Calculators and tools/Derivatives and Quant

Delta hedge simulator: why hedging an option pays the gap between implied and realised volatility.

Sell an option, hedge it with the future, and the result follows the difference between the volatility you sold it at and the volatility the index then delivered. The simulator runs seeded paths so you can change the hedging frequency, the implied volatility and the realised volatility and see the scatter.

Model, not market data
Seeded paths
Hedge frequency

ExampleStarts the forward, strike, days and volatilities from the fileOptionPositionForwardThe future of the exampleStrikeDays to expiryHedges over the lifeEvenly spaced, to the option's deltaImplied volatility, vol pointsPrices the option and sets the hedgeRealised volatility, vol pointsThe example's own: 12.82 over its 13 sessionsSimulated pathsSeedThe same seed gives the same paths

PremiumRs 321.32 a unitRs 20,886 a lot of 65 units
Average P&LRs 2,982a lot, over 2,000 paths
Spread of P&LRs 3,656one standard deviation, a lot
Paths with a loss21%5th to 95th: −2,533 to 9,375

Sold the 23,250 call, 20 days to expiry, forward 23,240.10, hedged 20 times over its life with the future. Priced at 15.03 vol points while the index moves at 12.82. The option was sold for more volatility than the index delivered, so the hedger keeps about the difference in what the option is worth at the two volatilities. That yardstick is Rs 3,118 a lot; the simulation averages Rs 2,982. The rest is path luck, which hedging more often removes only slowly.

Model, not market data

More hedges narrow the spread: Rs 13,859 with one hedge, Rs 1,691 with 160

The same paths and the same volatilities, hedged more and more often. The spread falls roughly with the square root of the number of hedges, and never to zero. Inputs: forward 23,240.10, strike 23,250, 20 days, implied 15.03 and realised 12.82 vol points, 20 hedges, 2,000 paths, seed 1, rate zero. Source: RupeeCase seeded simulation, Black 76. As of 10 Jun 2026.

Model, not market data

The seller's average P&L crosses zero where realised volatility meets the implied 15.0

Average over 1,000 paths for each realised volatility, hedged 20 times. P&L is rupees for one lot. Inputs: forward 23,240.10, strike 23,250, 20 days, implied 15.03 and realised 12.82 vol points, 20 hedges, 2,000 paths, seed 1, rate zero. Source: RupeeCase seeded simulation, Black 76. As of 10 Jun 2026.

Model, not market data

Five of the 2,000 paths the hedge is tested on

Each path is one random draw at the realised volatility. The hedge buys or sells the future as the option's delta changes along the path. Inputs: forward 23,240.10, strike 23,250, 20 days, implied 15.03 and realised 12.82 vol points, 20 hedges, 2,000 paths, seed 1, rate zero. Source: RupeeCase seeded simulation, Black 76. As of 10 Jun 2026.

Model, not market data

21% of the 2,000 paths lose money; the best is Rs 14,815 and the worst −Rs 17,216

Paths grouped by P&L for one lot, with the chosen number of hedges. Inputs: forward 23,240.10, strike 23,250, 20 days, implied 15.03 and realised 12.82 vol points, 20 hedges, 2,000 paths, seed 1, rate zero. Source: RupeeCase seeded simulation, Black 76. As of 10 Jun 2026.

Method: the forward moves in 160 equal steps as a driftless lognormal at the realised volatility. At each hedge time the hedger holds the future in an amount equal to the option's Black 76 delta at the implied volatility (the opposite sign for a sold option), and the future is held until the next hedge. The P&L is the premium less the payoff at expiry plus the gains on the future, per unit, for a sold option, and the mirror image for a bought one. Hedges are placed on the 160 step grid, so a count that does not divide it is rounded to the nearest one that does. The rate is zero, which keeps the arithmetic plain; a futures hedge needs no financing. Not shown: transaction costs, which grow with the number of hedges, jumps, and the margin the exchange blocks. This page cannot trade.

Starting values come from a dated NSE chain; the paths are a model, labelled Model, not market data. No costs, gaps or margin. Not a recommendation, and this page cannot trade.

How to read it

The set up

The simulator prices an option with Black 76 at an implied volatility and holds the future in an amount equal to its delta, resetting it a chosen number of times between the start and expiry. Between resets the future is left alone. The index moves along a random path at the realised volatility.

What decides the result

A hedged option makes or loses about the difference between what it is worth at the implied and at the realised volatility. Selling at 15 vol points when the index moves at 12 earns roughly that gap on average; the other way round, it loses. The yardstick on the page is that price difference.

Why the paths matter

On any one path luck dominates. Hedging more often shrinks the scatter, roughly with the square root of the number of hedges, but never removes it. The seed fixes the paths so a change you make is the effect of the change, not new luck.

Where it is taught

The dynamic delta hedging module derives the P&L identity and the gamma scalping module puts the same idea from the buyer's side.

What it leaves out

Real hedging pays costs on every trade, so more hedges are not free. The index gaps, volatility itself moves, and the option's implied volatility changes while it is held. Margin is blocked on a sold option and on the future. The simulator holds the rate at zero and the implied volatility constant, hedges at a fixed set of times and has no limit on position size. It shows the mechanism, not what any trader earns.

Go to the source

Does hedging remove the risk of selling an option?

It removes the first order exposure to the index, not the exposure to volatility. The result still follows the gap between implied and realised volatility, and the scatter on any one path stays large.

Why does hedging more often not give zero risk?

Because the hedge is reset only at the chosen times, and the index moves in between. The scatter shrinks with the square root of the number of hedges, so doubling it does not halve the risk.

Why is this called a model?

Because the paths are random draws from a lognormal model, not market data. Only the starting forward, strike, days and volatilities come from a dated chain.

LLearn it in depth

Module 14.15Dynamic delta hedging and the P&L identityAdvanced, Path 14 Module 14.16Gamma scalpingAdvanced, Path 14 Module 17.18P&L attribution by GreeksAdvanced, Path 17 ToolPayoff builderLegs from a dated chain Pillar guideDerivatives and QuantPaths 12 to 19

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Education only, not investment advice, a research report or a recommendation to buy, sell or hold any security. RupeeCase (QC Alpha Technologies Pvt Ltd) is not registered with SEBI as an investment adviser or research analyst. Contracts and prices shown are for illustration only, come from NSE end of day files at least 30 days old, and are not recommendatory. Derivatives trading carries a high risk of loss.