Martingale: A Fair Game, Formally Defined

A martingale is a process whose average future value, given everything known now, is exactly its current value. Three conditions define it: it is adapted, its average exists, and its conditional average equals its present value at every time. The definition says nothing about where any single path goes, only that the centre of the distribution never moves.

The definition is a conditional averageThe average taken over the outcomes still possible given what is known now. set equal to a present value, so everything about a martingale inherits from conditional expectation. A martingale depends on the information being conditioned on, it depends on which measure the average is taken under, and it is a statement about the centre of a distribution rather than about its spread. Every misreading of the word comes from forgetting one of those three.

What three conditions define a martingale?

Three, not one. Informal definitions usually give the third and drop the other two, and the second is a real condition rather than bookkeeping.

First, the process must be adaptedKnown by the time it happens, and never earlier.: its value at each time must be known by that time. A process whose value at a moment depends on something not yet knowable gives nothing to condition on. Its conditional average cannot be taken at all. Second, the average must exist. Third, the conditional average of any later value, given what is known now, must equal the present value.

The condition that gets dropped is the second, and it is the one that fails on real processes. There are perfectly ordinary looking processes whose averages are undefined, and for those the third condition is not false but meaningless: it sets one thing equal to a quantity that was never defined. IntegrabilityThe requirement that the average exists at all, which is a real condition rather than a formality. is the word for the second condition. Because the condition almost always holds, it reads like a technicality.

Why is the property about the centre and never about the spread?

Read the third condition again and notice what it constrains. The third condition fixes one number, the conditional average, at one value, the present level. The condition places no constraint whatsoever on how far the process might wander around that average, on how wide the distribution of future values is, or on how quickly that width grows.

A martingale pins the centre of the fan of future values and leaves the width of the fan entirely free. Both of those things are happening at once, and a reader who has heard the phrase fair game usually hears only the first. The centre standing still and the risk growing are not in tension; they are two facts about two different features of the same distribution.

Take the standard processThe single invented traded quantity that the stochastic calculus readings run on., invented, starting at Rs 100/-, drifting at 8 per cent a year and carrying a volatility of 20 per cent a year, and discount it at the risk-free rate of 5 per cent a year. Under the pricing rule that discounted processThe process multiplied by the price today of a rupee at that time. is a martingale. Its average is Rs 100/- at three months, at six months, at nine months and at a year. Meanwhile the spread of its logarithm is 0.100000 at three months, 0.141421 at six, 0.173205 at nine and 0.200000 at a year, so the band that holds roughly two thirds of the outcomes widens from Rs 90.03/- to Rs 109.97/- at three months out to Rs 80.25/- to Rs 119.72/- at a year.

What does a martingale not claim?

A martingale does not claim the process goes nowhere. The property does not claim the process is safe, still, low risk, or bounded. The phrase fair gameA game whose expected gain from any point onward is nil, which is all the word promises. is doing work here that the mathematics never authorised. Nor does it claim any path returns to where it started. The definition makes one statement about one feature of a distribution, and every other reassuring thing the phrase fair game suggests is imported by the reader.

The discounted locked path is the standard process along its published twelve step path, discounted continuously at 5 per cent. The path reads Rs 97.24/-, Rs 106.73/-, Rs 99.10/-, Rs 98.61/-, Rs 99.26/-, Rs 108.33/-, Rs 100.59/-, Rs 97.80/-, Rs 90.29/-, Rs 92.47/-, Rs 97.48/- and Rs 101.01/-.

Not one of those twelve readings is Rs 100/-, the path travels from Rs 90.29/- up to Rs 108.33/- during the year, and the martingale property holds exactly at every step. The lowest and highest readings are Rs 18.05/- apart, which is more than 18 per cent of where the process started. A property about the average constrains nothing about that. A single path is not evidence for or against the property, one way or the other.

How is the property checked on a process handed over?

The check is short and it is the same four steps every time. The step people skip is the first one, and skipping it makes the other three meaningless. Learn the four steps as a routine rather than reasoning them out afresh.

How to Check a Simple Martingale Property

1. Name the measure Before anything else, the rule the average is taken under has to be named. The same process is a martingale under one and not under another, so the question is incomplete until this is answered. This is called measure dependenceThe fact that the property holds or fails according to which rule the average is taken under. and it is the single most common source of confusion about the word.
   Skipping this is the commonest fault, and everything below is meaningless without it.
2. Check the process is adapted The value at each time has to be known by that time. Almost every process encountered is adapted to its own history by construction, so this usually passes immediately.
   It fails when somebody has built a quantity using a figure that arrives later.
3. Check the average exists Confirm the process is integrable. This is the condition that reads like bookkeeping and is not.
   A process with an undefined average makes step four a statement about nothing.
4. Take the conditional average one step ahead and compare Compute the average of the next value given what is known now, and see whether it returns the present value.
   Equal means martingale. Above means it drifts up. Below means it drifts down.

Running it on the discounted standard process over one half year step: from Rs 100/- the process moves to Rs 115.19/- or Rs 86.81/-. Under the pricing rule the weight on the up move is 0.553908. The average of the next value is then 0.553908 multiplied by Rs 115.1910/- plus 0.446092 multiplied by Rs 86.8123/-, or Rs 102.531512/-. Discounted by 0.975310, being one half year at 5 per cent, that comes to Rs 100.000000/-.

Step four returned the present value to six decimal places, so the discounted process is a martingale under the pricing rule, and the whole check took one line of arithmetic.

What happens when the same process is tested under the other measure?

Nothing about the standard process changes. The paths are the same paths, the discounting is the same discounting, and the outcome set has not gained or lost a single member. Only the rule used to take the average changes, and the property flips.

Under the pricing rule the discounted process averages Rs 100.000000/- at the horizon, exactly its starting value. The model's own rule gives the process its drift of 8 per cent a year. Under that rule the same discounted quantity averages Rs 103.045453/-. Rs 103.045453/- is Rs 100/- grown at 3 per cent for a year, and 3 per cent is exactly the drift of 8 per cent less the rate of 5 per cent.

One column stands still and the other climbs, from one process, one outcome set and one discounting, differing only in which measure the average is taken under. That is why the word martingale on its own is an incomplete sentence, and why the check routine puts naming the measure first rather than last.

The two measures are often described as two views of the world, and that framing puts the difference in the wrong place. Be concrete instead about what changed and what did not. The outcome set is one set of paths. The locked path is in both. Its twelve readings are the same twelve numbers under either rule, and the discounting applied to them is the same discounting. The weight attached to each path when an average is taken is what differs, and nothing else differs at all.

Notice that the volatility of 20 per cent a year appears nowhere in that calculation. The average of the discounted process depends on the drift and the rate and on nothing else. Whether a process is a martingale is settled entirely by its drift, and its volatility has no vote at all. The volatility decides how wide the fan is, and the fan can be arbitrarily wide without disturbing the property one way or the other.

Why is this the property the whole of pricing is built on?

Because a martingale is the one kind of process whose average can be taken across time without a correction term. Ask what a quantity will average at some later date and the answer is simply its value now. No growth rate has to be estimated, no adjustment for how long the horizon is, and no view about what is likely enters anywhere.

Everything that makes pricing tractable follows from arranging for the right quantity to be a martingale under the right measure. Its future average is then readable off the present without knowing anything else. That is why the second measure exists at all: it is not a description of what anybody believes, it is the rule that makes the discounted process a martingale, which is the property that makes the average computable.

How is this used by somebody checking a model rather than building one?

The four step routine is worth running on somebody else's work, and the useful thing about it is that it needs almost nothing from the model. Understanding how a figure was produced is not required in order to ask which measure it was averaged under, and that single question resolves a surprising share of disagreements about whether a number is right.

The pattern to watch for is a document that establishes the property under one measure and then uses it under another. The mix-up happens because the word martingale gets written down once, without its measure, and read later as though it were a property of the process. The tell is a growth rate appearing somewhere it should not: if a quantity really is a martingale under the measure being used, no growth rate is needed to project its average, so a projection that carries one has either changed measure or was never dealing with a martingale in the first place.

The standard process is invented.
Educational material. Not advice on any investment, tax, budget or market position.