The Binomial Tree
How Two Moves a Year Explain a Stock's Future
How Two Moves a Year Explain a Stock's Future
A share price that can only rise or fall 10% a year is enough to build a price forecast, measure the risk around it, update it as the years unfold, and show why long-run prices lean the way they do.
$100
Starting Price
±10%
Annual Move
60%
Good Year ↓ adjustable
📅 2026
⏱ 9 min read
📊 Interactive model
The Problem With a Single Forecast
Ask what a share will be worth in three years and you will usually get one number. A target price, a base case, a projection. It is the least useful honest answer available, because the one thing everybody knows about it is that it will be wrong.
A better answer is the full set of prices the share could reach, and how likely each one is. Once you have that, the single number falls out of it — along with the risk around it, the way the forecast should shift as the years actually unfold, and the reason long-run prices behave the way they do.
The binomial tree is the simplest model that produces all of it. A price starts somewhere and moves up or down each year. That is the entire model. From that one picture come the expected price, the risk estimate, the scenario weighting behind every piece of equity research, and the shape that sits underneath option pricing.
Here is the same model at three sizes.
One Year
Two outcomes, one decision point. The smallest honest model of an uncertain price.
Three Years
Four possible prices with uneven odds. Now there is a forecast, and a measure of the risk around it.
Many Years
The outcomes merge into a smooth curve — and for prices, that curve leans to the right.
What the Tree Gives You
Six things the model produces. The technical name for each is worth knowing, and the rest of the article uses them — but the job each one does matters more than what it is called.
The set of possible prices
Every price the share could reach, and nothing outside that list. Its technical name is a random variable.
How likely each one is
The odds attached to each outcome — either exactly, or cumulatively. The PMF and the CDF.
The price you would expect
The probability-weighted average of every outcome. Often a number that cannot actually happen. The expected value.
The risk around it
How widely the outcomes scatter around that forecast. Variance, and its readable form, the standard deviation.
Updating as news arrives
What the forecast becomes once a year has actually happened. The conditional expectation.
The shape prices take
Why long-run prices lean right instead of forming a neat bell. The normal and lognormal curves.
The Company
Company XYZ trades today at $100. In any given year, the economy is either good or bad:
Good Year — probability 0.60
The share price rises by 10%. A $100 price becomes $110.
Bad Year — probability 0.40
The share price falls by 10%. A $100 price becomes $90.
We hold the share for three years. That is the entire model — and two assumptions inside it are worth separating carefully, because they are easy to blur together.
Within a year, good and bad are mutually exclusive. They cannot both happen, and their probabilities must add to 1. At any point on the tree, the two branches leaving it are linked — fixing the chance of one fixes the chance of the other.
Across years, the outcomes are independent. Year 2 does not care what happened in year 1.
Mutually exclusive is not the same as independent, and the tree is the clearest place to see the difference.
Across years, the outcomes are independent. Year 2 does not care what happened in year 1.
Mutually exclusive is not the same as independent, and the tree is the clearest place to see the difference.
Before working anything out by hand, it helps to see the model move. The tool below is the live version of everything that follows — build your intuition here first, then come back to it as each idea is explained underneath.
The Live Model
Change how likely a good year is, how big the annual move is, and how many years you hold. Everything recalculates as you drag. Click any node on the tree to stand at that moment in time.
Good year chance
0.60
Years held
3
Annual move
10%
3 years — the worked example
13 years — the bell appears
50 years — the long run
Measure
Good years
Log return
Share price
Smooth curve
Off
Centred scale
Distribution of outcomes
Cumulative view — the chance of landing at or below each value
The tree — click any node to condition on it
Gold branches are good years, grey branches are bad years. Selecting a node greys out every path that can no longer happen.
Number of good years
Average1.80
Variance0.72
Standard deviation0.85
Lean of the distribution−0.24
Share price
Expected price$106.12
Variance314.6
Standard deviation$17.74
Typical path lands at$104.62
Why average beats typical
Growth of the average1.980%
Growth of a typical path1.504%
The gap0.476%
Half the log variance0.483%
Counting the Paths
Over three years there are two possibilities each year, so there are 2³ = 8 distinct journeys. But there are only four ending prices, because order does not matter to the final result. Up then down lands in the same place as down then up.
| Journey | Good years | Ending price | Chance |
| Good, Good, Good | 3 | $133.10 | 0.216 |
| Good, Good, Bad | 2 | $108.90 | 0.144 |
| Good, Bad, Good | 2 | $108.90 | 0.144 |
| Bad, Good, Good | 2 | $108.90 | 0.144 |
| Good, Bad, Bad | 1 | $89.10 | 0.096 |
| Bad, Good, Bad | 1 | $89.10 | 0.096 |
| Bad, Bad, Good | 1 | $89.10 | 0.096 |
| Bad, Bad, Bad | 0 | $72.90 | 0.064 |
Three journeys land on $108.90 and three land on $89.10, while the two extremes have only one route each. Those counts — 1, 3, 3, 1 — are what mathematicians call binomial coefficients. You did not need a formula to find them. You needed to notice that there are several ways to reach the middle and only one way to reach either end.
A single year is a Bernoulli trial — one attempt, two outcomes. Repeat it independently and count the successes and you have a binomial distribution. Every binomial probability is a count of routes multiplied by the chance of any one route.
Listing the Odds Two Ways
| Good years | Ending price | Chance of exactly this | Chance of this or less |
| 0 | $72.90 | 0.064 | 0.064 |
| 1 | $89.10 | 0.288 | 0.352 |
| 2 | $108.90 | 0.432 | 0.784 |
| 3 | $133.10 | 0.216 | 1.000 |
The third column is the probability mass function: the chance of landing on exactly a value. The fourth is the cumulative distribution function: the chance of landing at or below it. Once the number of years grows large enough for the bars to merge into a curve, the first idea has to change form — for a smooth curve, probability becomes the area under it rather than the height of a bar, and the chance of any single exact value drops to zero. That version is called a probability density function.
Two Different Quantities
The tree carries two random quantities, and they behave very differently. One is the count of good years. The other is the ending price. They share the same probabilities, but the price is a compounding function of the count, and compounding is not a straight line.
The average number of good years is straightforward — three years at a 60% chance each gives 1.80 good years, with a variance of 0.72. So it is tempting to compound at that average:
$100 × 1.101.80 = $118.72 — and this is wrong
Weighting the four real outcomes by their probabilities gives a very different answer:
$133.10(0.216) + $108.90(0.432) + $89.10(0.288) + $72.90(0.064) = $106.12
A gap of more than twelve dollars, from one apparently harmless shortcut. The principle underneath is worth keeping in plain words: you cannot push an average through a compounding calculation. The average of a compounded quantity is not the compounding of the average.
There is a much faster route to the right answer. In any single year the expected multiplier is:
0.60 × 1.10 + 0.40 × 0.90 = 1.02
$100 × (1.02)³ = $106.12
$100 × (1.02)³ = $106.12
Because the years are independent, expectations multiply. The expected price grows at 2% a year. Hold that number — it comes back shortly, and it is not the whole story.
The View From Outside
What we have just calculated is the unconditional expectation — the answer standing at day one, before a single year has been observed. Unconditional simply means conditioned on nothing at all.
Its companion measure is the unconditional variance, which weights how far each outcome sits from $106.12:
0.216($133.10 − $106.12)² + 0.432($108.90 − $106.12)²
+ 0.288($89.10 − $106.12)² + 0.064($72.90 − $106.12)²
= 314.63 → standard deviation $17.74
+ 0.288($89.10 − $106.12)² + 0.064($72.90 − $106.12)²
= 314.63 → standard deviation $17.74
An average nobody experiences
$106.12 appears nowhere on the tree. Neither does 1.80 good years. An expected value is a long-run weighted average, not a prediction of what will actually happen.
Why take the square root
The variance is 314.63 dollars squared, a unit nobody can picture. The square root converts it back into dollars, which is why $17.74 is the number worth quoting.
Stepping Inside the Tree
A year passes. The economy was good, and XYZ now trades at $110. You are no longer standing outside the model — three of the eight journeys have become impossible. The surviving branches keep their relative weights, rescaled so they add back to 1.
| Ending price | Chance before | Chance now that year 1 was good |
| $133.10 | 0.216 | 0.36 |
| $108.90 | 0.432 | 0.48 |
| $89.10 | 0.288 | 0.16 |
| $72.90 | 0.064 | 0.00 |
| Standing at | Expected price | Variance | Standard deviation |
| Day one, knowing nothing | $106.12 | 314.63 | $17.74 |
| Year 1, price at $110 | $114.44 | 242.82 | $15.58 |
| Year 1, price at $90 | $93.64 | 162.55 | $12.75 |
Two things changed. The expectation moved to track where you actually are — it did not drift back toward $106.12, because that figure was an average across worlds you now know you are not in. And the variance fell. One year of uncertainty has been resolved and cannot return. Variance measures what is still unknown, not uncertainty in general.
Notice too that the two conditional variances differ, 242.82 against 162.55. Because the moves are percentages rather than fixed dollar amounts, the dollar spread scales with the price level. Higher price, wider range of outcomes. It is one reason people who work with markets track returns rather than price changes.
Do not over-generalise this. Conditioning did not reduce the variance because conditioning always does. It reduced it because we conditioned on information — a year that has already happened. Condition instead on a scenario, such as assuming a severe downturn, and the range of outcomes can easily be wider than it was before.
The unconditional figure is not an independent quantity. It is the probability-weighted blend of the two conditional ones:
0.60 × $114.44 + 0.40 × $93.64 = $106.12
Which is exactly how forecasting works in practice. Build an optimistic case, a base case and a pessimistic case, attach probabilities to each, and blend them. That blending step has a name — the total probability rule — whether or not anyone calls it that out loud.
Up 10%, Down 10%, and You Are Poorer
Look at the middle of the tree. A good year followed by a bad one:
$100 × 1.10 × 0.90 = $99.00
Not $100. A 10% gain and a 10% loss do not cancel, because the loss is applied to a larger base than the gain was. This asymmetry is often called volatility drag, and it is the reason the next step abandons ordinary percentages in favour of logarithms.
Two Growth Rates for One Company
Ordinary returns multiply across time, which makes them awkward to average. Logarithmic returns add, which makes them easy. Take logs of the two possible multipliers:
Good year: ln(1.10) = +0.0953
Bad year: ln(0.90) = −0.1054
Bad year: ln(0.90) = −0.1054
The asymmetry is visible at once — a 10% gain and a 10% loss are not mirror images, and the loss is the larger of the two. Treat the annual log return as its own random quantity and take its average and variance the ordinary way:
Average: 0.60(0.0953) + 0.40(−0.1054) = 0.01504 → 1.504% a year
Variance: 0.009665 → half of it = 0.483%
Variance: 0.009665 → half of it = 0.483%
We now have two different growth rates for the same company, and the difference between them is the whole point.
| Growth rate | Per year | What it describes |
| Growth of the average outcome | 1.980% | Where the probability-weighted average ends up |
| Growth of a typical path | 1.504% | What a single ordinary journey compounds at |
| The gap between them | 0.476% | Almost exactly half the variance of the log return |
The gap between the two growth rates, 0.476%, is essentially half the variance, 0.483%. That is not a coincidence. It is where a term that appears throughout financial modelling comes from — the reason expected growth is written as the average log return plus half the variance, or equivalently why the typical path is the average minus half the variance.
XYZ's expected price compounds at 2.0% a year and reaches $106.12 after three years. XYZ's typical path compounds at 1.5% a year and reaches about $104.62.
Both are correct. The average is dragged upward by a small number of very good outcomes, because compounding is lopsided — the upside has no ceiling while the downside stops at zero. The more volatile the asset, the wider this gap grows, and the gap is half the variance.
This is why an arithmetic average return always exceeds a compound average return, and why a fund's advertised average return tends to overstate what an ordinary investor actually experienced.
Both are correct. The average is dragged upward by a small number of very good outcomes, because compounding is lopsided — the upside has no ceiling while the downside stops at zero. The more volatile the asset, the wider this gap grows, and the gap is half the variance.
This is why an arithmetic average return always exceeds a compound average return, and why a fund's advertised average return tends to overstate what an ordinary investor actually experienced.
A note on precision: the match here is close rather than exact — 0.476% against 0.483% — because a year that moves either exactly +10% or exactly −10% is a coarse model. As the time steps get smaller and more numerous, the two figures converge and the relationship becomes exact.
What Happens Over Many Years
Everything so far used three years because three years fits on a page. The last idea needs more. Push the years slider in the model above and two different things happen to the two quantities we have been tracking.
The count becomes normal
The number of good years is a sum of many independent yes-or-no results. Add enough of them together and the bars settle into a symmetric bell shape — the normal distribution. A common rule of thumb is that this works once the expected number of good years and the expected number of bad years both exceed five.
The price becomes lognormal
The price is a product of many independent multipliers, so its logarithm is a sum of many independent log returns. If the logarithm settles into a bell shape, the price itself does not: it is squashed against zero on the left and stretched out on the right. That shape is called lognormal.
This settles the most confused pair in the subject. Returns are modelled as normal; prices are modelled as lognormal. They are two views of the same tree, and the reason is structural rather than mathematical taste. Returns add up in log space and can be negative. Prices multiply, and cannot fall below zero.
Six Things You Did to One Picture
| What you do to the tree | What it gives you | In this example |
| Count the routes to each ending | Binomial coefficients | 1, 3, 3, 1 |
| Weight every ending by its chance | Unconditional expected value | $106.12 |
| Measure distances from that average | Unconditional variance | 314.63, or $17.74 |
| Stand at a node and rescale | Conditional expectation and variance | $114.44 and 242.82 |
| Blend the nodes by their chances | Total probability rule | Back to $106.12 |
| Stretch it over many years | Normal and lognormal distributions | A bell, and a right-leaning curve |
Check Your Understanding
Click each question to reveal the answer.
1. Three years produce eight possible journeys but only four possible ending prices. Why?
Because multiplication does not care about order. A good year followed by a bad year gives the same price as a bad year followed by a good one, so different journeys land on the same value. That is why three of the eight journeys end at $108.90 and three end at $89.10, while the two extremes have only one route each.
2. The average number of good years is 1.80. Why can the expected price not be found by compounding 10% for 1.80 years?
Because compounding is not a straight-line relationship. Averaging first and compounding second gives $118.72, while weighting the four real outcomes gives $106.12. The average of a compounded quantity is not the compounding of the average — an error worth more than twelve dollars here, and far more over longer horizons.
3. If a share is expected to grow at 2% a year, why does a typical holder end up with less than 2% compounded growth?
Because the 2% describes the average across all possible futures, and that average is pulled up by a few unusually good ones. The typical path compounds at 1.5%, and the difference is roughly half the variance of the log return. The more volatile the asset, the larger the gap between the advertised average and the ordinary experience.
RUQQI KEY TAKEAWAYS — DISTRIBUTIONS AND EXPECTATIONS
- One year is a Bernoulli trial. Repeat it and count the successes and you have a binomial distribution.
- Every binomial probability is a count of routes multiplied by the chance of one route.
- An expected value is a weighted average, not a forecast. It often lands on a number that cannot happen.
- Compounding is not a straight line, so you cannot push an average through it.
- Conditioning on information moves the expectation and shrinks the range still to be resolved. Conditioning on a scenario may widen it instead.
- The unconditional answer is always the probability-weighted blend of the conditional ones.
- Returns are normal, prices are lognormal, and the distance between the average outcome and the typical one is half the variance.