How it unfolds, step by step
The setup
An option is quoted at ₹80. Here are its sensitivities where it sits now. Delta is 0.50. Gamma is 0.01 for each ₹1 the underlying moves. Theta is −₹3 per calendar day. And vega is ₹5 per one percentage point of implied volatility. Implied volatility is the volatility that the market's own option prices point to, quoted as a percentage. Four numbers, and every one of them carries a unit. Per rupee of movement for delta. Per rupee of movement again for gamma. Per day for theta. Per percentage point for vega. The unit does as much work here as the number does.
What happens next
One day passes. The underlying rises ₹20. That is a large move, and in the favourable direction. At the same time, implied volatility falls from 20% to 18%. That fall can be described three ways, and all three are true. It is two percentage points. It is a ten percent relative decline. And on a decimal volatility scale it is 0.02. A vega quoted per percentage point takes −2. It does not take −10, and it does not take −0.02.
What it means — and does not
Add the four contributions one at a time. Delta gives 0.50 × 20 = +₹10. The gamma correction gives 0.5 × 0.01 × 20² = +₹2. Theta gives −3 × 1 = −₹3. Vega gives 5 × (−2) = −₹10. They add up to −₹1. So the estimate lands near ₹79 after a ₹20 move in the right direction. That does not show the contract is unfair. It shows that four sensitivities were pulling at once, and one of them pulled harder than the direction did. The result is also an approximation, and a local one. It leaves out higher-order and cross-input effects. And it gets worse as the inputs travel further from where the sensitivities were measured. It says nothing about the quote that would really have been available.
- Approximate premium (₹)
- Starting premium ₹80 · 80
The four contributions applied one at a time to a starting premium of ₹80 — delta +₹10, the gamma correction +₹2, theta −₹3, vega −₹10. Together they land near ₹79 after a favourable ₹20 move in the underlying. Step 1 is the starting premium. Steps 2 to 5 add delta, gamma, theta and vega in turn. The order is for display only: the four act at once. Illustrative arithmetic, not a price path.
Show the underlying data
| Step in the approximation | Approximate premium (₹) |
|---|---|
| 1 | 80 |
| 2 | 90 |
| 3 | 92 |
| 4 | 89 |
| 5 | 79 |
What this does not establish
The approximation stops describing the option as soon as the inputs have moved far enough to change the sensitivities themselves. Gamma is the obvious case. It exists precisely to say that delta is not constant. The same is true of all four. Close to and close to the they can change quickly. So a reading taken minutes ago is a snapshot, not a fixed property of the contract. What cannot happen is a Greek being refuted by an unexpected premium, because none of them predicted one. A sensitivity states what a model does when one input changes and everything else is held still. No real session ever works that way.
Things to check
6Is vega quoted per one percentage point of volatility, or against a decimal scale?
Which time convention is theta on — calendar days, trading days, or an annualised model figure?
Is a volatility change being stated in percentage points, or as a relative percentage?
Are all the units consistent before you multiply in quantity or lot size?
Is the output being described as an approximation, or as an expected result?
Remember: these are sensitivities measured now, not statements about what happens next.
Common mistakes
4Entering a 20%-to-18% fall as −10 because it is a ten percent relative decline. Against a vega quoted per percentage point the input is −2. The difference is a factor of five in the answer.
Treating delta as fixed across a large move. Gamma is the name for the fact that it is not, and that is why the correction term appears at all.
Reading theta as an amount that is simply taken off every day. It is a model sensitivity under a stated time convention, and its own value changes as the option ages.
Labelling the output of a Greek calculation an expected profit. It estimates a change in a theoretical price. A realised result comes from an entry fill and an exit fill.
Implied volatility falls from 20% to 18%, and the platform quotes vega per one percentage point of volatility. What is the input?
- −2, since the fall is two percentage points
- −10, since the fall is a ten percent relative decline
- −0.02
Show the answer
A. −2, since the fall is two percentage points
Both wrong answers are true statements about the same fall, wearing the wrong unit. Twenty to eighteen IS a ten percent relative decline, and it IS 0.02 on a decimal volatility scale. Neither one misreads the market. Each misreads the convention the vega is quoted in. Against a per-percentage-point vega the input is −2. The other two give an answer five times too large, or a hundred times too small. Nothing in the maths signals either. That is why you check the unit before you enter the number.
Frequently asked
3Is a fall from 20% to 18% two percentage points or ten percent?
Both statements are true of the same fall, and that is exactly why the unit matters. It is a fall of two percentage points. It is also a ten percent relative decline. A vega quoted per one percentage point of volatility uses −2.
Does theta remove the same amount from the premium every day?
No. It is a sensitivity under a stated time convention: calendar days, trading days or an annualised model figure. Its own value also changes as the option ages and as the underlying moves. Mixing those conventions changes the answer, and nothing warns you.
Do the Greeks say where the premium will be tomorrow?
No. Each one answers what a theoretical price does if one input changes while everything else is held still. Real sessions move several inputs at once. And nothing in the four says which way any of them will move.