Market Price and Model Price: Two Different Objects

A market price is a number at which something changed hands. A model price is a number a set of assumptions produces. The two numbers are different kinds of object: one is a record of an event and the other is the output of a calculation, and no amount of agreement makes them the same kind of thing.

Almost everything that goes wrong in this area goes wrong in the first five seconds, when a reader puts the two numbers side by side and silently decides they are rivals. Two numbers, one of them right. The two numbers are not answers to the same question, so the instinct to rank them is wrong before any arithmetic has been done. One of them reports; the other computes.

Here is the everyday version, and it is worth holding on to throughout this guide. A bus timetable says the bus reaches a particular stop at nine o'clock. The bus reaches that stop at seven minutes past. Nobody sane says the bus was wrong. The timetable is a calculation: somebody took a route, a set of assumed speeds and a set of assumed stopping times and worked out a number. The arrival is a record: it happened, it was observed, and it is now a fact about the morning. When the two differ, the interesting sentence is never that the bus misbehaved. The interesting sentence asks which of the timetable's assumptions did not hold. Transferred wholesale, that question is the whole of the subject.

One consequence follows immediately and it is the load-bearing claim here. Only one of the two can be wrong. Being mistaken requires having asserted something, and a record asserts nothing beyond its own occurrence, so a record of what happened cannot be mistaken about itself. A calculation asserts a great deal: that these assumptions hold, that these inputs are the right inputs, that this arithmetic was done correctly. Every one of those is a place where it can fail.

What is a market price?

A market priceA number at which something changed hands, and so a record of an event. is the record of a transaction. Two sides agreed, something moved, and a number was written down. The agreement, the movement and the number are the entire content of the object. A recorded price carries a time, it carries a quantity, and it carries a number, and beyond those three things it says nothing at all.

The absences matter just as much. A recorded price does not carry a reason. Nobody who records a price also records the state of mind that produced it, and no amount of staring at the number recovers one. A recorded price does not carry a claim about value; it is not asserting that the thing was worth that, only that it went for that. And it does not carry a model. The missing model is the point to keep returning to. There is no machine behind a recorded price to open, so it is not the output of anything that can be inspected.

The missing machine is also why a recorded price cannot be rerun. Ask what the price would have been at a slightly different setting and the question has no meaning: no setting was ever chosen. There is only what happened. Compare the weighing scale in a kitchen. The scale reads four hundred and twenty grammes. There is no asking what it would have read had the density been different. The scale did not compute anything from a density; it responded to an object being placed on it. The reading is a record of that placement and of nothing else.

Two further points, both of which readers routinely mishandle. First, a recorded price can be badly transcribed, mistimed or attached to the wrong contract. A bad transcription is a failure of the recording, not of the price, and it is repaired by going back to the record rather than by reasoning about value. Second, a recorded price is not thereby sensible, fair or informative. The transaction simply occurred. A record needs no defence to serve its purpose here: it merely has to be the thing the calculation is being compared against.

What is a model price?

A model priceA number a set of assumptions produces, and so the output of a calculation. is the value of a function at a point. The function takes a starting value, a strike, a rate, a horizon and a volatility setting, and returns a number. Everything the number knows, it was told. Nothing else went in, so nothing else can come out.

Three properties fall straight out of that, and each of them is a property no recorded price has. A model price is reproducible: the same arguments return the same number today, tomorrow and on any machine. A model price is decomposable: four arguments can be held still while the fifth varies, showing exactly what that one argument was contributing. And a model price is answerable: every number it returns can be traced to a stated assumption, so when it is questioned there is somewhere to look.

Reproducibility, decomposability and answerability are precisely what people are reaching for when they say a model price is more rigorous. A model price is not more rigorous. A model price is more inspectable, and inspectability is a different virtue and a smaller one. Inspectability shows where a number came from; it does not show whether the place it came from was right. The timetable is fully inspectable. Every assumed speed is written down. The timetable is still the timetable and not the bus.

Theoretical Value vs Observed Market Price: why are these two different kinds of object?

Because they answer different questions. The recorded price answers what happened. The computed price answers what this function returns under these assumptions. The two prices are not two attempts at one answer; they are two answers to two questions that happen to be denominated in the same units. Sharing units is what fools people. Kilometres per hour is a unit shared by a speed limit and a speedometer reading, and nobody thinks the limit is an estimate of the reading.

The phrase to hold on to is different objectsDifferent kinds of thing rather than two estimates of one thing.. If two numbers were both estimates of a single underlying quantity, then a difference between them would be informative about that quantity, and the usual machinery of comparing estimates would apply. The questions would then be about bias, about spread, and about how to combine the two. There is no common quantity for either of them to be an estimate of, so none of that machinery is available here. The record is not estimating anything. The record is the thing itself.

How to Separate Observed Prices From Model Outputs: what actually does the separating?

In a working file the two arrive mixed together, in the same column, to the same number of decimals, and the mixing is where the damage starts. Three questions separate them cleanly, and none of the three requires any knowledge of the model.

The first question is whether the number has a counterparty and a timestamp. A record does. Something happened, at a moment, between two sides, and a number whose moment and whose two sides cannot be named is not a record. The second question is whether the number moves when an assumption changes. A calculation does. A record has no assumptions to change, so a record does not. Turning the volatility setting moves the computed number at once. The recorded number was never listening. Turning the dial a hundred times leaves it exactly where it was. The third question is whether the number can be regenerated from scratch. A calculation can, and it will come back identical. A record cannot be regenerated at all; it can only be looked up.

The practical discipline that follows is to label the kind at the moment the number enters the file, not later. Two columns, headed by what the numbers are rather than by what they are for. Every number in the worked instance below sits in one column or the other, and a reader who keeps the columns apart never has to ask the question again. A reader who lets them merge will eventually average across them, and an average of a record and a calculation is a number with no meaning whatsoever.

What does it mean when the two disagree?

Far less than most readers assume, and the honest answer starts with arithmetic rather than with finance. Call the difference at one strike the missThe difference between the two at one strike, carrying a sign.. The miss carries a sign, and so it shows which way the calculation leaned as well as by how much.

Now the arithmetic that settles the whole block. The computed price at any one strike rises strictly as the setting rises, and it never turns back. The strict rise is not an empirical observation; it is a property of the function, and it is why the whole question has a clean answer.

Put those two together and the consequence is unavoidable. Each of the five strikes has its own single setting at which it is repriced exactly. A single number cannot simultaneously equal five distinct numbers, so if those five settings are five different numbers, no one setting can satisfy more than one of them. Nothing more is involved. The impossibility is arithmetic, and it was settled before anyone chose a model.

Here is the everyday version, and it is the one to carry away. One thermostat dial serves five rooms. The kitchen wants one temperature, the bedroom another, the store room a third. Any single room can be satisfied exactly by turning the dial to its number, and the moment that happens, the other four are wrong. Nobody concludes from this that the rooms are misbehaving, and nobody concludes that thermostats do not work. The dial has one degree of freedomHow many independent numbers there are to choose, one against five here. and the demand has five. The mismatch between one dial and five demands is the entire explanation, and it needs no theory of heat at all.

What happens when one number is asked to match five?

Take the five invented recorded prices and find the one setting that fits them best in the ordinary least squares sense. The fitted setting is called the best single settingThe one parameter value minimising the total of the squared misses, 0.198202 here., and here it is 0.198202. The result is worth sitting with, and it is not what most readers expect.

Here is the full result, and it is the centre of this guide. The setting minimising the squared misses is 0.198202 and it misses every single one of the five. Not four of five. All five.

Read the sign column across and the shape of the failure is legible. At the three low strikes the calculation came in cheap, by Rs 0.662462/-, Rs 0.606838/- and Rs 0.067452/-. At the two high strikes it came in dear, by Rs 0.324179/- and Rs 0.441183/-. A single number, forced to serve a set that wants a higher figure at the bottom and a lower one at the top, splits the difference and is wrong in one direction at one end and in the other direction at the other. The pattern is what a compromise looks like, and it is what a compromise always looks like.

One point of arithmetic hygiene. Every figure here is printed to six decimals. The minimising setting to twelve places is 0.198202243906, and every figure in the table above is computed at the six place figure 0.198202 so that each miss is exactly the computed price less the recorded price as printed. The two differ in the sixth decimal of the computed prices and nowhere that matters; working at the printed setting is what keeps the table internally reconcilable line by line.

Now compare that against the settings that do hit something. Turn the dial to 0.240000 and the Rs 80/- strike is repriced exactly. The other four blow out to as much as Rs 1.983458/-. Turn it to 0.200000 and the Rs 100/- strike lands perfectly, with a root mean squared miss of 0.474843 across the set. And here is the sting: the best single setting hits nothing at all and still beats it, at 0.471464. Fitting the set as a whole and hitting a member of the set are different achievements, and the second is worth less than it looks.

What does it mean when the two agree?

Agreement is the case almost nobody examines, and it is the one that matters most. Readers treat disagreement as needing an explanation and agreement as needing none, as though agreement were the natural state and difference the anomaly. The truth runs the other way round. Agreement is the surprising event, and it needs at least as much explaining as any gap.

Suppose the setting had been found that repriced all five strikes exactly. Ask what an exact fit across all five would establish. The formula above settles it: exact agreement across five strikes holds if and only if each strike's own setting is the same number. The lesson would be that the five recorded prices are mutually consistent with a single setting. Mutual consistency is a fact about the five numbers. A statement of internal consistencyWhether a set of prices can all come from one setting, the one thing agreement establishes. is genuinely worth having.

The one thing not learned is that the equation is true. The five records could sit on one setting for any number of reasons, and the fact that a function with one dial can reproduce them constrains the records far more than it endorses the function. A ruler that measures a table at exactly one metre has not proved that the metre is the right unit for tables; it has established only that the table is a round number of metres long. Agreement establishes a property of the set, never the truth of the equation.

Which of the two can be wrong?

Only the calculation, and it is worth being exact about why rather than treating it as a slogan. To be wrong, a thing has to have asserted something that fails to hold. A recorded price asserts nothing beyond its own occurrence: it does not claim the thing was worth that, it does not claim anyone was being sensible, and it does not claim the number will recur. There is no proposition inside it to falsify.

A computed price asserts a great deal, and each assertion is a separate place it can fail. A computed price asserts that its assumptions hold, and the assumptions here are strong: one constant setting, a particular shape of movement, a particular treatment of the horizon. The second assertion is that its inputs are the right inputs: a claim about the starting value, the rate and the horizon that were fed in. The third assertion is that its arithmetic was done correctly. Bad arithmetic is the least interesting failure and the most common one.

One qualification. Precision matters more than a neat slogan. A record can be badly captured. The digit can be transposed, the moment stamped wrongly, the strike attached wrongly. Each of those is a failure of the capture and it is repaired by going back to the source, not by reasoning about value. The distinction survives: the price did not become wrong, the copy of it did.

What goes wrong when a reader treats the gap as news?

The failure is the single most common error in this whole area and it never announces itself, so its precise shape is worth drawing. A reader sees the computed number below the recorded one at the Rs 80/- strike, by Rs 0.662462/-, and concludes that the recorded price was too high. Reading the reasoning back slowly makes the hidden step appear: to conclude anything about the record from the gap, the calculation must first be taken as sound. The hidden step is assuming the modelTaking the model as correct in order to judge something else, the failure at work here., and it is never stated, never checked and rarely even noticed.

What does a careful reader actually do with the distinction?

Four things, and all four are habits rather than techniques. The four habits cost nothing and they are what separates a working file that can be checked from one that cannot.

First, label the kind at entry. Every number that arrives in a file is either a record or an output, and the label goes on when it arrives rather than being reconstructed later from memory. Labelling at entry is the same discipline as writing down which readings on a survey came from the instrument and which were interpolated. Once the two are in one column, no amount of care afterwards recovers the distinction.

Second, never report a difference without naming the side taken as given. Saying the computed price is Rs 0.662462/- below the record is a complete and honest sentence. Saying the record is Rs 0.662462/- too high is not. The second wording has silently promoted the calculation to the status of a standard. The first sentence is a subtraction and says so; the second is a judgement pretending to be a subtraction.

Third, record the setting and the objective beside any fitted number. The figure 0.198202 is meaningless without both. The figure is the setting that minimises the squared misses in price across this particular set of five, and a different objective on the identical data returns a different number. The dependence on the objective is not a defect to be hidden; it is the character of a fitted figure and it belongs in the record beside the figure itself.

Fourth, treat exact agreement as an event to be explained rather than a target to be reached. If a calculation ever lands on every member of a set, the first question is what property of the set made that possible, not whether the calculation has been vindicated. Anyone who reaches for a better fit without asking what a fit would establish has already lost the thread of what they are doing.

The everyday version, one last time. A surveyor who writes down which distances were paced and which were computed from a bearing can be checked by anyone. A surveyor who writes only distances cannot be checked at all, even if every number is correct. Nobody can tell which numbers were claims and which were observations. The whole of good practice here is refusing to lose that column.