Delta Hedging: How a Directional Exposure Is Offset

Delta hedging means holding a quantity of the reference asset that moves the opposite way to an option position, so a small move in the reference asset price leaves the two together roughly where they were. The quantity to hold is a delta multiplied by what one contract stands for. A delta comes out of a pricing model, so the quantity itself depends on which model somebody chooses to run.

An option position answers to several things at once, and exactly one of those answers can be switched off without disturbing the rest. Because the reference asset price is the single input both positions have in common, the answer to it can be switched off by holding the reference asset itself the opposite way round. Nothing else about the option position is touched by doing it. The single shared input is the strength of the method and it is also the entire limit of the method.

What is a directional exposure, and which part of a position has one?

A directional exposure is the part of a position's value that moves when the reference asset price moves. A directional exposure is not the whole position and it is not the risk of the position. The exposure is one strand out of several, and the only reason it can be singled out at all is that a second thing exists which answers to that same price and to nothing else: the reference asset itself.

An everyday picture stays useful through several of the sections below, so it is worth carrying forward. A stall outside a single office gate sells umbrellas and cold drinks. Wet morning, the umbrellas clear and the drinks sit untouched. Hot dry morning, exactly the reverse. Over a month, the stallholder's takings have stopped depending on whether it rains, and that is a real achievement rather than a trick. The list of what has not changed matters as much. The takings still depend on whether the office is open. The takings still depend on whether a wedding down the road has pulled the crowd away. The price the wholesaler charged for stock that morning still matters. The weather has been neutralised. The day has not been made safe, and nobody who ran that stall would confuse the two.

To neutralise a directional exposure is to arrange that one answer, the answer to the reference asset price, cancels out, and the arrangement says nothing whatever about the others. The claim goes no further than that. The word is precise and it is also badly behaved in ordinary use. It sounds like a statement about the position when it is only ever a statement about one input.

The phrase delta neutral needs translating into plain words exactly once. Delta neutral means insensitive to a small move in one named input, at one moment. Not insensitive. Not neutral. Insensitive to one named thing, briefly. Every word that got dropped on the way to the short version was carrying something.

Both qualifiers come back later, and neither is decoration. Small: the offset was matched to a small move, and a large one behaves differently. At one moment: the response the offset was matched to is itself changing meanwhile. An account of the method that dropped those two words would hand over a tool with its instructions torn off.

What does the procedure actually do, step by step?

The offsetting procedure is five steps. Four of them can be carried out by anybody holding the position and the remaining one cannot be carried out from these figures at all, so a numbered sequence is the clearest way to show where the method is complete and where it is not.

Step one is to know the option position: which contracts, how many of them, and on which side of each one the holder sits. The answer to the reference asset price points the opposite way for a writer and a buyer of the same contract, so bought or written matters as much as the count. Step one needs no model and no data, only the position itself.

Step two is to obtain a delta for each contract, and obtaining one means choosing a pricing model, feeding it its inputs, and reading what comes back. A delta is not a property of the contract that somebody could look up on it. A delta is the output of a calculation somebody chose to run. Change the model and the number changes. Change one input and the number changes again. Without a pricing model in hand, step two is where the working stops.

Step three converts a sensitivity into a quantity, by multiplying that delta by what one contract stands for and then by the number of contracts held. This is where a decimal becomes a number of units of the reference asset, and it is also where an authority enters the arithmetic. The contract sizeHow much of the underlying thing a single contract stands for. It is a specification set by an authority rather than something the two sides agree. is not something the two sides settle between themselves. The Securities and Exchange Board of India (SEBI) sets it, at sebi.gov.in.

Step four takes that quantity in the opposite direction to the option position's answer. If the option position gains when the reference asset price rises, the offsetting holding must lose by the same amount, which means going shortHolding a position that gains when a price falls and loses when it rises. It is what taking a quantity in the opposite direction amounts to. the reference asset. If the option position loses when the price rises, the holding goes the other way. There is no cleverness in this step. Step four is arithmetic with a sign on it.

The answer matched in step two has already changed, so step five is to come back and do it again. Repetition is not a warning attached to the end of the procedure. Repetition is a step of the procedure, and a description that left it off would have described something else entirely.

Why does the offset stop working almost immediately?

Because the thing the offset was matched to is not a fixed property. A delta describes how an option position answers to the reference asset price at one price and at one moment. GammaThe measure of how quickly a response to price changes as the price itself changes. It is what makes a matched offset stop matching. is the measure of how quickly that answer changes as the price moves. So the moment the price does anything at all, the answer that was matched has become a slightly different answer, and the quantity being held was matched to the old one.

The offset was set against a response that begins changing the instant the price does, so the pair stops being neutral immediately rather than eventually. Read that word immediately as literally as it is written. Not by the close. Not by the end of the week. From the first tick that moves the reference asset price, the match is no longer exact, and it goes on becoming less exact for as long as nobody resets it.

One picture makes it obvious rather than merely believable. The option position's value plotted against the reference asset price is a line that bends. The offsetting holding's value plotted against the same price is a straight line, because a quantity of the reference asset moves in a straight line with its own price by construction. A straight line can be matched to a bending line at exactly one point. Everywhere else the two come apart, and how fast they come apart is what gamma measures.

Putting a scale on that bend would need a delta and a gamma at every price along it, and both come out of a pricing model that has to be fed a reading nobody took. The shape is therefore true in form and silent about size.

So the practical shape of this method is not what a first reading suggests. The method is not done once. The method is done again and again, and the interval between one doing and the next is a choice. Reset often and the match stays close, at the price of dealing again and again. Reset rarely and the dealing stops, at the price of holding a quantity that was matched to conditions that have gone. Choosing that interval needs an opinion about the distance the reference asset price is likely to cover between one reset and the next, so the interval belongs to whoever holds the position and to nobody else.

What does neutralising not mean?

Switching off the answer to the reference asset price leaves every other answer exactly where it was. Not reduced. Not partly addressed. Exactly where it was: nothing was done to any of them.

A position that has been delta hedged has not thereby been made safe. It is a position from which one answer has been taken out, and the taking out is temporary. Taking one answer out is a genuinely useful thing to have done. Taking one answer out is not what the shorthand suggests has been done.

Go back to the stall for a moment. The everyday version carries this better than the abstract one. The stallholder who sells umbrellas and cold drinks has done something real: the rain has stopped mattering. Nobody would look at that stall and say the stallholder can no longer have a bad month. The office could shut for a fortnight. The wholesaler could raise prices. A road could be dug up outside the gate. Every one of those was untouched by adding cold drinks to the umbrellas, and every one of them was untouched for the same reason the clock is untouched by holding the reference asset: the fix addressed one input and one input only.

What does running this procedure cost?

A method described without its running costs has been described dishonestly. Two costs attach to this one, and neither can be sized from these figures.

The first is dealing. Step five means going back into the reference asset and changing the quantity held. The quantity changes every time the answer the offset was matched to has moved, and that answer moves continuously. Each of those changes is a deal, and each deal costs something. So the running costWhat a method costs to keep going rather than to set up. For a method built on repetition, it is the repetition that costs, not the first action. of this method rises with how often the offset is reset, and falls with how rarely. The trade is a genuine one, and it has no right answer that holds in general.

The second is what has to be lodged behind the option position while it is held. A written option contract is an obligation to somebody, and the system that stands between the two sides does not take the obligation on trust. The system asks for collateralWhat a writer lodges with the party standing between the two sides so the obligation is backed by something. The level, and the working that arrives at it, are set by an authority.. Both the amount and the working that arrives at the amount are set by SEBI, sebi.gov.in, and both of them change. A figure written here would be a figure written from memory, and a plausible wrong number is worse than an honest gap.

Put together, the honest summary is short. The method has a running cost, and its size follows from how often the offset is reset and what each deal costs. Leaving the running cost out altogether makes the method look easier and cheaper than it is, and descriptions of this procedure usually mislead in exactly that way.

Why can the quantity to hold not be printed here?

Every step of the offsetting procedure can be written out except the step that decides how much. The offsetting amount is a delta multiplied by what one contract stands for, and that delta is beyond reach here for the same reason it is beyond reach in any account of the subject with no pricing model behind it. The method is therefore taught with a blank standing exactly where its central quantity belongs.

Locate the gap precisely: it is one step rather than a general haziness. Steps one, three, four and five are all available to anybody holding the position, and step two cannot be carried out here at all. The distinction matters more than it looks. A reader who thinks the whole method is vague will treat every part of it as approximate. A reader who knows the gap is at step two knows three useful things at once: what to ask for, who they are asking, and what to ask them about it.

The remainder is more than a consolation. The shape of the method is in hand, so somebody else's working can be followed rather than taken on trust. The reason it must be repeated is in hand, so a reader can ask how often theirs is repeated and what that costs. The list of what it never touched is in hand, so a reader can ask what happens to the position when the clock runs and nothing else does. And anybody quoting a delta has made a set of assumptions, chosen a model, and fed it inputs. Every one of those choices is open to question, and knowing that a number came out of a choice is the difference between reading it and believing it.

What does the worked pair settle, and where exactly does it stop?

Three things are given here and nothing else is. The reference asset carries a price of Rs 2,000.00/-, an exposure rather than money anybody has handed over. Borrowing costs 6.50 per cent across twelve months. The call was quoted at Rs 180.00/-. Both contracts finish on the same date one year out, and the reference asset hands its holder nothing across that year.

Rs 2,000.00/- is also the strike written into both contracts, matching the spot priceThe price of the thing itself for delivery now, as opposed to a price agreed today for a handover on some later date. exactly, and that agreement was arranged rather than copied across by accident. Struck at the money is the name for it, and having a strike land on the price ruling today is the whole of what the phrase has ever meant. The put premium is Rs 57.93/-.

First case: the bought call on its own

Step one is done in a line. The position is one bought call. Nothing about that needs a model, a rate or an assumption.

Step two asks for a delta, and this is where the working stops. A delta for this call would require a pricing model, and every pricing model built for this kind of contract asks first for a reading that was never taken of these contracts. So no delta is produced, and none is guessed at either.

Steps three, four and five can still be described, and they are worth describing with the blank left in. Seeing them work around a hole is what shows the hole to be one specific shape. Take the delta, multiply it by what one contract stands for, multiply by one because there is one contract, hold that quantity of the reference asset the opposite way round, and come back tomorrow to do it again. Four of the five steps survive intact. The bought call on its own is the ordinary case, and in the ordinary case the whole procedure can be shown and cannot be completed.

Second case: the bought call held with a written put

Now hold the bought call and write the put against it, both on one strike and one end date. The pair completes, and it completes without a model being consulted anywhere.

Check the arithmetic before leaning on it. Take the strike of Rs 2,000.00/- and bring it back one year at 6.50 per cent: it is worth Rs 1,877.9343/- today. Subtract the put premium from the call premium and Rs 122.07/- is left over. Now come at the same thing from the other side: take today's worth of the strike away from the price of the reference asset and Rs 122.0657/- is left over. The two routes land 0.43 paise apart, and they land apart because the put premium was written to two places. Rounding one premium to two places is the whole of that difference, so what these figures support is agreement down to the paisa and no claim past it. An exact agreement is not what the figures support.

Here is the result the second case exists to reach. On the last day, that pair hands over the reference asset price less Rs 2,000.00/-, at every price without exception. At Rs 2,400.00/- it hands over Rs 400.00/-. At Rs 1,600.00/- it hands over minus Rs 400.00/-. At Rs 2,130.00/- it hands over Rs 130.00/-. There is no bend anywhere in it. The choice that would have produced a bend has been cancelled: the buyer of the call chooses to act above the strike, and the writer of the put is made to act below it, so between them somebody acts at every price.

Because that payoff moves one rupee for every rupee the reference asset price moves, the offsetting quantity on the last day is exactly the whole of what those contracts stand for, and nothing was assumed to work it out. No model. No input the record does not hold. Just the observation that a line at forty five degrees is offset by holding the same amount the other way round. The reason this one case escapes is worth naming rather than leaving mysterious: the pair has had its optionalityThe part of a contract that exists only because one side may choose. It is the part a pricing model has to form a view about, and cancelling it leaves nothing for a model to do. cancelled out of it, so there is nothing left for a pricing model to have an opinion about.

There is a second thing this pair settles, and it is worth having because it shows the pair really is a forward obligation rather than merely something that resembles one. The net premium of Rs 122.07/- was paid on day one. Carry it forward at 6.50 per cent for the year and it becomes Rs 130.00/-. Now look at the forward priceThe price agreed today for a handover on a stated later date. It is reached by carrying the spot price forward at the financing rate, so it is arithmetic rather than an opinion about the future. of Rs 2,130.00/- and take the strike of Rs 2,000.00/- off it: Rs 130.00/- again. The two routes are the same arithmetic reached from opposite ends, and the agreement is forced rather than coincidental. The difference in the premiums carried forward is exactly what the price of the reference asset carried forward exceeds the strike by.

And now the condition, which belongs in the same breath

Everything above was said about the last day, and the last day is doing work in that sentence. Before the last day the cancelling is not rupee for rupee, and the same equality stated without that qualification would hold on exactly one date out of three hundred and sixty five.

Worked through with the case at hand: with a year still to run, suppose the price of the reference asset falls by Rs 80.00/-, to Rs 1,920.00/-. The fall is a declared illustration of a move rather than a reading taken off anything. Carry Rs 1,920.00/- forward at 6.50 per cent and the contract price for that date is Rs 2,044.80/-, where it had been Rs 2,130.00/-. So the contract price fell by Rs 85.20/- while the reference asset price fell by Rs 80.00/-. The carryWhat it costs to hold something with borrowed money from today until a stated date. Added to the price today, it gives the price for that date. applies to the new price just as it applied to the old one, so the move is passed through at 1.065 times its own size.

So the rule that carries across to every offset is to say at which moment the offset is being worked. A flat net line drawn with no moment attached to it is asserting an equality that holds on one day and fails on all the others, and the failure is a quiet one: a drawing with nothing written on it about timing looks like a general statement.

How does somebody working with these contracts actually use this?

Four sets of hands, and none of them is doing the same job with it.

Somebody who has written contracts is the one for whom this is a daily procedure rather than an idea. The writer took a premium and accepted an obligation, and the obligation answers to the reference asset price whether the writer has a view on that price or not. So they hold a quantity of the reference asset against it and reset that quantity as the answer moves. The interesting part is what they are left holding afterwards. Having taken the price answer out, the writer is left holding the clock, the implied number and the financing rate. The book is now a position in those three things. The remaining job is not a smaller one. The remaining job is a different one, and knowing that is the difference between calling a book hedged and describing what is actually held.

For an analyst working through a disclosure that other people wrote, the method runs backwards and becomes a list of questions instead of a list of steps. If a report says a book is delta neutral, the useful questions follow straight off the procedure. Neutral as at when. Neutral over what size of move. Which model produced the deltas, and what was fed into it. How often is the offset reset, and what is that costing. None of those questions needs a number to be worth asking, and a disclosure that cannot answer them has said less than it appeared to say.

A treasurer at a business that has written or bought contracts against a real commercial exposure has a narrower use for it, and a sharper one. The exposure they are trying to cover has a size and a date, and both are known to them. So the question in front of them is whether an offset matched to a small move at one moment is the right shape of protection for an obligation that runs for months. Often the honest answer is that the repetition is the problem: a method requiring continuous attention sits badly against a business that has other work to do. Which way that decision should go belongs to the treasurer. The things to be weighed are set out above.

And a household reading a statement that mentions the word hedged has exactly one thing to take from all this: the word is a claim about one input. No delta is needed to ask what it was neutral to. The question is available to anybody, costs nothing to ask, and is the one the shorthand was hiding. If the answer comes back as a description of a single input and a single moment, the word was being used correctly. If it comes back as a reassurance, it was not.

Is being able to work through this a reason to do it?

Should anybody run this method? The honest version of that question has an answer that cannot be given here. Not out of caution. Out of arithmetic.

Anybody answering would need three things laid out in front of them, and not one of the three is available here. Where is the reference asset liable to end up, with what weight attached to each landing place? The missing quantity is the identical one step two of the procedure went hungry for. Next, the circumstances of whoever holds the position, known to them and to nobody else. Last, the full bill, in two halves: what holding the arrangement all the way to its final date would cost, and what getting out of it early would cost instead. The repeated dealing this method requires is buried somewhere inside both halves.

The procedure described above has a blank in the middle of it, and a procedure that cannot be finished is not a procedure anybody should be talked into. Its worth lies elsewhere and it is real: somebody else working the method can now be followed, the step where their answer came out of a choice can be named, and that choice can be asked about.

The reference asset, its price of Rs 2,000.00/-, the financing of 6.50 per cent a year, the strike of Rs 2,000.00/- and the two premiums are invented.
Educational material. Not advice on any investment, tax, budget or market position.