The Option Greeks: Delta, Gamma, Vega, Theta and Rho
The Greeks are five rates of change. Each says how far an option premium moves per one unit of movement in something else: delta per unit of the reference asset price, gamma the rate at which delta itself changes, vega per unit of the implied volatility, theta per unit of time passing, and rho per unit of the financing rate. Every one is produced by a pricing model rather than read off the contract.
A premium is one number, and that one number has to answer to several different things at once. Move the price of the reference asset and the premium responds. Let a fortnight pass with nothing else changing and the premium responds again. Change what borrowing costs and it responds a third time. A single figure gives no way of telling which of the pushes did the moving, so anybody who wants to know how a premium behaves cannot get far while all of it is bundled together.
So the response is broken into parts, and each part is one of the five names in this guide. Holding everything else still, moving one input a little, and recording how far the premium moved: that is the whole idea. A sensitivityHow far one quantity moves per one unit of movement in another. Two quantities and a stated unit of movement in the second one. is one number describing the response of a premium to one named input while every other input is held still, and the holding still is not a condition attached to the definition but the definition itself.
Every complication in this subject comes out of the second half of that sentence. Holding everything else still is something anybody can do inside a pricing modelA set of stated assumptions that turns inputs into a premium, and that can be run backwards to turn a premium into one of the inputs., where the inputs are set by whoever runs it and nothing moves unless it is moved. Holding everything else still is not something anybody can do in a market, where the price of the reference asset, the time remaining and whatever anybody assumes about how far that price might travel all move together, and none of them waits for the others.
Before reading on. Somebody says an option is highly sensitive, and stops there. What is the first thing that has to be asked of them?
What is a sensitivity, and what does every one of them have to name?
Names are easy to learn and just as easy to learn wrongly, so start with the shape rather than with the five names. A rate of change says how far one quantity moves per one unit of movement in another. Two quantities, and a stated unit of movement in the second one. Take away any part of that and there is nothing left to read.
Every rate of change has to name its per unit in the same breath, exactly as a rate has to name its period, and a rate of change quoted without its per unit is not a small omission but an unfinished number. The habit is already familiar from rates. Somebody who says a rate is 6.50 per cent has said nothing until they name a year. A rate of 6.50 per cent a year and a rate of 6.50 per cent a month are not one arrangement wearing two labels. A sensitivity works the same way, and it fails in the same way.
The everyday version is worth holding on to for the rest of this guide. A tea stall stands outside one office building. Its takings change when it rains. The takings change when the office runs a late shift. The takings change again when the wholesale price of milk moves. Somebody then says the stall is sensitive, and nothing at all has been learned. Sensitive to which of the three? And per how much of it: per hour of rain, per late shift, per rupee on a litre of milk? Until both halves arrive, the word is decoration. The five names in this guide exist because somebody had precisely that problem with an option premium and solved it by splitting one word into five, each carrying its own per unit.
There is a third thing every sensitivity has to name, and it is the one readers drop first. A sensitivity is only true while everything except its one named input is held still, so a sensitivity is a statement about one moment rather than a description of how a premium behaves over a stretch of time. Nudge the price of the reference asset, and hold the clock, the financing rate and everything anybody assumes about travel exactly where they were. Clamping four inputs and moving one is a thought experiment, and it is also the only condition under which any of the five means anything.
Look at what the arrangement below actually requires. Five inputs go into a pricing model. One of them is nudged by a single unit and the other four are clamped where they stood, and the number that comes back out is the sensitivity to the one that moved. In a market nothing is clamped. Every one of the five therefore describes a moment and not a period.
What does delta measure, and per unit of what?
DeltaHow far an option premium moves per one unit of movement in the price of the reference asset. is how far the option premium moves per one unit of movement in the price of the reference asset. The definition ends there, and the per unit is the price of the thing the contract refers to. Most readers have half met delta already, usually as a number somebody quoted at them, so the useful work here is separating what delta is from what a quoted delta was.
Two properties of delta can be stated without attaching a figure to either, and both are worth carrying away.
The first is that delta is signed. The two contracts hand their buyers opposite entitlements at the same strike, so when the price of the reference asset rises, a call premium and a put premium do not move the same way. A call delta and a put delta written on the same reference asset carry opposite signs, and that is a property of what the two contracts oblige rather than a property of anything anybody has assumed. Nobody needs a model to know it. The sign can be read off what the two contracts are.
The second is that delta is bounded. The most any option can ever do is track the price of the reference asset one for one, so an option cannot respond to that price by more than the reference asset itself does. A delta lies between nil and one in size, and that bound comes out of the shape of the contract rather than out of any calculation. Again, no model is needed to see it.
The two properties above were stated carefully for a reason. Both of them describe the shape of the arrangement. Neither of them says where inside those bounds this particular contract sits, and where inside those bounds it sits is precisely what a delta is. Knowing that a delta is signed and bounded is not the same as knowing a delta, and this guide can establish the first but not the second.
What does gamma measure, and why does a single delta go stale?
GammaHow far the delta itself moves per one unit of movement in the price of the reference asset. A rate of change of a rate of change. is how far the delta itself moves per one unit of movement in the price of the reference asset. Gamma is a rate of change of a rate of change, and the second reading is where it lands. Read the definition once more before going on.
Here is why anybody bothered to name it. If delta were fixed, a delta written down once would describe the contract for as long as the contract existed, and there would be nothing further to say about it. Delta is not fixed. Move the price of the reference asset and the delta that described the premium a moment ago describes it no longer. Gamma is the measure of how quickly that happens.
Because delta itself moves, any delta is a statement about an instantThe moment a sensitivity describes, after which the reference asset price has moved and the sensitivity has changed., and it goes stale the moment the price of the reference asset changes. Not stale over a quarter. Stale on the next move. Everything a practitioner does with these numbers follows from delta being a perishable reading rather than a standing fact. Staleness is the point at which the subject stops being intuitive and starts being useful.
Gamma is the reason a position built on a delta has to be revisited, and the reason nobody working with these contracts speaks of a delta as a fixed property of the contract the way they speak of the strike. The strike is written into the contract and stays there. The delta was true when it was computed and has been quietly expiring ever since.
The everyday version is a speedometer and a pedal. A reading of forty gives the speed of the vehicle at this instant. Somebody has a foot on the accelerator and the reading is already changing, so the forty says nothing about how fast the vehicle will be going in ten seconds. Delta is the speedometer. Gamma is how hard the pedal is being pressed. A driver who quotes their speed from ten seconds ago is doing what a reader does when they reuse yesterday's delta.
Delta is the slope of the premium plotted against the price of the reference asset, and gamma is how fast that slope turns. The drawing below does the work in one glance that the definition never quite manages. Look at the drawing rather than at the definition. The line bends. Where it lies nearly flat, moving the price of the reference asset a little barely moves the premium at all. Further along, where the same line has steepened, the same move in the price carries the premium a long way. One line, two very different responses, and nothing about the contract changed in between.
Educational illustration. Not a quotation, not a price, and not a prediction of any price. The drawing above carries no scale on either axis, and no point on it may be read as a figure.
The one relationship a reader would want to move here is how far the premium responds when an input is nudged. Drawing that response needs a pricing model supplied with the very input the working record behind these contracts does not hold. A number that changes as it is dragged looks like evidence, so a manufactured figure at every position is worse than a single printed figure. So the shape is drawn, the axes are left bare, and the drawing is labelled for what it is.
A reader writes a delta down on Monday and uses the same figure on Friday, having watched the price of the reference asset move a long way in between. Which of the five explains why that is a mistake?
What does vega measure, and why is it the awkward one of the five?
VegaHow far an option premium moves per one unit of movement in the implied volatility. is how far the option premium moves per one unit of movement in the implied volatilityA number backed out of a premium quoted today by running a pricing model backwards until the premium it returns matches the premium being quoted.. The definition is that short, and vega is still the awkward one of the five.
Look at what the other four respond to. Delta responds to the price of the reference asset. Anybody can look that price up. Gamma responds to a movement in that same price. Theta responds to the clock. Rho responds to the financing rate. A financing rate is published and carries its period. Every one of those four inputs is a thing a person can go and observe, and two people who go and observe it will come back with the same answer.
An implied volatility is not measured from anything but backed out of a quoted premium by running a pricing model in reverse until the premium the model returns matches the premium being quoted, so vega responds to a number nobody ever observes. The number is not there before somebody performs that operation. Two people running different models over the same quoted premium come back with different numbers, and neither of them has made an error.
There is a second quantity wearing the same word, and the two must be kept apart every time either appears. A measured volatility is computed from a series of prices that have already happened, over a stated window and at stated intervals. An implied volatility is backed out of a premium being quoted now. A measured volatility and an implied volatility are different objects rather than two estimates of one quantity, and vega responds to the second of them and not to the first. A reader who blurs the two ends up thinking a longer run of history would pin vega down. No length of history will.
There is a coincidence here that matters. Vega is the sensitivity to exactly the quantity the working record behind these contracts does not hold, so vega is the one of the five whose definition names the missing thing out loud. That is uncomfortable, and it is more useful pointed at than tidied away. The reason no figure appears in this guide for any of the five is the same reason vega has nothing to respond to.
Four of the five measure the response to something a reader could go and observe. Which one does not?
What does theta measure, and in what direction does it point?
ThetaHow far an option premium moves per one unit of time passing. is how far the option premium moves per one unit of time passing. The per unit here is a stretch of the clock, and like every rate in this guide it is unfinished until somebody states which stretch: a day is not a week and a week is not a month.
Theta is unlike the other four in one respect that matters more than any of the rest: the input it responds to moves in one direction only, and it moves whether anybody trades or not. Nobody has to act for a day to pass. No order is entered, no counterparty agrees to anything, and the input has moved anyway. Every other input moves because somebody somewhere did something. Time moves because it is time.
Now the thing readers most often get wrong, and it is worth stating flatly because it is the commonest misreading of any of the five. Theta is not a fee, nothing is deducted from anybody, and no amount leaves any account when theta moves. There is no charge, there is no debit and there is no recipient. The premium moved once, at the start, from the buyer to the writer, and that was the last time anything travelled between them until the contract ends.
Theta describes the rate at which the price somebody would pay for the remaining time changes as the remaining time shortens. Theta is a statement about what a thing would fetch, not a statement about what anybody is being billed. The household version is a season pass with a month left on it against the same pass with a week left. Nobody debits the holder when a week goes by. The change is in what somebody else would hand over for what remains, and the shape of that change over the last stretch is what theta names.
A reader says that theta is the daily charge for holding an option. What has the reader got wrong?
What does rho measure, and why is it usually the quietest of the five?
RhoHow far an option premium moves per one unit of movement in the financing rate. is how far the option premium moves per one unit of movement in the financing rate. A financing rate that is not tied to a stretch of time is not a rate, so once again the per unit needs its period stated.
Rho is usually the quietest of the five in ordinary conditions because the financing rate reaches an option premium mainly through the present value of the strike, and over a short life that discount barely moves. Push a rate about a little on a contract with weeks to run and the amount being discounted has almost no time to be discounted over, so almost nothing happens. Stretch the life out and rho has more room to work in. Rho being quiet is a statement about where in the arrangement the financing rate enters, not a claim about a size.
The connection this needs comes from earlier work on these two contracts. The whole of the difference between a call premium and a put premium at one strike and one expiry is financing, and nothing else at all. The parity relationshipThe arithmetic tie between a call premium and a put premium at the same strike and expiry, holding whatever anybody assumes about how far the price might travel. says exactly that. Rho is the sensitivity that names how that financing difference responds, so rho is not an exotic extra bolted on at the end: it is attached to the one part of an option premium this guide can work out in full. No figure is attached to it.
The drawing below is built out of the arithmetic, so work it through once. The strike is Rs 2,000.00/-. Financing costs 6.50 per cent a year and the contracts run for one year. The present value of the strike is therefore Rs 2,000.00/- divided by one plus 6.50 per cent, and to four places that is Rs 1,877.9343/-. The strike less its present value is Rs 2,000.00/- less Rs 1,877.9343/-, being Rs 122.0657/-, and that quantity is the parity difference between the call premium and the put premium. Change the financing rate and every number in that build moves. Rho is the name for how far the premium moves when it does.
All five are now named. Setting them out together is what stops the five becoming five unrelated facts to be memorised. All five share one shape. How far the option premium moves per one unit of something else is the whole of it, with delta taking the reference asset price, gamma taking the delta itself, vega the implied volatility, theta the passage of time and rho the financing rate. Only the last column changes. Gamma is the one exception, the only one of the five that measures a movement in another sensitivity rather than a movement in the premium.
Before reading on. Theta and vega both move the same premium. Which of the two responds to something that is certain to happen?
Theta vs Vega: how do two sensitivities of one premium differ?
Both of them move the same premium, and readers blur them constantly. The blur usually shows up as a sentence in which the passing of time and the travel of a price are treated as two ways of saying one thing. The passing of time and the travel of a price are not one thing. Theta responds to a clock anybody can read and vega responds to a number nobody can. That single line separates them, and the three differences below are what it unpacks into.
The first difference is direction. Time passes one way. Time never runs backwards, and no arrangement anybody enters into changes that. The implied number vega responds to carries no such constraint: it can rise, it can fall, and it can sit exactly where it was. So one of the two sensitivities is attached to an input with a known direction of travel and the other is attached to an input with none.
The second difference is observability. The time left to run is a fact. Anybody can check it against a calendar, and two people who check will agree with each other. The implied number cannot be checked against anything. No such number exists until somebody takes a quoted premium and a pricing model and runs the model backwards. Two people using different models pull different numbers out of the same premium, and neither of them has made a mistake. The disagreement is not a flaw in anybody's arithmetic but the plain meaning of inferred.
The third difference is certainty. The passing of time is the one input in this subject that is certain to arrive, and it arrives without anybody's participation. A premium with time left in it is therefore exposed to something that will definitely happen. The same premium is also exposed to a number nobody observes and nobody can pin down. The difference between the two is not about which of them moves the premium further. The separation is about which of them is certain to move at all.
Two sensitivities of one premium, then, and the separation is clean: one responds to something certain and one responds to something inferred. Hold that and the two words stop swapping places.
What does the worked pair actually attach to?
One pair of contracts on one reference asset, both invented for teaching, and every figure below is either recomputed here or labelled as given. The reference asset has a price of Rs 2,000.00/-. The price is exposure, meaning the amount of the thing the contracts refer to, and it is not an amount either side has paid. Financing costs 6.50 per cent a year and the contracts run for one year. The reference asset pays nothing at all during the holding period. A payout would change the arithmetic below, and this record does not contain one.
Three figures in this guide are all Rs 2,000.00/- and a reader is entitled to wonder whether one of them was copied into the other two. No figure was copied into another. The price of the reference asset is Rs 2,000.00/- because that is where the price of the reference asset stands. The strike written into both contracts is Rs 2,000.00/- because the pair is struck at the money, and being struck at the money is exactly what it means for the strike to equal the price. And the exposure any one unit of this arrangement refers to is Rs 2,000.00/- because it is one unit of the same reference asset. Same number, three different quantities, agreeing on purpose.
The call premium is Rs 180.00/- and the put premium is Rs 57.93/-. Both are given. Computing an option premium from scratch needs a pricing model and the input that model requires. The working record behind these contracts does not hold that input. The relationship between them can be computed, and it follows in full.
| Line | Figure | Where it comes from |
|---|---|---|
| Price of the reference asset | Rs 2,000.00/- | Invented. Exposure the contracts refer to, not an amount anybody paid. |
| Financing, stated for one year | 6.50 per cent a year | Invented, and every rate in this guide carries its period. |
| Strike written into both contracts | Rs 2,000.00/- | Equal to the price because the pair is struck at the money. |
| Present value of the strike | Rs 1,877.9343/- | Rs 2,000.00/- divided by one plus 6.50 per cent, to four places. |
| The parity difference | Rs 122.0657/- | Rs 2,000.00/- less Rs 1,877.9343/-. Financing and nothing else. |
| Call premium | Rs 180.00/- | Given. Producing it would require a pricing model. |
| Put premium | Rs 57.93/- | Given, and consistent with Rs 180.00/- less Rs 122.0657/-. |
| Check the difference back | Rs 122.07/- | Rs 180.00/- less Rs 57.93/-, against exact parity of Rs 122.0657/-. |
The gap in that last row is the whole lesson about precision. Rs 180.00/- less Rs 57.93/- is Rs 122.07/-. Exact parity is Rs 122.0657/-. The two differ by Rs 0.0043/-, forty three hundredths of one paisa. Rounding the put premium to the paisa before the subtraction is where the difference comes from. The relationship holds to the paisa and it does not hold exactly, and claiming an exact equality that the rounded figures do not produce would teach a reader to stop checking.
The same discipline applies in the other direction, and it is worth one more line. Carry the present value of the strike forward a year at 6.50 per cent and Rs 1,877.9343/- becomes Rs 2,000.0000295/-, not Rs 2,000.00/-. The strike really is Rs 2,000.00/-; the tiny excess is the rounding of the present value to four places coming back out. Say to the paisa, print the gap, and move on.
Why can none of the five be given a value for these contracts?
Not one of the five sensitivities named above can be given a value for these contracts. Not delta, not gamma, not vega, not theta, not rho, and not one of them as an illustration either. The refusal is a fact about what the five are rather than a technicality.
The reason is a single chain and it is short. Each of the five is an output of a pricing model. Every such model wants the same input before it will return anything, and the input is a statement about how far the price of the reference asset might travel. The working record behind these contracts holds no figure of that kind, in any unit, over any period. So there is nothing to run, and nothing comes back.
Naming what each of the five measures, in what direction it points and what it responds to is one kind of knowing, and saying how large any of them is for these contracts is another. The two are not interchangeable, and only the first survives a missing input.
Printing five plausible looking values would mean manufacturing every one of them, so the absence is stated plainly where a reader expects the numbers. A manufactured delta looks exactly like a real one. A manufactured delta sits in the same column, carries the same number of decimal places, and nothing distinguishes the one sort from the other. The resemblance is precisely why no such figure is printed.
So the card below is drawn with five filled rows and five empty ones. Every definition is complete. Every value is blank, and it is blank for the same reason in all five cases.
The delta of the call is what is wanted. Known are the price of Rs 2,000.00/- of exposure, the strike of Rs 2,000.00/-, financing of 6.50 per cent a year, one year to run and the call premium of Rs 180.00/-. What is still missing?
Before reading on. Neither the call delta nor the put delta can be produced here. Can anything at all be said about the two of them together?
What can still be said about two of them without that value?
Something can, and it is a real result rather than a consolation prize offered at the end of a refusal.
The call and the put at the same strike and the same expiry are the pair worked with throughout. The parity relationship ties their two premiums together, and in the same breath it ties their two deltas together. The relationship that fixes the two deltas is arithmetic on financing and never consults a model, so the call delta less the put delta is one, and it is one whatever the missing input turns out to be.
The shape repeats, so sit with it for a moment. The difference between the two sensitivities is knowable and the level of either one is not. The two premiums have exactly that shape: the difference between them is fixed at Rs 122.0657/- by financing alone, and neither premium on its own could be produced here. In both cases the arithmetic reaches the gap and stops short of the levels. The repetition is not a coincidence. One relationship produced both, and one fact is being seen twice.
What goes wrong when a sensitivity is filed beside the strike?
The failure: treating a sensitivity as a property of the contract
A reader meets a delta quoted somewhere and files it away beside the strike, the expiry and the quantity one contract covers, as though it were one more term that the two sides had agreed between them. It is not. Nobody agreed it, nobody wrote it down, and the two sides may never have discussed it.
Set the two side by side and the difference is immediate. The strike of Rs 2,000.00/- is written into the contract. Anybody can read it off, two readers will agree, and it will say the same thing in a year. A delta is written nowhere at all. A delta has been produced by somebody's model, out of somebody's assumptions, and a different model fed different assumptions returns a different delta for the same contract on the same day.
Who makes this mistake: readers who meet the five as a list to be memorised rather than as outputs of a calculation. A list is very nearly always how the five are first presented. A list invites a reader to treat every item on it as the same kind of thing, and here four items on the list are definitions while the numbers people attach to them are not.
The cost: two people comparing sensitivities that were never comparable in the first place. A position sized off a figure whose assumptions nobody checked. And an argument about a number where the real disagreement is about a model. The last of the three is the most expensive: the two sides never reach the thing they actually disagree about.
The fix is a single habit stated in one line: when somebody hands over a Greek, the question is which model produced it and what that model assumed, in exactly the way a rate is asked for its period.
The reason two people can disagree without either being wrong is the chain the number came down. Assumptions go into a model, the model returns a premium, and the sensitivity is how far that returned premium moves when one input is nudged and the rest are held still, so a different set of assumptions produces a different sensitivity for the very same contract. Nothing in that chain is arithmetic anybody could check independently, until the assumptions at the front of it are on the table.
Two people quote different deltas for the same contract on the same day, and neither has made an arithmetic mistake. How is that possible?
Who decides the things that would turn a sensitivity into a quantity?
Three requirements are touched by the material above. Each of the three is set by the authority named beside it, and each of them changes. A printed value would be wrong rather than merely out of date on the day it moved.
The first matters more than the other two for this subject. A sensitivity is a response per one unit of the reference asset, so turning any sensitivity into a quantity of anything requires knowing what one contract covers and in what quantity. The input to that step is a requirement set by an authority rather than a figure, so the step is not arithmetic.
Where the reference is a rate or a currency rather than an equity, the equivalent arrangements sit with the Reserve Bank of India at rbi.org.in. The mechanism above is written without reference to any one market, so a second market becomes an addition to these three rows rather than a rewrite of the mechanism.
How does somebody working with these contracts actually use the five?
What the vocabulary is for, once the reader is the one in the conversation
Somebody quoting a book of these contracts does not use the five names to describe anything. The five names route a question. When a premium has moved and nobody knows why, the five names are the list of candidates: was it the reference asset price, was it the clock, was it the financing rate, or was it the number nobody observes? The five turn one vague question about a premium into four sharp questions with different answers, so the first practical use of the five is diagnostic.
A risk function uses them for something narrower and more sceptical. Every sensitivity that arrives on a risk report came out of a model, so the first thing a risk function does with a delta is not read it but trace it: which model, run with what assumptions, as at what moment. Two desks reporting sensitivities produced on different assumptions cannot be added together, and adding them anyway is one of the quieter ways a report becomes fiction.
An analyst reading somebody else's disclosure uses the five as a checklist for what is missing rather than for what is stated. A disclosure that gives a sensitivity without its per unit, or without saying as at when, has handed over an unfinished number, and the right response is a question rather than a calculation.
The habit is not specialised, and there is a household version. Somebody quotes a monthly instalment on a loan. The figure is not accepted as it stands: the question is what it responds to, over how many months, at what rate, and what happens to it if the rate moves. The five names make that same move. The instinct being trained here is to refuse an unfinished number politely, whether it arrives as a delta or as an instalment.
Is being able to name the five a reason to act?
The question has a plain answer rather than a silence. Nobody can answer it from a set of definitions, and the reason can be stated.
Three things would have to be known before anybody could answer it, and not one of them is available. The first is a view on how far the reference asset might travel and how likely each move is. The record behind these contracts holds no such view. The second is the circumstances of the particular holder. No reference work can see them. The third is what the arrangement would cost to hold to the end and what it would cost to unwind before then, neither of which is set out here.
The record behind these contracts carries no outcome, no track record, no probability and no distribution, so nothing about them can be ranked or compared on how it turned out. No payoff of theirs can honestly be called attractive and no position of theirs safe. Being able to name the five is a vocabulary, and a vocabulary is not a reason to take on an obligation.
The vocabulary is good for the sentence before the decision rather than for the decision. The vocabulary allows a better question to be put to whoever handed over a number, and it allows an unfinished number to be told from a finished one. Both are genuinely worth having, and both are a different thing from a reason to act.
References
| Source | Document | Where |
|---|---|---|
| SEBI | What one contract covers and in what quantity, which is the step that would turn any sensitivity into a quantity of the reference asset | sebi.gov.in |
| SEBI | The collateral a writer places against the obligation, and how it is worked out | sebi.gov.in |
| SEBI | How many contracts one participant may hold | sebi.gov.in |
| Reserve Bank of India | The equivalent arrangements where the reference is a rate or a currency rather than an equity | rbi.org.in |
| arXiv Quantitative Finance | Preprint repository, consulted for the standard naming and notation of the five sensitivities, for structure and notation only and with no text reproduced | arxiv.org |
| Social Science Research Network | Working paper repository for the same material, consulted before any construction was named | ssrn.com |
The reference asset, the price of Rs 2,000.00/-, the strike of Rs 2,000.00/-, the financing of 6.50 per cent a year, the call premium of Rs 180.00/- and the put premium of Rs 57.93/- are invented.
Educational material. Not advice on any investment, tax, budget or market position.