Risk-Neutral Probability: The Weight That Prices, Not a Belief

A risk-neutral probability is the weight that makes a discounted average of tomorrow's two values come back to today's value. The weight is solved out of one equation, from the two future values, the length of the step and the interest rate, and from nothing else. The weight carries no view about what will happen, and it exists only when cash grows by less than the up move and more than the down move.

Everything the panel just did is arithmetic that can be checked by hand. Behind it there is one equation with four numbers already fixed and one number missing. Solving for the missing one gives a risk-neutral weightThe number that makes a discounted average of tomorrow's values equal today's value.. Nothing about anybody's attitude was needed to write the equation down, so nothing about anybody's attitude to uncertainty appears in the solution. The absence of any attitude is the single hardest thing to hold on to about a risk-neutral weight.

Where does the formula for the weight come from?

Start with a setting small enough to write out completely. One quantity, one step forward in time, and two values it can take at the end of that step. Call today's level of the standard process S at time zero. Over the step the level either multiplies by an up factor or multiplies by a down factor. On one step of a lattice there are exactly two cases, so there is no third.

Now write the statement that defines the weight. Today's value equals the average of the two future values under some weight, discounted back over the step at the interest rate. That sentence is one equation. Today's value is fixed, the two future values are fixed, the discount is fixed, and the weight is the only symbol in the sentence not already pinned down. One equation, one unknown, one solution.

Notice what is absent. There is no term for how much uncertainty anybody dislikes, no term for how much extra return anybody demands for carrying it, and no term for how often the up move has happened before. The equation was written without any of those, so the number that solves it cannot contain any of them. A reader who has met expected returns elsewhere often looks for the missing ingredient here, and there is no missing ingredient. The absence is the point rather than an omission.

Rearranging the equation is two lines of school algebra. The level of the process appears in all three terms and is not zero, so it cancels from every one of them. Multiplying both sides by the growth of cash over the step leaves the growth of cash sitting between the two factors as a weighted average, and solving that for the weight gives the form everybody quotes.

Read as a position rather than as a probability, that ratio stops being mysterious. The down factor and the up factor mark the two ends of an interval. The growth of cash sits somewhere along it. The weight is simply how far along, expressed as a fraction of the whole distance. Mixing two liquids of different strengths to hit a target strength in the middle solves exactly this ratio; Indian school arithmetic teaches it as alligation, and the fraction it returns gives the mixing proportion and says nothing whatever about how often either bottle is reached for.

Which three numbers does the weight need, and where is each one found?

The formula has three inputs and they come from three different places, and confusing the sources is where most wrong weights come from.

The first input is the volatility, and it produces both factors at once. The up factorThe multiple the value takes if it moves up over one step. is the exponential of the volatility times the square root of the step, and the down factorThe multiple the value takes if it moves down over one step. is its reciprocal. Building them as reciprocals is a choice made by whoever specified the lattice, and it is the choice Cox, Ross and Rubinstein made in 1979, so the two factors sit symmetrically either side of one.

The second input is the interest rate, and it produces the growth of cashThe factor by which a certain rupee grows over one step at the interest rate. over the same step. Because the rate here is continuously compoundedQuoted so that growth over a period is the exponential of the rate times the period., that growth is the exponential of the rate times the step. Read the rate off the same specification that fixed the horizon. Read it as a decimal a year rather than as a percentage. The formula has no room for a stray factor of one hundred.

The third input is the step lengthThe slice of the horizon one move covers, written as a fraction of a year., and it is different in kind from the other two. The step length is a decision by whoever divided the horizon rather than a reading off anything, so nobody looks it up. A one year horizon divided into twelve steps gives a step length of one twelfth, which is 0.083333, whose square root is the 0.288675 that keeps reappearing. A different decision about the number of steps changes the weight, without one thing about the process having changed.

What does the whole computation look like on the standard process?

The standard process starts at Rs 100/-, carries a volatility of 20 per cent a year, and sits beside a risk-free rate of 5 per cent a year continuously compounded, over a horizon of one year. Cut the year into one step and the arithmetic runs in the six lines below.

Running the equation backwards is the only way to confirm the arithmetic. Weighting the two future values, 0.577493 of Rs 122.14/- plus 0.422507 of Rs 81.87/- comes to 105.127110. Discounting that over the year at 5 per cent multiplies by 0.951229 and lands on Rs 100.00/- exactly. The weighted average of the two future values, discounted back, returns today's value to six decimal places. Returning today's value is what the weight was constructed to do, and the only thing it was constructed to do.

What must be true for the weight to exist at all?

A weight that comes out at 1.4, or at minus 0.2, is not a weight. The formula will happily return one. A ratio of two differences has no way to refuse a negative answer. So there is a condition sitting underneath the arithmetic, and it is worth writing out as an inequality rather than as a caution.

The condition is this. The growth of cash over the step has to sit strictly between the down factor and the up factor. If cash grows by less than the down move, the growth of cash sits below the whole interval and the weight comes out negative. If cash grows by more than the up move, the growth of cash sits above the interval and the weight comes out above one. The existence conditionThe requirement that cash grows by less than the up factor and more than the down factor. is nothing more than the requirement that the growth of cash lies inside the interval the two factors mark out.

Put the locked numbers in and the edges stop being abstract. On one step of one year the step length is 1.0, its square root is 1.0, and the two edges are minus 20 per cent and plus 20 per cent a year. Twenty per cent is the volatility, exactly. At a rate of exactly 20 per cent a year, cash grows by 1.221403, precisely the up factor, and the weight reaches one. At minus 20 per cent, cash grows by 0.818731, precisely the down factor, and the weight reaches zero. Push past either edge and the formula returns a number that is no longer a weight.

The situation past the edge is worth naming. If cash grows by more than the best the process can do, then holding cash beats holding the process in every case the lattice allows, and the two together admit a position that costs nothing and pays something. A position costing nothing and paying something is what arbitrageA position costing nothing today whose worst outcome is nil and whose best is a gain. means, and the weight leaving the interval from zero to one is the arithmetic tell that the specification contains one. Whether that tell is what makes the weight price correctly is covered separately, under the no-arbitrage argument.

What happens to the weight as the step gets shorter?

Nothing about the standard process changes when the year is cut into more steps. The starting value is the same Rs 100/-, the volatility is the same 20 per cent, the rate is the same 5 per cent. Only the step length changes, and the weight moves anyway.

The two factor columns explain the last one. Look at them first. As the step shortens, the up factor falls toward one and the down factor rises toward one. The interval they mark out is closing. The growth of cash over the shorter step is closing on one as well, and it does so from the middle, so the fraction of the way along settles toward one half. The weight drifts toward one half as the step shortens because the two factors are closing in on each other, and that is a fact about how the lattice was built rather than news about anything becoming more evenly balanced.

How fast it drifts can be stated exactly. Expanding the formula for small steps, the leading correction to one half is proportional to the square root of the step length, with a coefficient built from the rate, the variance rate and the volatility.

With the locked numbers in, the rate less half the variance rate is 0.05 less 0.02, or 0.03 exactly. Divided by twice the volatility, the coefficient is 0.075. At a step of one year the square root of the step is 1.0 and the approximation gives 0.575 against the exact 0.577493. At a step of one month the square root is 0.288675 and the approximation gives 0.521651 against the exact 0.521710. The approximation is not the answer, but it gives the shape of the movement without computing anything: halving the step shrinks the distance from one half by a factor of the square root of two.

One more thing shifts with the step length, and it is easy to miss. The edges of the permitted band move too. The condition allows any rate between minus and plus the volatility divided by the square root of the step, so at one step of a year the edges are at plus and minus 20 per cent, at three months they are at plus and minus 40 per cent, and at one month they are at plus and minus 69.28 per cent. Cutting the horizon finer widens the range of rates the lattice will tolerate. The condition is therefore a statement about the lattice and not about the world.

What does the weight have to do with anybody's view of what will happen?

Nothing, and the cleanest way to see it is to put three numbers side by side that a careless reader would treat as the same thing. All three are about the standard process over the same year, and no two of them agree.

The first is the one step lattice weight, 0.577493. The second is 0.559618, the chance the process finishes above where it started, computed under the risk-neutral measure Q. The third is 0.617911, the chance of the same event under the physical measure P, with the process carrying its drift of 8 per cent a year. Three numbers, one process, one year, and the gaps between them are not rounding. The second and third are 5.83 percentage points apart, and the first is not equal to either.

Here is the everyday version. A lift in a building knows which floor it is standing on, and that is a complete description of where it is. The floor number says nothing about how often the lift visits the fourth floor. The weight is a position, not a frequency, and asking it how often the process rises is asking the wrong kind of question of the right kind of number.

How is a weight checked once it has been handed over?

Most of the time the weight was computed by somebody else. Another person built the lattice, in a workbook or in a few lines of code, and only the output is at hand. Four checks, in this order, settle whether that number is what it claims to be, and each one catches a different kind of error.

1. Does it lie strictly between zero and one?A weight at 1.08 or at minus 0.04 is not a weight, and no amount of interpretation rescues it. What it shows is that the three inputs are not consistent with each other: the growth of cash has escaped the interval the two factors mark out.
   Catches an inconsistent specification, before anything is built on top of it.
2. Does the discounted weighted average return today's value?The two future values, weighted and then discounted over one step, are compared with the starting value. On the standard process that returns Rs 100.00/- to six decimal places.
   Catches an arithmetic slip, a rate applied over the year instead of the step, or a factor entered upside down.
3. Does it move when the interest rate moves?Changing the rate by two percentage points and recomputing settles it. On the one step lattice, 5 per cent gives 0.577493 and 7 per cent gives 0.630234. If nothing moves, the number is not being recomputed from the inputs.
   Catches a hard-coded figure, a stale cell, or a formula pointing at the wrong reference.
4. Does it stay put when the drift moves?Change the drift and recompute. The weight must not move by anything. If it moves, whatever produced the number used the drift, and a pricing weight never does.
   Catches a probability wearing a pricing weight's label, the commonest failure of all.

The four checks are not a classroom exercise, and it is worth saying who runs them. The four together separate a lattice that was implemented from a lattice that was described, so somebody reviewing a valuation workbook they did not build runs all four before signing anything that depends on it. The fourth check is the only one that can be run without the workbook, so an analyst handed a number and a one line explanation runs that one on its own. And anybody teaching this material runs the second check in front of the room. A weight that returns today's value to six decimals in public is more persuasive than any amount of argument about what the weight means. The checks cost about a minute each and they are the whole of the quality control available on a quantity that is invisible once it has been used.

No jurisdiction sets a probability axiom and no regulator publishes a lattice weight, so no country specific rule applies. The arithmetic holds identically everywhere. The only things that vary between settings are the conventions used to quote a rate, and those belong with the instrument rather than with the mathematics.

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