Complete Market vs Incomplete Market: One Price or a Range

A complete market is one in which every payoff can be assembled out of things already carrying a price, so each payoff has exactly one price. An incomplete market is one in which some payoffs cannot be assembled, and those carry a range instead. Completeness is a counting condition, its absence costs a computable interval, and the incomplete case is the ordinary one.

The whole comparison turns on one question that can be settled before any pricing is attempted: are there enough independent things to trade to reach every way the horizon can turn out? Counting is the whole of it. The count returns a yes or a no, the check takes a minute, and everything else in this guide follows from which answer comes back.

Here is the everyday version, and it uses nothing but a balance and a box of weights. Suppose the box holds a one kilogram weight and a two kilogram weight. One kilogram can be made. So can two, and so can three. One and a half cannot. There is no arrangement of those two weights that lands on it, and the honest answer stops being a number: the amount asked for sits somewhere between one kilogram and two, and the box cannot narrow it further. Nothing about the request was unreasonable and nothing about the balance was faulty; the box simply did not contain enough distinct weights to reach the amount asked for.

A market is that box. The weights are the things whose prices are already known. The amounts that might be asked for are payoffs. When the box can make the amount exactly, the cost of making it is the price, and there is nothing to argue about. When it cannot, what comes back is a bracket, and no amount of care in the arithmetic turns a bracket into a number.

Everything below is that picture written carefully enough to check, on the standard process: a single traded quantity starting at Rs 100/-, carrying a volatility of 20 per cent a year and a drift of 8 per cent a year, watched for one year, with cash growing at 5 per cent a year continuously compounded. The four parameters are assumed rather than measured, and every figure computed from them follows by arithmetic rather than from any observation.

What is a complete market, defined from scratch?

A complete marketOne where every payoff can be built out of the things that are already priced, so nothing that can be asked for falls outside what the available holdings can produce. is a setting in which, for every payoff that could be handed over, there exists a holding of the available things that delivers exactly that payoff in every outcome. Not approximately. Not on average. In every outcome, to the last paisa.

Because the rest depends on it, a payoff needs saying precisely here. A payoff is a list of amounts, one entry for each way the horizon can turn out. The list is not a contract, and where the list came from does not matter here. Somebody hands over the list and asks what it costs to arrange for exactly those amounts to arrive. Completeness is the property that this question always has an answer, whatever list is handed over.

The available things here are two. The first is the standard process itself, starting at Rs 100/- and finishing at whatever value the outcome delivers. The second is cash, starting at one rupee and finishing at e to the rate times the horizon, or 1.051271 rupees over the year. A holding is a pair: how many units of the standard process, and how many rupees of cash. Both may be negative. A negative entry means the holding was sold or the cash was borrowed.

The last clause of that statement is the whole of this guide. As many equations as there are outcomes. Two unknowns. A system like that has a solution for every right hand side only when the equations are not more numerous than the unknowns can handle, and that is arithmetic rather than finance.

What is an incomplete market, defined from scratch?

An incomplete marketOne where at least one payoff cannot be built out of the things already priced, so that payoff carries an interval of permitted prices rather than a single one. is a setting in which at least one payoff that could be handed over has no holding that delivers it in every outcome. The system above has more equations than the two unknowns can satisfy, so for some right hand sides there is simply no pair that works.

Notice what has and has not happened. The assumption underneath everything is untouched: no position costing nothing may deliver something for certain while never delivering a loss. The no-arbitrage assumption still rules out a great deal. The assumption has merely stopped picking one figure. Instead of a price it returns a price rangeThe interval of figures the no-arbitrage assumption permits when a payoff cannot be built, with everything outside it ruled out and nothing inside it preferred., and the assumption gives no reason to prefer any figure inside that interval over any other.

The temptation is to read incompleteness as a defect, either of the market or of the mathematics. Incompleteness is neither. Go back to the box of weights. Being unable to make one and a half kilograms is not a fault in the balance and not a fault in arithmetic; it is a true report about how many distinct weights are in the box. An incomplete market is not a market that has gone wrong, it is a description that has reported honestly how much it can and cannot pin down.

The honesty is uncomfortable to live with. A range is harder to put in a report than a number, and the pressure to produce a number does not go away because the mathematics stopped supplying one. The pressure, and what people do under it, is taken up below, where a single number is needed and the mathematics supplies a range.

How is completeness checked by counting, before pricing anything?

Here the comparison becomes checkable rather than atmospheric. Completeness is a counting conditionA check that compares the number of independent things available to trade with the number of outcomes that can occur, settled before any pricing is attempted.: line up the horizon values of everything available to trade as the columns of a table, put one row for each outcome, and ask how many independent columns there are against how many rows.

Two columns. Rank at most two. So with the standard process and cash as the only things available, completeness survives up to two outcomes and fails from three onward, and no amount of skill in the pricing changes that. The check is finished before any pricing begins. A count of the independent things available to hold, set against a count of the outcomes the description admits, has already decided whether a single price is available at all.

The word doing the work is independent. Adding a third thing to hold that is itself a fixed combination of the first two adds a column but no rank, and buys nothing. Adding it is the same trap as putting a second one kilogram weight in the box: the box now has three weights and still cannot make one and a half. SpanningReaching every possible outcome pattern using combinations of what is available, which is what the independent columns of the payoff table can and cannot do. is about independent directions, never about how many items are on the shelf.

What does completeness actually buy?

Uniqueness. Uniqueness is the entire purchase, and it is worth being blunt about how narrow it is. Completeness does not make a price accurate. Completeness does not make a description true. Completeness does not reduce anybody's exposure to anything. Completeness buys one thing, that the argument returns a point rather than an interval, and it buys nothing else at all.

The mechanism behind that purchase is the same counting again in a different costume, and it is worth seeing directly. Instead of asking which payoffs can be built, ask which sets of weights on the outcomes are admissible: strictly positive, adding to one, and repricing the standard process itself. The admissible weight sets form a set of their own, and the size of that set decides whether the price is a point.

Now read the dimension. At two outcomes it is nought, so the set is a single point and the measure is the only one there is. At three it is one, so the set is a segment and there are infinitely many admissible measures. At four it is two, at five it is three, and the set only grows. The counting of instruments against outcomes and the counting of free directions in the measure set are the same statement written twice.

The counting is why the phrase that completeness makes a model better is worth resisting. Look at what actually moved. The description of how the standard process can behave did not become more faithful when the third outcome was struck out; it became narrower. A complete description is a description that has been made small enough that two instruments can reach all of it, and smallness is not the same as truth.

What does its absence cost, as a number rather than as a worry?

Here is the worked instance, small enough to check by hand. One step, one year. The standard process finishes at Rs 122.140276/- in the up outcome and Rs 81.873075/- in the down outcome, those being Rs 100/- multiplied by e to plus and minus the volatility times the square root of the horizon. Cash grows by 1.051271. The payoff handed over is Rs 22.140276/- in the up outcome and nothing in the down one, and its origin does not matter here.

With two outcomes the two equations in two unknowns solve once. The holding is Rs 22.140276/- divided by Rs 40.267201/-, or 0.549834 units of the standard process, and the cash is minus Rs 42.821115/-, meaning that much is borrowed. Building the payoff therefore costs 0.549834 times Rs 100/- less Rs 42.821115/-, or Rs 12.162285/-. Rs 12.162285/- is the price, it is the only price, and the counting is why.

Now admit a third outcome at Rs 100/-, and change nothing else. The two instruments are the same two. The new outcome sits at the level where the handed-over list already returns nothing, so the payoff there is nothing. Three equations, two unknowns, and the system has no solution: the payoff cannot be built. No-arbitrage now returns an interval, and both of its ends can be derived rather than asserted.

On the three outcome case that cheapest bundle is the same 0.549834 units and the same borrowing of Rs 42.821115/-, costing Rs 12.162285/-. The bundle matches the payoff exactly at the two extreme outcomes and overshoots at the middle one, where it is worth Rs 9.966799/- against a payoff of nothing. SuperreplicationBuilding a holding that is never worth less than the payoff in any outcome, whose cost gives the top of the permitted interval. is exactly that overshoot, and the overshoot is what is paid for at the top of the interval.

Working that closed form through lands on Rs 100/- less Rs 95.122942/-, or Rs 4.877058/-. The holding that achieves it is one whole unit of the standard process, with Rs 95.122942/- borrowed, and it checks outcome by outcome: at Rs 122.140276/- it is worth Rs 22.140276/- against a payoff of the same, at Rs 100/- it is worth nothing against a payoff of nothing, and at Rs 81.873075/- it is worth minus Rs 18.126925/- against a payoff of nothing, a value below the payoff and therefore permitted.

The interval is Rs 7.285227/- wide and its top is 2.493775 times its bottom. The width is the cost of incompleteness on this payoff, and it is not a rounding at the edge of an answer, it is most of the answer. One precision point belongs here and it is easy to check on the table: both of the weight sets at the ends place nothing on one outcome, and admissible weights have to be strictly positive. So the ends themselves are ruled out along with everything outside them, and what the assumption permits is every figure strictly between. Selling at exactly Rs 12.162285/- and holding the superreplicating bundle would cost nothing to set up, lose in no outcome, and gain Rs 9.966799/- at the middle one. Such an arrangement is precisely what the assumption exists to forbid.

Why is the incomplete case ordinary and the complete case special?

Because every honest addition to a description adds outcomes, and almost none of them adds an instrument. One sentence carries the whole argument, and it is worth walking through the additions one at a time to feel how one sided the traffic is.

Admitting that the process can finish somewhere between the two extremes rather than only at them is the mildest thing anybody could ask for, and it moves the count from two outcomes to three. Admitting that it moves at more than one date multiplies the outcomes with the dates. Admitting jumps, in the sense of Merton, 1976, adds outcomes that differ by how many arrivals occurred and how large each was, none of which is reachable by holding the process and cash. Admitting that the volatility itself moves, in the sense of Heston, 1993, adds something worse than an outcome: a second source of movement with nothing traded against it, and that is unhedgeable riskA source of movement that no combination of the traded things can offset, so exposure to it survives every holding available. in the plain sense of the phrase.

Now count what went the other way. To restore the balance, each of those additions would need a matching independent thing to hold, with a price already known, whose horizon value moves with exactly the new source and not with the old ones. Sometimes such a thing exists. Usually it does not, and where it does, its price came from somewhere and that somewhere has to be accounted for too.

Completeness is therefore not a property that descriptions tend to have, it is a property that descriptions are built to have, by keeping the outcome count down to what the available instruments can reach. The two outcome case survives in teaching, this guide included, for one reason: it is the only setting where the argument produces a single number cleanly enough to check by hand. Being checkable by hand is a good reason to keep the case and a bad reason to believe it.

Here is the counting example again, from the balance and the box. Every extra amount somebody might ask to have weighed out is another row. Weights are what would have to be added to keep up, and nothing about being asked for a new amount puts a new weight in the box. The requests grow on their own; the box only grows when somebody buys a weight.

What is done when a single number is needed and the model gives a range?

Three responses are in common use, and they are honest in different ways. None of them recovers a single number from the mathematics, and the reason to name all three is so that a reader can tell which one they are looking at when a report hands them a figure.

1. Quote two prices rather than one The interval is reported as an interval. A figure at which the payoff would be taken on and a higher figure at which it would be passed on, sitting at or inside the two ends. Reporting the interval hides nothing, and the report says on its face that the argument returned a bracket. The cost is that a bracket cannot be added up, averaged or put in a single cell.
2. Take one end of the interval and say which end The top, Rs 12.162285/- on the worked payoff, is the cost of a bundle that is never worth less in any outcome, so taking it is a conservative choice with a clear meaning. The bottom, Rs 4.877058/- here, has the mirror meaning. Neither end was selected by the assumption. Both were selected by somebody deciding which way to lean.
3. Add an assumption that picks a point inside the interval Choose one measure out of the many admissible ones, on some ground the assumption did not supply: a preference, a penalty for how far the chosen measure sits from the physical measure P, a fitted parameter, a convention. The output is a single number again. The number is single because of the added assumption, and the number moves when the assumption does.

The one thing that must survive from this section is that all three are choices, and a report that presents any of them as a result of the argument has mislabelled its own output. Naming which response was used costs one sentence and tells the reader everything about how much weight the figure carries.

A bid and offerTwo prices rather than one, which is one honest way to report a payoff whose price the argument returns as an interval. pair is the first response wearing its everyday clothes, and the balance makes the same point one last time. Asked for one and a half kilograms, a person can say between one and two, can hand over two and say it is at least what was asked for, or can decide on some other ground to call it one and a half. All three are usable. Only the first is a statement about the box.

How does somebody reading a model output actually use this?

Concretely, and it takes about two minutes. Someone handed a valuation, a risk figure or a reserve number that came out of a model does the counting themselves, on the model's own description rather than on its results.

First, find the outcome count. The count is rarely called that. The count appears as the number of states in a lattice, the number of factors driving a diffusion, whether jumps are admitted, whether the volatility is a constant or a process of its own. Second, find the instruments the model treats as already priced and independent: the underlying quantity, cash, and whatever else the model is told the price of. Third, compare the two numbers.

Where the first exceeds the second and one number came out anyway, ask the only question that matters: which assumption picked that number, and how far does the answer move when it changes? A model whose author can name the selecting assumption in one sentence is being read correctly, and a model whose author cannot has produced a figure nobody can attribute.

The same counting works in reverse as a design check. A clean answer arriving early is usually the shape of the description talking rather than the mathematics, so somebody building a model who finds that a single price falls out should look at the outcome count before feeling pleased.

None of this asks for market access, data or software. The check is two counts and a subtraction, done on the document the model is described in. Stating completeness as a counting condition rather than as a property that a setting either has or lacks by nature is what makes the check that cheap.

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