Physical and Risk-Neutral Measures Compared

The physical measure is what a model says is likely. The risk-neutral measure is the weighting under which discounted prices behave consistently. Both are ordinary probability measures on the same set of paths, and no formal property tells them apart. The difference is entirely what each is for, and using one to answer the other's question is the error that follows.

Here is the uncomfortable part, and it is worth having in view before anything else. Nothing inside the mathematics marks one of these two as real and the other as artificial. Both assign every event a number between zero and one. Both total exactly one. Both obey every rule a probability measure has to obey. Both sit on the same outcome setOne set of paths carrying both measures, with no path added or removed by either of them., weighing the very same paths, with not a single path created by one or deleted by the other. The distinction lives in the question being asked. No check can enforce it, so the person doing the work has to.

Everything below is worked on one invented object. The standard process starts at Rs 100/-, drifts at 8 per cent a year, moves with a volatility of 20 per cent a year, pays nothing out, and is watched for exactly one year against a risk-free rate of 5 per cent continuously compounded.

What is the physical measure, defined from scratch?

The physical measureThe rule describing what the model says is likely, written P., written P, is the rule that says how much weight each possible path of the process carries. Ask it any question about what the process might do and it answers with a number: how often, how likely, how much of the weight sits in that region. The physical measure is the model's own account of what is probable. Nothing more mysterious than that.

On the standard process, P is fixed by the drift of 8 per cent a year. The drift is the parameter that says which way the process leans, and it is the only parameter that carries a claim about direction. Change it and every statement P makes about the future changes with it. Depending on the drift alone is what makes every number produced under P a forecast rather than a fact.

Worked out, the whole shape falls into place. The middle of the distribution, the value with half the weight on each side, sits at Rs 100/- multiplied by the exponential of 0.06, giving Rs 106.18/-. The average sits at Rs 100/- multiplied by the exponential of 0.08, giving Rs 108.33/-. The two are Rs 2.15/- apart, and that gap is the half variance correction made visible: an average is pulled up by a long right tail that a middle value ignores. And the question the whole comparison turns on, does the process finish above Rs 100/-, comes out at 0.617911 under P.

Now the honest sentence about that 0.617911. The physical figure is a forecastA statement about what will probably happen, which only one of the two measures makes., and it inherits everything that is wrong with the drift it came from. Nobody observes a drift. A drift is estimated, argued over, and revised. Move it from 8 per cent to 12 and the same figure becomes 0.691462. Move it to 4 and it becomes 0.539828. The physical figure is only ever as trustworthy as the least trustworthy parameter that feeds it, and here that is the drift.

An everyday version, and it is a measurement example rather than anything to do with a business. A kitchen scale sitting on a shelf reads a weight. The number on the dial is not the weight of the object; it is the reading of that instrument, and if the instrument has been set up carelessly the reading is confidently wrong. The physical measure is a reading of the model, not a reading of the world. The physical measure is what the model says is likely, and the sentence stops there.

What is the risk-neutral measure, defined from scratch?

The risk-neutral measureThe weighting under which discounted prices behave consistently, written Q., written Q, is also a rule assigning weight to each possible path. Q is a probability measure in exactly the same technical sense as P. But it is not arrived at the same way, and that is where the whole distinction lives. P is arrived at by asking what is likely. Q is arrived at by demanding one property and then accepting whatever weighting delivers it.

The property is this: under Q, the process discounted at the risk-free rate must have no tendency to rise or fall. Today's discounted value must equal the average of tomorrow's discounted value. Q is not chosen because it describes what will happen; it is chosen because it is the one weighting under which prices computed as discounted averages cannot be arbitraged against each other. The no-arbitrage property is a statement about the internal consistency of a set of prices, and it is not a statement about the future at all.

Read the second half of that block again and notice what is missing. The letter for the drift does not appear. Under Q the standard process grows at the risk-free rate of 5 per cent a year, and the 8 per cent has gone entirely. Nobody decided that 5 per cent was a better guess than 8 per cent. The 5 per cent arrived because it is the only growth rate for which the discounted process sits still, and sitting still is the property that was demanded.

Checked, the arithmetic closes. The average finish under Q is Rs 100/- multiplied by the exponential of 0.05, giving Rs 105.13/-. Discounting that at 5 per cent for one year means multiplying by 0.951229, and it gives Rs 100.00/- back, exactly the starting value, to every decimal place. The same operation on the physical average of Rs 108.33/- gives Rs 103.05/-, not the starting value and never going to be. The failure of that check is the cleanest demonstration that P does not have the property Q was built to have, and it takes one multiplication.

The same question asked of Q gives 0.559618. Into that figure went the starting value, the volatility, the horizon and the rate. Left out was any view about which way the process is heading. The risk-neutral figure is available to somebody with no opinion whatever about the future, and a price built from it is therefore defensible to a person who disagrees about the future.

What is each measure for, which is the only thing that separates them?

With the two set side by side, every formal test available returns the same verdict. Does each assign every event a number between zero and one? Both do. Do the weights total exactly one? Both do, and the worked table further down shows both totalling 1.000000 across the same five regions. Are both countably additive, so that the weight of a union of separate events is the sum of their weights? Both are. Can an average be taken under either? Both allow one, and four averages have already been taken above.

The two measures are also equivalentAgreeing about which paths are possible, and disagreeing only about weight., a formal property they share rather than one that separates them. Equivalence means the two agree completely on which events are impossible. If P says a region carries no weight at all, Q says the same, and the other way round. Neither measure invents a path the other rules out. So even the one property with a technical name is a point of agreement.

So what does separate them? Only what each is for. P answers what is likely. Q answers what a payoff should cost. The separation is not a mathematical difference, it is a difference of use, and the mathematics cannot see it. The absence of a mathematical difference has a consequence worth sitting with: no gate, no formula, no line of code and no reviewing colleague can tell that a number came from the wrong measure. The number carries no label. Only the working that produced it does, and only if somebody bothers to read that working.

Here is the everyday version, and again it comes from measurement. Two maps of the same country make the point. One preserves angles so that a straight bearing on the paper is a straight bearing on the ground. The other preserves area so that two regions covering the same paper cover the same ground. Neither map is wrong. Neither map is more real than the other. Every place exists on both, and neither map creates a place or removes one. But using the angle-preserving map to compare sizes produces something false stated with total confidence. The two measures are two projections of one set of paths, and the mistake has exactly the same shape.

One number puts a size on the separation. Take the drift less the risk-free rate and divide by the volatility: 0.08 less 0.05 is 0.03, divided by 0.20 gives 0.15 exactly. The 0.15 is the market price of riskThe drift less the rate, over the volatility, which is what separates the two measures here., and it is the whole distance between P and Q on this process. With the drift set equal to the rate the two measures coincide, the separation vanishes and there is nothing left to compare. Every gap between the two measures is that 0.15 showing up in a different costume.

Risk-Neutral Valuation vs Real-World Forecasting: which question does each answer?

The comparison is cleanest set out block by block. On one side is forecasting: the question is what will probably happen to the standard process over the next year. On the other side is valuationA statement about what a payoff should cost, which only one of the two measures makes.: the question is what a payoff should cost today. Forecasting and valuation are different questions with different inputs, different outputs and different failure modes, and it is only the shared vocabulary of probability that ever made them look alike.

The last row is the whole comparison in eight words. The physical figure must never be quoted as a price and the risk-neutral figure must never be quoted as a chance, and the second of those errors is the one people actually make. Nothing about either number announces its row. Both are six decimal places between zero and one, produced by machinery labelled probability from end to end.

There is a second asymmetry hiding in that table and it is worth naming. A forecast can be checked against what happens. After a year, with the finishing value observed and enough repetitions gathered, something can be said about whether the drift was sensible. A price cannot be checked that way at all. The risk-neutral figure is not a prediction that failed or succeeded when the year ended; it was never a prediction. The two questions are not only answered differently, they are settled differently, and one of them is never settled by waiting.

Why can one event carry two different numbers without either being wrong?

Take one event, as plain as it comes: does the standard process finish above Rs 100/- after one year? One process. One outcome set. One threshold. No ambiguity anywhere in the wording. Ask P and the answer is 0.617911. Ask Q and the answer is 0.559618. The two are 5.8294 percentage points apart on an event whose statement did not change by a single word between the two questions.

Neither is an approximation to the other and neither is a corrected version of the other. The two figures answer two different questions that happen to be expressed in the same words. The English sentence "the chance it finishes above Rs 100/-" is ambiguous in a way the mathematics is not, and that ambiguity is where every version of this error begins.

To see that no sleight of hand happened, cut the outcome set into five regions and ask both measures to weigh all five. Every region gets weight from both. No region is empty under one and full under the other. Both columns total exactly 1.000000. The one difference is the distribution of weight across the same regions, and the weight moves in one direction consistently. Dropping the growth from 8 per cent to 5 per cent has to put more weight low and less weight high under Q.

The split is not a quirk of the threshold either. Move the question to Rs 110/- and ask again: P gives 0.429931, Q gives 0.372004, a gap of 5.7927 percentage points in the same direction. At Rs 90/- the answers are 0.795826 and 0.750734, a gap of 4.5091 points, again the same way round. The drift of 8 per cent is above the rate of 5 per cent, so the physical figure is above the risk-neutral figure at every threshold, and the whole shape of the disagreement is that one comparison.

Which of the two averages is a forecast, and which is not?

The probabilities are not the only quantity that splits. The averages separate in the same direction and for the same reason. The split is systematic rather than an artefact of one threshold. Under P the average finish is Rs 108.33/-. Under Q it is Rs 105.13/-. Same process, same year, same starting value of Rs 100/-, two averages Rs 3.20/- apart.

Which of the two is the forecast? The Rs 108.33/-, and only that one. The physical average is what the model claims about where the process is heading, and like every claim of that kind it is a statement about the future that could turn out badly. The Rs 105.13/- forecasts nothing. The risk-neutral average comes from the growth rate the pricing property forced, and its job is finished the moment a payoff has been averaged and discounted. Anybody who reports Rs 105.13/- as where the process is expected to be has reported a number that nobody ever intended as an expectation.

What happens to each figure when the drift moves and when the rate moves?

Here is the test that settles the whole distinction, and it takes about fifteen seconds. Move one parameter at a time and watch which figure responds. The physical figure contains the drift and does not contain the rate. The risk-neutral figure contains the rate and does not contain the drift. So moving the drift moves one of them and moving the rate moves the other, and there is no setting of either control at which both respond together.

Two moments in that control are worth stopping on. At a rate of 2 per cent the rate equals half the variance rate, so the pricing weighting puts the middle of the distribution exactly at the starting value and the risk-neutral figure reads exactly 0.500000. And at a rate of 8 per cent the rate has been set equal to the drift, so the two measures collapse into one another and the two figures land on the same 0.617911. The two figures agreeing is not a sign that the numbers have become correct; it is a sign that the separation was set to zero.

Where does confusing the two produce a specific, nameable error?

Quoting a risk-neutral figure as the chance of something happening. The error is exactly that, it is the commonest one in this whole subject, and it is easy for reasons that have nothing to do with carelessness. The number sits between zero and one. The number was produced by machinery labelled probability at every step. The number appears in a column headed with the word probability in a great many sets of working. And it is the figure that falls out of a pricing calculation, the calculation most people are running.

Note what makes this failure different from most. The working is correct, so no review catches the error. The number is a valid probability, so no test catches it. And a single outcome says almost nothing about whether a figure of 0.559618 or 0.617911 was the right one to quote, so the passage of time does not catch it either. The only thing standing between the correct working and the false sentence is somebody knowing which question the number was built to answer.

How does somebody reading a valuation actually use this?

The distinction is not decoration, and using it needs no software, no data and no access to anything. The distinction is a reading test run on a set of working, and it has three steps.

The first step is to find out which question the document was answering. A document that ends in a price was answering the valuation question and every probability inside it is a pricing weight. A document that ends in a projection of where something will be was answering the forecasting question and every probability inside it is a claim about likelihood. Documents that end in both are common, and they contain both kinds of number and almost never label which is which.

Second, find the drift. If the working contains a growth rate that somebody estimated, argued for or chose, the numbers around it are physical, and the quality of everything downstream is limited by the quality of that one estimate. If the working contains no such rate anywhere, and grows the process at the risk-free rate instead, the numbers are risk-neutral and no estimate of direction entered them at all.

Third, check for the discount factor. A price is an average taken with risk-neutral weights and then discounted. If a number has been discounted, it was built as a price. If it was not, and it is being described as a chance, the physical measure is the only measure whose numbers can honestly be described that way.

Run those three steps and the interpretationWhat a number is taken to mean, which is where this distinction lives. of every probability in a document is settled in a couple of minutes. The person who reads a set of working for its question before reading it for its answers is the person this error never reaches. The three steps are the whole practical content of the comparison, and they survive the reader having no market access, no data and no tools.

The same discipline applies to anybody building the working rather than reading it. Nothing downstream will be able to tell the two apart, so where both kinds of number are produced in one set of working, each should be labelled where it is produced. A household deciding how much to keep aside for a large expense next year is asking the forecasting question, and reaching for a pricing weight there would be an error of the same shape, in a setting with no mathematics in it at all.

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