Brownian Motion vs Geometric Brownian Motion

Brownian motion moves by adding a random amount, so it can take any value including a negative one, and its level is normally distributed. Geometric Brownian motion moves by multiplying by a random factor, so it cannot reach zero from above, and it is the logarithm of its level that is normally distributed. Everything else separating the two follows from additive against proportional.

One of these two processes builds itself by adding the randomness to the level. The other builds itself by multiplying the level by it. The choice between adding and multiplying is the whole of the difference. Every other contrast between them, in what values they reach, in what shape their outcomes take, in what is normally distributed and in what each one is used for, is a consequence of that one choice rather than a separate fact to memorise.

The distinction is familiar from ordinary life without the names. A thermometer that reads two degrees high reads two degrees high at forty degrees and at minus five degrees alike: that error is additiveA change added to the level, so its size does not depend on the level.. A shop offering twenty per cent off takes Rs 200/- off a Rs 1,000/- item and Rs 20/- off a Rs 100/- one: that discount is proportionalA change applied as a multiple, so its size scales with the level.. The two behave differently everywhere, and the mathematics below is that same difference stated precisely and then followed to its consequences.

What is Brownian motion, defined from scratch?

Brownian motion is written W with a time subscript, and W at time zero is zero. Four statements fix it completely, and there is nothing else to it. The process starts at zero. Its path is continuous, so it has no gaps and no sudden jumps. Its incrementsThe change in a process from one time to a later one, taken as a quantity in its own right. over intervals that do not overlap are independent of one another, so the next move does not consult the last one. And the increment over an interval of any length is normally distributed with an average of zero and a variance equal to the length of that interval.

Read those four statements again and notice what is absent: not one of them mentions the level the process is currently sitting at. The size of the next move depends on how much time passes and on nothing else. A Brownian motion at plus fifty and a Brownian motion at minus fifty face exactly the same distribution of next moves. The absence of the level from all four statements is where everything in this guide comes from.

Put numbers on it using the partition this subject works with throughout. The year is cut into twelve equal steps, so each step has a length of one twelfth of a year and the square root of that length is 0.288675. The driving valuesThe published numbers that generate the locked path, identical for both processes here. are the twelve published numbers minus 0.5, 1.6, minus 1.3, minus 0.1, 0.1, 1.5, minus 1.3, minus 0.5, minus 1.4, 0.4, 0.9 and 0.6, and each one is multiplied by 0.288675 to give that month's increment. The increments added up in sequence give the path.

Doing that gives readings of minus 0.144338, 0.317543, minus 0.057735, minus 0.086603, minus 0.057735, 0.375278, zero, minus 0.144338, minus 0.548483, minus 0.433013, minus 0.173205 and zero. The twelve driving values were constructed to sum to zero, so the path sits below zero at eight of its twelve monthly readings, touches zero exactly twice, and finishes the year at exactly zero. The twelve values are fixed rather than sampled afresh, and they are carried to six decimal places, so every path drawn in this subject rests on the same twelve numbers.

What is geometric Brownian motion, defined from scratch?

Geometric Brownian motionA process whose proportional changes are driven by Brownian motion. is written S with a time subscript. Its level at any time is its starting level multiplied by an exponential whose exponent has two pieces: a straight-line term in elapsed time, and the volatility multiplied by the Brownian motion at that time. Equivalently, and this is the version worth holding, the logarithm of the level is a Brownian motion with a straight-line trend added to it.

The standard process used throughout this subject is one of these. The process, an invented one, starts at Rs 100/- exactly, its drift is 8 per cent a year, its volatility is 20 per cent a year and the horizon is one year. Its variance rate, being the volatility squared, is 0.04 exactly, and half of that is 0.02 exactly.

Two things about that exponent are worth pausing on. The first is that the whole Brownian motion sits inside an exponential, and an exponential of any real number is a positive number, so the level is a positive number multiplied by a positive number and can never be anything but positive. The second is the correction of half the variance rateThe gap between the growth of a process and the growth of its logarithm. subtracted from the drift. The growth of the logarithm carries a correction of half the variance rate. The derivation is the chain rule result for processes of this kind, covered separately later in this subject. Take the correction as given and watch what it does.

The definition also reads perfectly well as a recipe. The path starts at Rs 100/-. Each month, the exponent for that month is worked out and the level is multiplied by the resulting factor. Twelve repetitions give the path. The factors for the first three months of the locked path, computed from the same driving values, are 0.976415, 1.102275 and 0.932342.

What single choice separates the two processes?

Set beside each other, the two definitions differ in one word. The first adds its random quantity to the level. The second multiplies the level by a factor built out of the same random quantity. Everything else in this guide is downstream of that.

Writing the two as one step on the twelve step partition makes the difference visible without any machinery at all. On the additive process the step is a quantity in the units of the process, subtracted or added regardless of where the process stands. On the proportional process the step is a ratio, and the level is multiplied by it.

The left hand side of the first expression is a difference and the left hand side of the second is a ratio, and that is the entire root of the comparison. A difference does not care what it is a difference from. A ratio only means anything relative to what it is applied to.

The abstraction becomes concrete on the locked path, in the sharpest small illustration available. The driving value minus 1.3 appears twice, in month 3 and again in month 7. The additive process receives an identical step both times, to six decimal places. The proportional process receives an identical percentage step both times, also to six decimal places. But the rupee move differs. The process was standing at Rs 107.63/- the first time and at Rs 111.08/- the second.

What values can each of the two reach?

The additive process can reach any number at all. Give it enough time and there is no value, however far above or below zero, that it does not reach with certainty. Nothing in its definition refers to the level, so nothing in it places a floor or a ceiling anywhere.

The proportional process can reach any value strictly above zero and nothing else. The process cannot reach zero and cannot pass below it. The reason is arithmetic rather than deep: the level at any time is a positive starting value multiplied by an exponential, an exponential is always positive, and a positive number multiplied by a positive number is positive. No factor in the sequence is ever zero, so no sequence of steps reaches zero.

The proportional process does not reach zero and then stop there. Zero is not on the list of values it can reach at all. A level that can be reached and never left is called absorbingA level that, once reached, can never be left, which this process never reaches., and the proportional process does not have one. The level can become arbitrarily small. Zero itself stays out of reach.

Set it against two everyday measurements. A temperature reading can sit below zero perfectly sensibly, and a scale that reads two degrees high does so at every temperature: that is an additive quantity. The number of people waiting in a queue cannot be negative, and it cannot be halved indefinitely into a negative either: quantities that are counted or measured as a size behave more like the second process. Choosing a process means deciding which of those two a quantity is.

Which of the two is normally distributed?

The question of which process is normally distributed is answered wrongly more often than any other in the comparison, and the wrong answer is not a slip of the tongue. The wrong answer changes the arithmetic downstream.

The additive process is normally distributed at every horizon, straight from its definition. The level at one year is the increment from time zero to one year, and that increment is normal with an average of zero and a variance of one, hence a standard deviation of exactly 1.000000.

The proportional process is not normally distributed at any horizon. The logarithm of its level is what is normally distributed. At the one year horizon that logarithm is normal with an average of 4.665170 and a standard deviation of 0.20 exactly. A quantity whose logarithm is normal is called lognormalThe distribution of a quantity whose logarithm is normal., and a lognormal shape is not a normal shape stretched a bit; it is a different shape with different properties.

The difference between the two shapes is not cosmetic. One is symmetricA shape whose two sides are mirror images, which the normal is and the lognormal is not.. Its two sides are mirror images, and its average, its middle outcome and its most likely outcome are one and the same number. The other leans right, stops dead at zero on the left, and has an average, a middle outcome and a most likely outcome that are three different numbers.

Saying only that the shape leans right gives nothing to check, so the three numbers are worth naming. The most likely outcome is Rs 102.02/-, the middle outcome is Rs 106.18/- and the average is Rs 108.33/-. The three run in that order, and the order is a property of the shape rather than a coincidence of these parameters.

For a symmetric shape those three markers would sit on top of one another, and here they are spread across Rs 6.31/- of the scale. The gap between the middle outcome and the average is Rs 2.15/- exactly, and that gap is the half variance rate correction made visible: Rs 100/- grown at the drift of 8 per cent gives Rs 108.33/-, and Rs 100/- grown at the drift less half the variance rate, which is 6 per cent, gives Rs 106.18/-.

Why do the same twelve driving values produce two different shapes?

Here is the claim this guide exists to make. Both paths drawn here come from one set of twelve numbers. Not two sets, not two samples, not two runs of anything. The identical twelve driving values are fed into both constructions, and what comes out looks nothing alike.

The shared randomness can be checked by looking at where each path turns. The additive path turns upward or downward in months 2, 4, 6 and 9. A turn happens when the driving value changes sign, and both processes see the same driving values, so the proportional path turns at exactly the same four months. The turning points are shared. Everything else is not.

The randomness feeding those two pictures is not merely similar but identical, so every difference between them belongs to the construction. Choosing between these two processes is not choosing between two kinds of randomness. The choice is what is done with one kind.

Where does the growth come from when the randomness cancels?

The twelve driving values were constructed to sum to exactly zero. The additive path therefore returns to exactly where it started. The proportional path, fed the same twelve values, finishes at Rs 106.18/-. So the randomness contributed nothing over the year, and the process still gained Rs 6.18/-. Where did that come from?

From the straight-line term in the exponent, and nowhere else. With the Brownian motion at zero at the horizon, the exponent is the drift less half the variance rate multiplied by one year. The arithmetic is 0.08 less 0.02, giving 0.06 exactly. Rs 100/- multiplied by the exponential of 0.06 is Rs 106.1837/-, the published finishing value to four decimal places.

Here are both paths month by month, from the same twelve driving values. The two number columns, read against each other, hold the whole comparison.

Which of the two should be reached for?

One question settles it in almost every case, and it is not a question about mathematics at all. The question is whether the quantity being modelled can sensibly take a negative value.

If it can, the additive process fits, and its ability to go below zero is a feature rather than a defect to be worked around. If it cannot, the proportional process fits. A floor at zero is built into it rather than bolted on afterwards. Prices are commonly modelled with the proportional process for exactly that reason. A process is a model and never a description, so commonly modelled with is not the same as what prices do.

The additive process is not a poor relation left over once the proportional one has taken the interesting work. There is a whole later part of this subject built on quantities where going below zero is exactly what the model has to be able to do, and forcing such a quantity through a construction with a floor at zero would be the error rather than the fix. The two processes are tools for two different jobs, and neither is more advanced than the other.

How does somebody reviewing a model use this distinction?

Using any of this requires no rebuilding of anybody's model. Two checks, both cheap, catch most of what goes wrong with these two processes in practice, and both can be run on a printed sheet of output with nothing else to hand.

1. Ask what the quoted range is symmetric about Find any range or interval in the output and see whether the number in the middle is exactly halfway between the two ends. If it is, and the quantity has a floor at zero, the range was almost certainly built the symmetric way.
   On the standard process the symmetric build reaches Rs 6.31/- too far down and stops Rs 5.92/- short at the top.
2. Ask whether the average or the middle outcome is being quoted For a symmetric shape the question is empty because they are the same number. For the proportional process they are Rs 108.33/- and Rs 106.18/-, and which of the two is meant changes the answer by Rs 2.15/- before anything else is discussed.
   A document that says the expected level without saying which of the two it means has left a Rs 2.15/- ambiguity unresolved.

Both checks ask one question twice: has the person treated a shape that leans right as though it were symmetric? The question is worth carrying away even if every formula here fades. It catches the error long after the arithmetic has been forgotten.

There is a household version of the second check that makes it stick. Told that a shop's prices fell by 50 per cent and then rose by 50 per cent, most people will say the price is back where it started. The price is not back where it started: Rs 100/- falls to Rs 50/- and rises to Rs 75/-. Proportional changes do not add up the way additive ones do, the average of the moves is not the move of the average, and the whole of the half variance rate correction lives in that gap. Anybody who has been caught by the 50 per cent example has already met the idea this guide is about.

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