Girsanov, Radon-Nikodym and Change of Numeraire

Girsanov's result says a change of measure shifts the drift of a process and leaves its volatility exactly where it was. The Radon-Nikodym derivative is the weight that carries out that change, path by path. A numeraire is the unit prices are quoted in, and changing it changes which measure makes discounted prices behave. The three are one operation described three ways.

The pricing argument already established that a second measure exists and that valuation happens under it. The argument did not establish how to get there from the description actually available: a process with a drift somebody estimated and a volatility somebody estimated. Three names stand between those two things, and they are almost always taught as three separate items to memorise. The list is the wrong shape. The three names are one operation seen from three angles, and once the operation is in view they stop being a list and become three ways of asking about the same move.

The operation is this. There is a set of paths. Every path carries a weight, and the weights add up to one. A change of measure leaves every path exactly as it is and changes only the weights. Nothing is created, nothing is deleted, nothing is bent. A path that finished at Rs 106.18/- still finishes at Rs 106.18/-. The weight changes: how much that particular path counts when an average is taken over all of them.

The everyday version is worth holding on to. Every result that follows is a consequence of it. A hall contains three hundred people and every single height has been measured. Somebody then reports that the sample over-represents tall people and supplies a table of corrections: count this person as 0.92 of a person, count that one as 1.07. Every measured height is untouched. Nobody grew or shrank, nobody left the hall, nobody walked in. The average height changes. The heights actually present in the hall and how far apart they sit are facts about the people. The corrections were facts about the counting. The spread of the heights does not change at all.

GirsanovThe result saying a change of measure shifts the drift and leaves the volatility alone. says what that reweighting does to the drift. The Radon-Nikodym derivativeThe weight carrying out a change of measure, path by path. is the table of corrections itself. A numeraireThe unit prices are quoted in, usually cash but not necessarily. fixes what prices are quoted in and therefore what has to behave. The choice of unit is what decides which reweighting is wanted in the first place. The three separate cleanly one at a time, then collapse back into one worked change on an invented process.

What are the three tools, and what does each one do?

The three tools are best taken in the order a calculation meets them. A calculation starts with a description of a process under one rule for weighting paths and needs its description under a different rule. The numeraire is what names which different rule. Girsanov gives what the new description looks like. The Radon-Nikodym derivative is the object that connects the two rules, and it is the only one of the three that is a number computable for a single named path.

What does the change actually leave alone?

Two rules for weighting paths are called equivalent measuresTwo rules agreeing about which paths are possible and disagreeing about weight. when they agree completely about which paths are possible and disagree only about how much each one counts. The word equivalent is doing real work. If one rule said a path could happen and the other said it could not, no table of corrections could get from the first to the second. No finite weight turns something into nothing. Agreement about the possible is exactly the condition that lets a weight exist for every path.

Everything in this guide follows from that. A quantity computed from a single path, using nothing but the numbers along that path, cannot notice a change of measure. The change never reached the path. A quantity computed by averaging across paths can notice nothing else. Averaging is exactly where the weights enter. So the whole question of what survives has a one line answer: whether the quantity is a property of the paths or a property of the weights decides whether it moves.

The description is handed over in exactly that form, written under the physical measure P. P is simply the rule under which somebody estimated the 8 per cent. The pricing argument wants the same process described under the risk-neutral measure Q. Nothing about the process is going to change. Only the description is.

What does Girsanov's result change, and what does it leave alone?

The bridge between the two descriptions is a single number, and it is a number already computable from the four locked parameters. The market price of riskThe drift less the rate, over the volatility, being 0.15 here. is the excess of the drift over the rate, divided by the volatility. The ratio asks how much extra rate of return the process is described as carrying, measured in units of the risk it is described as carrying.

Girsanov's result now says something very specific, and the specificity is the point. Under the new rule for weighting, the driving Brownian motion is not W any more. The driver is W with theta times the elapsed time added to it, and that new object is a perfectly ordinary standard Brownian motion under the new rule. Notice what kind of statement that is: it does not modify the process, it relabels the thing driving the process.

Substitute that into the process and watch what happens. The whole result is two lines of algebra. The process carries the term sigma times the increment of W. Replace the increment of W by the increment of W-tilde less theta times the time step. The volatility multiplies both terms, so one stays attached to the Brownian increment and the other becomes a deterministic term that merges into the drift.

Read the second line of that block once more, slowly. The substitution replaced one deterministic quantity by another deterministic quantity. The substitution never went near the coefficient multiplying the random term. The volatility is not preserved as an approximation or as a convenient result: it is untouched because nothing in the operation ever reaches it. That is the load-bearing fact of the whole apparatus, and it is why a pricing model can borrow a volatility estimated under one rule and use it under the other without apology.

The shift deserves its own name and is the number actually computed. The drift shiftThe amount the drift moves under the change, 0.03 here. is the volatility multiplied by the market price of risk: 0.20 times 0.150000, exactly 0.030000. Subtracting it from 0.080000 lands on 0.050000, the rate. The arithmetic reconciles in both directions: 0.030000 divided by 0.20 gives back 0.150000, and 0.150000 multiplied by 0.20 gives back 0.030000.

One consequence deserves saying out loud before anything else is built on it. The locked path, twelve steps long and fixed throughout, is a path. The path is a sequence of twelve numbers and it does not know which rule it is being weighted under. So it reads the same twelve numbers under P and under Q, and it finishes at Rs 106.18/- either way. The match is not a coincidence of the parameters. Reading the same either way is what a reweighting means.

Why can the volatility not move?

The algebra already showed that the substitution never reaches the volatility. The algebra is a proof, and it is also slightly unsatisfying. The result feels like a fact about how the symbols were arranged rather than a fact about the world. So here is the same conclusion reached from the other end, and this version is the one worth carrying away.

The volatility of the standard process is recoverable from a single path, with no probability anywhere in the calculation. The recipe takes the logarithm of the path, chops the year into steps, squares each step and adds up the squares. The sum of those squares is the quadratic variation, and it converges to the variance rate multiplied by the elapsed time. Dividing by the elapsed time and taking the square root gives the volatility. At no point does that recipe consult a rule for weighting paths. The recipe reads numbers off one path and does arithmetic.

Now the argument closes itself. A change of measure does not touch the path. The recipe above uses nothing but the path. So the recipe returns the same answer before and after, and the volatility it returns is the same 0.200000 in both cases. There is no room for it to be otherwise. The volatility survives a change of measure for the same reason a person's height survives a change in how people are counted.

A useful honesty check on the same locked path: the realised sum at its twelve step partition is 0.040300 rather than 0.040000. Each monthly move of the logarithm carries a drift component of 0.005, and twelve of those squared contribute 0.000300. The 0.040000 is the limit, reached as the partition refines, giving 0.040014 at 252 steps. The point stands in both readings. The realised 0.040300 is identical under P and under Q as well. The drift component is a different number under the two rules, but it belongs to the same single path, and the sum reads the same either way.

What is the Radon-Nikodym derivative weighing?

So far the change has been described by its effect. The Radon-Nikodym derivative is the change itself, written down as an object. For each path it returns one positive number, and that number is the factor by which the weight on that path is multiplied when moving from the first rule to the second. Named after Radon and Nikodym, whose result guarantees such an object exists whenever the two rules agree about which paths are possible.

For the change from P to Q on the standard process, the factor depends on the path through exactly one thing: where the Brownian motion finished. Not the route it took, not its highest point, not how many times it crossed zero. Only the finishing value.

Put the locked path through it. Its twelve driving values were built to sum to zero exactly, so its Brownian motion finishes at 0.000000. The multiplier is then the exponential of minus 0.011250, or 0.988813. The locked path is weighted very slightly down. Now try two neighbours. A path finishing at plus 0.5 gets the exponential of minus 0.075 less 0.011250, or 0.917365. A path finishing at minus 0.5 gets the exponential of plus 0.075 less 0.011250, or 1.065826.

That is the reweightingChanging how much each path counts, without changing any path. in one sentence: paths that ran up are weighted down and paths that ran down are weighted up, and that is exactly how the average finishing value falls without any path falling. There is a crossover, and it is worth locating precisely rather than guessing. The multiplier equals 1.000000 exactly where minus theta times the finishing value equals half theta squared. The crossover therefore sits at minus theta over two, being minus 0.075000. Slightly below zero, not at zero. The correction term shifts it.

There is a second way to see the same thing, and it makes the volatility invariance visible rather than argued. Draw the distribution of the finishing Brownian value under each rule. Under P it is centred at 0.000000 with a spread of 1. Under Q it is centred at minus 0.150000 with a spread of 1. The multiplier curve above is precisely what turns the first shape into the second, and what it does is slide the shape sideways. The curve does not squeeze the shape, stretch it or lean it over.

What is a numeraire, and why would it be changed?

A numeraire is the unit of accountWhat a price is expressed in; a change of numeraire changes it.: the thing prices are quoted in. Almost always it is cash, in the specific form of one rupee put into the risk-free rate and left there. One such rupee grows to 1.051271 over the year. Quoting in that unit is so automatic that most people never notice they are doing it, in the same way that most people quoting a distance never mention that they are quoting it in metres.

But the choice is a choice, and it can be made differently. Every price could instead be quoted in units of the standard process itself. A position worth Rs 106.18/- becomes a position worth 1.000000 units of the process. Nothing about the position changed. The number changed because the unit changed, exactly as a two metre length becomes a 6.56 foot length without the object moving.

Changing the numeraire is a change of unit, and the reason it matters is that which quantity holds steady on average depends on the unit it is quoted in. Under the cash unit, it is the process divided by the cash account that holds steady, and the rule that makes that true is Q. Under the process unit, it is the cash account divided by the process that has to hold steady, and that needs a different rule again.

The formula is the general rule, it is short, and it is worth pausing on. To change from one unit to another, weight each path by how much better the new unit did than the old one on that path. Nothing about a market price of risk appears. Nothing about a drift appears. The weight is a ratio of growth factors, and that is why a change of numeraire and a change of measure are not two subjects.

Under its own unit the process looks different again. Run Girsanov with the new weight and the drift moves a second time. The Brownian motion picks up another shift, this time of the volatility itself, so the drift of the standard process under its own unit becomes the rate plus the variance rate.

Three descriptions of one process now sit side by side with drifts of 0.080000, 0.050000 and 0.090000. Every one of them carries a volatility of 0.200000. One more demonstration that the drift is a statement about weights and the volatility is a statement about paths sits there, in three lines.

How do the three fit together on one worked change?

All three now run end to end on the standard process, and the numbers meet. The reweighting is not a story about the drift falling but arithmetic that can be carried out and checked, and when it is carried out the drift falls as a consequence.

Take twelve outcomes at the horizon, built by a deterministic construction: cut the probability scale into twelve equal slices and take the finishing Brownian value at the middle of each. The construction gives twelve finishing levels for the standard process from Rs 75.10/- to Rs 150.13/-, and one weight multiplier for each. No random draw is involved and the table reproduces identically every time.

Read the bottom row. The plain average of the twelve finishing levels is Rs 108.11/-, the physical average, and the exact figure it is approaching is Rs 108.33/-. The average once each level is counted by its weight is Rs 105.11/-, and the exact figure that is approaching is Rs 105.13/-, namely Rs 100/- grown at the risk-free rate. The reweighting moved the average by roughly Rs 3/- without moving a single one of the twelve levels by a single paisa. Look down the third column: every entry is exactly what it was before any of this started.

Two honesty notes on that table. Hiding them would teach a false neatness. First, the twelve weights average 0.998855 rather than 1.000000, and they should average exactly one. The twelve slice construction slightly compresses the spread it is standing in for, and the gap is that known understatement, closing as the number of slices rises. Second, the same understatement is why the plain average reads Rs 108.11/- rather than Rs 108.33/-. Neither gap is a property of the reweighting; both are a property of using twelve slices to stand in for a whole distribution.

One last reconciliation ties the three tools into a single line. The market price of risk gave 0.150000. Girsanov turned that into a drift shift of 0.030000 and a drift of 0.050000. The Radon-Nikodym derivative turned the same 0.150000 into a weight per path, and averaging with those weights produced Rs 105.11/-, namely Rs 100/- grown at 0.050000. The drift Girsanov predicted and the drift the weights actually delivered are the same number, reached two different ways. The agreement between the two is the check worth running on any measure change.

How does somebody checking a model use these three?

Somebody handed a valuation and a description of the process behind it rarely gets to see the code. Usually two paragraphs of description and a number arrive. The three tools give that reviewer three checks that need no data, no software and no agreement with whoever built the thing.

The first check is the cheapest and catches the most. Read the volatility out of both descriptions and confirm they are the same number. If a document describes the process one way and then prices it with a different volatility, something has been done that is not a change of measure, and the reviewer can say so without knowing anything else. On the standard process both readings must be 0.200000. A document showing 0.20 in one place and 0.17 in the other has either made an error or has quietly recalibrated, and the difference between those two possibilities is worth an hour of anybody's time.

The second check is the arithmetic of the shift. Take the drift in the first description, take the drift in the second, and divide the gap by the volatility. On the standard process that is 0.030000 divided by 0.20, giving 0.150000, and it must match whatever market price of risk the document says it used. If the document names no market price of risk at all, that quantity is still implied by the two drifts, and computing it is often the fastest way to find out what a model is assuming that nobody wrote down.

The third check is the one that catches a broken reweighting. The weights are corrections to a rule that already added to one. Averaging them across all paths must therefore give exactly one. Anybody implementing a reweighting by hand can verify that the average of their weight multipliers comes out at 1.000000, and if it comes out at 0.94 or 1.08 the correction term has been mishandled. Notice from the twelve slice table above that this check is sensitive: it read 0.998855 there, and the reason was not a broken reweighting but too few slices, so a reviewer running it needs to know what accuracy the construction can deliver before calling a fault.

A fourth use belongs to whoever is building rather than reviewing, and it is where the change of numeraire earns its keep. When a valuation involves a ratio of two things, quoting in units of one of them can make the whole calculation collapse to something simpler. The divisor becomes 1.000000 by construction and drops out of the algebra. Collapsing the algebra is the entire practical motivation for ever changing the unit, and it is why the technique appears far more often in rate and currency work than in a first course. The mathematics is identical to what is set out in this guide; only the choice of unit differs.

All four checks share a property worth naming. None of them needs the reviewer and the builder to agree on what is going to happen. The checks only need agreement on a drift, a rate and a volatility, the three numbers any description has to state anyway. Needing no shared view is what makes these checks portable between people who share no code.

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