Calibrated Parameters: Fitted Numbers, Not Facts

A calibrated parameter is the output of a search over a bounded range against a chosen loss on a chosen set of targets at a particular moment. A calibrated parameter is not a property of anything. The fitted value is the number that made a model agree with some other numbers as closely as that arrangement allowed, and it moves when any part of the arrangement moves.

Building the fit, and the five steps it takes, is set out under calibration. A different question decides what may afterwards be done with the answer: what kind of object is the number that comes out? The fitted number looks like a measurement. The number is written to six decimal places, it is stable to the last of them, and it arrives in a cell beside other numbers that really are measurements. But four separate choices go into it, none of them made by the data, and moving any one of the four moves the number. A quantity that shifts when a person changes their mind is not describing anything outside that person's decisions.

Everything below is worked on the same invented instance the rest of this subject area uses. The quantity being modelled is the standard process, written S with a time subscript, starting at Rs 100/- with a rate of 5 per cent a year over a horizon of one year and paying nothing out. The numbers being fitted to are five invented call prices at strikes of Rs 80/-, Rs 90/-, Rs 100/-, Rs 110/- and Rs 120/-, quoted as volatilities of 0.240000, 0.220000, 0.200000, 0.190000 and 0.185000. The five quoted volatilities are set down for teaching rather than read off a market, and volatilities actually observed at those strikes would sit somewhere else.

What is a calibrated parameter?

A calibrated parameterThe output of a search over a bounded range against a chosen loss on chosen targets, at a stated moment. is the value a search returned. The whole definition stops there. Every problem with calibrated parameters comes from the urge to make that definition grander. The search was given three things by a person and one thing by arithmetic. The person supplied the numbers to aim at, the rule for measuring how far away the model was, and the window inside which the value was allowed to lie. The arithmetic then swept that window and handed back whichever value made the rule read smallest.

The everyday version is exactly right rather than roughly right. A kitchen scale reads a little heavy, and a small screw underneath it shifts the reading. Somebody turns the screw until five known weights read as close to their printed values as one screw can manage, then writes down the screw position. The screw position is a real number, it is reproducible to whatever precision the fingers allow, and it is entirely a fact about that afternoon. A different set of five objects would have stopped those fingers somewhere else, so the screw position is not a fact about the scale. Nor is the position a fact about the objects. The screw position is the resting place of a procedure, and the procedure had a person in it.

Four inputs feed that resting place, and naming all four at once builds the habit worth having: ask for all four whenever a fitted figure arrives on a desk. The first is the target setThe numbers the model is being asked to match. Five here, one at each strike., which is the list of numbers the model is asked to match. The second is the loss functionThe rule that turns the several gaps between model and targets into one number to be made small.. The loss function is the rule that turns several gaps into one number to be made small. The third is the pair of boundsThe window the search is allowed to look inside. Between 0.05 and 0.50 on this instance. that fence the search in. The fourth is the moment the whole thing was run. The targets came from outside the model, and outside numbers are dated.

The parameters used on every earlier occasion in this subject area stand in contrast. The volatility of 0.20 a year on the standard process was stated. The rate of 5 per cent a year was stated. The drift of 8 per cent a year was stated. A stated parameter is an assumption, and an assumption is honest about being one: nobody reading it imagines it was discovered. A fitted parameter is an assumption too, but it arrives dressed as a result, and the dressing is what does the damage.

Why is a calibrated parameter not a fact?

Because holding the data completely still and changing one rule about how to measure closeness returns a different number. Not a slightly different number after rounding. A different number, at the second decimal place, from arithmetic that is exactly correct both times.

Take the five invented prices and leave every one of them untouched. Now write down three rules for what close means, all three of which are in ordinary use and none of which is wrong. The first says the gaps are gaps in rupees, so add up their squares. The second says a gap in rupees means different things at different strikes, because a far out contract barely moves when the parameter moves, so divide each gap by that contract's vegaHow much a contract's price moves for a one point move in volatility. Larger near the middle strike, smaller at the wings. before squaring it, which converts every gap into volatility points. The third says the targets were quoted as volatilities in the first place, so compare volatilities directly and forget prices altogether.

On the invented instance those three rules return 0.198202, 0.204420 and 0.207000. The gap between the outermost two is 0.008798, close to nine tenths of a volatility point. In derivative pricing that is not a rounding difference but a real disagreement about what to feed the next calculation. All three rules are ways of measuring closeness rather than statements about the process, so nothing observable chooses between them, and closeness has no meaning until somebody defines it.

From the inside the three feel identical. Whichever one is used, the experience is the same. The rule is stated, the search is run, the search converges, and out comes a figure stable to six places. There is no wobble to give warning, no wide interval, no flag saying that a colleague two desks away is about to report a different number. The precision is real and it is precision about the procedure, not about the process.

What does the date of the fit do to the number?

Observation Date

The observation dateThe moment the targets were taken and the fit was run. Without it, a fitted number cannot be told apart from a current one. is the moment the target set was taken and the search was run. On this instance it is 23 August 2026. The date line does more work than any of the six decimal places beside it, and the date line is the one most often left off.

Start with what a date is not. A date is not a formality, not a version stamp, and not a courtesy to whoever files the note. The date is the only thing in the record that connects the parameter to the numbers that produced it. The targets are numbers that came from outside the model, and outside numbers move. A fit run against yesterday's target set and a fit run against today's are two different searches over two different sets, and they will land in two different places even with the loss and the bounds held identical.

Now the part that makes an undated number dangerous rather than merely untidy. There is nothing inside a figure that reveals its age. The digits 0.198202 look exactly the same whether the search ran this morning or three years ago. Almost anything else carries its own age. A photograph shows the season. A printed sheet yellows. A spoken sentence carries a voice that can be dated. A six digit decimal carries nothing at all, so an undated fitted number cannot be distinguished from a current one by any inspection of the number itself.

The everyday version is a lift that only knows which floor it is on. The indicator reports, correctly and instantly, that the lift is on the fourth floor. The indicator cannot report whether the lift arrived a second ago or has been sitting with the doors shut since Tuesday. The age of the reading has to come from somewhere other than the reading. StalenessA fitted number being older than it appears. No undated figure can rule it out, because nothing inside the digits records their age. is exactly this: the reading is right and the reading is old, and the two facts are invisible to each other.

What do the bounds do to the number?

Parameter Constraint

A parameter constraintAnother name for a bound. A limit on where the search may look, chosen by the analyst rather than by the data. is a limit on where the search is permitted to look. On this instance the volatility was searched between 0.05 and 0.50, and there are perfectly good reasons for a window like that. A window keeps the search away from settings that are arithmetically permitted and physically absurd. The window stops a numerical routine wandering off into a region where the pricing formula loses accuracy. And the window encodes what somebody already knows about the plausible range before any fitting starts.

All of which is sensible, and none of which changes the fact that a bound is a decision. The data did not supply 0.05. The data did not supply 0.50. A person did, and that person could have chosen differently. The window is one of the four levers rather than part of the furniture.

Most of the time the window does nothing at all. The search lands somewhere in the middle, well clear of both walls, and the answer would have been identical had the walls been twice as far apart. An answer of that kind is an interior solutionAn answer sitting away from both bounds, where the loss has genuinely flattened out., and the fit on this instance is one: 0.198202 sits comfortably between 0.05 and 0.50. Narrowing the upper wall from 0.50 all the way down to 0.30 returns the very same 0.198202, and the wall was never in the way.

Sometimes the window does everything. Move the lower wall up to 0.20 and the search returns 0.200000. The search converges, it reports six decimal places, and it says nothing whatsoever about the five prices. An answer of that kind is a boundary solutionAn answer sitting exactly on a bound. It reports where the wall was put rather than where the loss was smallest., and it is the single most important idea on the subject.

Here is the part that turns a technicality into a live problem. Both cases produce a converged search, a stable figure and a clean report. Neither produces a warning. From the search's point of view nothing has gone wrong: it was asked for the best value in a window and it found one. The distinction between an answer found in a valley and an answer stopped at a wall lives entirely in the record, and if the record does not carry it, it is gone.

What do the choices give on the invented instance?

The claim being made is arithmetic rather than attitude, so here is every combination on one sheet. Two target sets, three losses, six searches, all against the same five invented observations of 23 August 2026, all inside the same window of 0.05 to 0.50. Every one of the six converged cleanly. Every one is reproducible to the last digit shown.

Six numbers, one data set, spread 0.008798 from lowest to highest. Read down the first column and the loss alone moves the answer. Read across any row and the target set alone moves it. The bottom row is the honest summary of what the five prices support, and it is a span rather than a point.

Dropping the wing strikes sounds like an act of tidiness rather than a decision, so the middle three column deserves a sentence of its own. Dropping them is a decision, and a consequential one. The strikes at Rs 80/- and Rs 120/- are the two the flat model misses by most, so removing them removes the pull that was dragging the squared price fit downward, and the answer rises from 0.198202 to 0.199868. The rise of 0.001666 is produced entirely by choosing to attend to three numbers instead of five. Every target included gets a vote in where the answer lands, and every target excluded gets none.

What happens when the date or the bounds go missing?

Both omissions produce the same outcome: a number that cannot be interrogated. The two omissions get there by different routes, and the fixes differ, so the routes are worth separating.

Dropping the date removes the ability to ask whether the number is still about anything. The targets it was fitted to belong to a particular morning. Without the morning, the parameter floats free of the numbers that made it, and every downstream use of it inherits an unknown age. Nobody can say whether it is a week old or three years old. Nobody can even say how wrong they might be: the size of the error depends on how much the targets moved, and that question cannot be asked without a start date.

Dropping the bounds removes the ability to ask whether the number is an answer at all. Dropping the bounds is the sharper of the two failures, and it is the one worth building a habit around.

How should a calibrated parameter be written down?

Six items. Not five, the number most records manage, and not four, the number most notes manage. Six, and the sixth is the one that decides whether the other five mean anything.

Walk the six in order and notice what each one rescues. The value on its own is a digit string. Add the targets and the reader knows what the model was asked to match, and any downstream use has to respect that list. Add the loss and a second analyst reporting 0.207000 stops looking like a disagreement about the world and starts looking like what it is, a difference of measurement rule. Add the bounds and the answer becomes checkable against them. Add the date and the number stops floating. Add the boundary status and, for the first time, the reader can tell a finding from a constraint.

The sixth is the one nobody records, and the reason is human rather than technical. The other five are inputs, so they are sitting there in the setup and copying them out is clerical. The sixth is an observation about the output, so somebody has to look at the answer, compare it against the walls, and write down what they saw. Looking at the answer is a small extra act of attention at the very end of a job that already feels finished, and the end of a finished-feeling job is exactly where things fall out of a workflow.

What does a reader do with a fitted number that arrives bare?

Most people who meet a calibrated parameter will never run a search. Nearly all of them will at some point be handed the output of one, in a note, a slide or a spreadsheet cell, and asked to carry it forward. The six items turn into six requests, and a bad answer to an early one makes the later ones idle, so the six are worth making in order.

Two closing habits make the six stick. The first is to treat a number arriving without item six as unanswered rather than as answered badly. The digits do not reveal it, and asking costs nothing. The second is to say the number aloud in its full form whenever it is passed on. Not 0.198202, but this: 0.198202, fitted to five prices at strikes from Rs 80/- to Rs 120/- under squared price error, searched between 0.05 and 0.50, on 23 August 2026, landing inside the window rather than on a wall. That sentence is longer than the number and it is the only version of the number that can be checked by anybody.