1. What you will learn
By the end of this lesson you will be able to say what delta and gamma actually measure, estimate an option's value change for a small move using both, explain why that estimate fails for a large move, aggregate Greeks correctly across contracts using the lot multiplier, and describe what a delta-neutral position is still exposed to. This is education about option sensitivities and is not advice to trade or to use any instrument.
2. The idea explained
Delta is the rate at which an option's price changes when the underlying moves by one unit. A call delta of zero point five means that a one-rupee rise in the underlying is expected to raise the option's price by about fifty paise, for a small move, holding everything else fixed.
Gamma is the rate at which delta itself changes as the underlying moves. A gamma of zero point zero four per rupee means that a one-rupee rise raises the delta by about zero point zero four.
Together they give a local approximation. The estimated change in the option's value is delta multiplied by the move, plus one half of gamma multiplied by the square of the move. The first term is a straight line; the second is the curvature correction. Because the correction involves the square of the move, it is always positive for a long option position, which is why a long option gains more on a rise than it loses on an equal fall.
The word local is the whole caveat. Both delta and gamma are computed at today's price, today's volatility and today's time to expiry, and all three change as soon as the market moves. For a small move the approximation is good; for a large one it is not merely imprecise but structurally wrong, because gamma itself will have changed substantially along the way. For a large move the correct procedure is to reprice the option at the new inputs, not to extend the formula.
Aggregation is where practical errors happen. Greeks are quoted per unit of the underlying, so a position's delta is the option's delta multiplied by the lot size multiplied by the number of lots. A portfolio delta of nine hundred means the position behaves, for small moves, like nine hundred units of the underlying.
Finally, what delta-neutral does not mean. A position with zero net delta is insensitive to a small immediate move in the underlying, and to nothing else. It can retain substantial gamma, so its delta will move as soon as the price does; it retains vega, so a change in implied volatility will move it; it retains theta, so the passage of time will move it. Neutrality is one local condition at one instant, not an absence of risk, and maintaining it costs money in spreads and charges every time the hedge is adjusted.
3. The market, the regulator and the rulebook
Exchange-traded options in India operate under the Securities and Exchange Board of India, the statutory securities regulator constituted by an Act of Parliament, which approves contracts and supervises exchanges, brokers and clearing members.
The exchanges publish each contract's specification, including the lot size that every aggregation depends on, the strike intervals, the expiry calendar, the exercise style and the settlement method. They also publish the option chain and, in some cases, model-derived figures.
The clearing corporations publish the margin framework, which for a hedged options position is computed on the portfolio rather than leg by leg. Never state a lot size, margin percentage, implied volatility level or charge as a fixed fact; describe the mechanism and cite the page with a date.
Worked example
4. Worked example
All figures are invented. A fictional call has a delta of zero point five zero and a gamma of zero point zero four per rupee.
For a two-rupee rise in the underlying, the delta term is zero point five zero multiplied by two, which is one rupee. The gamma term is one half multiplied by zero point zero four multiplied by two squared, which is one half multiplied by zero point zero four multiplied by four, or zero point zero eight rupees. The estimated change in the option's value is one rupee and eight paise.
Note also what happened to delta itself. After a two-rupee rise, the delta is about zero point five zero plus zero point zero four multiplied by two, which is zero point five eight. The position has become more sensitive to the underlying simply because the underlying moved.
Now attempt the same formula for a fifty-rupee move. The delta term would be twenty-five rupees and the gamma term one half multiplied by zero point zero four multiplied by two thousand five hundred, which is fifty rupees, for a total of seventy-five. That answer is not merely imprecise; it is meaningless, because a call's delta cannot exceed one, so the true change is bounded, and gamma would have fallen away long before the underlying had travelled fifty rupees. The correct procedure is to reprice at the new inputs.
Finally, aggregation. Suppose the lot size is one hundred units and twenty lots are held. The position delta is zero point five zero multiplied by one hundred multiplied by twenty, which is one thousand units of underlying equivalent, and the position gamma is zero point zero four multiplied by one hundred multiplied by twenty, which is eighty units per rupee. To be delta-neutral, one thousand units of the underlying would be sold. After a two-rupee rise the position delta becomes one thousand plus eighty multiplied by two, which is one thousand one hundred and sixty, so a further one hundred and sixty units would have to be sold to restore neutrality, and that adjustment costs spread and charges.
5. Common mistakes and how to fix them
The first mistake is treating delta as a probability. It is a rate of change from a model,.
The second is extending the local approximation to large moves. Set a threshold beyond which you reprice rather than approximate, and say what it is.
The third is aggregating without the lot multiplier. Multiply by lot size and number of lots, and state the result in units of the underlying.
The fourth is believing delta neutrality removes risk. It removes sensitivity to one small immediate move; gamma, vega and theta all remain.
The fifth is ignoring the cost of rehedging. Each adjustment pays a spread and charges, and a theoretical benefit smaller than that cost does not exist. Most active traders lose money, option writing exposes the writer to losses far exceeding the premium received, leverage magnifies losses as much as gains, past performance does not indicate future results, and a SEBI-registered investment adviser is the person to consult about an individual's own money.
Key takeaways
6. Board summary
Delta is the rate of change of the option price with respect to the underlying; gamma is the rate of change of delta. The local estimate is delta times the move, plus one half of gamma times the square of the move. Because the correction uses the square of the move, it is positive for a long option position in both directions. Greeks are per unit, so a position's Greek is the quoted figure times lot size times number of lots. Delta neutrality is one local condition at one instant, and maintaining it costs spread and charges every time.
Check your understanding
7. Practice and self-check
One. Delta is zero point four and the underlying rises three rupees. What is the delta term? One rupee twenty paise.
Two. Gamma is zero point zero two per rupee. What is the gamma term for that move? One half times zero point zero two times nine, which is nine paise.
Three. Estimated change in the option's value? One rupee twenty-nine paise.
Four. What is the new delta after the move? Zero point four six.
Five. The lot size is fifty and eight lots are held. What is the position delta before the move? Zero point four times fifty times eight, which is one hundred and sixty units.
Six. And the position gamma? Zero point zero two times fifty times eight, which is eight units per rupee.
Seven. After the three-rupee rise, what is the position delta? One hundred and sixty plus twenty-four, which is one hundred and eighty-four.
Eight. To stay delta-neutral, what must be done? Sell twenty-four further units of the underlying.
Nine. Why is the same formula unsuitable for a forty-rupee move? Because gamma itself changes substantially over such a move and delta is bounded, so the option must be repriced at the new inputs.