Replication vs Hedging: An Argument and an Activity

Replication is an argument. A payoff can be built exactly out of things that already have prices, so it cannot cost anything other than what building it costs. Hedging is an activity. Somebody holds positions meant to offset a risk they already carry. One is a step in a proof, the other is something a person does, and a result about the first says less about the second than its name suggests.

One observation carries the whole comparison. Replication is used to establish a price, and then it is put away. Nobody has to carry the bundle out for the price to be right, and the price does not become less right if nobody ever does. Every convenience the argument grants itself is a convenience it can afford precisely because the thing it describes never has to happen. The comparison starts there, and the convenience is why the two words are not two settings of one dial.

What is replication, defined from scratch?

ReplicationBuilding a payoff exactly out of things that already have prices, as a step in an argument rather than as something anybody performs. starts with a payoff handed over as a set of numbers, one number for each way the world is allowed to turn out. Whatever produced those numbers does not matter. The question is narrow and mechanical: is there some combination of things that already carry prices which delivers exactly those numbers, in every one of the stated outcomes? If such a combination exists, then the payoff has a price, and that price is the cost of putting the combination together today.

The step that converts a matching combination into a price is set out under the law of one price: two things that deliver identical numbers in every outcome cannot carry two different prices. So once the match is exact, the price is forced. Not estimated, not approximated, not fitted. Forced, by the fact that anything else would let somebody take one price and give the other and finish with more than they started with in every outcome at once.

A balance makes the same point in everyday terms. An object of unknown weight sits on one pan. Beside it stands a box of calibrated standard weights. Weights go onto the other pan until the beam sits level. The beam sits level rather than nearly level, so the weight of the object is then known exactly rather than roughly. Then the weights come off the pan and go back in the box. The weights were never the answer. They were how the answer was reached, so the reading survives their removal.

Two shapes of the same argument get separate names, and both are arguments. Static replicationBuilding the payoff once at the start and never touching the holding again until the horizon. builds the bundle once and never touches it. Dynamic replicationBuilding the payoff in a way that requires the holding to be changed as the process moves, up to changing it at every instant. builds it in a way that requires the holding to be adjusted as the process moves. The dynamic shape is taken up below, and the adjustment is part of the argument rather than part of anybody's day.

So the last move deserves its own name. The bundle is scaffoldingSomething put up in order to establish a result and then taken down, with the result standing on its own afterwards.. Scaffolding goes up so that the price can be established, and it comes down as soon as the price is written. A reader who finishes the argument holding nothing but the number has taken away exactly what the argument was built to give them.

What is hedging, defined from scratch?

HedgingHolding positions intended to offset a risk somebody already carries, as an activity performed through time by a person with an account. starts somewhere else entirely. Hedging starts with somebody who already carries an exposure they would rather not carry. The person takes on a further position whose value tends to move the other way, and the combination is then less sensitive to whatever they were exposed to. Notice what the definition contains and how little it is: a person, an exposure that already exists, and a purpose expressed as less rather than as none.

Three things belong to that activity and are not optional parts of it. The offset has to be corrected as things move, so there is a frequency, and somebody has to decide how often. Putting the offset on and changing it is paid for every time, so there is a cost. What is left over after the offset is never nothing, so there is a residual, and it has to be judged against a standard that somebody chose. Hedging is scored on a matter of degree. A matter of degree has no exact answer and never claimed one.

The everyday version is a tally rather than a balance. Somebody stands at the door of a hall keeping a running count of how many people are inside. Every time a person enters or leaves, the count has to be corrected. Between corrections the count is wrong by however many people moved in the meantime. Correcting more often keeps the count wrong for shorter stretches and takes more of the counter's attention; correcting less often is easier and lets the count drift further. There is no setting of that dial at which the tally is simply right and stays right, and none of that is a defect in the idea of counting. Drift between corrections is what counting a moving thing is.

The correction has a name of its own. RebalancingChanging a holding as conditions move, so that the offset keeps doing what it was put on to do. is the act of changing the offsetting holding after the thing being offset has moved. Every rebalancing is a decision, it happens at a moment somebody chose, and it costs something. A description of hedging that leaves out the frequency and the cost has left out most of the activity.

Where do the two part company, and is the gap one of degree?

Both descriptions contain a holding. Both name units of the standard process. Both put a number on how much of it there is. The surface similarity is the entire trap. The difference is one of kind rather than of degree: one is a sentence inside a proof and the other is a thing done in the world, and no amount of adjusting either turns it into the other.

Take the balance and the tally back out. The standard weights on the pan and a bag of sand carried in a small boat to keep it steady are both weights. Both are chosen, both are measured, both are put somewhere. But the standard weights exist to produce a reading and then leave the pan, and the reading does not care whether they are still there. The bag of sand has to stay in the boat, has to be shifted when the boat is loaded differently, and stops working the moment nobody attends to it. Nobody would confuse the two, and the only reason the confusion arises in pricing is that both are written down as a number of units.

What does the argument look like on one payoff?

The whole argument fits on one set of numbers. The standard process starts at Rs 100/-. Over the one year horizon it is allowed exactly two destinations, Rs 122.140276/- or Rs 81.873075/-. A payoff is handed over as two numbers and nothing else is said about it: Rs 22.140276/- if the process finishes at the higher value, and nothing if it finishes at the lower one. Nothing is said about what produced that pair, and the argument does not need to know.

The holding follows from dividing the spread of the payoff by the spread of the process. The payoff spread is Rs 22.140276/- and the process spread is Rs 40.267201/-, being the difference between Rs 122.140276/- and Rs 81.873075/-. Dividing one by the other gives 0.549834 units. The borrowing follows from the lower outcome: 0.549834 units are worth Rs 45.016600/- there, and the payoff needs nothing there, so the borrowing has to grow to exactly Rs 45.016600/- by the horizon. At a rate of 5 per cent continuously compounded that means borrowing Rs 42.821115/- today.

Assembling that bundle today costs 0.549834 times Rs 100/-, or Rs 54.983400/-, less the Rs 42.821115/- that was borrowed rather than paid, leaving Rs 12.162285/-. Rs 12.162285/- is the price of the payoff. Read the other way, by weighting the up outcome at 0.577493 and discounting at 0.951229, the same price comes to Rs 12.162285/- again, agreeing to six decimal places. Two routes to the same number, and neither of them requires anybody to hold anything. The bundle has now done its work and does not appear again in the argument.

The pattern in that question is what makes the whole argument work, so take it apart. Hold the cost of the bundle fixed at Rs 12.162285/- and let the holding vary. Every rupee of extra holding has to be paid for with extra borrowing, and the borrowing costs the same in both outcomes while the extra units are worth much more in the up outcome than in the down one. So raising the holding lifts what the bundle delivers in the up outcome and lowers what it delivers in the down one. The two gaps move in opposite directions, and that is exactly why one and only one value can close both at once.

What is the argument allowed to assume?

Five things carried that worked instance, and each one is something the activity does not get. The argument assumes that the process moves to exactly one of two stated values. The argument assumes the bundle can be held in fractional unitsHolding any amount of something at all, including 0.549834 of a unit, with no smallest tradeable size., including 0.549834 of a unit. A single rate is assumed for both borrowing and lending. Trading is assumed frictionlessCosting nothing at all to trade, which the argument assumes and which the activity never gets., so nothing at all is paid to put the bundle on. And it assumes the holding never has to change once it is set.

Five for five, and none of the five is a small idealisation of something a person nearly has. Each one is the difference between a sentence that can be written down and a thing that can be done. None of that is a criticism of the argument. An argument that had to pay to trade and could only hold whole units would not establish anything cleanly, and the price it produced would be a range rather than a number. The assumptions are load bearing, and the load they bear is the exactness of the conclusion.

What does a replication result promise, and what does it not?

A replication result promises a price, and promises that the price is the only one consistent with the assumptions it stated. The promise is a strong one and worth having. The result does not promise that the holding inside it can be achieved. Nothing in the result promises that a nearby holding would deliver anything near the payoff. Nothing in the result promises that the holding is stable if the argument is changed. And the result never considered whether holding it would be a sensible thing for anybody to do, so it very definitely does not promise that either.

The last point is easy to miss, so take it head on. Refine the same construction and the price improves while the demand on the holder gets steadily worse. At one step there is one moment at which a holding has to be chosen, and the price comes out at Rs 12.16/-. At two steps there are three such moments and the price falls to Rs 9.54/-. At four steps there are ten moments and Rs 9.97/-. At twelve steps there are seventy eight moments and Rs 10.29/-, against the value the construction is converging to, Rs 10.45/-. The lattice construction here is the one associated with Cox, Ross and Rubinstein, 1979, and the limiting value is the one associated with Black, Scholes and Merton, 1973. The approach is not a straight line: it overshoots high, comes back too far, and then closes in from below.

There is one more reason not to read the holding as an instruction, and it is the sharpest one available. The same payoff rule, priced by the one step argument, carries a holding of 0.549834 units. Priced by the continuous argument, it carries a holding of 0.636831 units at time zero. Same process, same rule for the payoff, two correct arguments, two different numbers. A quantity that changes when the argument changes is a feature of the argument rather than a property of the world, and nobody should be told to hold it.

Why does the distinction decide how a result should be read?

Because the distinction changes which claims the result has to defend. Ask one question of any passage that contains a holding: what did it conclude with? If it concluded with a price, then whatever holding appears inside it is a means, and the assumptions around it were chosen for the convenience of reaching that price. Nothing in it is addressed to anybody, and nothing in it is a suggestion. If it concluded with an instruction, then it is a claim about action, and it needs assumptions that survive contact with action. The assumptions that survive contact with action are different ones, and somebody else usually has to state them.

There is a second test and it is quicker. Would the result still be true if nobody ever carried it out? For a replication result the answer is yes, unambiguously, and that is precisely why the bundle can be discarded at the end. A hedge that nobody puts on offsets nothing at all, so for any claim about hedging the answer is no. A result that survives nobody doing it was never an instruction, whatever it looks like written down.

How does somebody reading a result actually use this?

The reader this matters most to is not the person writing the argument. The reader who matters is the person on the receiving end of working somebody else produced, whether a preprint from a repository, a model note handed across a desk, or a stretch of derivation in a set of course materials. Such a reader has to decide which claims the passage must defend before deciding whether to believe it, and the two questions above do that job in about fifteen seconds.

1. Find the last line and see what kind of thing it isA number with a currency on it is a price. A sentence with a holding as its object is an instruction. Almost every replication passage ends in the first kind, however much holding language appears in the middle of it.
   Here the last line of the argument is Rs 12.162285/-.
2. Ask whether the result survives nobody doing itIf the conclusion would still be true in a world where nobody carried out a single trade, the passage was establishing something rather than proposing something. If it would not, it was proposing something and needs the assumptions that go with proposing.
   The price stands whether or not the bundle is ever assembled.
3. List the conditions and ask which of them holdTwo outcomes, fractional units, one rate, free trading, a fixed holding. The situation at hand scores against each. Where the score is zero out of five, nothing in the passage was addressed to that situation, and that is a statement about the passage rather than about the reader.
   Nobody scores above zero on the second, fourth or fifth.
4. Check whether the number moves when the argument doesPrice the same payoff rule a second way and see whether the holding changes. If it does, the holding belonged to the method rather than to the payoff, and it was never a candidate for anybody to carry.
   0.549834 at one step against 0.636831 in the continuous version.

The same discipline runs the other way, and it is worth saying so. A genuine claim about hedging, one that really is about what somebody should hold and how often they should correct it, is not evidenced by a replication result. The frequency, the cost of the corrections, and the standard the leftover is being judged against are the three things to ask for. The frequency, the cost and the standard are the whole of the activity, and a replication result answers none of the three because it was never asked.

The standard process and its four parameters are invented.
Educational material. Not advice on any investment, tax, budget or market position.