Geometric Brownian Motion: Why Prices Are Modelled This Way

Geometric Brownian motion is four assumptions wearing an equation: that changes are proportional rather than absolute, that they carry no memory of what came before, that their size never moves, and that the path never jumps. Each one buys something that can be named and fails somewhere that can be pointed at. Geometric Brownian motion is a modelling choice with consequences, not a description of anything.

Calling the model a list of assumptions does not say the model is right, and does not say the model is wrong. A list is the kind of thing that can be argued with one line at a time. An equation is not: there is nowhere on an equation to put a finger. A list of four assumptions has four places to put a finger, and every property this model has, every clean answer it produces and every objection anybody has ever raised against it lands on exactly one of those four lines.

The equation is short, and it is an output rather than a starting point: it is what four decisions look like once they have been written down together. Each decision was made one at a time and could have been made differently, so each gets its own block below, with what it buys set beside where it fails. Any answer this model produces traces back to the assumption it rests on, and that, not the arithmetic, is what settles whether to trust it.

What is geometric Brownian motion, stated without starting from the equation?

Start with the thing being modelled. Every part of this reading order runs on one invented quantity called the standard process, written S with a time subscript. The standard process starts at Rs 100/-, carries a drift of 8 per cent a year, written mu, and a volatility of 20 per cent a year, written sigma, so its variance rate is 0.04 exactly. The horizon over which it is watched is one year.

Writing down how that quantity moves from one instant to the next requires a rule, and no rule is given; one has to be chosen. Geometric Brownian motionThe model made of the four assumptions set out in this guide, and nothing more than those four. is the name for one particular set of four choices, taken here in order.

1. Changes are proportional, not absoluteThe random push and the steady drift are both applied as a multiple of wherever the process currently stands, rather than as a fixed number of rupees added on top.
   Choice: multiply, do not add.
2. Changes carry no memoryWhatever the process did in the last instant says nothing whatsoever about what it does in the next one. The increments are independent of everything that has already happened.
   Choice: forget, do not remember.
3. The size of the random term never movesOne number, sigma, set at 20 per cent a year, and it is the same number at the start of the year, in the middle and at the end, at any level and after any history.
   Choice: one number, not many.
4. The path never jumpsThe process gets everywhere it gets by moving continuously. There is no instant that carries a move of its own, however violent the shaking becomes.
   Choice: walk, do not teleport.

Those four choices are the entire model. Nothing else has been assumed and nothing else is needed. Every one of the four is a decision with an alternative, and none of them was forced on anybody by evidence. The equation below is not the model; it is what the four choices look like when they are written down together in one line.

Two of the four assumptions are not visible in that line at all. Look at where each one actually sits. Assumption one is the two appearances of S on the right hand side; take them away and the model is a different one. Assumption three is the fact that sigma is a letter with no time subscript. Assumptions two and four are not in the equation at all: they are properties of W, imported wholesale from the earlier work on Brownian motion, and a reader who reads the line without knowing what W is has read two assumptions without noticing them.

Solving that equation, already done earlier in this reading order, gives a statement about where the process stands at the horizon rather than about how it moves. The solved form is where the four assumptions cash out into numbers that can be checked, and it is worth stating even though the derivation belongs elsewhere.

From that one line the checkable figures follow immediately. Multiply Rs 100/- by the exponential of 0.08 and the expected value at the horizon is Rs 108.33/-. Multiply the same Rs 100/- by the exponential of 0.06 instead and the median is Rs 106.18/-. The distance between them is Rs 2.15/-, and that distance is the half variance correction made visible. Hold on to those three numbers. The whole argument of the failure block below is that they are entirely correct and still not enough.

Assumption one, proportional changes: what does it buy and where does it fail?

Here is the everyday version first. Imagine measuring things with two different rulers. The first ruler has marks a fixed centimetre apart, so a mark means the same physical distance whether the thing measured is a matchbox or a room. The second ruler has marks that are always one per cent of whatever is being measured, so a mark is a hair on the matchbox and a hand span in the room. Neither ruler is wrong. The two rulers answer different questions, and choosing between them is a decision about what kind of change is being watched.

A proportional changeA change applied as a multiple, so its size scales with the level it is applied to. is the second ruler. An absolute changeA change added to the level, whose size does not depend on where the level currently stands. is the first. Geometric Brownian motion picks the second ruler, and that single pick is assumption one.

What it buys

Two things, and both are large. The first is a floor. Multiplying a positive number by a positive number can never produce nil or a negative number, so the solved form above has a process that stays strictly above zero at every instant, at every volatility, over any horizon. The floor is not a bound anybody imposed. It falls out of the choice to multiply.

The second is that the volatility becomes a scale free number. Under this assumption, a volatility of 20 per cent a year means the same thing at Rs 50/- as at Rs 500/-, so one number can be quoted and it travels. Multiply 20 per cent by the square root of one twelfth and the typical move over one month is 5.7735 per cent of wherever the process stands. The same 5.7735 per cent is Rs 2.89/- at a level of Rs 50/- and Rs 28.87/- at a level of Rs 500/-. Under an absolute model the size would have to be requoted every time the level moved.

The floor is the more striking of the two purchases. Push the same comparison further. Take the absolute model with its shake fixed at Rs 20/- a year and start it at three different levels. At Rs 100/- the chance it finishes the year at or below nil is about three parts in a hundred million, small enough to shrug at. At Rs 50/- it is 0.3467 per cent, no longer nothing. At Rs 20/- it is 14.0071 per cent, and a model that puts a seventh of its probability on a level that cannot exist has stopped being usable. The proportional model gives exactly nil at all three levels, not by a fix bolted on afterwards but because multiplying positives cannot produce a negative.

Where it fails

Wherever the change genuinely is absolute rather than proportional. The assumption is not a law; it is a claim about what kind of thing is doing the moving, and there are quantities whose movements are naturally measured in units rather than in percentages. A model that multiplies when the underlying change adds will misstate the size of every move away from the one level where the two were made to agree, and the table above shows exactly how badly: by a factor of ten across a range of levels that is only ten times wide.

Notice the honest shape of that failure. The proportional choice is not worse. The proportional choice was a choice, made for two named benefits, and it is the right one only for a quantity whose changes really do scale with its level. Assumption one buys a floor at zero and a percentage that travels, and it costs every situation where the change being modelled does not scale with the level.

Assumption two, independence: what does it buy and where does it fail?

The everyday picture for this one is a queue that forgets who joined it. Everybody who arrives is drawn fresh, with no reference at all to who arrived a moment earlier. Nothing is passed along. There is no crowd that attracts a crowd, and no lull that begets a lull. Each arrival is a clean slate.

The clean slate is what independent incrementsChanges that carry no information about each other, so knowing one says nothing about the next. means for the model, and the assumption arrives through W rather than through anything written in the equation. Brownian motion has independent increments by construction, so choosing W as the driver is choosing this assumption, whether or not the choice was noticed.

What it buys

Independence buys the arithmetic. Independence is precisely what lets variances add along the horizon, and variances adding is what makes the standard deviation grow with the square root of elapsed time. From the same monthly figure of 5.7735 per cent, three months gives 10 per cent, a year gives 20 per cent and four years gives 40 per cent, and every one of those follows from adding variances rather than from anything observed.

Independence also buys everything the earlier work established. The stochastic integral, the multiplication table and the chain rule that carries Ito's name were all built on a driver with independent increments. Undo assumption two and a great deal of settled machinery has to be rebuilt rather than reused. The rebuilding is a real cost, and one honest reason the assumption is popular.

Where it fails

Wherever a move now makes a move later more or less likely. Here is what that would do to the arithmetic, computed rather than asserted. Take the same twelve monthly increments with the same monthly volatility of 5.7735 per cent, and let each one carry a correlation with the one before it instead of nothing at all. The annual variance is no longer twelve monthly variances; it is twelve of them plus twenty two correlation terms. At a correlation of positive 0.25 the annual volatility becomes 24.1523 per cent. At negative 0.25 it becomes 14.7196 per cent. The identical monthly number can support an annual volatility anywhere from 14.7196 to 24.1523 per cent, and the model's answer of exactly 20 comes entirely from setting the dependence to zero.

The spread from 14.7196 to 24.1523 per cent is the honest measure of what assumption two is doing. Independence is not a small technical convenience. Independence holds the entire relationship between short horizons and long ones in place, and if it is wrong then every scaling this model performs is wrong by an amount nobody has bounded. The volatility process, covered separately, exists in large part because there is a specific, well studied way in which this assumption is questioned: models in which the size of the shake carries memory even when its direction does not.

Assumption three, constant size: what does it buy and where does it fail?

Picture a measuring tape that reads the same at every temperature. Reading the same at every temperature is a wonderful property in a tape, and it is exactly what constant volatilityThe assumption that the size of the random term does not move through time, so one number covers the whole horizon. claims for the size of the random term. One number, 20 per cent a year, covering the first month and the last, every level and every history.

What it buys

Constant size buys the whole of the model's tractability, and it is worth being blunt about how much rests on it. Because sigma is a constant, the solved form above has a closed shape rather than an integral that has to be worked out. Because sigma is a constant, the logarithm at the horizon is exactly normal with mean 4.665170 and standard deviation 0.20, rather than approximately something. Because sigma is a constant, the model is fully specified by five numbers rather than by a second process with parameters of its own. Every clean answer above traces back through this line.

Where it fails

Wherever that number is visibly not constant through the horizon, and the failure is subtler than it first looks. The obvious cost is that a single number cannot describe a size that moves. The less obvious cost sits in the average. Volatilities do not average. Variances do, and taking the average of two volatilities in an attempt to be fair about it gives the wrong answer.

Here is that worked out. Suppose a model in which the variance rate is 0.0625 for the first half of the year, being a volatility of 25 per cent, and 0.0225 for the second half, being 15 per cent. The total variance over the year is the average of the two rates, 0.0425, and the horizon volatility is therefore 20.6155 per cent. Now take the naive route and average the two volatilities instead: 25 and 15 average to 20, giving a variance of 0.0400. The two routes differ by 0.0025 of variance, 5.8824 per cent of the true total, and the constant-size model set at the average volatility understates the spread at the horizon even though it looks careful.

The averaging gap is the specific place assumption three fails, stated as sharply as it can be stated. The Heston model exists to relax exactly this line, replacing the single number with a process of its own that has a level, a speed of return, a size and a correlation. Every one of those extra parameters describes something assumption three threw away in exchange for a closed form.

Assumption four, no jumps: what does it buy and where does it fail?

The everyday version is pacing out a coastline with a ruler that must stay on the ground. The ruler can be as short as anyone likes, and the coast can be as ragged as it likes, but the ruler may never be lifted and set down further along. Everywhere the pacing reaches, it reaches by passing through everywhere in between. Pacing that never lifts the ruler is a continuous pathA path with no jumps, so it passes through every level between any two levels it visits, which is what makes the chain rule usable., and assumption four says the process has one.

What it buys

Continuity is what makes the calculus of the earlier reading order usable at all. The chain rule that carries Ito's name is derived by expanding a function around a small move and keeping the terms that survive. The whole derivation assumes the move is small, and a small move is another way of saying the path did not jump. Give up continuity and the expansion has to be replaced by something that handles a finite move directly, and the tidy correction term acquires company.

Continuity also buys the quadratic variation result the earlier work is built on. Over the one year horizon the quadratic variation of the logarithm of the standard process is 0.04 in the limit, being the variance rate multiplied by the elapsed time. The quadratic variation result is a limit statement about a continuous path, and it is what makes a variance rate readable off a path at all.

Where it fails

Wherever a move occurs that is too large to have been reached one small step at a time, and this is where the numbers get uncomfortable in the model's favour and then against it. Consider a downward arrival whose logarithm is minus 0.25, a proportional move of minus 22.1199 per cent in an instant. Ask the continuous model how surprising that is. Over a month, whose log standard deviation is 0.0577350, it is a move of 4.3301 standard deviations. Over a day, whose log standard deviation is 0.0125988, it is a move of 19.8432 standard deviations.

The model does not say a move of 19.8432 standard deviations is unlikely. The model says instead that a single instant carries no move at all, and the only way to reach that level is to pass through every level in between. That is the content of assumption four, and it is the one readers find hardest to feel, which is why the simulation below is built around it.

How Geometric Brownian Motion Models Asset Prices: what does each assumption buy?

Seeing the four as a ledger rather than as four separate arguments is what turns the model from a description into a decision. Put all four side by side on the standard process. Every row has a purchase and a price, and no row is free.

Three of those four rows carry a small red line saying the assumption is relaxed by later work, and that line is the strongest evidence available anywhere that these are choices rather than facts. Nobody builds a whole body of technique to undo something that was simply true.

The control below moves along one formula, so it is worth stating that formula rather than leaving it only in the readout. Add a jump component to the continuous one and the two variances simply add, with the jump contribution being the intensity multiplied by the mean squared jump size. Setting the intensity to nil deletes the second term and returns the model this guide is about, exactly.

What does the model get right that a simpler one does not?

Against those four failures stands what the model earns, and that deserves stating just as plainly. Three things, and they are not small.

First, the floor. A model whose process cannot reach nil from above is describing a quantity with the right shape, and it gets there without any patch. Second, the shape of the distribution at the horizon. Because the logarithm is normal, the level itself is skewed to the right, and the mean of Rs 108.33/- sits above the median of Rs 106.18/- by exactly Rs 2.15/-. A model that puts a symmetric distribution on the level would say the mean and the median are the same. For a multiplicatively growing quantity they are not. Third, tractability. Five numbers determine everything, the answers have closed forms, and every later model in this subject area can be written as this one plus a named departure.

The model earns a floor at zero, a right skewed horizon distribution whose mean and median differ by exactly the half variance correction, and a description short enough to argue with, and no simpler model gives all three. That is a real achievement and it is why the model has lasted. The achievement is completely compatible with every criticism above. None of those criticisms said the model was useless; they said it was a choice.

How to Specify a Geometric-Brownian-Motion Model: which five lines does it take?

A specificationThe set of lines that fix a model completely, so that nothing about its answers is left open. is the point at which the model stops being a shape and starts being a thing that produces numbers. Five lines do it, and every one of the five is needed. Miss one and the model determines nothing at all. Determining nothing is a stronger statement than merely being incomplete.

The last two lines are the ones people leave out. A horizon is left out because the drift and the volatility are quoted per year, so the year feels as though it is already in there. The year is not in there: a rate a year says nothing about a date until a date is named. The measure is left out because under the physical measure P the drift is the 8 per cent above, and under the risk-neutral measure Q it would be something else entirely for the same process, so a drift with no measure attached is a number with no meaning. A model given a starting value, a drift and a volatility looks specified and is not. The horizon and the measure are both missing, and the model determines nothing without them.

Why is it still the starting point, given all of that?

Because a base case has a job that has nothing to do with being correct, and this model does that job better than anything else available. Three honest reasons.

The first is that it is the smallest model with the three properties named above, and smallest matters. Every parameter added is a parameter somebody has to pin down, and every parameter that has to be pinned down is a place where a wrong number can enter without anybody noticing. Five numbers can be listed on one line and checked by a reader.

The second is that it is the thing every later model is a departure from. Relaxing constant size gives a model with a variance process of its own; relaxing continuity gives a model with a counting process and a size distribution; relaxing independence in the size gives another. In each case the extra apparatus is described as what has been added to this model, and the parameter that switches the addition off returns exactly this model. Being reachable again is a real service. A base case recovered by setting one parameter to nil is worth more than a slightly better model with no way back.

The third is this. Because the model is four assumptions rather than a description, which of the four a particular answer leans on can always be asked, and that question is answerable only against a model whose assumptions are separated and named. A model presented instead as a description of how things behave offers nowhere to stand from which to challenge one of its answers.

How is a list of four assumptions actually used?

By running the four as a checklist over somebody's output. Anyone reviewing a model's number earns their keep that way. Whether the model is right in general is a question with no answer, and never the one to ask. The question is which of the four assumptions this particular number leans on, and how hard.

Here is the practice. Somebody hands the analyst a figure produced by a model of this shape. The arithmetic is almost always fine. Instead of checking it, the analyst asks four questions in order. Does this number depend on the change being proportional? Does it depend on increments having no memory? Does it depend on the size being one constant across the horizon? Does it depend on the path never jumping? The answers that come back yes show precisely which later model, if any, the question needs.

The value of the checklist is that it produces different verdicts for different questions off the same model. A number about the shape of the horizon distribution leans hard on constant size and hardly at all on jumps at the level of the median. A number about the extreme left of the horizon distribution leans on the no jumps line above everything else, as the simulation above showed: moving the intensity from nil to 0.5 barely touched the right side and roughly doubled the weight below Rs 70/-. The same model is adequate for one question and inadequate for another, and only a separated list of assumptions can tell the two apart.

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