How Delta, Gamma, Theta and Vega Describe Option Sensitivity

Each of the four describes how an option premium answers one input, on the condition that no other input budges at all. The condition that nothing else budges is the whole difficulty. Each statement is local, so it covers a small move; each is instantaneous, so it covers a moment; and each is produced by a model rather than written into the contract. None can be read off the contract at all.

Most treatments of the subject start somewhere else. The five measurements by name settle what each of the four measures and what each one is counted in. The far more useful thing almost nobody is told is what KIND of statement a sensitivity is when somebody hands one over. A price is one kind of statement. A strike is another. A sensitivity is a third kind entirely, and mistaking it for either of the first two is the single most common way the four get misused.

The reason there is anything to say at all comes down to one gap. A premium responds to several things at the same time, and no reader alive can think about several moving things at once, so the subject splits the response into pieces and inspects one piece at a time. The split is genuinely useful. The split is also a fiction: pinning three inputs down while a fourth walks about can be arranged inside a model and cannot be arranged in a market. Every limitation of the four falls out of that one gap between the tool and the premium the tool describes.

What kind of statement is a sensitivity, exactly?

Every one of the four has the same shape, and the shape is a conditional. Stated in full, it reads: if this one input moved a little, and nothing else moved at all, the premium would move about this far. Three clauses. Readers hold on to the first and the third and drop the middle one. The middle clause is the only one doing any work.

The middle clause is not a simplification that may quietly be discarded; it is the condition under which the whole statement is true. Dropping it does not make the statement rougher. Dropping it makes the statement one about something that did not happen. There is a difference between an approximation and an answer to a different question, and the middle clause is where that difference lives.

The everyday version is worth carrying about. A food stall stands outside one office building. The person running it works out, carefully and correctly, that takings fall by a certain amount on a wet day. The rain finding is real and it is not wrong. Then a day arrives that is wet and is also a public holiday, so the office is shut and there is no one on the street at all. The rain finding was still correct. The finding was simply answering a question nobody was in a position to ask that morning. The condition it carried, everything else about the day being ordinary, had quietly failed. A sensitivity fails in exactly that way, and it fails silently. Nothing about the number announces that its condition has stopped holding.

Notation makes the middle clause impossible to lose. Keeping a condition attached to a measurement is the one thing notation is genuinely good for. Written out properly, the condition is not a footnote to the measurement. The condition is part of the measurement.

The bar is worth reading carefully. The treatment that names the five measurements settles what delta counts and per unit of what. The bar adds three things: delta arrives with an instruction attached, the instruction cannot be obeyed anywhere outside a model, and a reader who accepts the number without the instruction has accepted a statement about a world nobody lives in. Three of the four measurements swap out which symbol sits to the left of the bar; not one of them removes the bar.

What does it cost that a sensitivity is only local?

Local is the first of the two limitations, and it concerns DISTANCE. A sensitivity describes a SMALL move of its input. The further the input actually travels, the worse the description becomes, and it becomes worse smoothly rather than at some point where a warning goes off.

The reason is that a premium does not answer the price of the reference asset along a straight line, so a straight line drawn at any one point walks steadily away from the truth the further it is followed. The widening gap is the entire content of the word local, and no definition of the word delivers it as well as a picture does.

Gamma stops being an extra definition at exactly this point and becomes the useful thing in the room. Gamma is the correction term: it measures how quickly the straight line stops being a good line. Where the response bends gently, a straight line survives a long walk. Where the response bends sharply, the same straight line is out of date after a few steps. Gamma is what tells the two situations apart, and it is why a reader who holds only a delta is holding half the description of a price move.

Now the uncomfortable part, and it is worth being blunt about it rather than leaving it implied. Any sensitivity is a good description of a small move and a poor description of a large one, and no rule anywhere fixes where small stops. There is no threshold to look up. Where small stops depends on how sharply the response is bending at the point in question, and the bending in turn depends on the one input the working record behind these contracts does not carry. So the honest instruction is not a number. The honest instruction is a habit: the description is reliable close in and decorative far out, and the boundary between the two is nowhere on display.

Why is being instantaneous a second fault rather than the same one twice?

Local restricts how FAR the input may travel. Instantaneous restricts WHEN the statement is true. The two restrictions are separate, and a reader who collapses them into a single vague sense that sensitivities are approximate has lost the ability to tell which of the two just went wrong.

A sensitivity describes the premium at a moment. The moment then passes, as moments do. By the time anyone has read the number, thought about it and acted on it, the price of the reference asset has moved, the clock has advanced, and the measurement itself is no longer the measurement that applies. Nothing about the written figure changed. The world it described did.

A sensitivity written down is a photograph rather than a property, and a photograph taken half an hour ago is a photograph of something that is no longer standing there. A property travels with the object: the strike stays what it was written as, the expiry stays what it was written as. A photograph does not travel with anything. A photograph travels with the instant it was taken, and the instant is gone.

Here is the origin of a word that turns up constantly in this subject and sounds, at first hearing, like an admission of sloppiness. The subject speaks of rebalancingReturning to a position already set up and adjusting it, because the measurement it was built around is no longer the measurement that applies. rather than of setting a position up correctly once, and that is a statement about the measurements rather than about anybody being careless. If a sensitivity were a property of the contract, it would be consulted once, acted on, and never thought about again, the way a strike is consulted once. Because it is a photograph, the reader goes back and takes another one, and then another. The going back is not a failure to plan. Going back is what the measurement, honestly understood, requires. How that going back is actually carried out is worked through separately.

Notice, too, that of the inputs on the list, one of them moves whether anybody does anything or not. The price of the reference asset needs somebody to deal before it moves. The inferred number needs somebody to quote a premium before it moves. The clock needs nobody at all. A clock that needs nobody is why the measurement answering it goes stale with the most reliable regularity, and why the passage of time on its own is enough to retire a written sensitivity even on a day when nothing whatever traded.

What is a sensitivity conditional on, once the conditions are listed out properly?

Ask a reader what a delta depends on and the usual answer is the reference asset price. The reference asset price is what a delta MEASURES. The list a delta DEPENDS ON is considerably longer, and every item on it is capable of changing the number by itself.

Because a different model produces a different number from the same inputs, a delta depends on the pricing modelA set of stated assumptions plus the arithmetic that turns them into a premium. Alter one assumption and the number coming out alters with it. that produced it. A delta depends on every assumption sitting inside that model, and there are more of them than a summary ever lists. A delta depends on the inputs actually fed to the model being right, and for at least one of those inputs that means somebody estimated it. And it depends on the premium the model was matched to being a premium at which somebody would genuinely deal, rather than a figure printed on a screen that nobody stands behind.

Put plainly: a sensitivity is a statement about a model that has been tuned to a market, rather than a statement about the market. The sentence is worth reading twice. Nothing in it says the number is useless. A model tuned to a market is a serious object and it carries real information. The number simply carries information of a particular kind, and the kind is not the kind most readers assume they are holding.

The habit that follows is one already owned from a different subject. Nobody accepts a rate without asking over what period. The rate habit needs no teaching; a rate with no period attached is visibly incomplete. Which model produced a sensitivity, and what it assumed, is asked in exactly the same breath and with exactly the same reflex as the period of a rate. The two questions are the same question wearing different clothes: both ask what kind of number is being handed over.

There is a second thing this habit protects against, and it is subtler than a wrong number. When two people disagree about a sensitivity, the argument usually gets conducted as though one of them has made an arithmetic error. Almost always neither has. The two chose differently somewhere up the chain, and the disagreement is a disagreement about assumptions wearing the costume of a disagreement about arithmetic. Naming the chain out loud turns an unresolvable argument about a figure into a tractable argument about a choice. A tractable argument is a considerably better argument to be having.

What do the four describe between them, and what falls through the gaps?

Line them up and the division of labour is tidy. Delta and gamma both answer the price of the reference asset, at two levels of detail: delta gives the slope of the response and gamma gives how fast that slope is turning. Theta answers the clock. Vega answers the inferred number. Between them the four cover the inputs that actually shift a premium in the ordinary run of things, with financing left to a fifth measurement that is covered separately.

Because the tidiness is precisely what hides it, turn now to what that tidy division leaves out. Every one of the four is a one-at-a-time statement, and the day on which a reader actually wants them is a day when the price moved, time passed and the inferred number changed, all at once. Not one of the four was built for that day. Each was built for a day on which exactly one thing happened, and no such day has ever occurred.

So what does using them anyway produce? An approximation. Approximation is the honest word and it should not be softened into a synonym that sounds firmer. Adding the four responses together produces an approximation; the error in that approximation has a real size; and no arithmetic that stops short of the missing input can say what that size is. The size depends on how far each input travelled and on how sharply the response was bending along the way, and the bending runs back to the one input the working record behind these contracts does not carry. A typical error would have to be manufactured, and a manufactured comfort figure is worse than leaving the reader uncomfortable.

The opening state is worth having in plain words as well as in a picture, so it survives if the picture is stripped away: with only the price of the reference asset allowed to move and everything else held down, delta is the measurement that describes the response, and it describes a small move. Push the same input a long way and delta on its own describes it badly, with gamma standing as the correction. Let only the clock run and theta is the measurement, and the clock is the one input on the frame that advances whether anybody deals or not. Let only the inferred number shift and vega is the measurement, and that number is the one thing on the frame nobody ever observes. The inferred number is always backed out of a premium rather than read off anything.

Why is none of the four written anywhere on the contract?

The next point is mechanism rather than a warning tacked on at the foot, and it explains all the rest. A strikeThe one price written into the option itself, at which its buyer may choose to deal. Once the contract is struck that figure does not move, whatever else does. can be read off the contract. A sensitivity can be read off nothing whatever.

Each of the four is manufactured. A model makes it, out of inputs that include the one quantity missing from the working record behind these contracts from the very start. The absence is exactly why what each of the four measures is stated with a figure against none of them. The blank is not a stylistic choice about tone. The blank is a consequence of what is and is not in that working record.

Here is a test that applies to any number anybody ever hands over: would it still exist if every model in the world were deleted overnight? The four things worked here run through it as follows. The strike of Rs 2,000.00/- is written on the contract in ink, so it would still exist. The date it runs to would still exist, for the same reason. The premium of Rs 180.00/- would still exist too. Somebody actually handed that money over, and the handing over is a fact about the world rather than a fact about a calculation. So would the payoff the contract produces on its last day, and so would the profit that follows once the premium has been counted in. Not one of the four sensitivities would exist at all. The four sensitivities would simply be gone, with nothing left behind. There was never anything there apart from the model that made them.

The test is more useful than it looks, and it generalises well beyond these four measurements. The test sorts the things that were AGREED from the things that were PRODUCED, and a good deal of confusion in this subject starts with a produced number being filed in the drawer marked agreed. A strike is agreed. A premium is agreed, at the moment it is paid. A sensitivity is produced, by somebody, using something, under assumptions, and it belongs in a different drawer with a different label on it.

What does the worked pair actually settle?

The same invented pair this sequence has been running on all along carries the argument here, and the figures are worth setting out once more to show exactly which of them a measurement could attach to. The reference asset stands at a PRICE of Rs 2,000.00/-, and that figure is EXPOSURE rather than money either side has handed over. Financing runs at 6.50 per cent a year. The contracts run for one year. Across that year the reference asset hands its holder nothing at all. A payout arriving partway through would shift every carry figure, and there is no payout.

Where the reference asset stands and what the two options are struck at read the same figure, Rs 2,000.00/-, and that agreement was arranged rather than stumbled into: the pair is struck at the moneySaid of an option whose strike sits at the same figure as the current price of the thing it references., and matching the strike to the price is the whole of what at the money means. Neither number was copied out of the other. Rs 180.00/- was paid as the call premiumThe money handed over at the outset for the option itself. It goes across whether the contract ends up being used or not, and it is never returned. and Rs 57.93/- as the put premium. Neither figure was produced here; both arrived ready-made. The writerThe side that takes the premium at the start and carries the obligation afterwards. It holds no right to walk away from what it agreed to. on each side took that money at the outset and has carried the obligation ever since.

Four words get kept strictly apart from here on. A PRICE is where the reference asset stands, Rs 2,000.00/-, and it measures exposure. A PREMIUM is what actually changes hands for an option at the start: Rs 180.00/- for the call and Rs 57.93/- for the put. A PAYOFF is whatever the contract hands over on its final day, reckoned before the premium enters the reckoning at all. A PROFIT is that same payoff once the premium has been counted and carried forward to the same date. Substitute any one of those four for any other and what follows stops being true.

Now the arithmetic, reproducible with a calculator and nothing else. The strike of Rs 2,000.00/- set back to today at financing of 6.50 per cent a year becomes Rs 1,877.9343/-. Taken away from Rs 2,000.00/-, Rs 122.0657/- is what is left over. The two given premiums run against it: Rs 180.00/- for the call, less Rs 57.93/- for the put, arrives at Rs 122.07/-. The gap of 0.43 of a paisa from the exact figure is the put rounded to two places. The relationship holds to the paisa, and calling it exact would teach the reader to stop checking.

The parity relationship is arithmetic on financing. Nothing in it consults a model, nothing in it needs the missing input, and nothing in it can be argued with. Which means that if the same relationship is asked how each side answers a move in the reference asset price, the answer has to be arithmetic too.

Then the argument stops, and what it stops short of is worth saying out loud. The call delta is not stated. The put delta is not stated. No gamma, no theta and no vega is stated for either contract, in any unit, including any unit dressed up as illustrative or typical. Every one of those is a LEVEL rather than a relationship, and a level would have to be manufactured out of material the working record behind these contracts has never held. The line that is not crossed is the line between a relationship that can be argued for and a level somebody would have to invent.

What survives when there is no model at all?

More than a reader expects, and it sits after the refusal rather than in front of it so that nobody mistakes it for a sweetener attached to one. Three statements survive, and every one of them is argued from the contract rather than calculated from anything.

The first is about SIGNS. A call premium and a put premium sharing one strike move in opposite directions as soon as the reference asset price shifts: the call becomes worth more and the put becomes worth less, or the reverse. Opposite directions are not a modelling result. The direction falls straight out of what the two contracts oblige, and the obligation is settled before any measurement enters. So the two deltas carry opposite signs, and they carry opposite signs whatever anybody assumes about anything.

The second is about BOUNDS. Nothing in the contract lets the option out-move the thing it is written on. Whatever distance the reference asset covers, the option cannot answer it with more. No clause anywhere manufactures extra movement. So a delta stays inside a band running from nil to one in size, a band exactly one wide. Again, nothing was calculated. The limit is argued from what the contract can and cannot oblige.

The third is a DIFFERENCE, and it is the one just worked above: the call delta less the put delta comes to one, fixed by arithmetic on financing rather than by anybody's assumption. The difference is settled in full where the five measurements are named, so it appears here as an example rather than as a result.

The shape the three share is the lesson rather than the three items: every one of them is a statement about a RELATIONSHIP between the measurements or about their LIMITS, and not one of them is a level. The distinction between a relationship and a level is worth carrying into every corner of this subject. Relationships and limits can often be argued from the contract and from arithmetic. Levels almost always have to be manufactured. Where one is available and the other is not, the blank is a property of the material rather than a matter of discretion.

The same division works elsewhere in this material, a useful cross-check that it is a real division rather than a convenience. Applying 6.50 per cent a year to Rs 2,000.00/- of exposure piles on Rs 130.00/- of financing, so a forward on that reference asset stands at Rs 2,130.00/-. The forward figure is arithmetic and it is a COST rather than an opinion about where anything is going. But that same arithmetic will not give either premium on its own; it gives only the gap of Rs 122.0657/- between them. Arithmetic reaches gaps and relationships. Arithmetic stops short of levels. The measurements behave the same way for the same reason.

Written out as notation, the missing piece stops being an abstraction and becomes a term that can be pointed at.

The fix is one habit and it fits in a single line. Before adding two sensitivities together, the middle clause of each is worth saying out loud, and if the two clauses contradict each other then the sum is an approximation whose error nobody here can size. Delta says nothing else moved. Theta says nothing else moved. Put them in the same sum and each is being asked to accept that the other one moved. Each of them ruled exactly that out. The contradiction is not hidden. The contradiction is never spoken, for the same reason the middle clause is never spoken.

How does somebody working with these contracts actually handle one of these?

Here is the practical picture, and it looks less like arithmetic than a newcomer expects. Somebody responsible for a position that includes options does not treat a sensitivity as a fact they have acquired. The number is a reading they have taken, with a timestamp attached and a shelf life they can feel rather than look up.

The first thing done with a number that arrives from somebody else is to ask where it came from. Not out of suspicion, but because the answer changes what the number can be used for. A sensitivity produced by one model and a sensitivity produced by another are two different objects that happen to be printed in the same font. If the two have to be compared, they have to be put through the same chain first, and if they cannot be, the comparison does not happen.

The second is that these numbers are used to size an ACTION rather than to make a FORECAST, and the distinction is the whole of the professional discipline around them. A delta is not consulted to learn where the premium is going. A delta is consulted to work out how much of something else would have to be arranged for the position to stop answering to the price of the reference asset in the way it currently does. Where the answer to a question is going to be acted on within minutes, the fact that the reading goes stale within minutes matters much less. Trouble starts when a reading taken this morning is quoted this afternoon, and the trouble is a governance problem rather than a mathematical one.

The third thing, and it is the one most worth borrowing even by a reader who never goes near a contract, is that gamma is read as a statement about HOW OFTEN, not about how much. A response that bends sharply means the delta reading expires quickly, and a reading that expires quickly means going back to look more often. A response that bends gently means a reading survives longer. An experienced reader treats gamma as a schedule rather than as a quantity: it says how frequently everything else has to be redone.

Take that to a household and it lands immediately. Somebody working out a monthly budget knows the rule of thumb that an extra guest at dinner adds a certain amount to the food bill. The guest rule is a sensitivity and it behaves exactly like one. The rule is local: it holds for one extra guest and falls apart for thirty. The rule is instantaneous: it was worked out at last month's prices. The rule is conditional: it assumed the usual menu and the usual shop. Nobody in that household would dream of multiplying it by thirty and calling the answer a wedding budget. The discipline being taught here is one that most people already apply perfectly well to their own money and then abandon the moment the same kind of statement arrives with a Greek letter attached to it.

Is being able to name the four a reason to act?

Should one of these be taken on? Nothing above settles that, and the reason is a shortage of material rather than a shortage of nerve. Three things would have to be in hand first.

One is a settled view about the distances the reference asset might cover before the last day, together with some sense of which of those distances deserve to be taken seriously. The working record behind these contracts holds nothing of that kind, and here is the part worth sitting with: each of the four measurements is calculated out of exactly that missing material. The absence is not off to one side of the subject. The missing material is the input the subject runs on.

Two is the reader's own situation, and no written material can look at that. Three is the full cost of carrying the arrangement through to its end date, alongside the separate cost of getting out of it before then. Neither of those is available here either, and neither could be.

Being able to name what a measurement would describe falls a long way short of holding the measurement, and it falls further still short of a reason to accept an obligation. A reader who reaches this point knows what kind of statement a sensitivity is. Most people who quote them do not. Knowing that is a good place to be standing. The place is not one from which anything follows about what anybody should do.

The reference asset, the two contracts struck on it, the premiums attached to them and the financing rate are invented.
Educational material. Not advice on any investment, tax, budget or market position.