its work. More resets narrow the scatter and add cost.
The delta line steepens as expiry nears, so the hedge needs resetting more often
Worked example: a sold call, hedged on three dates
Three NSE end of day files let me rebuild a hedge by hand: 10 Jun, 16 Jun and 29 Jun 2026, all for the 30 Jun expiry, plus the settlement on 30 Jun. Sell one unit of the 23,250 call. On each date, set the long future to the call's delta. Everything is per unit, and a lot is 65 units. The call had 20 days left on 10 Jun and 14 on 16 Jun. The 30 Jun contract settled on the index close, the rule for expiries before 7 Dec 2026.
| Date | Future | Call |
|---|---|---|
| 10 Jun | 23,240.10 | 321.65 |
| 16 Jun | 24,001.00 | 785.65 |
| 29 Jun | 23,972.90 | 723.55 |
| 30 Jun | 23,865.75 | 615.75 |
| Step | Option | Hedge | Net |
|---|---|---|---|
| 10 Jun to 16 Jun | −464.00 | 381.08 | −82.92 |
| 16 Jun to 29 Jun | 62.10 | −24.83 | 37.27 |
| 29 Jun to 30 Jun | 107.80 | −106.33 | 1.47 |
| Total | −294.10 | 249.92 | −44.18 |
The hedge was reset after a sharp first week, then the index drifted
Read 3 more paragraphs, 194 words
The call delta, which sets the hedge, was 0.50 on 10 Jun, 0.88 on 16 Jun and 0.99 on 29 Jun. From those marks, the profit and loss of each step in rupees per unit sold:
Read it row by row. In the first six days the future ran 761 points while the call's implied volatility of 15.1% had priced a one standard deviation move of only about 450 points over the six days. The identity, with the gamma of 0.000485 on 10 Jun, predicts a hedged result of −Rs 91.30. The ledger shows −Rs 82.92. The two are close. The difference is made of vega, which helped the seller because implied volatility fell to 13.7%, and of higher order terms from a move that is large for one step.
The next two rows are quiet. The future moved −28 points, then −107 to the settlement, and the hedge, long 0.88 then 0.99, lost a little on each. Meanwhile the call's time value melted away, and that was the seller's gain. The net of these two rows is the theta the seller was paid, less what the hedge gave up.
How a hedger runs it
There are two common rules for when to reset. A time rule resets at fixed times, for example every close. A band rule resets when the delta has drifted beyond a set width, for example whenever the position is off by a whole future lot. A band rule trades more in fast markets and less in quiet ones, and a time rule is easier to follow and to check. Both are choices about the trade off between scatter and cost, not about being right. The ledger above used a time rule with only the three dates the files give me, which is a coarse hedger. I do not claim it is a sensible one.
What the total says, and what it does not
The option alone ended at −Rs 294.10 a unit by settlement, or −Rs 19,117 a lot. With the three resets the result was −Rs 44.18 a unit, or −Rs 2,872 a lot. Hedging cut the loss because the index rose. It would have cut a gain by the same logic if the index had fallen. One option, one expiry and three reset dates are one path, not a result. A daily hedge on the same expiry would have given a different number, and the simulator below lets you see how different.
A hedged short call is a tent: it keeps the premium near the strike and loses on a big move
Lots, costs and the shape of the hedge
The hedge trades in whole lots of 65 units. One lot of the sold call has a delta of 32.6 units, about half a lot of the future, so one call lot cannot be hedged exactly. Two call lots against one future lot is the usual rounding (module I12 shows it). Each reset also costs money. STT on futures is 0.05% on the sell side, which on one lot at the 16 Jun price is Rs 780, before the exchange charge, stamp duty and brokerage (module F06). A hedge that is reset daily pays that cost daily. The costs are why the choice of hedge frequency is a trade off and not a rule.
The maths: how the scatter shrinks with the number of hedges
Each step adds an error equal to half gamma times F squared times (r squared less its expected value). Its size is in proportion to the step's share of the time, 1/n, and the n errors are close to independent. Their sum therefore has a spread of the square root of n times 1/n, which is 1 over the square root of n. In words: four times as many hedges halve the hedging error, and the mean stays where the identity puts it. That is the whole scatter only when the index moves at the implied pace. When it does not, a second part depends on where the index spent its time against the strike, and no number of resets removes it. The costs rise with n, so there is an n beyond which resetting more often only loses money.
What the identity leaves out
- Vega. The identity holds at a fixed implied volatility. Implied volatility itself moves between resets, and a seller is short vega (module I10 has the units). In the ledger it fell from 15.1% to 13.7%, which flattered the seller.
- Gaps. The identity assumes small steps. An index that opens far from the last close skips every price in between, and no reset can be made at those prices. Module A14 takes the gap risk of a selling book.
- Prices. The ledger uses end of day settlement prices, which NSE sets and which are not fills. Real trades meet a bid ask spread and some impact (module I06).
- Whole lots. A future trades in lots of 65 units. A position that needs a hedge of a third of a lot cannot be hedged to a third.
The common mistake
The claim: If I hedge delta every day, I keep the theta and the risk is gone.
The ledger above is a short call that was hedged, and it still ended at −Rs 44.18 a unit. Theta is a price quoted at a given implied volatility. The seller keeps it only when the index moves less than that volatility implied. At 15.1% on 10 Jun the call priced a daily move of about 184 points, and the first week delivered more than that on its biggest day. Hedging removes the direction. It does not remove the volatility. The question page What is delta neutral and delta hedging? has the short version.
Key takeaways
- A delta hedge removes direction, not gamma, theta or vega.
- Hedged P&L is half gamma times the gap between squared moves.
- The seller keeps theta only if the index moves less than implied.
- The sold 23,250 call, hedged on three dates, still ended at −Rs 44.18 a unit.
- More resets narrow the hedging error by the square root of the count.
- Futures trade in whole lots, so a one lot hedge is rounded.
The term this section turns on: Delta hedge
Before you trade. Derivatives trading carries a high risk of loss. SEBI's study "Profitability of Individual Traders in the Equity Derivatives Segment (FY25 to FY26)" (August 2026) found that 87.7% of individual traders in equity F&O made net losses in FY26, with aggregate net losses of Rs 91,685 crore. Read the SEBI study